The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups James Arthur Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario M5S 2E4 E-mail address:
[email protected]
2000 Mathematics Subject Classification. Primary
Supported in part by NSERC Discovery Grant A3483. Abstract.
Contents Foreword: A selective overview
vii
Chapter 1. Parameters 1.1. The automorphic Langlands group 1.2. Self-dual, finite dimensional representations 1.3. Representations of GLpN q 1.4. A substitute for global parameters 1.5. Statement of three main theorems
1 1 6 15 26 40
Chapter 2. Local Transfer 2.1. Langlands-Shelstad-Kottwitz transfer 2.2. Characterization of the local classification 2.3. Normalized intertwining operators 2.4. Statement of the local intertwining relation 2.5. Relations with Whittaker models
51 51 62 79 96 109
Chapter 3. Global Stabilization 3.1. The discrete part of the trace formula 3.2. Stabilization 3.3. Contribution of a parameter ψ 3.4. A preliminary comparison 3.5. On the vanishing of coefficients
121 121 130 137 145 152
Chapter 4. The Standard Model 4.1. Statement of the stable multiplicity formula 4.2. On the global intertwining relation 4.3. Spectral terms 4.4. Endoscopic terms 4.5. The comparison 4.6. The two sign lemmas 4.7. On the global theorems 4.8. Remarks on general groups
165 165 171 184 191 199 212 227 238
Chapter 5. A Study of Critical Cases 5.1. The case of square integrable ψ 5.2. The case of elliptic ψ 5.3. A supplementary parameter ψ 5.4. Generic parameters with local constraints
251 251 262 270 285
v
vi
CONTENTS
Chapter 6. The Local Classification 6.1. Local parameters 6.2. Construction of global representations π9 6.3. Construction of global parameters φ9 6.4. The local intertwining relation for φ 6.5. Orthogonality relations for φ 6.6. Local packets for composite φ 6.7. Local packets for simple φ 6.8. Resolution
299 299 307 319 330 343 353 365 374
Chapter 7. Local Nontempered Representations 7.1. Local parameters and duality 7.2. Construction of global parameters ψ9 7.3. The local intertwining relation for ψ 7.4. Local packets for composite and simple ψ 7.5. Some remarks on characters
385 385 394 401 413 427
Chapter 8. The Global Classification 8.1. On the final step 8.2. Proof by contradiction 8.3. Reflections on the results 8.4. Refinements for even orthogonal groups 8.5. An approximation of the Langlands group
441 441 451 467 482 500
Chapter 9. Inner Forms 9.1. Inner twists 9.2. Parameters and centralizers 9.3. On the normalization of transfer factors 9.4. Statement of the local classification 9.5. Statement of a global classification
511 511 524 533 545 557
Bibliography
567
Foreword: A selective overview This preface contains a summary of the contents of the volume. We start with a rough description of the main theorems. We then give short descriptions of the contents of the various chapters. At the end of the preface, we will add a couple of remarks on the overall structure of the proof, notably our use of induction. The preface can serve as an introduction. The beginning of the actual text, in the form of the first two or three sections of Chapter 1, represents a different sort of introduction. It will be our attempt to motivate what follows from a few basic principles. Automorphic representations for GLpN q have been important objects of study for many years. We recall that GLpN q, the general linear group of invertible pN N q-matrices, assigns a group GLpN, Rq to any commutative ring R with identity. For example, R could be a number field F , or the ring A AF of ad`eles over F . Automorphic representations of GLpN q are the irreducible representations of GLpN, Aq that occur in the decomposition 2 of its regular representation on L GLpN, F qzGLpN, Aq . This informal definition is made precise in [L6], and carries over to any connected reductive group G over F . The primary aim of the volume is to classify the automorphic representations of special orthogonal and symplectic groups G in terms of those of GLpN q. Our main tool will be the stable trace formula for G, which until recently was conditional on the fundamental lemma. The fundamental lemma has now been established in complete generality, and in all of its various forms. In particular, the stabilization of the trace formula is now known for any connected group. However, we will also require the stabilization of twisted trace formulas for GLpN q and SOp2nq. Since these have yet to be established, our results will still be conditional. A secondary purpose will be to lay foundations for the endoscopic study of more general groups G. It is reasonable to believe that the methods we introduce here extend to groups that Ramakrishnan has called quasiclassical. These would comprise the largest class of groups whose representations could ultimately be tied to those of general linear groups. Our third goal is expository. In adopting a style that is sometimes more discursive than strictly necessary, we have tried to place at least some of the techniques into perspective. We hope that there will be parts of the volume that are accessible to readers who are not experts in the subject.
vii
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Automorphic representations are interesting for many reasons, but among the most fundamental is the arithmetic data they carry. Recall that GLpN, Aq
¹
GLpN, Fv q
v
is a restricted direct product, taken over (equivalence classes of) valuations v of F . An automorphic representation of GLpN q is a restricted direct product π
â
πv ,
v
where πv is an irreducible representation of GLpN, Fv q that is unramified for almost all v. We recall that πv is unramified if v is nonarchimedean, and πv contains the trivial representation of the hyperspecial maximal compact subgroup GLpN, ov q of integral points in GLpN, Fv q. The representation is then parametrized by a semisimple conjugacy class cv pπ q cpπv q in the complex dual group GLpN q^
GLpN, Cq
of GLpN q. (See [Bo, (6.4), (6.5)] for the precise assertion, as it applies to a general connected reductive group G.) It is the relations among the semisimple conjugacy classes cv pπ q that will contain the fundamental arithmetic information. There are three basic theorems for the group GLpN q that together give us a pretty clear understanding of its representations. The first is local, while others, which actually predate the first, are global. The first theorem is the local Langlands correspondence for GLpN q. It was established for archimedean fields by Langlands, and more recently for padic (which is to say nonarchimedean) fields by Harris, Taylor and Henniart. It classifies the irreducible representations of GLpN, Fv q at all places v by (equivalence classes of) semisimple, N -dimensional representations of the local Langlands group LFv
#
WF v , v archimedean, WFv SU p2q, otherwise.
In particular, an unramified representation of GLpN, Fv q corresponds to an N -dimensional representation of LFv that is trivial on the product of SU p2q with the inertia subgroup IFv of the local Weil group WFv . It therefore corresponds to a semisimple representation of the cyclic quotient LFv {IFv
SU p2q WF {IF Z, and hence a semisimple conjugacy class in GLpN, Cq, as above. v
v
FOREWORD: A SELECTIVE OVERVIEW
ix
The first of the global theorems is due to Jacquet and Shalika. If π is any smooth representation of GLpN, Aq, one can form the family of semisimple conjugacy classes cpπ q lim ÝÑ cv pπq cpπv q : v
RS
(
S
in GLpN, Cq, defined up to a finite set of valuations S. In other words, cpπ q is an equivalence class of families, two such families being equivalent if they are equal for almost all v. The theorem of Jacquet and Shalika asserts that if an automorphic representation π of GLpN q is restricted slightly to be isobaric [L7, §2], it is uniquely determined by cpπ q. This theorem can be regarded as a generalization of the theorem of strong multiplicity one for cuspidal automorphic representations of GLpN q. The other global theorem for GLpN q is due to Moeglin and Waldspurger. It characterizes the automorphic (relatively) discrete spectrum of GLpN q in terms of the set of cuspidal automorphic representations. Since Langlands’ general theory of Eisenstein series characterizes the full automorphic spectrum of any group G in terms of discrete spectra, this theorem characterizes the automorphic spectrum for GLpN q in terms of cuspidal automorphic representations. Combined with the first global theorem, it classifies the full automorphic spectrum of GLpN q explicitly in terms of families cpπ q, constructed from cuspidal automorphic representations of general linear groups. Our goal is to generalize these three theorems. As we shall see, however, there is very little that comes easily. It has been known for many years that the representations of groups other than GLpN q have more structure. In particular, they should separate naturally into L-packets, composed of representations with the same L-functions and ε-factors. This was demonstrated for the group G SLp2q Spp2q by Labesse and Langlands, in a paper [LL] that became a model for Langlands’s conjectural theory of endoscopy [L8], [L10]. The simplest and most elegant way to formulate the theory of endoscopy is in terms of the global Langlands group LF . This is a hypothetical global analogue of the explicit local Langlands groups LFv defined above. It is thought to be a locally compact extension 1
ÝÑ
KF
ÝÑ
LF
ÝÑ
WF
ÝÑ
1
of WF by a compact connected group KF . (See [L7, §2], [K3, §9].) However, its existence is very deep, and could well turn out to be the final theorem in the subject to be proved! One of our first tasks, which we address in §1.4, will be to introduce makeshift objects to be used in place of LF . For simplicity, however, let us describe our results here in terms of LF . Our main results apply to the case that G is a quasisplit special orthogonal or symplectic group. They are stated as three theorems in §1.5. The proof of these theorems will then take up much of the rest of the volume.
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Theorem 1.5.1 is the main local result. It contains a local Langlands parametrization of the irreducible representations of GpFv q, for any p-adic valuation v of F , as a disjoint union of finite L-packets Πφv . These are indexed by local Langlands parameters, namely L-homomorphisms
ÝÑ LGv LG G p GalpF v {Fv q of G. v
φv : LFv
from LFv to the local L-group The theorem includes a way to index the representations in an L-packet Πφv by linear characters on a finite abelian group Sφv attached to φv . Since similar results for archimedean valuations v are already known from the work of Shelstad, we obtain a classification of the representations of each local group GpFv q. Theorem 1.5.1 also contains a somewhat less precise description of the representations of GpFv q that are local components of automorphic representations. These fall naturally into rather different packets Πψv , indexed according to the conjectures of [A8] by L-homomorphisms (1)
ψv : LFv
SU p2q ÝÑ
L
Gv ,
with bounded image. The theorem includes the assertion, also conjectured in [A8], that the representations in these packets are all unitary. Theorem 1.5.2 is the main global result. As a first approximation, it gives a rough decomposition (2)
Rdisc
à
Rdisc,ψ
ψ
of the representation Rdisc of GpAq on the automorphic discrete spectrum L2disc GpF qzGpAq . The indices can be thought of as L-homomorphisms (3)
ψ : LF
SU p2q ÝÑ
L
G
of bounded image that do not factor through any proper parabolic subgroup p GalpF {F q. They have localizations (1) deof the global L-group L G G fined by conjugacy classes of embeddings LFv LF , or rather the makeshift analogues of such embeddings that we formulate in §1.4. The localizations ψv of ψ are unramified at almost v, and consequently lead to a family cpψ q lim ÝÑ cv pψq cpψv q : v
RS
(
S
of equivalence classes of semisimple elements in L G. The rough decomposition (2) is of interest as it stands. It implies that G has no embedded eigenvalues, in the sense of unramified Hecke operators. In other words, the family cpψ q attached to any global parameter ψ in the decomposition of the automorphic discrete spectrum is distinct from any family obtained from the continuous spectrum. This follows from the nature of the parameters ψ in (3), and the application of the theorem of Jacquet-Shalika to the natural image of cpψ q in the appropriate complex general linear group.
FOREWORD: A SELECTIVE OVERVIEW
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Theorem 1.5.2 also contains a finer decomposition (4)
Rdisc,ψ
à π
mψ pπ q π,
for any global parameter ψ. The indices π range over representations in the global packet of ψ, defined as a restricted direct product of local packets provided by Theorem 1.5.1. The multiplicities mψ pπ q are given by an explicit reciprocity formula in terms of the finite abelian groups Sψv , and their global analogue Sψ . We thus obtain a decomposition of the automorphic discrete spectrum of G into irreducible representations of GpAq. We shall say that a parameter φ ψ is generic if it is trivial on the factor SU p2q. The representations π P Πφ , with φ generic and mφ pπ q 0, are the constituents of the automorphic discrete spectrum that are expected to satisfy the analogue for G of Ramanujan’s conjecture. It ψ is not generic, the formula for mψ pπ q has an extra ingredient. It is a sign character εψ on Sψ , defined (1.5.6) in terms of symplectic root numbers. That the discrete spectrum should be governed by objects of such immediate arithmetic significance seems quite striking. Theorem 1.5.2 has application to the question of multiplicity one. Suppose that π is an irreducible constituent of the automorphic discrete spectrum of G that also lies in some generic global packet Πφ . We shall then show that the multiplicity of π in the discrete spectrum equals 1 unless p SOp2n, Cq, in which case the multiplicity is either 1 or 2, according to an G explicit condition we shall give. In particular, if G equals either SOp2n 1q or Spp2nq, the automorphic representations in the discrete spectrum that are expected to satisfy Ramanujan’s conjecture all have multiplicity 1. Local results of Moeglin [M4] on nontempered p-adic packets suggest that similar results hold for all automorphic representations in the discrete spectrum. Theorems 1.5.1 and 1.5.2 are founded on the proof of several cases of Langlands’ principle of functoriality. In fact, our basic definitions will be derived from the functorial correspondence from G to GLpN q, relative to the standard representation of L G into GLpN, Cq. Otherwise said, our construction of representations of G will be formulated in terms of representations of GLpN q. The integer N of course equals 2n, 2n 1 and 2n as G ranges over the groups SOp2n 1q, Spp2nq and SOp2nq in the three infinite families Bn , p being equal to Spp2n, Cq, SOp2n 1, Cq and Cn and Dn , with dual groups G SOp2n, Cq, respectively. The third case SOp2nq, which includes quasisplit outer twists, is complicated by the fact that it is really the nonconnected group Op2nq that is directly tied to GLpN q. This is what is responsible for the failure of multiplicity one described above. It is also the reason we have not yet specified the equivalence relation for the local and global parameters p (1) and (3). Let us now agree that they are to be taken up to G-conjugacy if G equals SOp2n 1q or Spp2nq, and up to conjugacy by Op2n, Cq, a group whose quotient r N pGq O p2n, Cq{SOp2n, Cq Out
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p SOp2n, Cq, if G equals SOp2nq. This acts by outer automorphism on G is the understanding on which the decomposition (2) holds. r pGv q for the set of equivalence classes of In the text, we shall write Ψ r ψ attached to any ψv P Ψ r pGv q will then local parameters (1). The packet Π v r N pGv q-orbits of equivalence classes of irreducible repbe composed of Out r N pGq being trivial in case G equals SOp2n 1q or resentations (with Out r pGq for the set of equivalence classes of general Spp2nq). We will write Ψ r 2 pGq for the subset of classes with the suppleglobal parameters ψ, and Ψ r pGq will then be restricted mentary condition of (3). The packet of any ψ P Ψ direct product rψ Π
â
rψ Π v
v
of local packets. These are the objects that correspond to (isobaric) aur ψ, tomorphic representations of GLpN q. In particular, the global packets Π r rather than the individual (orbits of) representations π in Πψ , are the objects that retain the property of strong multiplicity one from GLpN q. Similarly, r ψ attached to parameters ψ P Ψ r 2 pGq retain the qualithe global packets Π tative properties of automorphic discrete spectrum of GLpN q. They come with a sort of Jordan decomposition, in which the semisimple packets correspond to the generic global parameters ψ, and contain the automorphic representations that are expected to satisfy the G-analogue of Ramanujan’s conjecture. In view of these comments, we see that Theorem 1.5.2 can be regarded as a simultaneous analogue for G of both of the global theorems for GLpN q. Theorem 1.5.3 is a global supplement to Theorem 1.5.2. Its first assertion r pGq that are both generic and simple, in applies to global parameters φ P Ψ the sense that they correspond to cuspidal automorphic representations πφ of p is orthogonal (resp. GLpN q. Theorem 1.5.3(a) asserts that the dual group G symplectic) if and only if the symmetric square L-function Lps, πφ , S 2 q (resp. the skew-symmetric square L-function Lps, πφ , Λ2 q) has a pole at s 1. Theorem 1.5.3(b) asserts that the Rankin-Selberg ε-factor ε 21 , πφ1 πφ2 r pGi q such that G p1 equals 1 for any pair of generic simple parameters φi P Φ p 2 are either both orthogonal or both symplectic. These two assertions and G are automorphic analogues of well known properties of Artin L-functions and ε-factors. They are interesting in their own right. But they are also an essential part of our induction argument. We will need them in Chapter 4 to interpret the terms in the trace formula attached to compound parameters r p G q. ψPΨ This completes our summary of the main theorems. The first two sections of Chapter 1 contain further motivation, for the global Langlands group LF in §1.1, and the relations between representations of G and GLpN q in §1.2. In §1.3, we will recall the three basic theorems for GLpN q. Section 1.4
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xiii
is given over to our makeshift substitutes for global Langlands parameters, as we have said, while §1.5 contains the formal statements of the theorems. As might be expected, the three theorems will have to be established together. The unified proof will take us down a long road, which starts in Chapter 2, and crosses many different landscapes before coming to an end finally in §8.2. The argument is ultimately founded on harmonic analysis, which is represented locally by orbital integrals and characters, and globally by the trace formula. This of course is at the heart of the theory of endoscopy. We refer the reader to the introductory remarks of individual sections, where we have tried to offer guidance and motivation. We shall be content here with a minimal outline of the main stages. Chapter 2 is devoted to local endoscopy. It contains a more precise formulation (Theorem 2.2.1) of the local Theorem 1.5.1. This provides for a r ψ in terms of twisted characters canonical construction of the local packets Π v on GLpN q. Chapter 2 also includes the statement of Theorem 2.4.1, which we will call the local intertwining relation. This is closely related to Theorem 1.5.1 and its refinement Theorem 2.2.1, and from a technical standpoint, can be regarded as our primary local result. It includes a delicate construction of signs, which will be critical for the interpretation of terms in the trace formula. Chapter 3 is devoted to global endoscopy. We will recall the discrete part of the trace formula in §3.1, and its stabilization in §3.2. We are speaking here of those spectral terms that are linear combinations of automorphic characters, and to which all of the other terms are ultimately dedicated. They are the only terms in the trace formula that will appear explicitly in this volume. In §3.5, we shall establish criteria for the vanishing of coefficients in certain identities (Proposition 3.5.1, Corollary 3.5.3). We will use these criteria many times throughout the volume in drawing conclusions from the comparison of discrete spectral terms. In general, we will have to treat three separate cases of endoscopy. They are represented respectively by pairs pG, G1 q, where G is one of the groups to which Theorems 1.5.1, 1.5.2 and 1.5.3 apply and G1 is a corresponding r pN q, G in which G r pN q is the twisted general endoscopic datum, pairs G r 0 pN q GLpN q and G is a corresponding twisted endoscopic linear group G r G r 1 q in which G r is a twisted even orthogonal group datum, and pairs pG, 0 1 r SOp2nq and G r is again a corresponding twisted endoscopic datum. G r G r1 q The first two cases will be our main concern. However, the third case pG, is also a necessary part of the story. Among other things, it is forced on us by the need to specify the signs in the local intertwining relation. For the most part, we will not try to treat the three cases uniformly as cases of the general theory of endoscopy. This might have been difficult, given that we have to deduce many local and global results along the way. At any rate, the separate treatment of the three cases gives our exposition a more concrete flavour, if at the expense of some possible sacrifice of efficiency.
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In Chapter 4, we shall study the comparison of trace formulas. Specifically, we will compare the contribution (4.1.1) of a parameter ψ to the discrete part of the trace formula with the contribution (4.1.2) of ψ to the corresponding endoscopic decomposition. We begin with the statement of Theorem 4.1.2, which we will call the stable multiplicity formula. This is closely related to Theorem 1.5.2, and from a technical standpoint again, is our primary global result. Together with the global intertwining relation (Corollary 4.2.1), which we state as a global corollary of Theorem 2.4.1, it governs how individual terms in trace formulas are related. Chapter 4 represents a standard model, in the sense that if we grant the analogues of the two primary theorems for general groups, it explains how the terms on the right hand sides of (4.1.1) and (4.1.2) match. This is discussed heuristically in Sections 4.7 and 4.8. However, the purpose of this volume is to derive Theorems 2.4.1 and 4.1.2 for our groups G from the standard model, and whatever else we can bring to bear on the problem. This is the perspective of Sections 4.3 and 4.4. In §4.5, we combine the analysis of these sections with a general induction hypothesis to deduce the stable multiplicity formula and the global intertwining relation for many ψ. Section 4.6 contains the proof of two critical sign lemmas that are essential ingredients of the parallel Sections 4.3 and 4.4. Chapter 5 is the center of the volume. It is a bridge between the global discussion of Chapters 3 and 4 and the local discussion of Chapters 6 and 7. It also represents a transition from the general comparisons of Chapter 4 to the study of the remaining parameters needed to complete the induction hypotheses. These exceptional cases are the crux of the matter. In §5.2 and §5.3, we shall extract several identities from the standard model, in which we display the possible failure of Theorems 2.4.1 and 4.1.2 as correction r 2 p G q, terms. Section 5.3 applies to the critical case of a parameter ψ P Ψ and calls for the introduction of a supplementary parameter ψ . In §5.4, we shall resolve the global problems for families of parameters ψ that are assumed to have certain rather technical local properties. Chapter 6 applies to generic local parameters. It contains a proof of the local Langlands classification for our groups G (modified by the outer automorphism in the case G SOp2nq). We will first have to embed a given local parameter into a family of global parameters with the local constraints of §5.4. This will be the object of Sections 6.2 and 6.3, which rest ultimately on the simple form of the invariant trace formula for G. We will then have to extract the required local information from the global results obtained in §5.4. In §6.4, we will deduce the generic local intertwining relation from its global counterpart in §5.4. Then in §6.5, we will stabilize the orthogonality relations that are known to hold for elliptic tempered characters. This will allow us to quantify the contributions from the remaining elliptic tempered characters, the ones attached to square integrable representations. We will use the information so obtained in §6.6 and §6.7. In these sections, we shall establish Theorems 2.2.1 and 1.5.1 for the remaining “square integrable”
FOREWORD: A SELECTIVE OVERVIEW
xv
r 2 pGq. Finally, in §6.8, we shall resolve the Langlands parameters φ P Φ various hypotheses taken on at the beginning of Chapter 6. Chapter 7 applies to nongeneric local parameters. It contains the proof of the local theorems in general. In §7.2, we shall use the construction of §6.2 to embed a given nongeneric local parameter into a family of global parameters, but with local constraints that differ slightly from those of §5.4. We will then deduce special cases of the local theorems that apply to the places v with local constraints. These will follow from the local theorems for generic parameters, established in Chapter 6, and the duality operator of Aubert and Schneider-Stuhler, which we review in §7.1. We will then exploit our control over the places v to derive the local theorems at the localization ψ ψ9 u that represents the original given parameter. We will finish the proof of the global theorems in the first two sections of Chapter 8. Armed with the local theorems, and the resulting refinements of the lemmas from Chapter 5, we will be able to establish almost all of the global results in §8.1. However, there will still to be one final obstacle. It is r sim pGq, which among other things, will the case of a simple parameter ψ P Ψ be essential for a resolution of our induction hypotheses. An examination of this case leads us to the initial impression that it will be resistant to all of our earlier techniques. However, we will then see that there is a way to treat it. We will introduce a second supplementary parameter, which appears ungainly at first, but which, with the support of two rather intricate lemmas, takes us to a successful conclusion. Section 8.2 is the climax of our long running induction argument, as well as its most difficult point of application. Its final resolution is what brings us to the end of the proof. We will then be free in §8.3 for some general reflections that will give us some perspective on what has been established. In §8.4, we will sharpen our results for the groups SOp2nq, in which the outer automorphism creates some ambiguity. We will use the stabilized trace formula to construct the local and global L-packets predicted for these groups by the conjectural theory of endoscopy. In §8.5, we will describe an approximation LF of the global Langlands group LF that is tailored to the classical groups of this volume. It could potentially be used in place of the ad hoc global parameters of §1.4 to streamline the statements of the global theorems. We shall discuss inner forms of orthogonal and symplectic groups in Chapter 9. The automorphic representation theory for inner twists is in some ways easier for knowing what happens in the case of quasisplit groups. In particular, the stable multiplicity formula is already in place, since it applies only to the quasisplit case. However, there are also new difficulties for inner twists, particularly in the local case. We shall describe some of these in Sections 9.1–9.3. We shall then state analogues for inner twists of the main theorems, with the understanding that their proofs will appear elsewhere.
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Having briefly summarized the various chapters, we had best add some comment on our use of induction. As we have noted, induction is a central part of the unified argument that will carry us from Chapter 2 to Section 8.2. We will have two kinds of hypotheses, both based on the positive integer N that indexes the underlying general linear group GLpN q. The first kind includes various ad hoc assumptions, such as those implicit in some of our r pGq defined in §1.4 are definitions. For example, the global parameter sets Ψ based on the inductive application of two “seed” Theorems 1.4.1 and 1.4.2. The second will be the formal induction hypotheses introduced explicitly at the beginning of §4.3, and in more refined form at the beginning of §5.1. They assert essentially that the stated theorems are all valid for parameters of rank less than N . In particular, they include the informal hypotheses implicit in the definitions. We do not actually have to regard the earlier, informal assumptions as inductive. They really represent implicit appeals to stated theorems, in support of proofs of what amount to corollaries. In fact, from a logical standpoint, it is simpler to treat them as inductive assumptions only after we introduce the formal induction hypotheses in §4.3 and §5.1. For a little more discussion of this point, the reader can consult the two parallel Remarks following Corollaries 4.1.3 and 4.2.4. The induction hypotheses of §5.1 are formulated for an abstract family Fr of global parameters. They pertain to the parameters ψ P Fr of rank less than N , and are supplemented also by a hypothesis (Assumption 5.1.1) for certain parameters in Fr of rank equal to N . The results of Chapter r These are the 5 will be applied three times, to three separate families F. family of generic parameters with local constraints used to establish the local classification of Chapter 6, the family of nongeneric parameters with local constraints used to deduce the local results for non-tempered representations in Chapter 7, and the family of all global parameters used to establish the global theorems in the first two sections of Chapter 8. In each of these cases, r In the case of the assumptions have to be resolved for the given family F. Chapter 6, the induction hypotheses are actually imposed in two stages. The local hypothesis at the beginning of §6.3 is needed to construct the family r on which we then impose the global part of the general hypothesis of F, Chapter 5 at the beginning of §6.4. The earlier induction hypotheses of §4.3 apply to general global parameters of rank less than N . They are used in §4.3–§4.6 to deduce the global theorems for parameters that are highly reducible. Their general resolution comes only after the proof of the global theorems in §8.2. Our induction assumptions have of course to be distinguished from the general condition (Hypothesis 3.2.1) on which our results rely. This is the stabilization of the twisted trace formula for the two groups GLpN q and SOp2nq. As a part of the condition, we implicitly include twisted analogues of the two local results that have a role in the stabilization of the
FOREWORD: A SELECTIVE OVERVIEW
xvii
ordinary trace formula. These are the orthogonality relations for elliptic tempered characters of [A10, Theorem 6.1], and the weak spectral transfer of tempered p-adic characters given by [A11, Theorems 6.1 and 6.2]. The stabilization of orthogonality relations in §6.5, which requires twisted orthogonality relations for GLpN q, will be an essential part of the local classification in Chapter 6. The two theorems from [A11] can be regarded as a partial generalization of the fundamental lemma for the full spherical Hecke algebra [Hal]. (Their global proof of course depends on the basic fundamental lemma for the unit, established by Ngo.) We will use them in combination with their twisted analogues in the proofs of Proposition 2.1.1 and Corollary 6.7.4. The first of these gives the image of the twisted transfer of functions from GLpN q, which is needed in the proof of Proposition 3.5.1. The second gives a relation among tempered characters, which completes the local classification. There is one other local theorem whose twisted analogues for GLpN q and SOp2nq we shall also have to take for granted. It is Shelstad’s strong spectral transfer of tempered archimedean characters, which is to say, her endoscopic classification of representations of real groups. This of course is major result. Together with its two twisted analogues, it gives the archimedean cases of the local classification in Theorems 2.2.1 and 2.2.4. We shall combine it with a global argument in Chapter 6 to establish the p-adic form of these theorems. The general twisted form of Shelstad’s endoscopic classification appears to be within reach. It is likely to be established soon by some extension of recent work by Mezo [Me] and Shelstad [S8]. Finally, let me include a word on the notation. Because our main theorems require interlocking proofs, which consume a good part of the volume, there is always the risk of losing one’s way. Until the end of §8.2, assertions as Theorems are generally stated with the understanding that their proofs will usually be taken up much later (unless of course they are simply quoted from some other source). On the other hand, assertions denoted Propositions, Lemmas or Corollaries represent results along the route, for which the reader can expect a timely proof. Theorems stated in §8.4 and §8.5 are not part of the central induction argument. Their proofs, which are formally labeled as such, follow relatively soon after their statements. The theorems stated in §9.4 and §9.5 apply to inner twists, and will be proved elsewhere. The actual mathematical notation might appear unconventional at times. I have tried to structure it so as to reflect implicit symmetries in the various objects it represents. With luck, it might help a reader navigate the arguments without necessarily being aware of such symmetries. The three main theorems of the volume were described in [A18, §30]. I gave lecture courses on them in 1994–1995 at the Institute for Advanced Study and the University of Paris VII, and later in 2000, again at the Institute for Advanced Study. Parts of Chapter 4 were also treated heuristically in the earlier article [A9]. In writing this volume, I have added some topics to my original notes. These include the local Langlands classification
xviii
FOREWORD: A SELECTIVE OVERVIEW
for GLpN q, the treatment of inner twists in Chapter 9 and the remarks on Whittaker models in §8.3. I have also had to fill unforeseen gaps in the notes. For example, I did not realize that twisted endoscopy for SOp2nq was needed to formulate the local intertwining relation. In retrospect, it is probably for the best that this second case of twisted endoscopy does have a role here, since it forces us to confront a general phenomenon in a concrete situation. I have tried to make this point explicit in §2.4 with the discussion surrounding the short exact sequence (2.4.10). In any case, I hope that I have accounted for most of the recent work on the subject, in the references and the text. There will no doubt be omissions. I most regret not being able to describe the results of Moeglin on the structure of p-adic packets r ψ ([M1]–[M4]). It is clearly an important problem to establish analogues Π v of her results for archimedean packets. Acknowledgments: I am greatly indebted to the referee, for a close and careful reading of an earlier draft of the manuscript (which went to the end of §8.2), and for many very helpful comments. I also thank Chung Pang Mok, who read a later draft of the manuscript, and offered further helpful suggestions.
CHAPTER 1
Parameters 1.1. The automorphic Langlands group We begin with some general motivation. We shall review some of the fundamental ideas that underlie the theoretical foundations laid by Langlands. This will help us put our theorems into perspective. It will also lead naturally to a formulation of some of the essential objects with which we need to work. We take F to be a local or global field of characteristic 0. In other words, F is a finite extension of either the real field R or a p-adic field Qp , or it is a finite extension of Q itself. Suppose that G is a connected reductive algebraic group over F , which to be concrete we take to be a classical matrix group. For example, we could let G be the general linear group GpN q GLpN q of invertible matrices of rank N over F . In his original paper [L2], Langlands introduced what later became known as the L-group of G. This object is a semidirect product L
p ΓF GG
p of G with the Galois group of a complex dual group G
ΓF
GalpF {F q
p (called an L-action) is of an algebraic closure F of F . The action of ΓF on G determined by its action on a based root datum for G and a corresponding p according to the general theory of algebraic groups. (See splitting for G, [K3, §1.1–§1.3].) It factors through the quotient ΓE {F of ΓF attached to any finite Galois extension E F over which G splits. We sometimes formulate the L-group by the simpler prescription
Gp ΓE{F , since this suffices for many purposes. If G GpN q, for example, the action L
G L GE { F
p is trivial. Since G p is just the complex general linear group of ΓF on G GLpN, Cq in this case, one can often take L
p GLpN, Cq. GG 1
2
1. PARAMETERS
Langlands’ conjectures [L2] predicate a fundamental role for the L-group in the representation theory of G. Among other things, Langlands conjectured the existence of a natural correspondence φ
ÝÑ
π
between two quite different kinds of objects. The domain consists of (continuous) L-homomorphisms φ : ΓF
ÝÑ
L
G,
p (An L-homomorphism between two groups taken up to conjugation by G. that fibre over ΓF is a homomorphism that commutes with the two projections onto ΓF .) The codomain consists of irreducible representations π of GpF q if F is local, and automorphic representations π of GpAq if F is global, taken in each case up to the usual relation of equivalence of irreducible representations. Recall that if F is global, the adele ring is defined as a restricted tensor product
A AF
¹
Fv
v
of completions Fv of F . In this case, the Langlands correspondence should satisfy the natural local-global compatibility condition. Namely, if φv denotes the restriction of φ to the subgroup ΓFv of ΓF (which is defined up to conjugacy), and π is a restricted tensor product π
â v
πv ,
φv
Ñ πv ,
of representations that correspond to these localizations, then π should correspond to φ. We refer the reader to the respective articles [F] and [L6] for a discussion of restricted direct products and automorphic representations. The correspondence φ Ñ π, which remains conjectural, is to be understood in the literal sense of the word. For general G, it will not be a mapping. However, in the case G GLpN q, the correspondence should in fact reduce to a well defined, injective mapping. For local F , this is part of what has now been established, as we will recall in §1.3. For global F , the injectivity would be a consequence of the required local-global compatibility condition and the theorem of strong multiplicity one, or rather its generalization in [JS] that we will also recall in §1.3. However, the correspondence will very definitely not be surjective. In our initial attempts at motivation, we should not lose sight of the fact that the conjectural Langlands correspondence is very deep. For example, even though the mapping φ Ñ π is known to exist for G GLp1q, it takes the form of the fundamental reciprocity laws of local and global class field theory. Its generalization to GLpN q would amount to a formulation of nonabelian class field theory.
1.1. THE AUTOMORPHIC LANGLANDS GROUP
3
Langlands actually proposed the correspondence φ Ñ π with the Weil group WF in place of the Galois group ΓF . We recall that WF is a locally compact group, which was defined separately for local and global F by Weil. It is equipped with a continuous homomorphism WF
ÝÑ
ΓF ,
with dense image. If F is global, there is a commutative diagram
ÝÑ
Ñ
Ñ
WF
ΓFv ã
ã
ÝÑ
WF v
ΓF
for any completion Fv of F , with vertical embeddings defined up to conjugation. (See [T2].) In the Weil form of the Langlands correspondence, φ represents an L-homomorphism from WF to L G. The restriction mapping of continuous functions on ΓF to continuous functions on WF is injective. For this reason, the conjectural Langlands correspondence for Weil groups is a generalization of its version for Galois groups. For G GLpN q, the Weil form of the conjectural correspondence φ Ñ π again reduces to an injective mapping. (In the global case, one has to take π to be an isobaric automorphic representation, a natural restriction introduced in [L7] that includes all the representations in the automorphic spectral decomposition of GLpN q.) If G GLp1q, it also becomes surjective. However, for nonabelian groups G, and in particular for GLpN q, the correspondence will again not be surjective. One of the purposes of Langlands’ article [L7] was to suggest the possibility of a larger group, which when used in place of the Weil group, would give rise to a bijective mapping for GLpN q. Langlands formulated the group as a complex, reductive, proalgebraic group, in the spirit of the complex form of Grothendieck’s motivic Galois group. Kottwitz later pointed out that Langlands’ group ought to have an equivalent but simpler formulation as a locally compact group LF [K3]. It would come with a surjective mapping LF
ÝÑ
WF
onto the Weil group, whose kernel KF should be compact and connected, and (in the optimistic view of some [A17]) even simply connected. If F is local, LF would take the simple form (1.1.1)
LF
#
WF ,
if F is archimedean,
WF
if F is nonarchimedean.
SU p2q,
In this case, LF is actually a split extension of WF by a compact, simply connected group (namely, the trivial group t1u if F is archimedean and the three dimensional compact Lie group SU p2q SU p2, Rq if F is p-adic.) If F is global, LF remains hypothetical. Its existence is in fact one of the deepest
4
1. PARAMETERS
ÝÑ
WF
ÝÑ
Ñ
Ñ
LF
ΓFv ã
ã
ÝÑ
LFv
Ñ
ÝÑ
WFv
ã
problems in the subject. Whatever form it does ultimately take, it ought to fit into a larger commutative diagram
ΓF
for any completion Fv , the vertical embedding on the left again being defined up to conjugation. The hypothetical formal structure of LF is thus compatible with an extension of the Langlands correspondence from WF to LF . This is what Langlands proposed in [L7] (for the proalgebraic form of LF ). The extension amounts to a hypothetical correspondence φ Ñ π, in which φ now represents an L-homomorphism φ : LF ÝÑ L G, p taken again up to G-conjugacy. Here it is convenient to use the Weil form of the L-group L p WF , GG
p inherited from ΓF . An L-homomorphism between for the action of WF on G two groups over WF is again one that commutes with the two projections. There are some minor conditions on φ that are implicit here. For example, since WF and LF are no longer compact, one has to require that for any p be semisimple. If G is not quasisplit, one λ P LF , the image of φpλq in G generally also requires that φ be relevant to G, in the sense that if its image lies in a parabolic subgroup L P of L G, there is a corresponding parabolic subgroup P of G that is defined over F . Suppose again that G GLpN q. Then the hypothetical extended correspondence φ Ñ π again reduces to an injective mapping. However, this time it should also be surjective (provided that for global F , we take the image to be the set of isobaric automorphic representations). If F is local archimedean, so that LF WF , the correspondence was established (for any G in fact) by Langlands [L11]. If F is a local p-adic field, so that LF WF SU p2q, the correspondence was established for GLpN q by Harris and Taylor [HT] and Henniart [He1]. For global F , the correspondence for GLpN q is much deeper, and remains highly conjectural. We have introduced it here as a model to motivate the form of the theorems we seek for classical groups. If G is more general than GLpN q, the extended correspondence φ Ñ π will not reduce to a mapping. It was to account for this circumstance that Langlands introduced what are now called L-packets. We recall that Lpackets are supposed to be the equivalence classes for a natural relation that is weaker than the usual notion of equivalence of irreducible representations. (The supplementary relation is called L-equivalence, since it is arithmetic in nature, and is designed to preserve the L-functions and ε-factors of representations.) The extended correspondence φ Ñ π is supposed to project to
1.1. THE AUTOMORPHIC LANGLANDS GROUP
5
a well defined mapping φ Ñ Πφ from the set of parameters φ to the set of L-packets. For G GLpN q, L-equivalence reduces to ordinary equivalence. The L-packets Πφ then contain one element each, which is the reason that the correspondence φ Ñ π reduces to a mapping in this case. The general construction of L-packets is part of Langlands’ conjectural theory of endoscopy. It will be a central topic of investigation for this volume. We recall at this stage simply that the L-packet attached to a given φ will be intimately related to the centralizer (1.1.2)
Sφ
Cent Impφq, Gp
p of the image φpLF q of φ, generally through its finite quotient in G
Sφ
(1.1.3)
Sφ{Sφ0 Z pGpqΓ.
Following standard notation, we have written Sφ0 for the connected com-
p q for the center of G, p ponent of 1 in the complex reductive group Sφ , Z pG p qΓ for the subgroup of invariants in Z pG p q under the natural action and Z pG of the Galois group Γ ΓF . For G GLpN q, the groups Sφ will all be connected. Each quotient Sφ is therefore trivial. The implication for other groups G is that we will have to find a way to introduce the centralizers Sφ , even though we have no hope of constructing the automorphic Langlands group LF and the general parameters φ. There is a further matter that must also be taken into consideration. Suppose for example that F is global and that G GLpN q. The problem in this case is that the conjectural parametrization of automorphic representations π by N -dimensional representations
φ : LF
ÝÑ
p GLpN, Cq G
is not compatible with the spectral decomposition of L2 GpF qzGpAq . If φ is irreducible, π is supposed to be a cuspidal automorphic representation. Any such representation is part of the discrete spectrum (taken modulo the center). However, there are also noncuspidal automorphic representations in the discrete spectrum. These come from residues of Eisenstein series, and include for example the trivial one-dimensional representation of GpAq. Such automorphic representations will correspond to certain reducible N dimensional representations of LF . How is one to account for them? The answer, it turns out, lies in the product of LF with the supplementary group SU p2q SU p2, Rq. The representations in the discrete automorphic spectrum of GLpN q should be attached to irreducible unitary N dimensional representations of this product. The local constituents of these automorphic representations should again be determined by the restriction of parameters, this time from the product LF SU p2q to its subgroups LFv SU p2q. Notice that if v is a p-adic valuation, the localization LFv
SU p2q WF SU p2q SU p2q v
6
1. PARAMETERS
contains two SU p2q-factors. Each will have its own distinct role. In §1.3, we shall recall the general construction, and why it is the product LF SU p2q that governs the automorphic spectrum of GLpN q. Similar considerations should apply to a more general connected group G over any F . One would consider L-homomorphisms ψ : LF
SU p2q ÝÑ
L
G,
p (the analogue of the unitary condition with relatively compact image in G for GLpN q). If F is global, the parameters should govern the automorphic spectrum of G. If F is local, they ought to determine corresponding local constituents. In either case, the relevant representations should occur in packets Πψ , which are larger and more complicated than L-packets, but which are better adapted to the spectral properties of automorphic representations. These packets should in turn be related to the centralizers
Sψ
Cent Impψq, Gp
and their quotients Sψ
Sψ {Sψ0 Z pGpqΓ.
For G GLpN q, the groups Sψ remain connected. However, for other classical groups G we might wish to study, we must again be prepared to introduce parameters ψ and centralizers Sψ without reference to the global Langlands group LF . The objects of study in this volume will be orthogonal and symplectic groups G. Our general goal will be to classify the representations of such groups in terms of those of GLpN q. In the hypothetical setting of the discussion above, the problem includes being able to relate the parameters ψ for G with those for GLpN q. As further motivation for what is to come, we shall consider this question in the next section. We shall analyze the self-dual, finite dimensional representations of a general group ΛF . Among other things, this exercise will allow us to introduce endoscopic data, the internal objects for G that drive the classification, in concrete terms. 1.2. Self-dual, finite dimensional representations We continue to take F to be any local or global field of characteristic 0. For this section, we let ΛF denote a general, unspecified topological group. The reader can take ΛF to be one of the groups ΓF , WF or LF discussed in §1.1, or perhaps the product of one of these groups with SU p2q. We assume only that ΛF is equipped with a continuous mapping ΛF Ñ ΓF , with connected kernel and dense image. We shall be looking at continuous, N -dimensional representations r : ΛF
ÝÑ
GLpN, Cq.
Any such r factors through the preimage of a finite quotient of ΓF . We can therefore replace ΛF by its preimage. In fact, one could simply take a large
1.2. SELF-DUAL, FINITE DIMENSIONAL REPRESENTATIONS
7
finite quotient of ΓF in place of ΛF , which for the purposes of the present exercise we could treat as an abstract finite group. We say that r is self-dual if it is equivalent to its contragredient representation r_ pλq t rpλq1 , λ P ΛF , t where x Ñ x is the usual transpose mapping. In other words, the equivalence class of r is invariant under the standard outer automorphism θ p xq x_
t
x1 ,
x P GLpN q,
of GLpN q. This condition depends only on the inner class of θ. It remains the same if θ is replaced by any conjugate θg pxq g 1 θpxqg,
g
P GLpN q.
We shall analyze the self-dual representations r in terms of orthogonal and symplectic subgroups of GLpN, Cq. We decompose a given representation r into a direct sum r
`1r1 ` ` `r rr ,
for inequivalent irreducible representations rk : ΛF
ÝÑ
and multiplicities `k with N
GLpNk , Cq,
`1N1
1¤k
¤ r,
`r Nr .
The representation is self-dual if and only if there is an involution k Ø k _ on the indices such that for any k, rk_ is equivalent to rk_ and `k `k_ . We shall say that r is elliptic if it satisfies the further constraint that for each k, k _ k and `k 1. We shall concentrate on this case. Assume that r is elliptic. Then r
r1 ` ` rr ,
for distinct irreducible, self-dual representations ri of ΛF of degree Ni . If i is any index, we can write 1 ri_ pλq Ai ri pλqA i ,
λ P ΛF ,
for a fixed element Ai P GLpNi , Cq. Applying the automorphism θ to each side of this equation, we then see that
_ _ 1 ri pλq A_ i ri pAi q
pA_i AiqripλqpA_i Aiq1.
Since ri is irreducible, the product A_ i Ai is a scalar matrix. We can therefore write t Ai ci Ai , ci P C . If we take the transpose of each side of this equation, we see further that c2i 1. Thus, ci equals 1 or 1, and the nonsingular matrix Ai is either symmetric or skew-symmetric. The mapping x
ÝÑ pAi 1q tx Ai,
x P GLpN q,
8
1. PARAMETERS
of course represents the adjoint relative to the bilinear form defined by Ai . Therefore ri pλq belongs to the corresponding orthogonal group OpAi , Cq or symplectic group SppAi , Cq, according to whether ci equals 1 or 1. Let us write IO and IS for the set of indices i such that ci equals 1 and 1 respectively. We then write rε pλq Aε
à
P
ri pλq,
λ P ΛF ,
i Iε
à
P
Ai ,
i Iε
and Nε
¸
P
Ni ,
i Iε
for ε equal to O or S. Thus AO is a symmetric matrix in GLpNO , Cq, AS is a skew-symmetric matrix in GLpNS , Cq, and rO and rS are representations of ΛF that take values in the respective groups OpAO , Cq and SppAS , Cq. We have established a canonical decomposition r
rO ` rS
of the self-dual representation r into orthogonal and symplectic components. It will be only the equivalence class of r that is relevant, so we are free to replace rpλq by its conjugate B 1 r pλ qB
by a matrix B
P GLpN, Cq. This has the effect of replacing the matrix A AO ` AS
by t BAB. In particular, we could take AO to be any symmetric matrix in GLpNO , Cq, and AS to be any skew-symmetric matrix in GLpNS , Cq. We may therefore put the orthogonal and symplectic groups that contain the images of rO and rS into standard form. It will be convenient to adopt a slightly different convention for these groups. As our standard orthogonal group in GLpN q, we take OpN q OpN, J q,
where
0
J
J pN q
1
..
.
1
0
is the “second diagonal” in GLpN q. This is a group of two connected components, whose identity component is the special orthogonal group (
SOpN q x P OpN q : detpxq 1 .
1.2. SELF-DUAL, FINITE DIMENSIONAL REPRESENTATIONS
As the standard symplectic group in GLpN q, defined for N take the connected group
9
2N 1 even, we
SppN q SppN, J 1 q
for the skew-symmetric matrix J 1 J 1 pN q
0 J pN 1 q
J pN 1 q
0
.
The advantage of this formalism is that the set of diagonal matrices in either SOpN q or SppN q forms a maximal torus. Similarly, the set of upper triangular matrices in either group forms a Borel subgroup. The point is that if t t 1 , x P GLpN q, tx J x J J x J denotes the transpose of x about the second diagonal, the automorphism IntpJ q θ : x ÝÑ JθpxqJ 1
t x 1
of GLpN q stabilizes the standard Borel subgroup of upper triangular matrices. Notice that there is a related automorphism θrpN q IntpJrq θ : x
defined by the matrix (1.2.1)
ÝÑ
0
Jr JrpN q
p1qN
r pxqJr1 Jθ
. ..
1
1
1
,
0
which stabilizes the standard splitting in GLpN q as well. Both of these automorphisms lie in the inner class of θ, and either one could have been used originally in place of θ. Returning to our discussion, we can arrange that A equals J pNO q ` J 1 pNS q. It is best to work with the matrix
JO,S
J pNO , NS q
0
J pN 1 q
J pNO q
S
J pNS1 q 0
,
NS
2NS1 ,
obtained from the obvious embedding of J pNO q ` J 1 pNS q into GLpN, Cq. The associated elliptic representation r from the given equivalence class then maps ΛF to the corresponding subgroup of GLpN, Cq, namely the subgroup OpNO , Cq SppNS , Cq
defined by the embedding
px, yq ÝÑ where yij are the four y P SppNS , Cq.
pNS1
y11 0 y12 0 x 0 , y21 0 y22
NS1 q-block components of the matrix
10
1. PARAMETERS
The symplectic part rS of r is the simpler of the two. Its image is contained in the connected complex group pS G
SppNS , Cq.
This in turn is the dual group of the split classical group GS
SOpNS
1 q.
The orthogonal part rO of r is complicated by the fact that its image is contained only in the disconnected group OpNO , Cq. Its composition with the projection of OpNO , Cq onto the group OpNO , Cq{SOpNO , Cq Z{2Z
of components yields a character η on ΛF of order 1 or 2. Since we are assuming that the kernel of the mapping of ΛF to ΓF is connected, η can be identified with a character on the Galois group ΓF of order 1 or 2. This in turn determines an extension E of F of degree 1 or 2. Suppose first that NO is odd. In this case, the matrix pI q in OpNO q represents the nonidentity component, and the orthogonal group is a direct product OpNO , Cq SOpNO , Cq Z{2Z. We write
pO , SOpNO , Cq G
where GO is the split group SppNO 1q over F . We then use η to identify the direct product L pGqE{F GpO ΓE{F
with a subgroup of OpNO , Cq, namely SOpNO , Cq or OpNO , Cq, according to whether η has order 1 or 2. We thus obtain an embedding of the (restricted) L-group of GO into GLpNO , Cq. Assume next that N is even. In this case, the nonidentity component in OpN q acts by an outer automorphism on SOpNO q. We write pO , SOpNO , Cq G
where GO is now the corresponding quasisplit orthogonal group SOpNO , η q over F defined by η. In other words, GO is the split group SOpNO q if η is trivial, and the non-split group obtained by twisting SOpNO q over E by the given outer automorphism if η is nontrivial. Let w rpNO q be the permutation matrix in GLpNO q that interchanges the middle two coordinates, and leaves the other coordinates invariant. We take this element as a representative of the nonidentity component of OpNO , Cq. We then use η to identify the semidirect product L pGO qE{F GpO ΓE{F
with a subgroup of OpNO , Cq, namely SOpNO , Cq or OpNO , Cq as before. We again obtain an embedding of the (restricted) L-group of GO into GLpNO , Cq.
1.2. SELF-DUAL, FINITE DIMENSIONAL REPRESENTATIONS
11
We have shown that the elliptic self-dual representation r factors through the embedded subgroup L
GE { F
L pG O qE { F L pG S qE { F
of GLpN, Cq attached to a quasisplit group
G GO GS
over F . The group G is called a θ-twisted endoscopic group for GLpN q. It is determined by r, and in fact by the decomposition N NO NS and the character η ηG (of order 1 or 2) attached to r. The same is true of the L-embedding ξ
ξO,S,η :
L
p ΓF GG
Ñ
ã
L
GpN q GLpN, Cq ΓF ,
obtained by inflating the embedding above to the full L-groups. It is convenient to form the semidirect product
pN q GLpN q xθy GpN q xθrpN qy, where xθy and xθrpN qy are the groups of order 2 generated by the automorphisms θ and θrpN q. We write r 0 pN q GLpN q 1 GpN q 1 G r G
for the identity component, which we can of course identify with the general linear group GLpN q, and (1.2.2)
r pN q GLpN q θ G
GpN q θrpN q
for the other connected component. Given r, and hence also the decomposition N NO NS , we can form the semisimple element
1 θ, JO,S p r pN q of complex points GLpN, Cq θ. The complex group in the “dual set” G p p p G GO GS , attached to r as above, is then the connected centralizer of s sO,S
s in the group
p pN q G r 0 pN q GLpN, Cq. G p
r pN q, since it becomes The triplet pG, s, ξ q is called an endoscopic datum for G r pN q with the a special case of the terminology of [KS, p. 16] if we replace G 0 r r pair G pN q, θpN q . The endoscopic datum pG, s, ξ q we have introduced has the property of being elliptic. This is a consequence of our condition that the original selfdual representation r is elliptic. A general (nonelliptic) endoscopic datum r pN q is again a triplet pG, s, ξ q, where G is a quasisplit group over F , s for G p r pN q of which G p is the connected centralizer in is a semisimple element in G
r 0 pN q, and ξ is an L-embedding of L G into the L-groups L G r 0 pN q L G pN q G of GLpN q. (In the present setting, we are free to take either the Galois or p and Weil form of the L-groups.) We require that ξ equal the identity on G, p
12
1. PARAMETERS
r 0 pN q of the image of ξ lie in the full centralizer that the projection onto G r pN q then said to be of s. The endoscopic group G (or datum pG, s, ξ q) for G Γ p p q of G p invariant elliptic if Z pGq , the subgroup of elements in the center Z pG under the action of the Galois group Γ ΓF , is finite. The notion of isomorphism between two general endoscopic data is defined in [KS, p. 18]. In the case at hand, it is given by an element g in the p pN q GLpN, Cq whose action by conjugation is compatible in dual group G a natural sense with the two endoscopic data. We write p
r N pGq Aut r pG q Aut GpN q
for the group of isomorphisms of the endoscopic datum G to itself. The main role for this group is in its image r N pGq Aut r N pGq{rIntN pGq Out
in the group of outer automorphisms of the group G over F . (Following standard practice, we often let the endoscopic group G represent a full endoscopic datum pG, s, ξ q, or even an isomorphism class of such data.) If G r N pGq is represents one of the elliptic endoscopic data constructed above, Out trivial if the integer NO is odd or zero. In the remaining case that NO is even r N pGq is a group of order 2, the nontrivial element being and positive, Out the outer automorphism induced by the nontrivial connected component of OpNO , Cq. We write r pN q ErpN q E G r pN q, and for the set of isomorphism classes of endoscopic data for G r pN q Erell pN q Eell G
for the subset of classes in ErpN q that are elliptic. The data pG, s, ξ q, attached to equivalence classes of elliptic, self-dual representations r as above, form a complete set of representatives of Erell pN q. The set Erell pN q is thus parametrized by triplets pNO , NS , η q, where NO NS N is a decomposition of N into nonnegative integers with NS even, and η ηG is a character of ΓF of order 1 or 2 with the property that η 1 if NO 0, and η 1 if NO 2. (The last constraint is required in order that the datum be elliptic.) The goal of this volume is to describe the representations of the classical groups G in terms of those of GLpN q. The general arguments will p is either purely orthogonal be inductive. For this reason, the case in which G or purely symplectic will have a special role. Accordingly, we write r pN q Ersim pN q Esim G
for the set of elements in Erell pN q that are simple, in the sense that one of the integers NO or NS vanishes. We then have a chain of sets (1.2.3)
Ersim pN q Erell pN q ErpN q,
1.2. SELF-DUAL, FINITE DIMENSIONAL REPRESENTATIONS
13
which are all finite if F is local, and all infinite if F is global. The elements G P ErpN q are usually called twisted endoscopic data, since they are attached to the automorphism θ. We shall have to work also with ordinary (untwisted) endoscopic data, at least for the quasisplit orthogonal and symplectic groups G that represent elements in Ersim pN q. An endoscopic r pN q. It datum G1 for G is similar to what we have described above for G 1 1 1 1 amounts to a triplet pG , s , ξ q, where G is a (connected) quasisplit group p of which G p 1 is the connected centralover F , s1 is a semisimple element in G 1 L 1 p and ξ is an L-embedding of G into L G. We again require that izer in G, p 1 , and that its image lie in the centralizer of s1 in ξ 1 equal the identity on G L G. (See [LS1, (1.2)], a special case of the general definition in [KS], which we have specialized further to the case at hand.) There is again the notion of isomorphism of endoscopic data, which allows us to form the associated finite group OutG pG1 q AutG pG1 q{IntG pG1 q of outer automorphisms of any given G1 . We write E pGq for the set of isomorphism classes of endoscopic data G1 for G, and Eell pGq for the subset p 1 qΓ is finite. We then have a of data that are elliptic, in the sense that Z pG second chain of sets Esim pGq Eell pGq E pGq,
(1.2.4)
where Esim pGq tGu is the subset consisting of G alone. Similar definitions apply to groups G that represent more general data in ErpN q. It is easy to describe the set Eell pGq, for any G P Ersim pN q. It suffices to p with eigenvalues 1. For example, in the consider diagonal matrices s1 P G p SppN, Cq (with N NS even), it first case that G SOpN 1q and G is enough to take diagonal matrices of the form
s1
I12
I21
0
,
0 I12 where I12 is the identity matrix of rank N12 N11 {2, and I21 is the identity matrix of rank N21 . The set Eell pGq is parametrized by pairs pN11 , N21 q of even integers with 0 ¤ N 1 ¤ N 1 and N N 1 N 1 . The corresponding 1
2
1
2
1q SOpN21
1 q,
endoscopic groups are the split groups G1
SOpN11
with dual groups
p1 G
p SppN11 , Cq SppN21 , Cq SppN, Cq G. The group OutG pG1 q is trivial in this case unless N11 N21 , in which case it
has order 2. The other cases are similar. In the second case that G SppN 1q p SOpN, Cq (with N NO odd), Eell pGq is parametrized by pairs and G 1 pN1, N21 q of nonnegative even integers with N N11 pN21 1q, and characters
14
1. PARAMETERS
η 1 on ΓF with pη 1 q2 quasisplit groups
1. The corresponding endoscopic groups are the G1
SOpN11 , η1q SppN21 q,
with dual groups p1 G
p SOpN11 , Cq SOpN21 1, Cq SOpN, Cq G. p SOpN, Cq (with N NO In the third case that G SOpN, η q and G even), Eell pGq is parametrized by pairs of even integers pN11 , N21 q with 0 ¤ N11 ¤ N21 and N N11 N21 , and pairs pη11 , η21 q of characters on ΓF with pη11 q2 pη21 q2 1 and η η11 η21 . The corresponding endoscopic groups
are the quasisplit groups G1
SOpN11 , η11 q SOpN21 , η21 q,
with dual groups p1 G
p SOpN11 , Cq SOpN21 , Cq SOpN, Cq G.
In the second and third cases, each character η 1 has to be nontrivial if the corresponding integer N 1 equals 2, while if N 1 0, η 1 must of course be trivial. In these cases, the group OutG pG1 q has order 2 unless N is even and N11 N21 ¥ 1, in which case it is a product of two groups of order 2, or N11 0, in which case the group is trivial. p SOpN, Cq with N Observe that in the even orthogonal case, where G 1 even, the endoscopic data G P Eell pGq will not be able to isolate constituents ri of r of odd dimension. The discrepancy is made up by a third kind of endoscopic datum. These are the twisted endoscopic data for the even orthogonal groups G SOpN, η q in Ersim pN q. For any such G, let r r G θ, G
(1.2.5)
be the nonidentity component in the semi-direct product of G with the group of order two generated by the outer automorphism θr of SOpN q. In r 1 , sr1 , ξr1 q, where G r1 this setting a (twisted) endoscopic datum is a triplet pG 1 is a quasisplit group over F , sr is a semisimple element in the “dual set”
r G p θr of which G r 1 is the connected centralizer in G, p and ξr1 is an G r 1 into L G, all being subject also to further conditions L-embedding of L G r q E pG r q of isomorphism classes and definitions as above. The subset Eell pG r is parametrized by pairs of odd integers of elliptic endoscopic data for G 1 1 1 1 r r r r pN1, N2q, with 0 ¤ N1 ¤ N2 and N Nr11 Nr21 , and pairs of characters pηr11 , ηr21 q on ΓF , with pηr11 q2 pηr21 q2 1 and η ηr11 ηr21 . The corresponding endoscopic groups are the quasisplit groups p
p
r1 G
SppNr11 1q SppNr21 1q,
with dual groups r1 G p
p SOpNr11 , Cq SOpNr21 , Cq SOpN, Cq G.
p q
1.3. REPRESENTATION OF GL N
15
r q will have to be part of our analysis. However, its role The family Eell pG will be subsidiary to that of the two primary families Erell pN q and Eell pGq. We have completed our brief study of elliptic self-dual representations r. Remember that we are regarding these objects as parameters, in the spirit of §1.1. We have seen that a parameter for GLpN q factors into a product of two parameters for quasisplit classical groups. The products are governed by twisted endoscopic data G P Erell pN q. They can be refined further according to ordinary endoscopic data G1 P Eell pGq. Thus, while the parameters are not available (for lack of a global Langlands group LF ), the endoscopic data that control many of their properties are. Before we can study the ramifications of this, we must first formulate a makeshift substitute for the parameters attached to our classical groups. We shall do so in §1.4, after a review in §1.3 of the representations of GLpN q that will serve as parameters for this group. We have considered only the self dual representation r that are elliptic, since it is these objects that pertain directly to our theorems. Before going on, we might ask what happens if r is not elliptic. A moment’s reflection reveals that any such r factors through subgroups of GLpN, Cq attached to several data G P Erell pN q, in contrast to what we have seen in the elliptic case. This is because r also factors through a subgroup attached to a datum M in the complement of Erell pN q in E pN q, and because any such M can be identified with a proper Levi subgroup of several G. The analysis of general self-dual representations r is therefore more complicated, though still not very difficult. It is best formulated in terms of the centralizers r pN q Srr pN q Sr G
Cent Imprq, GprpN q
and
pq Sr Sr pGq Cent Imprq, G of the images of r. We shall return to this matter briefly in §1.4, and then more systematically as part of the general theory of Chapter 4.
1.3. Representations of GLpN q A general goal, for the present volume and beyond, is to classify representations of a broad class of groups in terms of those of general linear groups. What makes this useful is the fact that much of the representation theory of GLpN q is both well understood and relatively simple. We shall review what we need of the theory. Suppose first that F is local. In this case, we can replace the abstract group ΛF of the last section by the local Langlands group LF defined by (1.1.1). The local Langlands classification parametrizes irreducible representations of GLpN, F q in terms of N -dimensional representations φ : LF
ÝÑ
GLpN, Cq.
Before we state it formally, we should recall a few basic notions.
16
1. PARAMETERS
Given a finite dimensional (semisimple, continuous) representation φ of LF , we can form the local L-function Lps, φq, a meromorphic function of s P C. We can also form the local ε-factor εps, φ, ψF q, a monomial of the form abs which also depends on a nontrivial additive character ψF of F . If F is archimedean, we refer the reader to the definition in [T2, §3]. If F is p-adic, we extend φ analytically to a representation of the product of WF with the complexification SLp2, Cq of the subgroup SU p2q of LF . We can then form the representation
χφ pwq φ w,
|w|
1 2
0
|w|
0
1 2
,
w
P WF ,
of WF , where |w| is the absolute value on WF , and the nilpotent matrix
1 1 Nφ log φ 1, 0 1
.
The pair Vφ pχφ , Nφ q gives a representation of the Weil-Deligne group [T2, (4.1.3)], for which we define an L-function Lps, φq Z pVφ , qFs q
and ε-factor
εps, φ, ψF q εpVφ , qFs q, following notation in [T2, §4]. (We have written qF here for the order of the residue field of F .) Of particular interest are the tensor product L-function Lps, φ1 φ2 q Lps, φ1 b φ2 q
and ε-factor
εps, φ1 φ2 , ψF q εps, φ1 b φ2 , ψF q, attached to any pair of representations φ1 and φ2 of LF . We expect also to be able to attach local L-functions Lps, π, rq and εfactors εps, π, r, ψF q to any connected reductive group G over F , where π ranges over irreducible representations of GpF q, and r is a finite dimensional representation of L G. For general G, this has been done in only the simplest of cases. However, if G is a product GpN1 q GpN2 q of general linear groups, there is a broader theory [JPS]. (See also [MW2, Appendice].) It applies to any representation π π1 π2 , in the case that r is the standard representation (1.3.1)
rpg1 , g2 q : X
ÝÑ
g1 X
tg2,
gi
P GpNiq,
of
p GLpN1 , Cq GLpN2 , Cq G on the space of complex pN1 N2 q-matrices X. The theory yields functions
Lps, π1 π2 q Lps, π, rq
and
εps, π1 π2 , ψF q εps, π, r, ψF q,
p q
1.3. REPRESENTATION OF GL N
17
known as local Rankin-Selberg convolutions. The local classification for GLpN q is essentially characterized by being compatible with local Rankin-Selberg convolutions. It has other important properties as well. Some of these relate to supplementary conditions we can impose on the parameters φ as follows. Suppose for a moment that G is any connected group over F . We p write ΦpGq for the set of G-orbits of (semisimple, continuous, G-relevant) L-homomorphisms φ : LF
ÝÑ
L
G,
and ΠpGq for the set of equivalence classes of irreducible (admissible) representations of GpF q. (See [Bo].) These sets come with parallel chains of subsets Φ2,bdd pGq Φbdd pGq ΦpGq
and
Π2,temp pGq Πtemp pGq ΠpGq.
In the second chain, Πtemp pGq denotes the set of tempered representations in ΠpGq, and Π2,temp pGq Π2 pGq X Πtemp pGq,
where Π2 pGq is the set of representations in ΠpGq that are essentially square integrable, in the sense that after tensoring with the appropriate positive character on GpF q, they are square integrable modulo the centre of GpF q. Recall that Πtemp pGq can be described informally as the set of represen tations π P ΠpGq that occur in the spectral decomposition of L2 GpF q . Similarly, Π2,temp pGq is the set of π that occur in the discrete spectrum (taken modulo the center). In the first chain, Φbdd pGq denotes the set of φ P ΦpGq whose image in L G projects onto a relatively compact subset of p and G, Φ2,bdd pGq Φ2 pGq X Φbdd pGq,
where Φ2 pGq is the set of parameters φ in ΦpGq whose image does not lie in any proper parabolic subgroup L P of L G. In the case G GpN q GLpN q of present concern, we write ΦpN q Φ GLpN q and ΠpN q Π GLpN q , and follow similar notation for the corresponding subsets above. Then ΦpN q can be identified with the set of equivalence classes of (semisimple, continuous) N -dimensional representations of LF . The subset Φsim pN q Φ2 pN q
consists of those representations that are irreducible, while Φbdd pN q corresponds to representations that are unitary. On the other hand, the set Πunit pN q of unitary representations in ΠpN q properly contains Πtemp pN q, if N ¥ 2. It has an elegant classification [V2], [Tad1], but our point here is
18
1. PARAMETERS
that it is not parallel to the set of N -dimensional representations φ that are unitary. We do observe that Π2,temp pN q Π2 pN q X Πunit pN q Π2,unit pN q, so the notions of tempered and unitary are the same for square integrable representations. If F is p-adic, we can write Πscusp,temp pN q (resp. Πscusp pN q) for the set of supercuspidal representations in Π2,temp pN q (resp. Π2 pN q). We can also write Φscusp,bdd pN q (resp. Φscusp pN q) for the set of φ in Φsim,bdd pN q (resp. Φsim pN q) that are trivial on the second factor SU p2q of LF . If F is archimedean, it is natural to take Πscusp,temp pN q and Πscusp pN q to be empty unless GLpN, F q is compact modulo the center (which is a silly way of saying that N 1), in which case we can take them tobe the correspond ing sets Π2,temp GLp1q Πtemp GLp1q and Π2 GLp1q Π GLp1q . We thus have two parallel chains (1.3.2)
Φscusp,bdd pN q Φsim,bdd pN q Φbdd pN q ΦpN q,
and (1.3.3)
Πscusp,temp pN q Π2,temp pN q Πtemp pN q ΠpN q,
for our given local field F . The local classification for G GLpN q can now be formulated as follows. Theorem 1.3.1 (Langlands [L11], Harris-Taylor [HT], Henniart [He1]). There is a unique bijective correspondence φ Ñ π from ΦpN q onto ΠpN q such that
piq
φbχ
ÝÑ π b pχ detq, for any character χ in the group Φp1q Πp1q, piiq det φ ÝÑ ηπ , for the central character ηπ of π, and
ÝÑ π_, for the contragredient involutions _ on ΦpN q and ΠpN q, and such that if φi ÝÑ πi , φi P ΦpNi q, i 1, 2,
piiiq
φ_
then
pivq
Lps, π1 π2 q Lps, φ1 φ2 q
and
pv q
εps, π1 π2 , ψF q εps, φ1 φ2 , ψF q.
Furthermore, the bijection is compatible with the two chains (1.3.2) and (1.3.3), in the sense that it maps each subset in (1.3.2) onto its counterpart in (1.3.3).
p q
1.3. REPRESENTATION OF GL N
19
We now take F to be global. This brings us to the representations that will be the foundation of all that follows. They are the objects in the set Acusp pN q Acusp GLpN q
of (equivalence classes of) unitary, cuspidal automorphic representations of GLpN q. Assume first that G is an arbitrary connected group over the global field F . To suppress the noncompact part of the center, one often works with the closed subgroup (
G pA q1
x P GpAq : |χpxq| 1, χ P X pGqF of GpAq, where X pGqF is the additive group of characters of G defined over F . We recall that GpF qzGpAq1 has finite volume, and that there is a sequence
L2cusp GpF qzGpAq1
L2disc GpF qzGpAq1 L2 GpF qzGpAq1
,
of embedded, rightGpAq1 -invariant Hilbert spaces. In particular, the space L2cusp GpF qzGpAq1 of cuspidal functions in L2 GpF qzGpAq1 is contained in the subspace L2disc GpF qzGpAq1 that decomposes under the action of GpAq1 into a direct sum of irreducible representations. We can then introduce a corresponding chain of subsets of irreducible automorphic representations Acusp pGq A2 pGq ApGq.
By definition, Acusp pGq, A2 pGq and ApGq denote the subsets of irreducible unitary representations π of GpAq whose restrictions to GpAq1 are irreducible constituents of the respective spaces L2cusp , L2disc and L2 . We shall also write Acusp pGq and A2 pGq for the analogues of Acusp pGq and A2 pGq defined without the condition that π be unitary. (The definition of ApGq here is somewhat informal, and will be used only for guidance. It can in fact be made precise at the singular points in the continuous spectrum where there might be some ambiguity.) We specialize again to the case G GLpN q, taken now over the global field F . We write ApN q A GLpN q , with similar notation for the corresponding subsets above. Observe that GLpN, Aq1 is the group of adelic matrices x P GLpN, Aq whose determinant has absolute value 1. In this case, the set ApN q is easy to characterize. It is composed of the induced representations π
IP pπ1 b b πr q,
πi
P A2pNiq,
where P is the standard parabolic subgroup of block upper triangular matrices in G GLpN q corresponding to a partition pN1 , . . . , Nr q of N , with the standard Levi subgroup of block diagonal matrices MP
GLpN1q GLpNr q.
This follows from the theory of Eisenstein series [L1], [L5], [A1] (valid for any G), and the fact [Be] (special to G GLpN q) that an induced
20
1. PARAMETERS
representation IPG pσ q is irreducible for any representation σ of MP pAq that is irreducible and unitary. Let us also write A pN q for the set of induced representations π as above, but with the components πi now taken from the larger sets A2 pNi q. These representations can be reducible at certain points, although they typically remain irreducible. We thus have a chain of sets (1.3.4)
Acusp pN q A2 pN q ApN q A
pN q
of representations, for our given global field F . This is a rough global analogue of the local sequence (1.3.3). We have dropped the subscript “temp” in the global notation, since the local components of a constituent of L2 GLpN, F qzGLpN, Aq1 need not be tempered, and added the superscript to remind ourselves that A pN q contains full induced representations, rather than irreducible quotients. There are two fundamental theorems on the automorphic representations of GLpN q that will be essential to us. The first is the classification of automorphic representations by Jacquet and Shalika, while the second is the characterization by Moeglin and Waldspurger of the discrete spectrum in terms of cuspidal spectra. We shall review each in turn. Suppose again that G is an arbitrary connected group over the global field F . An automorphic representation π of G is among other things, a weakly continuous, irreducible representation of GpAq. As such, it can be written as a restricted tensor product π
â
πv
v
of irreducible representations of the local groups GpFv q, almost all which are unramified [F]. Recall that if πv is unramified, the group Gv G{Fv is unramified. This means that Fv is a p-adic field, that Gv is quasisplit, and p factors through the infinite cyclic quotient that the action of WFv on G WFv {IFv
xFrobv y
of WFv by the inertia subgroup IFv with canonical generator Frobv . Recall also that the local Langlands correspondence has long existed in this very p particular context. The G-orbit of homomorphisms φv : LFv
ÝÑ
L
Gv
Gp xFrobv y
in ΦpGv q to which πv corresponds factors through the quotient LFv { IFv
SU p2q WF {IF v
v
of LFv . The resulting mapping πv
ÝÑ cpπv q φv pFrobv q
is a bijection from the set of unramified representations of GpFv q (relative to any given hyperspecial maximal compact subgroup Kv GpFv q) and the p set of semisimple G-orbits in L Gv that project to the Frobenius generator
p q
1.3. REPRESENTATION OF GL N
21
of WFv {IFv . Let cv pπ q be the image of cpπv q in L G under the embedding of L Gv into L G that is defined canonically up to conjugation. In this way, the automorphic representation π of G gives rise to a family of semisimple conjugacy classes ( cS pπ q cv pπ q : v R S in L G, where S is some finite set of valuations of F outside of which G is unramified. S pGq be the set of families Let Caut cS
tcv : v R S u
of semisimple conjugacy classes in L G obtained in this way. That is, cS cS pπ q, for some automorphic representation π of G. We define Caut pGq 1 to be the set of equivalence classes of such families, cS and pc1 qS being equivalent if cv equals c1v for almost all v. We then have a mapping π
ÝÑ cpπq
from the set of automorphic representations of G onto Caut pGq. The families cS pπ q arise most often in the guise of (partial) global Lfunctions. Suppose that r is a finite dimensional representation of L G that is unramified outside of S. The corresponding incomplete L-function is given by an infinite product ¹
LS ps, π, rq
Lps, πv , rv q
R
v S
of unramified local L-functions
Lps, πv , rv q det 1 r cpπv q qvs
1
,
v
R S,
which converges for the real part of s large. It is at the ramified places v P S that the construction of local L-functions and ε-factors poses a challenge. As the reader is no doubt well aware, the completed L-function Lps, π, rq
¹
Lps, πv , rv q
v
is expected to have analytic continuation as a meromorphic function of s P C that satisfies the functional equation (1.3.5)
Lps, π, rq εps, π, rqLp1 s, π, r_ q,
where r_ is the contragredient of r, and εps, π, rq
¹
R
εps, πv , rv , ψFv q.
v S
Here, ψFv stands for the localization of a nontrivial additive character ψF on A{F that is unramified outside of S. We return again to the case G GLpN q. The elements in the set Caut pN q Caut GLpN q
22
1. PARAMETERS
can be identified with families of semisimple conjugacy classes in GLpN, Cq, defined up to the equivalence relation above. It is convenient to restrict the domain of the corresponding mapping π Ñ cpπ q. Recall that general automorphic representations for GLpN q can be characterized as the irreducible constituents of induced representations ρ I P pπ 1 b b π r q ,
πi
AcusppNiq.
(See Proposition 1.2 of [L6], which applies to any G.) With this condition, the nonunitary induced representation ρ can have many irreducible constituents. However, it does have a canonical constituent. This is the irreducible representation â π πv , v
P Π GLpN qv is obtained from the local Langlands parameter φv φ1,v ` ` φr,v , φi,v ÝÑ πi,v , in Φ GLpN qv by Theorem 1.3.1 (applied to both GLpN qv and its subgroups GLpNi qv ). Isobaric representations are the automorphic representations π of GLpN q obtained in this way. The equivalence class of π does not change if we reorder the cuspidal representations tπi u. We formalize this property where πv
by writing (1.3.6)
π
π1
`
`
πr ,
πi
P AcusppNiq,
in the notation [L7, §2] of Langlands, where the right hand side is regarded as a formal, unordered direct sum. Theorem 1.3.2 (Jacquet-Shalika [JS]). The mapping π
π1
`
`
πr
ÝÑ cpπq,
from the set of equivalence classes of isobaric automorphic representations π of GLpN q to the set of elements c cpπ q in Caut pN q, is a bijection.
Global Rankin-Selberg L-functions Lps, π1 π2 q and ε-factors εps, π1 π2 q are defined as in the local case. They correspond to automorphic representations π π1 π2 of a group G GLpN1 q GLpN2 q, p The analytic behaviour of and the standard representation (1.3.1) of G. these functions is now quite well understood [JPS], [MW2, Appendice]. In particular, Lps, π1 π2 q has analytic continuation with functional equation (1.3.5). Moreover, if π1 and π2 are cuspidal, Lps, π1 π2 q is an entire function of s unless N1 equals N2 , and π2_ is equivalent to the representation π1 pg1 q | det g1 |s1 1 ,
g1
P GLpN1, Aq,
for some s1 P C, in which case Lps, π1 π2 q has only a simple pole at s s1 . It is this property that is used to prove Theorem 1.3.2. Theorem 1.3.2 can be regarded as a characterization of (isobaric) automorphic representations of GLpN q in terms of simpler objects, families of semisimple conjugacy classes. However, it does not in itself characterize
p q
1.3. REPRESENTATION OF GL N
23
the spectral properties of these representations. For example, it is not a priori clear that the representations in ApN q, or even its subset A2 pN q, are isobaric. The following theorem provides the necessary corroboration. Theorem 1.3.3 (Moeglin-Waldspurger [MW2]). The representations π P A2 pN q are parametrized by the set of pairs pm, µq, where N mn is divisible by m, and µ is a representation in Acusp pmq. If P is the standard parabolic subgroup corresponding to the partition pm, . . . , mq of N , and σµ is the representation p n 1 q n 1 n3 x ÝÑ µpx1 q| det x1 | 2 b µpx2 q| det x2 | 2 b b µpxn q| det xn | 2 of the Levi subgroup
MP pAq x px1 , . . . , xn q : xi
P GLpm, Aq
(
,
then π is the unique irreducible quotient of the induced representation IP pσµ q. Moreover, the restriction of π to GLpN, Aq1 occurs in the discrete spectrum with multiplicity one.
Consider the representation π P A2 pN q described in the theorem. For any valuation v, its local component πv is the Langlands quotient of the local component IP pσµ,v q of IP pσµ q. It then follows from the definitions that πv is the irreducible representation of GLpN, Fv q that corresponds to the Langlands parameter of the induced representation IP pσµ,v q. In other words, the automorphic representation π is isobaric. We can therefore write (1.3.7)
π
µ
n 1 2
`
µ
n 3 2
`
`
µ
pn2 1q
,
in the notation (1.3.6), for cuspidal automorphic representations µpiq : xi
ÝÑ µpxq|x|i,
x P GLpm, Aq.
Theorem 1.3.3 also provides a description of the automorphic representations in the larger set ApN q. For as we noted above, ApN q consists of the set of irreducible induced representations
IP pπ1 b b πr q, πi P A2pNiq, where N N1 Nr is again a partition of N , and P is the corresponding standard parabolic subgroup of GLpN q. It is easy to see from its π
irreducibility that π is also isobaric. We can therefore write (1.3.8)
π
π1
`
`
πr ,
πi
P A2pNiq.
Observe that despite the notation, (1.3.8) differs slightly from the general isobaric representation (1.3.6). Its constituents πi are more complex, since they are not cuspidal, but the associated induced representation is simpler, since it is irreducible. There is another way to view Theorem 1.3.3. Since F is global, the Langlands group LF is not available to us. If it were, we would expect its set ΦpN q of (equivalence classes of) N -dimensional representations φ : LF
ÝÑ
GLpN, Cq
24
1. PARAMETERS
to parametrize the isobaric automorphic representations of GLpN q. A very simple case, for example, is the pullback |λ| to LF of the absolute value on the quotient WFab GLp1, F qzGLp1, Aq
of LF . This of course corresponds to the automorphic character on GLp1q given by the original absolute value. We would expect the subset Φsim,bdd pN q Φ2,bdd pN q
of irreducible unitary representations in ΦpN q to parametrize the unitary cuspidal automorphic representations of GLpN q. How then are we to account for the full discrete spectrum A2 pN q? A convenient way to do so is to take the product of LF with the group SU p2q SU p2, Rq. Let us write ΨpN q temporarily for the set of (equivalence classes of) unitary N -dimension representations ψ : LF
SU p2q ÝÑ
GLpN, Cq.
According to Theorem 1.3.3, it would then be the subset Ψsim pN q Ψ2 pN q
of irreducible representations in ΨpN q that parametrizes A2 pN q. Indeed, any ψ P Ψsim pN q decomposes uniquely as a tensor product µ b ν of irreducible representations of LF and SU p2q. It therefore gives rise to a pair pm, µq, in which µ P Acusppmq represents a unitary cuspidal automorphic representation of GLpN q as well as the corresponding irreducible, unitary, m-dimensional representation of LF . Conversely, given any such pair, we again identify µ P Acusp pmq with the corresponding representation of LF , and we take ν to be the unique irreducible representation of SU p2q of degree n N m1 . We are identifying any finite dimensional representation of SU p2q with its analytic extension to SLp2, Cq. With this convention, we attach an N dimensional representation φψ : u
ÝÑ
ψ u,
of LF to any ψ P ΨpN q. If ψ decomposes as a direct sum
|u| 0
1 2
0
|u|
1 2
,
u P LF ,
µ b ν belongs to the subset ΨsimpN q, φψ
` ` µpuq|u|p q, to which we associate the induced representation IP pσµ q of Theoreom 1.3.3. According to the rules of the hypothetical global correspondence from ΦpN q to isobaric automorphic representations of GLpN q, it is the unique irreducible quotient π πψ of IP pσµ q that is supposed to correspond to the u
ÝÑ µpuq|u|
n 1 2
n 1 2
parameter φψ . This representation is unitary, as of course is implicit in Theorem 1.3.3. The mapping ψ Ñ πψ is thus an explicit realization of the
p q
1.3. REPRESENTATION OF GL N
25
bijective correspondence from Ψsim pN q to A2 pN q implied by the theorem of Moeglin and Waldspurger (and the existence of the group LF ). More generally, suppose that ψ belongs to the larger set ΨpN q. Then the isobaric representation πψ attached to the parameter φψ P ΦpN q belongs to ApN q. It can be described in the familiar way as the irreducible unitary representation induced from a unitary representation of a Levi subgroup. The mapping ψ Ñ πψ becomes a bijection from ΨpN q to ApN q. In general, the restriction ψ Ñ φψ of parameters is an injection from ΨpN q into ΦpN q. The role of the set ΨpN q we have just defined in terms of the supplementary group SU p2q is thus to single out the subset of ΦpN q that corresponds to the subset ApN q of “globally tempered” automorphic representations. We are also free to form larger sets Ψ
pN q ΨpN q
and
Ψsim pN q Ψsim pN q of representations of LF SU p2q by removing the condition that they be unitary. Any element ψ P Ψ pN q is then a direct sum of irreducible representations ψi P Ψsim pNi q. The components ψi should correspond to automorphic representations πi πψi in A2 pNi q. The corresponding induced representation πψ IP pπ1 b b πr q then belongs to the set of A pN q of (possibly reducible) representations of GLpN, Aq introduced above. Notice that the extended mapping ψ Ñ φψ from Ψ pN q to ΦpN q is no longer injective. In particular, πψ will not in general be equal to the automorphic representation corresponding to φψ . However, the mapping ψ Ñ πψ will be a bijection from Ψ pN q to A pN q. In the interests of symmetry, we can also write Ψcusp pN q Φsim,bdd pN q
for the set of representations ψ that are trivial on the factor SU p2q. This gives us a chain of sets (1.3.9)
Ψcusp pN q Ψsim pN q ΨpN q Ψ
pN q
that is parallel to (1.3.4). We will then have a bijective correspondence ψ Ñ πψ that takes each set in (1.3.9) to its counterpart in (1.3.4). The global parameter sets in the chain (1.3.9) are hypothetical, depending as they do on the global Langlands group LF . However, their local analogues are not. They can be defined for the general linear group G0v pN q GLpN qv over any completion Fv of F . Replacing LF by LFv in the definitions above, we obtain local parameter sets (1.3.10)
Ψcusp,v pN q Ψsim,v pN q Ψv pN q Ψv pN q.
We also obtain a restriction mapping ψ Ñ ψv from the hypothetical global set Ψ pN q to the local set Ψv pN q. The generalized Ramanujan conjecture
26
1. PARAMETERS
for GLpN q implies that this mapping takes ΨpN q to the subset Ψv pN q of Ψv pN q. However, the conjecture is not known. For this reason, we will be forced to work with the larger local sets Ψv pN q in our study of global spectra. Our purpose in introducing the hypothetical families of parameters (1.3.9) has been to persuade ourselves that they correspond to well defined families of automorphic representations (1.3.4). This will inform the discussion of the next section. There we shall revisit the constructions of the last section for orthogonal and symplectic groups, but with the objects (1.3.4) in place of the parameter sets (1.3.9). Notice that the left hand three sets in (1.3.4) contain only isobaric automorphic representations. They therefore correspond bijectively with three subsets (1.3.11)
Csim pN q C2 pN q C pN q Caut pN q
of Caut pN q under the mapping of Theorem 1.3.2. We have written Csim pN q Ccusp pN q
for the smallest of these sets, in part because it would be bijective with the hypothetical family Φsim,bdd pN q Ψcusp pN q of irreducible unitary, N -dimensional representations of the group LF . The largest set Caut pN q would of course be bijective with the family ΦpN q of all N -dimensional representations of LF . The sets (1.3.11) can all be expressed in terms of the smallest set Csim pN q (or rather its analogue Csim pmq for m ¤ N ). This is a consequence of Theorem 1.3.3, or if one prefers, its embodiment in the left hand three parameter in (1.3.9). The sets Csim pN q thus contain all the global information for general linear groups. The notation we have chosen is meant to reflect the role of their elements as the simple building blocks. Our ultimate goal is to show that this fundamental role extends to orthogonal and symplectic groups. 1.4. A substitute for global parameters We can now resume the discussion from §1.2. We shall use the cuspidal automorphic representations of GLpN q as a substitute for the irreducible N dimensional representations of the hypothetical global Langlands group LF . This will allow us to construct objects that ultimately parametrize families of automorphic representations of classical groups. Assume that F is global. In the last section, we wrote Ψ2 pN q temporarily for the set of equivalence classes of irreducible unitary representations of the group LF SU p2q. From now on, we let Ψsim pN q Ψsim GpN q
Ψsim
GLpN q
stand for the set of formal tensor products ψ
µ
b
ν,
µ P Acusp pmq,
1.4. A SUBSTITUTE FOR GLOBAL PARAMETERS
27
where N mn and ν is the unique irreducible representation of SU p2q of degree n. Elements in this set are in bijective correspondence with the set of pairs pm, µq of Theorem 1.3.3. We have therefore a canonical bijection ψ Ñ πψ from Ψsim pN q onto the relative discrete, automorphic spectrum A2 pN q of GLpN q. For any such ψ, we set cpψ q equal to cpπψ q, an equivalence class of families in C2 pN q. Then cpψ q cv pψ q : v
RS
(
,
for some finite set S S8 of valuations of F outside of which µ is unramified, and for semisimple conjugacy classes n1 n 1 cv pψ q cv pµq b cv pν q cv pµqqv 2 ` ` cv pµqqv 2
in GLpN, Cq. Here S8 denotes the set of archimedean valuations of F as usual, while qv is again the order of the residue field of Fv . In §1.3, we also wrote ΨpN q temporarily for the set of all unitary representations of LF SU p2q. Following the notation (1.3.8), we now let ΨpN q Ψ GpN q
Ψ GLpN q
denote the set of formal, unordered direct sums (1.4.1)
ψ
`1ψ1 `
`
`r ψr ,
for positive integers `k and distinct elements ψk µk b νk in Ψsim pNk q. The ranks here are positive integers Nk mk nk such that N
`1N1
`r Nr
`1 m 1 n 1
`r mr nr .
For any ψ, we take πψ to be the irreducible induced representation IP π ψ1 b b πψ1 loooooooomoooooooon
b b πloooooooomoooooooon ψ b b πψ
r
r
`r
`1
of GLpN, Aq, where P is the parabolic subgroup corresponding to the partition N 1 , . . . , N1 , . . . , N r , . . . , Nr looooomooooon looooomooooon `1
`r
of N . We then have a bijection ψ Ñ πψ from ΨpN q onto the full automorphic spectrum ApN q of GLpN q. Again we set cpψ q equal to the associated class cpπψ q in C pN q. Then ( cpψ q cv pψ q : v R S , for a finite set S S8 outside of which each µi is unramified, and semisimple conjugacy classes cv pψ q cloooooooooooomoooooooooooon v pψ1 q ` ` cv pψ1 q `1
` `
cloooooooooooomoooooooooooon v pψr q ` ` cv pψr q `r
in GLpN, Cq. Suppose that v is any valuation in F . If ψ µ b ν belongs to Ψsim pN q, the local component µv of µ is an irreducible representation of GLpm, Fv q.
28
1. PARAMETERS
It corresponds to an m-dimensional representation of the local Langlands group LFv , whose tensor product ψv with ν belongs to the set Ψv pN q of N -dimensional representations of LFv b SU p2q. More generally, if ψ is any element (1.4.1) in the larger set ΨpN q, the direct sum ψv
ψ 1,v ` ` ψ1,v looooooooomooooooooon
` `
ψ r,v ` ` ψr,v looooooooomooooooooon
`1
`r
also belongs to the set Ψv pN q. There is thus a mapping ψ Ñ ψv from ΨpN q to Ψv pN q. We emphasize again that we cannot say that ψv lies in the smaller local set Ψv pN q, since the generalized Ramanujan conjecture for GLpN q has not been established. We now have a set ΨpN q of formal parameters to represent the automorphic spectrum ApN q of GLpN q. By Theorem 1.3.2, the elements in ApN q are faithfully represented by the (equivalence classes of) families cpψ q in C pN q. As we noted at the end of §1.3, and have seen explicitly here, these families are easily constructed in terms of cuspidal families c. Reiterating the conclusion from §1.3, the sets Csim pN q Ccusp pN q of cuspidal families are to be regarded as our fundamental data, in terms of which everything we hope to establish for classical groups can be formulated. We are now ready to formalize the constructions of §1.2. The outer automorphism θ: x Ñ x_ of GLpN q acts on the set ΨpN q. It transforms a general parameter (1.4.1) in this set to its contragredient ψ_
`1ψ1_ `r ψr_ `1pµ_1 ν1_q `r pµ_r `1pµ_1 ν1q `r pµ_r `
`
b
`
b
`
`
`
b
b
νr_ q
νr q,
where µ_ k is the contragredient of the cuspidal automorphic representation µk of GLpmk q. We have written νk_ νk , since any irreducible representation of SU p2q is self-dual. The contragredient πψ_ of the associated automorphic representation πψ P ApN q is then equal to πψ_ . We introduce the subset (1.4.2)
r pN q Ψ G r pN q Ψ
ψ
P ΨpN q :
ψ_
ψ
(
of self-dual parameters in ΨpN q, a family that is naturally associated to the connected component r pN q GpN q θrpN q G
r pN q GLpN q xθ y. Elements in this subset correspond to the repin G r pN, Aq generated by resentations πψ P ApN q that extend to the group G r pN, Aq. There is also an action G
tc_v u r pN q of ΨpN q of θ on the set of automorphic families C pN q. The subset Ψ r corresponds to the subset C pN q of self-dual families in C pN q. c tcv u
ÝÑ
c_
1.4. A SUBSTITUTE FOR GLOBAL PARAMETERS
29
It is sometimes convenient to write Kψ for the set that parametrizes the simple constituents ψk in (1.4.1) of a general element ψ P ΨpN q. We shall usually think of Kψ as an abstract indexing set, of which (1.4.1) represents r pN q of ΨpN q, a (noncanonical) enumeration. If ψ belongs to the subset Ψ there is an involution k Ñ k _ on Kψ such that ψk_ ψk_ and `k_ `k . We can then write the indexing set as a disjoint union Kψ
Iψ ² Jψ ² Jψ_,
Jψ_
tj _ : j P Jψ u,
where Iψ is the set of fixed points of the involution, and Jψ is some complementary subset that represents the orbits of order 2. The union of Iψ and Jψ of course represents the set tKψ u of all orbits of the involution on Kψ . r pN q, we can form the subset Having defined the set Ψ r sim pN q Ψ r pN q X Ψsim pN q, Ψ
r pN q that are simple, in the sense that r 1 and of parameters (1.4.1) in Ψ r sim pN q is thus a self-dual parameter µ b ν in the `1 1. An element ψ in Ψ set Ψsim pN q. We shall also write r sim pN q Ψ r cusp pN q Ψ r pN q X Ψcusp pN q Φ
r sim pN q that are generic, in the sense that ν for the subset of parameters in Ψ r pN q is thus a self-dual element in is trivial. A simple generic parameter in Ψ the basic set Acusp pN q of unitary cuspidal automorphic representations of GLpN q. These will be our fundamental building blocks for orthogonal and symplectic groups. Recall that the term simple was applied in §1.2 to endoscopic data. It was used to designate the subset Ersim pN q of elliptic endoscopic data G P Erell pN q that are not composite. We note here that any G in Ersim pN q, or indeed in the larger set Erell pN q, has the property that the split component of its center is trivial. This means that the group GpAq1 equals the full adelic group GpAq. r sim pN q is a simple generic (formal) Theorem 1.4.1. Suppose that φ P Φ global parameter. Then there is a unique Gφ P Erell pN q (taken as an isomorphism class of endoscopic data pGφ , sφ , ξφ q) such that
cpφq ξφ cpπ q ,
for some representation π in A2 pGq. Moreover, Gφ is simple. The assertion here is quite transparent. From among all the twisted, r pN q, there is exactly one that is elliptic endoscopic data G P Erell pN q for G the source of the conjugacy class data of φ. This is of course what we would expect from the discussion of §1.2. Theorem 1.4.1 is the first of a collection of theorems for orthogonal and symplectic groups, the main body of which will be formulated in the next section. We have stated it here as a “seed theorem”, which will be needed
30
1. PARAMETERS
to define the objects on which the others depend. All of the theorems will be established together. The proof will be based on a long induction argument, which is to be carried throughout the course of the volume. The implication for Theorem 1.4.1 is that the definitions it yields will ultimately have to be inductive. We remind the reader that our results will also be conditional on the stabilization of the discrete part of the twisted trace formula, a hypothesis we will state formally in §3.2. Suppose that N is fixed, and that Theorem 1.4.1 holds with N replaced by any integer m ¤ N . What can we say about an arbitrary parameter ψ r pN q Ψ G r pN q ? in the set Ψ We can certainly write (1.4.3)
ψ
ð
P
`i ψi
ð
`
P
i Iψ
`j pψj
`
ψj _ q .
j Jψ
If i belongs to Iψ , we apply Theorem 1.4.1 to the simple generic factor r sim pmi q of ψi µi b νi . This gives a canonical datum pGµ , sµ , ξµ q in µi P Φ i i i Erpmi q, which it will be convenient to denote by Hi . If j belongs to Jψ , we simply set Hj GLpmj q. We thus obtain a connected reductive group Hk over F for any index k in either Iψ or Jψ , or equivalently, in the set tKψ u of orbits of the involution on Kψ . Let L Hk be the Galois form of its L-group. We can then form the fibre product Lψ
(1.4.4)
¹
k
PtKψ u
p LHk ÝÑ
ΓF q
of these groups over ΓF . This is to be our substitute for the global Langlands group in our study of automorphic representations attached to ψ. To make matters slightly more transparent, we have formulated it in algebraic form, as an extension of the profinite group ΓF by a complex reductive group rather than an extension of WF by a compact topological group. For this reason, we shall work with the Galois forms of L-groups throughout much of the volume, rather than their Weil form. This leads to no difficulty in the framework of orthogonal and symplectic groups we will study. If an index k i in (1.4.1) belongs to Iψ , we have the standard embedding µ ri ξµi : L Hi ÝÑ L GLpmi q GLpmi , Cq ΓF that is part of the endoscopic datum Gi . If k a standard embedding rj : µ
by setting rj phj µ
L
Hj
ÝÑ
L
GLp2mj q
σq phj ` thj 1q σ,
hj
j belongs to Jψ , we define
GLp2mj , Cq ΓF P Hpj GLpmj , Cq, σ P ΓF .
We then define an L-embedding ψr : Lψ SLp2, Cq
ÝÑ
L
GLpN q GLpN, Cq ΓF
1.4. A SUBSTITUTE FOR GLOBAL PARAMETERS
as the direct sum (1.4.5)
ψr
à
P
`i pµ ri b νi q
`
i Iψ
à
P
`j pµ rj
b νj q
31
.
j Jψ
Our use of SLp2, Cq here in place of SU p2q is purely notational, and is in keeping with our construction of Lψ as a complex proalgebraic group. We are of course free to interpret the embedding ψr also as an N -dimensional representation of Lψ SLp2, Cq. With either interpretation, we shall be prir as a GLpN, Cq conjugacy class marily interested in the equivalence class of ψ, of homomorphisms from Lψ SLp2, Cq to either GLpN, Cq or L GLpN q . r pGq for the set of eleSuppose that G belongs to Ersim pN q. We write Ψ r factors through L G. By this, we mean that r pN q such that ψ ments ψ P Ψ there exists an L-homomorphism such that (1.4.6)
ψrG : Lψ SLp2, Cq
ÝÑ
L
G
ξ ψrG
r ψ, L G into L GpN q that is part of the twisted
where ξ is the embedding of endoscopic datum represented by G. Since ψr and ξ are to be regarded as GLpN, Cq-conjugacy classes of homomorphisms, ψrG is determined up to the p The quotient of stabilizer in GLpN, Cq of its image, a group that contains G. p equals Out r N pGq, the group of outer automorphisms of the this group by G endoscopic datum G, described in §1.2. It is a finite group, which is actually trivial unless G is an even orthogonal group, in which case it equals Z{2Z. p In the latter case though, there can be two G-orbits of homomorphisms ψrG r r pGq in place of the in the class of ψ. It is for this reason that we write Ψ more familiar symbol ΨpGq. More generally, suppose that G belongs to the larger set Erell pN q, or even r pN q. As a group over F , G equals the full set ErpN q of endoscopic data for G a product ¹ G Gι ι
of groups Gι that can range over (quasisplit) connected orthogonal and symplectic groups and (split) general linear groups. We define the set of parameters for G as the product r pG q Ψ
¹
r ι pG ι q , Ψ
ι
r pGι q if Gι is orthogonal or symplectic, and equals where Ψι pGι q equals Ψ ΨpGι q if Gι is a general linear group. After a little reflection, we see that rG q, for a parameter r pGq can be identified with a pair pψ, ψ an element in Ψ r ψ P ΨpN q and an L-embedding
ψrG : Lψ SLp2, Cq
ÝÑ
L
G
32
1. PARAMETERS
p that satisfies (1.4.6), and is defined as a G-orbit only up to the action of r OutN pGq. The projection
pψ, ψrGq ÝÑ
ψ
is not generally injective. This is in contrast to the injective embedding of r pGq into Ψ r pN q for simple G, which is an implicit aspect of our original Ψ definition. (It is also in contrast to the special case of elliptic parameters r ell pN q and elliptic endoscopic data G P Erell pN q, which we will describe ψPΨ in a moment.) However, we shall still sometimes allow ourselves to denote r pGq by ψ when there is no danger of elements in the more general sets Ψ confusion. r pN q. Let Suppose that ψ is any parameter in Ψ r pN q Srψ pN q Sψ G
Cent Impψrq, GprpN q
and r 0 pN q Srψ0 pN q Sψ pG
Cent Impψrq, Gpr0pN q Cent Impψrq, GLpN, Cq
be the centralizers of the image Impψrq ψr Lψ SLp2, Cq
p r pN q of ψr in the respective components G
p q
p q
Gpr0pN q θ and Gpr0pN q.
Then
p0 r N , which acts simply transitively by Srψ0 N is a reductive subgroup of G left or right translation on Srψ N . Both Srψ N and Srψ0 N are connected. r G , the On the other hand, if G is a datum in Er N and ψ belongs to Ψ
p q
centralizer
Sψ
p q
p q p q
Sψ pGq Cent ImpψrGq, Gp
p q
of the image of ψrG need not be connected. Its quotient Sψ
Sψ pGq Sψ {Sψ0 Z pGpqΓ
F
is a finite abelian group, which plays an essential role in the theory. Notice that there is a canonical element sψ
ψrG
1,
1 0
0 1
in Sψ . Its image in Sψ (which we denote also by sψ ) will be a part of the description of nontempered automorphic representations of G. The centralizers Srψ pN q and Sψ are in obvious analogy with the centralizers Srr pN q and Sr described at the end of §1.2. Their definition settles the question posed in §1.1 of how to introduce centralizers of parameters without recourse to the global Langlands group LF . We write r pN q r ell pN q Ψell G Ψ
1.4. A SUBSTITUTE FOR GLOBAL PARAMETERS
33
r pN q such that the indexing set Jψ is for the subset of parameters ψ P Ψ empty, and such that `i 1 for each i P Iψ . These objects are analogous to the self-dual representations r we called elliptic in §1.2. The discussion r ell pN q has of §1.2 applies here without change. It tells us that any ψ P Ψ r pGq. More precisely, let Ψ r 2 pGq be the a unique source in one of the sets Ψ r ell pN q in Ψ r pGq, for any G P Erell pN q. The mapping from Ψ r pG q preimage of Ψ r r r to ΨpN q then takes Ψ2 pGq injectively onto a subset of Ψell pN q, which we r 2 pGq, and Ψ r ell pN q is then the disjoint union identify with Ψ r ell pN q Ψ
º
P p q
G Erell N
r 2 p Gq Ψ
of these subsets. We thus have parallel chains of parameter sets r sim pN q Ψ r ell pN q Ψ r pN q Ψ
and (1.4.7)
r sim pGq Ψ r 2 pG q Ψ r pGqq, Ψ
G P Erell pN q,
r sim pGq denotes the intersection of Ψ r sim pN q with Ψ r 2 pGq. Observe where Ψ r r that Ψ2 pGq is the subset of parameters ψ P ΨpGq such that the centralizer r sim pGq consists of those ψ such that Sψ equals the Sψ is finite, while Ψ p q ΓF . minimal group Z pG Suppose that ψ ψ1 ` ` ψr r belongs to Ψell pN q. How do we determine the group G P Erell pN q such that r 2 pGq? To answer the question, we have to be able to write ψ lies in Ψ ψ ψO ` ψS , where ð ψO ψi
P
i IO
is the sum of those components of orthogonal type, ψS
ð
P
ψi
i IS
is the sum of components of symplectic type, and
Iψ,O ² Iψ,S is the corresponding partition of the set Iψ Kψ of indices. Iψ
Consider a general component ψi
µi
b
νi
r sim pmi q has a central of the given ψ. The irreducible cuspidal element µi P Ψ character ηi ηµi of order 1 or 2, which we identify with a character on ΓF . It also gives rise to a datum Hi P Ersim pmi q, according to Theorem 1.4.1. This p i GLpmi , Cq. The ni provides a complex, connected classical dual group H dimensional representation νi of SLp2, Cq determines the complex connected p i GLpni , Cq that contains its image. By considering classical group K
34
1. PARAMETERS
p i is principal unipotent elements, for example, the reader can see that K symplectic when ni is even, and orthogonal when ni is odd. The tensor product embedding pi K pi H
ÝÑ
GLpNi , Cq,
mini, p i GLpNi , Cq. In concrete is then contained in a canonical classical group G Ni
p i is symplectic if one of H p i or K p i is symplectic and the other is terms, G p p orthogonal, and is orthogonal if Hi and Ki are both of the same type. This allows us to designate ψi as either orthogonal or symplectic, and thus gives rise to the required decomposition ψ ψO ` ψS . As in §1.2, we set
Nε
¸
ε O, S,
Ni ,
P
i Iψ,ε
r 2 pGS q of Ψ r ell pNS q, for the datum GS P Ersim pNS q Then ψS lies in the subset Ψ p r 2 pGO q of with dual group GS SppNS , Cq, while ψO lies in the subset Ψ r ell pNO q, for the datum GO P Ersim pNO q with dual group G p O SOpNO , Cq Ψ and character ¹ ¹ ηO pη i qn i ηi .
P
P
i Iψ,O
i Iψ,O
r 2 pGq, where G is the product The original element ψ therefore lies in Ψ datum GO GS in Erell pN q. More generally, suppose that ψ is a parameter (1.4.3) in the larger set r pN q. Then ψ can have several preimages in the various families Ψ r pG q , Ψ r G P Eell pN q. How does one determine them explicitly? For a given datum G GO GS , there are several ways to divide the components ψk among the two factors GO and GS . To describe the possibilities, it suffices to determine r pGq of Ψ r pN q attached to a fixed whether a given ψ belongs to the subset Ψ simple datum G P Ersim pN q. p of G P Ersim pN q is purely orthogonal or symplectic. The dual group G The parameter ψ comes with a partition of its index set Kψ into subsets Iψ , Jψ and Jψ_ . If j belongs to Jψ , the corresponding component `j pψrj ` ψrj _ q
of the embedding ψr takes values in a factor L
Mj
L
GL pNj q GLpNj q looooooooooooooomooooooooooooooon
`j
of a Levi subgroup M of L G. As we have just seen (at least in the special r 2 pGq), the complementary set Iψ has its own partition into case that ψ P Ψ subsets Iψ,O and Iψ,S of orthogonal and symplectic type. It also indexes characters ηi of order 1 or 2. If an index i P Iψ has even multiplicity `i 2`1i , the component `i ψri of ψr takes values in another factor L
Mi
L
p q
p q
GL Ni GL Ni loooooooooooooomoooooooooooooon `1i
1.4. A SUBSTITUTE FOR GLOBAL PARAMETERS
35
of L M , whether i belongs to Iψ,O or Iψ,S . The indices j P Jψ and i P Iψ with `i even therefore impose no constraints. However, if i P Iψ with `i odd, there is a supplementary copy of ψi to content with. In this case ψi p must be of the same type, either both symplectic or both orthogonal. and G p is orthogonal, these factors also impose a constraint on the Moreover, if G character η ηG that goes with G. It must satisfy the identity η
¹
¹
pη i q ` n i i
P
pη i q` . i
P
i Iψ,O
i Iψ,O
The last two conditions are both necessary and sufficient for ψ to belong to r pG q . the subset Ψ It is also easy to describe the centralizers Sψ Sψ pGq. Suppose that G P Ersim pN q, and that ψ is a parameter (1.4.3) that lies in the subset r pGq of Ψ. r Every index j P Jψ contributes a factor GLp`j , Cq to Sψ . The Ψ complementary indexing set has a separate partition
Iψ pGq ² IψpGq,
Iψ
p where Iψ pGq (resp. Iψ pGq) is the set of indices i P Iψ such that ψi and G are of the same (resp. different) type. In other words, Iψ pGq Iψ,O and
Iψ pGq
Iψ,S
p is orthogonal, while I pGq if G ψ
Iψ,S
and Iψ pGq
Iψ,O
p is symplectic. It follows from what we have just observed that an if G index i P Iψ pGq has even multiplicity `i . It contributes a factor Spp`i , Cq to Sψ . Each index i P Iψ pGq contributes a factor Op`i , Cq to Sψ , with the stipulation that a product (with multiplicities) of the determinants of these factors be 1. Let ¹
P p q
Op`i , Cq
ψ
i Iψ G
be the kernel of the character ξψ
ξ ψ pG q :
¹
gi
ÝÑ
¹
i
pdet giqN ,
gi
i
P O p`i , C q, i P I ψ pG q .
i
The centralizer Sψ is then given by (1.4.8) Sψ
¹
P p q
Op`i , Cq
ψ
¹
G i Iψ
P p q
i Iψ G
Spp`i , Cq
¹
P
j Jψ
r 2 pGq. Then Suppose in particular that ψ belongs to Ψ
Iψ pGq Iψ
t1, . . . , ru,
and each `i equals 1. In this case, the centralizer specializes to (1.4.9)
#
p Z{2Zqr , Sψ O p1 q ψ pZ{2Zqr1, i1 r ¹
if each Ni is even, otherwise.
GLp`j , Cq .
36
1. PARAMETERS
Similar constructions apply to (ordinary) endoscopic data G1 P E pGq, or for that matter, iterated data G2 P E pG1 q, G3 P E pG2 q, and so on. As a group over F , G1 is again a product of elementary groups G1ι for which the r pG1 q to be the corresponding sets Ψι pG1ι q have been defined. We define Ψ r pG1 q can be identified with product of these sets. A parameter ψ 1 ψG1 in Ψ 1 r pGq, and an L-embedding a pair pψ, ψr q, for a parameter ψ in Ψ ψr1
ψrG1 :
that satisfies
Lψ SLp2, Cq ξ 1 ψr1
(1.4.10)
ÝÑ
L
G1
ψrG,
p 1 -orbit only up to the action of the finite group and is defined as a G r N pG1 q OutG pG1 q Out r N pG q. Out
Here, ξ 1 : L G1 Ñ L G is the L-embedding attached to G1 . Notions introduced r pG1 q here. For earlier for ψ have obvious meaning for parameters ψ 1 P Ψ example, we have the centralizer Sψ 1 and its abelian quotient for any such ψ 1 . Any pair
Sψ1
Sψ1 pG1q Cent Impψ1q, Gp1 Sψ1 pG1q Sψ1 {Sψ0 1 Z pGp1qΓ
pG 1 , ψ 1 q ,
G1
gives rise to a second pair
pψ, sq,
P E pG q,
ψ1
F
,
P Ψr pG1q,
P Ψr pGq, s P Sψ , r pGq attached to ψ 1 , and s is the image where ψ ψG is the parameter in Ψ p of the semisimple element s1 P G p 1 that is part of the endoscopic datum in G 1 r pGq, and that s is a G . Conversely, suppose that ψ is any parameter in Ψ ψ
p 1 be the connected centralizer of s in G, p semisimple element in Sψ . Let G p 1 . The product and let s1 be the preimage of s in G
G1
Gp1 ψrG
Lψ SLp2, Cq
rG is an L-subgroup of L G, for which the identity p 1 with the image of ψ of G 1 embedding ξ is an L-homomorphism. We take G1 to be a quasisplit group p 1 , with the L-action induced by G 1 , is a dual group. In the for which G present situation, there is a natural way to identify G 1 with L G1 . The triplet pG1, s1, ξ1q represented by G1 is then an endoscopic datum for G1, in the restricted sense of §1.2. Since s lies in the centralizer of the image of ψrG , ψrG factors through L G1 . We obtain an L-embedding ψr1 of Lψ SLp2, Cq r pG1 q. The pair into L G1 that satisfies (1.4.10), and hence an element ψ 1 P Ψ 1 1 pψ, sq thus gives rise to a pair pG , ψ q of the original sort.
1.4. A SUBSTITUTE FOR GLOBAL PARAMETERS
The correspondence (1.4.11)
37
pG1, ψ1q ÝÑ pψ, sq
is a general phenomenon. It applies to arbitrary endoscopic data, at least in the context of Langlands parameters. In particular, it has an obvious variant r pN q, G . This for our other primary case in which pG, G1 q is replaced by G leads to a more systematic way to view the discussion of §1.2. In general, the correspondence reduces many questions on the transfer of characters to a study of groups Sψ . It will be part of the foundations we are calling the standard model in Chapter 4. The discussion of this section applies also to our supplementary case, r (1.2.5) for an even orthogonal group G P Ersim pN q. We that of a bitorsor G r rq Ψ r pG r q for the subset of θ-fixed r pGq. These are write ΨpG elements in Ψ r pGq such that the Aut r N pGq-orbit of L-embeddings the parameters in Ψ ψrG : Lψ SLp2, Cq
ÝÑ
L
G
p contains only one G-orbit, or equivalently, such that ψrG factors through the L-embedding r 1 ÝÑ L G ξr1 : L G r 1 P Eell pG r q. They comprise the minimal of some twisted endoscopic datum G set in a chain rq Ψ r pG q Ψ r pN q ΨpN q ΨpG of subsets of our original family of parameters ΨpN q for GLpN q. Observe that the subset rq Ψ r 2 pGq X ΨpG rq Ψ2 pG r q consists of those parameters (1.4.1) that lie in Ψ r 2 pGq, and for which of ΨpG one of the irreducible ranks Ni is odd. In particular, the subset rq Ψ r sim pGq X ΨpG rq Ψ r sim pGq X Ψ2 pG rq Ψsim pG
r q is empty, since N is even. For any ψ of simple elements in ΨpG the centralizer p rG q, G r r q Cent Impψ Srψ Sψ pG is a bitorsor under Sψ , whose quotient
Srψ
Sψ pGrq Srψ {Srψ0 Z pGpqΓ
P ΨpGrq,
F
is a bitorsor under the quotient Sψ of Sψ . It gives rise to an analogue of the r G r 1 q in place of pG, G1 q. correspondence (1.4.11), with pG, We have so far devoted this section to global parameters. We have shown that their study is parallel to that of the finite dimensional representations in §1.2. We have also tried to demonstrate that objects attached to any ψ can be computed explicitly, even if the details may be somewhat complicated. The details themselves need not be taken too seriously. When we resume this discussion in Chapter 4, we shall treat the relations among our global parameters in a more systematic fashion.
38
1. PARAMETERS
We shall now discuss the local parameters attached to a completion Fv of F . These are simpler, since they really do represent homomorphisms defined on the group LFv SU p2q. In particular, the remarks in §1.3 pertaining to the chain (1.3.10) apply here. Observe that the local analogue of the assertion of Theorem 1.4.1 is r sim,v pN q, elementary. For we have seen in §1.2 that any parameter ψv P Ψ r and in particular, any ψv φv in the subset Φsim,v pN q of generic parameters r sim,v pN q, factors through a unique local endoscopic datum Gv in the set in Ψ
r v pN q . Erell,v pN q Eell G
Moreover, this datum is simple. In these more concrete terms, all of the global discussion above carries over to the completion Fv of F . In fact, it extends without change to the larger parameter set r pN q ψv Ψ v
P Ψv pN q :
of self-dual, N -dimensional representations of necessarily unitary. We thus attach chains of parameter sets (1.4.12)
(
ψv_
ψv LF SU p2q v
r sim pGv q Ψ r 2 p Gv q Ψ r pG v q Ψ r Ψ
that are not
pG v q
to local endoscopic data Gv P Erv pN q, following the local analogues of the global notation (1.4.7). They consist of L-homomorphisms ψv : LFv
SU p2q ÝÑ
L
Gv ,
which can be composed with the endoscopic embedding ξv
ξG
v
:
L
Gv
ÝÑ
G pN q .
L
r pGv q to Ψ r pN q is injective if Gv is simple, The resulting mapping from Ψ v r r pGv q. This allows us in each or if ψv belongs to the subset Ψ2 pGv q of Ψ of the two cases to identify the domain of the mapping with a subset of r pN q. For example, if Gv is simple, any parameter set in (1.4.12) equals Ψ v r pGv q with the relevant set of parameters for G r v pN q in the intersection of Ψ the chain r sim,v pN q Ψ r ell,v pN q Ψ r v pN q Ψ r pN q. Ψ v
pGv q, the local centralizer Sψ pGv q Cent Impψv q, Gpv ,
r For any parameter ψv in Ψ
Sψv
v
and its abelian quotient
Sψ pGv q Sψ {Sψ0 Z pGpv qΓ , r pG1 q attached to any are defined in the usual way. We also have the set Ψ v 1 Gv P E pGv q, as well as the local analogue of the correspondence (1.4.11). We Sψv
v
v
v
v
shall apply all of these local constructions at will, sometimes with the local field being F itself (and the notation modified accordingly) rather than the completion Fv here of the global field F .
1.4. A SUBSTITUTE FOR GLOBAL PARAMETERS
39
Finally, we will need to define a localization mapping ψ Ñ ψv from the r pGq attached to any G P Ersim pN q to the associated local set global set Ψ r Ψ pGv q. It would not be hard to formulate the localization as a mapping r pN q to Ψ r pN q, or if we wanted to follow the general remarks for the from Ψ v hypothetical group LF at the end of the last section, as a mapping between the larger sets ΨpN q and Ψv pN q. To complete the definition, however, we r pGq of Ψ r pN q to the subset would need to know that it takes the subset Ψ r pGv q of Ψ r v pN q. This property is not elementary. It is a consequence of Ψ a second “seed theorem”, which we state here as a complement to Theorem 1.4.1, but which like Theorem 1.4.1, will have to be proved inductively at the same time as our other theorems. r sim pN q is simple generic, as in Theorem Theorem 1.4.2. Suppose that φ P Φ 1.4.1. Then the localization φv of φ at any v, a priori an element in the r v pN q of generic parameters in Ψ r pN q, lies in the subset Φ r pGφ,v q of subset Φ v r v pN q attached to the localization Gφ,v of the global datum Gφ P Ersim pN q of Φ Theorem 1.4.1.
The theorem asserts that the N -dimensional representation φv of LFv , which is attached by the local Langlands correspondence to the v-component of the cuspidal automorphic representation of GLpN q given by φ, factors through the local endoscopic embedding ξφ,v :
L
Gφ,v
ÝÑ
L
G pN qv .
It allows us to identify φv with an L-homomorphism φv : LFv
ÝÑ
L
Gφ,v .
Here Gφ is the endoscopic datum attached to φ by Theorem 1.4.1, and φv is r N p Gφ q determined as a mapping into L Gφ,v up to the action of the group Aut p on Gφ . Like Theorem 1.4.1, this theorem will be proved by a long induction argument that includes the proof of our other results. Suppose that Theorems 1.4.1 and 1.4.2 hold with N replaced by any integer m ¤ N . Let us consider the group Lψ attached to a general paramr pN q, and its relation to the local Langlands group LF at eter (1.4.1) in Ψ v L v. If Hk is one of the factors (1.4.4) of Lψ , Hk represents either a simple endoscopic datum in Ersim pmk q, or a general linear group GLpmk q. In the first case, Theorem 1.4.2 gives a conjugacy class of L-homomorphisms
(1.4.13)
LFv
ÝÑ
ΓFv
LH k
ÝÑ
ΓF
,
r m pHk q on H p k . In the second which is determined up to the action of Aut k case, we obtain a similar assertion from the local Langlands correspondence
40
1. PARAMETERS
r m pHk q equal to Int GLpmk , Cq . The fibre product for GLpmk q, with Aut k (1.4.4) then yields a conjugacy class of L-homomorphisms
LFv
ÝÑ
ΓFv
Lψ
ÝÑ
ΓF
(1.4.14)
which is determined up to the L-action of the group (1.4.15)
r pL ψ q Aut
¹
r m pH k q Aut k
k
on Lψ . r pGq, for some G P Ersim pN q, or indeed, for Suppose that ψ belongs to Ψ r any G in the general set E pN q. It then follows from this discussion that we can identify the localization of ψ with an L-homomorphism ψv : LFv
SU p2q ÝÑ
L
Gv ,
which fits into a larger commutative diagram of L-homomorphisms
(1.4.16)
LFv SU p2q
ψ ÝÑ
Lψ SLp2, Cq
ÝψÝÑ
v
rG
LG
v
LG
ÝÑ
ΓFv
ÝÑ
ΓF .
The left hand vertical arrow is given by the mapping of LFv into Lψ in (1.4.14) and the usual embedding of SU p2q into SLp2, Cq. Since ψv is essentially the restriction of the global embedding ψrG to the image of LFv SU p2q, the global centralizer Sψ is contained in Sψv . It follows that there is a canonical mapping x ÝÑ xv , x P Sψ , of Sψ into Sψv .
1.5. Statement of three main theorems We shall now state our main theorems. They apply to the quasisplit orthogonal and symplectic groups we have introduced, and provide what amounts to a classification of the representations of these groups in terms of representations of general linear groups. They will be conditional on the stabilization of the twisted trace formula, an assumption we shall state formally when we come to discuss the relevant part of the trace formula in Chapter 3. Recall that the groups under consideration represent simple endoscopic r pN q attached to a general linear group data G P Ersim pN q, for the torsor G
1.5. STATEMENT OF THREE MAIN THEOREMS
41
GpN q GLpN q. By presenting them in standard form in §1.2, we have implicitly attached extra structure to these groups. Each G P Ersim pN q comes with a scheme structure over the ring of integers oF . It also comes with a standard Borel subgroup B, with maximal torus T , and a standard splitr N pGq of outer autoting for pT, B q. We have also identified the group Out morphisms of G over F with the subgroup of F -automorphisms of G that preserve the given splitting. This group seems innocuous enough, being of order 1 or 2, but it does play an essential role. The first theorem concerns the case that F is local. It classifies representations of GpF q in terms of a construction defined by the endoscopic transfer of characters. We shall postpone the formal description of this construction until §2.1, as part of our general discussion of local endoscopic transfer. In the meantime, we shall be content to give a somewhat less precise statement of the theorem. We are assuming for the moment that F is local. We fix a maximal compact subgroup KF of the group GF , which we take to be special if F is p-adic, and hyperspecial if G is unramified over F . The oF -scheme structure on G actually leads to a natural choice of KF . For example, if F is p-adic and G is split, we can take KF GpoF q. On the other hand, if F is padic, but G SOpN, ηG q is quasisplit but not split (so that N 2n is even), KF is a little more complicated. (See the diagrams in [Ti, §1.16].) In any case, given KF , we write HpGq for the corresponding Hecke algebra of smooth, left and right KF -finite functions of compact support on GpF q. r pGq for the subalgebra of Out r N pGq-invariant functions in We also write H HpGq. Because our results are ultimately tied to GLpN q, they will apply r pGq, which is slightly weaker than that of to the representation theory of H H p G q. r pGq for the set of orbits of Out r N pGq in ΠpGq, which we We write Π r pGq equals recall is the set of irreducible representations of GpF q. Then Π r p Gq ΠpGq unless G is an even orthogonal group SOp2n, ηG q, in which case Π r contains orbits of order 2 as well as of order 1. Any element π P ΠpGq gives rise to a well defined character
fG pπ q tr π pf q ,
P HrpGq, r pGq. Similar notation applies to any subset of ΠpGq, such as for example on H the set Πunit pGq of unitary representations in ΠpGq. We shall state the local r unit pGq”. theorem in terms of packets that are “sets over Π f
By a set over S, or an S-set, or even an S-packet, we mean simply a set S1 with a fibration S1 ÝÑ S
over S. Equivalently, S1 is a set that can be represented as a disjoint union r pGq in of subsets of S. Any function on S, such as the character fG pπ q on H r case S equals ΠpGq, will be identified with its pullback to a function on S1 .
42
1. PARAMETERS
The order m1 : S
ÝÑ
N Y t0, 8u
of the fibres in S1 represents a multiplicity function, which makes S1 into a multiset on S in the sense of [Z, p. 169]. If S1 is multiplicity free, in that every element in S has multiplicity at most 1, it is just a subset of S. Theorem 1.5.1. Assume that F is local and that G P Ersim pN q. r pGq, there is a finite set Π r ψ over (a) For any local parameter ψ P Ψ r unit pGq, constructed from ψ by endoscopic transfer, and equipped with a Π canonical mapping r ψ, π ÝÑ x, π y, πPΠ
r ψ into the group Spψ of characters on Sψ such that x, π y 1 if G and from Π π are unramified (relative to KF ). r bdd pGq of parameters in Ψ r pGq that (b) If φ ψ belongs to the subset Φ r φ are tempered and mulare trivial on the factor SU p2q, the elements in Π r φ to Spφ is injective. tiplicity free, and the corresponding mapping from Π r temp pGq belongs to exactly one packet Π r φ . FiMoreover, every element in Π r p nally, if F is nonarchimedean, the mapping from Πφ to Sφ is bijective.
r φ of Π r temp pGq in (b) represent the tempered LThe finite subsets Π r N pGq-orbits of) tempered packets for GpF q. They are composed of (Out representations, which are parametrized by characters in Spφ . Since the ther temp pGq is a disjoint union of these packets, it can be orem asserts that Π regarded as a classification of the irreducible, tempered representations of G pF q . r ψ of (a) represent local factors of automorphic The more general sets Π r N pGq-orbits of) nontempered reprepresentations. They can contain (Out resentations, and typically only fibre over Spψ . It seems likely that these r unit pGq rather than packets are also multiplicity free (that is, subsets of Π multi-subsets). For nonarchimedean F , Moeglin has recently established rφ this fact [M4], using some of the properties of the tempered L-packets Π we will establish in the course of proving the theorem. However, the genr ψ to Spψ is still not injective, so the elements in Π r ψ are eral mapping from Π not parametrized by characters on Sψ . One could rearrange the definition r ψ that map to a given character. simply by combining all the elements in Π r ψ would no longer be With this equivalent formulation, the elements in Π irreducible, but they would map injectively to Spψ . This was the point of view in the announcement of [A18, §30]. The objects of the theorem will be defined in a canonical way. We shall state the precise form of the construction in the next chapter. For archimedean F , the resulting strong form of the theorem includes special cases of results of Shelstad [S3] and [S4]–[S7] for tempered representations,
1.5. STATEMENT OF THREE MAIN THEOREMS
43
and of Adams, Barbasch and Vogan [ABV] for nontempered representations. In case F is nonarchimedean, part (b) of the theorem gives the local r N pGq 1, and a slightly Langlands correspondence for GpF q in case Out r N pGq 1. weakened form of the correspondence if Out In all cases, the theorem yields local Rankin-Selberg L-functions and ε-factors for representations of general quasisplit orthogonal and symplectic groups. For any pair of representations πi
P Πr temppGiq,
Gi
P ErsimpNiq, i 1, 2,
we have only to take the corresponding pair of parameters φi
P Φr bddpGiq : πi P Πr φ , i 1, 2
(
i
for the associated local packets. These objects can of course also be regarded as local parameters for general linear groups. They allow us to define the local L-functions Lps, π1 π2 q Lps, φ1 φ2 q and ε-factors
εps, π1 π2 , ψF q εps, φ1 φ2 , ψF q
in terms of what is already known for general linear groups. We note that similar definitions apply if one of the groups Gi is replaced by a general linear group. In this case, they have been studied in terms of the LanglandsShahidi method. The remaining two theorems are global. However, their statement requires an extension of the local construction. We could easily formulate the r pGq. However, objects of Theorem 1.5.1 for parameters in the larger set Ψ it is instructive to introduce an intermediate set r pG q Ψ r r Ψ unit pGq Ψ
p G q,
which serves the purpose and is perhaps more natural. We first introduce an analogous intermediate set ΨpN q Ψunit pN q Ψ
for GLpN q. Recall that the local set Ψ representations
pN q
pN q consists of N -dimensional
`1ψ1 ` ` `r ψr , ψi P ΨsimpNiq of the locally compact group LF SU p2q. Any such ψ is an irreducible ψ
representation of the Levi subgroup GLpN1 , F q`1
GLpNr , F q`
r
of GLpN, F q, which is unitary modulo the centre. We write Ψunit pN q for the set of ψ P Ψ pN q such that the associated induced representation of GLpN, F q is irreducible and unitary. This set can be described explicitly in terms of the parametrization of unitary representations in [V2] and [Tad1].
44
1. PARAMETERS
It is the natural domain for the set of local factors of automorphic representations in the discrete spectrum of GLpN q. The associated set for G is then defined as the intersection r r Ψ unit pGq Ψ
pGq X ΨunitpN q.
r Suppose that ψ P Ψ Any such parameter gives rise to an unit pGq. r OutN pGq-orbit of standard parabolic subgroups P M N of G, a parameter r pM q and a point λ in the open chamber of P in the real vector space ψM P Ψ
aM
X pM qF b R.
These objects are characterized by the property that ψ equals the comr pM q of ψM by λ with the L-embedding position of the twist ψM,λ P Ψ L M L G. Recall that there is a canonical homomorphism HM : M p F q
ÝÑ
aM
Hom X pM qF , R .
The twist ψM,λ is the product of ψM with the central Langlands parameter that is dual to the unramified quasicharacter
ÝÑ eλpH pmqq, m P M pF q. The Levi subgroup M G attached to ψ is a product of several general linear groups with a group G P Ersim pN q, for some N ¤ N . The r N pP q of P in Out r N pGq is isomorphic to Out r N pG q. The stabilizer Out r ψ and x, πM y to obvious variant of Theorem 1.5.1(a) attaches objects Π r ψ is then an Out r N pP q-orbit of the parameter ψM . An element πM P Π representations σ P Πunit pM q. Let πM,λ be the corresponding orbit of repr N pGq-orbit of resentations πM b χλ , and let IP pπM,λ q be the associated Out χλ : m
M
M
M
induced representations. With this understanding, we define the packet of ψ as the family rψ Π
(1.5.1)
π
IP pπM,λq :
πM
P Πr ψ
M
(
,
r ψ . We are not assuming at this point a finite set that is bijective with Π M that the induced representations in question are irreducible or unitary. In r ψ is an Out r N pGq-orbit of (possibly reducible, other words, an element π P Π possibly nonunitary) representations of GpF q. However, the numbers tr π pf q r pG q . and the operators π pf q are nonetheless well defined for functions f P H As functions of λ, these objects extend analytically to entire functions on the complexification aM,C of aM . If λ P iaM is purely imaginary and in general position, the induced representations in the corresponding packet are tempered and irreducible, and are therefore among the packets treated in Theorem 1.5.1. x. It The G-centralizer Sψ Sψ pGq of the given ψ is contained in M follows that Sψ SψM , from which we deduce that Sψ SψM . We can therefore define the character
(1.5.2)
s
ÝÑ xs, πy xsM , πM y,
π
P Πr ψ ,
1.5. STATEMENT OF THREE MAIN THEOREMS
45
for any element s sM in the group Sψ SψM . We thus obtain the r local objects of Theorem 1.5.1(a) for any parameter ψ P Ψ unit pGq. The assertion (a) for these objects, which is to say the canonical construction by endoscopic transfer that we will make precise in §2.2, will follow immediately by analytic continuation. Suppose now that the field F is global. Our group G P Ersim pN q then represents a global endoscopic datum. The first global theorem is the central result. It gives a decomposition of the automorphic discrete spectrum of G r 2 pGq and the local objects of Theorem in terms of global parameters ψ P Ψ 1.5.1(a) (or rather their extensions (1.5.1) and (1.5.2)). It is best stated in terms of the global Hecke algebra HpGq on GpAq with respect to the maximal compact subgroup K
¹
Kv .
v
Recall that HpGq is the space of linear combinations of products ¹
fv ,
fv
v
P H p G v q,
such that fv is the characteristic function of Kv for almost all v. In other words, HpGq is the restricted tensor product of the local Hecke algebras r pGq for the restricted tensor product of the local HpGv q. Let us write H r pGv q. This is the subspace of functions in HpGq symmetric Hecke algebras H r N pGv q. For any function f that are invariant under each of the groups Out r in HpGq, and any admissible representation of GpAq of the form π
â
πv ,
v
r N pGv q-orbit of any component the operator π pf q depends only on the Out r pGq-module. πv . The point is to describe the discrete spectrum as an H We are assuming the seed Theorems 1.4.1 and 1.4.2. The second of these r pGq to implies that the localization mapping ψ Ñ ψv takes the global set Ψ r r the subset Ψ unit pGv q of ΨpN q. The local theorem we have just stated then r attaches a local packet Πψv to ψ and v. We can thus attach a global packet
(1.5.3)
rψ Π
!â
πv : πv
v
P Πr ψ , x, πv y 1 for almost all v
)
v
r of (orbits  of) representations of GpAq to any ψ P ΨpGq. Any representation π πv in the global packet determines a character
(1.5.4)
xx, πy
¹ v
on the global quotient Sψ .
xxv , πv y,
x P Sψ ,
46
1. PARAMETERS
Theorem 1.5.2. Assume that F is global and that G P Ersim pN q. Then there r pGq-module isomorphism is an H (1.5.5)
L2disc GpF qzGpAq
à
à
P p q P p q
mψ π,
˜2 G π Π ˜ ψ εψ ψ Ψ
where mψ equals 1 or 2, while
ÝÑ t1u
εψ : Sψ
is a linear character defined explicitly in terms of symplectic ε-factors, and r ψ such that r ψ pεψ q is the subset of representations π in the global packet Π Π the character x, π y on Sψ equals εψ . We need to supplement the statement of Theorem 1.5.2 with a description of the integer mψ and the character εψ . The first of these is easy enough. The parameter ψ comes with the L-embedding ψrG : Lψ SLp2, Cq
ÝÑ LG, r N pGq by conjugation on G. p We determined up to the action of the group Aut r p r define mψ for any ψ P ΨpGq to be the number of G-orbits in the AutN pGqorbit of ψrG . The theorem applies to a parameter ψ
ψ1 `
`
ψr
r 2 pGq. In this case, it is easy to check that mψ equals 1, unless N is in Ψ p equals SOpN, Cq, and the rank Ni of each of the components ψi of even, G ψ is also even, in which case mψ equals 2. The sign character εψ is more interesting. But first we make an observar pN q is an arbitrary global tion on general L-functions. Suppose that ψ P Ψ parameter, and that r is an arbitrary finite dimensional representation of Lψ , subject only to the condition that its equivalence class is stable under r pLψ q. Then r pulls back to a well defined representation rv of the group Aut LFv , for any v. We can therefore define the global L-function
Lps, rq
¹
Lps, rv q
v
by an Euler product that converges for the real part of s large. Of course we do not know in general that it has analytic continuation and functional equation. We can still define the global ε-factor as a finite product εps, r, ψF q
¹ v
εps, rv , ψFv q,
where ψF is a nontrivial additive character on A{F . We cannot say that this function is independent of ψF in general. But if r is symplectic, by which we mean that it takes values in the symplectic subgroup of the underlying general linear group, the value of the local factor εps, rv , ψFv q at s 12 is
1.5. STATEMENT OF THREE MAIN THEOREMS
known to equal a global sign
1 or
47
1, and to be independent of ψF . We therefore have v
ε
1 2, r
ε
1 2 , r, ψF
1
in this case, which is independent of ψF . r pGq. We first define We shall define εψ if ψ is a general parameter in Ψ a representation τψ : Sψ Lψ SLp2, Cq
ÝÑ
ˆq GLpg
p by setting ˆ of G on the Lie algebra g
τψ ps, g, hq Ad s ψrG pg, hq ,
s P Sψ , pg, hq P Lψ SLp2, Cq,
where Ad AdG is the adjoint representation of L G. Notice that since it ˆ, this representation is orthogonal, is invariant under the Killing form on g and hence self-dual. Let τψ
à
τα
α
à α
pλα b µα b ναq
be its decomposition into irreducible representations λα , µα and να of the respective groups Sψ , Lψ and SLp2, Cq. We then define (1.5.6)
εψ pxq
¹1
det λα psq ,
α
where x is the image of s in Sψ , and indices α with µα symplectic and (1.5.7)
ε
1 2 , µα
±1
s P Sψ ,
denotes the product over those
1.
(The definition (1.5.6) is equivalent to [A18, (30.16)], and is conjecturally equivalent to the slightly different formulations [A8, (8.4)] and [A9, (4.5)] in the earlier papers in which the general multiplicity formula (1.5.5) was postulated.) Theorem 1.5.2 thus asserts that there is an intimate relationship between the automorphic discrete spectrum of G and symplectic root numbers. Before we state the second global theorem, we recall a property of certain Rankin-Selberg L-functions. Suppose that π P Acusp pN q is a cuspidal automorphic representation of GLpN q. Then there are two formal products (1.5.8) and (1.5.9)
Lps, π π q Lps, π, S 2 q Lps, π, Λ2 q
εps, π π, ψF q εps, π, S 2 , ψF q εps, π, Λ2 , ψF q,
where S 2 (resp. Λ2 ) is the representation of GLpN, Cq on the space of symmetric (resp. skew-symmetric) pN N q-complex matrices. Each of S 2 and p , for a Λ2 is associated with a Siegel maximal parabolic subgroup Pp G split, simple endoscopic group G P Ersim pN q with N 2N . The dual p equals SppN , Cq or SOpN , Cq, according to which of the two group G
48
1. PARAMETERS
representations S 2 are Λ2 we are considering. In each case, the represenx GLpN, Cq tation is given by the adjoint action of the Levi subgroup M on the Lie algebra of the unipotent radical. Viewed in this way, one sees that the L-functions on the right hand side of (1.5.8) are among the cases of the Langlands-Shahidi method treated in [Sha4]. In both cases, the local L-functions and ε-factors have been constructed so that the formal products (1.5.8) and (1.5.9) become actual products, and so that the global L-functions have analytic continuation with functional equation (1.3.5). r sim pN q is cuspidal generic. In other words, φ Suppose now that φ P Φ is given by a representation π P Acusp pN q that is self dual. Theorem 1.4.1 asserts that φ belongs to the subset r sim pGq Φ r sim pN q X Ψ r pG q , Φ
for a unique G P Ersim pN q. We will need to understand how G is related to L-functions and ε-factors. The Rankin-Selberg L-function Lps, φ φq Lps, π π q Lps, π π _ q
has a pole of order 1 at s 1. It is known that neither of the corresponding factors Lps, φ, S 2 q and Lps, φ, Λ2 q on the right hand side of (1.5.8) has a zero at s 1. It follows that exactly one of them has a pole at s 1 (which will be of order 1). This motivates the first assertion (a) of the second global theorem, which we can now state. Theorem 1.5.3. Assume that F is global. r sim pGq. Then G p (a) Suppose that G P Ersim pN q and that φ belongs to Φ 2 is orthogonal if and only if the symmetric square L-function Lps, φ, S q has p is symplectic if and only if the skew-symmetric a pole at s 1, while G 2 L-function Lps, φ, Λ q has a pole at s 1. r sim pGi q, for simple endo(b) Suppose that for i 1, 2, φi belongs to Φ scopic data Gi Ersim pNi q. Then the corresponding Rankin-Selberg ε-factor satisfies ε 12 , φ1 φ2 1,
p 1 and G p 2 are either both orthogonal or both symplectic. if G
The assertion (a) of this theorem was suggested by the corresponding property for L-functions of representations of Galois groups and Weil groups. It seems to have been first conjectured for automorphic representations by Jacquet and Shalika, at the early stages of their work on Rankin-Selberg L-functions. The assertion (b) is suggested by the corresponding property for ε-factors of orthogonal representations of Galois groups [FQ] and Weil groups [D1]. It was conjectured for automorphic representations in [A8, §8] and [A9, §4]. Observe that (a) gives an independent characterization of the group Gφ of Theorem 1.4.1. It has been established for representations with Whittaker models by Cogdell, Kim, Piatetskii-Shapiro and Shahidi [CKPS1],
1.5. STATEMENT OF THREE MAIN THEOREMS
49
[CKPS2]. In fact, apart from the uniqueness assertion of the first seed Theorem 1.4.1, the statements of Theorem 1.4.1 and 1.4.2, together with that of (a) above, follow from the combined results of [CKPS2] and [GRS]. However, we will not be able to use these results. From our standpoint, the essence of the two seed theorems will be in two further statements that we have temporarily suppressed in the interests of simplicity. These give further characterizations of Gφ and its completions Gφv in terms of harmonic analysis, and will be what really drives the general argument. They will be included in the assertions of Theorems 2.2.1 and 4.1.2, which we will come to in due course. The broader significance of (b) will become clear later. This assertion has been proved in special cases by Lapid [Lap]. Its role for us will be rather similar to that of (a). Both assertions will be established in the course of proving the other theorems. In fact, they are both an inextricable part of the general induction argument by which all of the theorems will eventually be proved.
CHAPTER 2
Local Transfer 2.1. Langlands-Shelstad-Kottwitz transfer We now take the field F to be local, a condition that will remain in place throughout Chapter 2. In the first section, we shall review the endoscopic transfer of functions. This is based on the transfer factors of Langlands and Shelstad, and the twisted transfer factors of Kottwitz and Shelstad. It will serve as the foundation for the local classification we stated in §1.5. r pN q. The twisted transfer factors of KotConsider our GLpN q-coset G r pN q depend on a choice of F -rational automorphism twitz and Shelstad for G of GLpN q within the given inner class. For this object, we choose the automorphism θrpN q introduced in §1.2 that fixes the standard splitting of r pN q to stand for the pair G r 0 pN q, θrpN q , as well GLpN q. We then allow G as the connected component r 0 pN q θrpN q GLpN q θ pN q, G
obtained from the pair. We recall that θrpN q lies in the inner class of the original automorphism θ θpN q, and therefore gives the same coset in the r pN q. semidirect product G The datum r pN q G r 0 pN q, θrpN q G
r 0 pN q, θrpN q, 1 G
is a special case of a general triplet pG0 , θ, ω q, where G0 is a connected reductive group over F , θ is an automorphism of G over F , and ω is a character on G0 pF q. The group generated by θ need not be finite. Its semidirect product with G0 therefore need not be an algebraic group. However, we can still form the connected variety G G0 θ over F , equipped with the two-sided G0 -action x 1 py θ qx 2
x1 yθpx2 q
θ,
x P G0 ,
of G0 on G. The restriction of this action to the diagonal image of G0 can evidently be identified with the familiar action of G0 on itself by θr pN q, we shall also write conjugation. As in the case of G G pG0 , θ, ω q. 51
52
2. LOCAL TRANSFER
In other words, we allow G to represent the underlying triplet as well as the associate G0 -bitorsor over F . For the moment, let us take G to be a general triplet over F . We can then form the general local Hecke module HpGq of G. It consists of the smooth, compactly supported functions on GpF q that are left and right Kfinite, relative to the two-sided action on GpF q of a suitable (fixed) maximal compact subgroup K of G0 pF q. A semisimple element γ P G will be called strongly regular if its G0 -centralizer Gγ
x P G0 : x1 γx γ
pG0qγ
(
is an abelian group, whose group Gγ pF q of rational points lies in the kernel of the character ω on G0 pF q. For any such γ, and any function f P HpGq, we form the invariant orbital integral fG pγ q |Dpγ q|
1 2
»
p qz p q
Gγ F G0 F
f px1 γxqω pxqdx.
It is normalized here by the usual Weyl discriminant Dpγ q det
p1 Adpγ qg {g 0
0 γ
,
where g0 and g0γ denote the Lie algebras of G0 and G0γ respectively. We regard fG as a function on the set of strongly regular points, and write I pGq fG : f
P H pG q
(
for the G0 -invariant Hecke space of such functions. The functions in I pGq also have a spectral interpretation. Suppose that π is a unitary extension to GpF q of an irreducible unitary representation pπ0, V q of G0pF q. In other words, π is a function from GpF q to the space of unitary operators on V such that (2.1.1)
π px1 xx2 q π 0 px1 qπ pxqπ 0 px2 qω px2 q,
x1 , x2
P G 0 pF q .
The set of such extensions of π 0 is a U p1q-torsor, on which the character tr π pf q
is equivariant. We set
»
tr
p q
G F
f pxqπ pxqdx ,
f
P H pG q ,
fG pπ q tr π pf q . It can then be shown that either of the two functions tfG pγ qu or tfG pπ qu attached to f determines the other. We can therefore regard fG as a function of either γ or π. The spectral interpretation is in some ways preferable. This is because the trace Paley-Wiener theorem ([BDK], [CD], [Ro1], [DM]) leads to a simple description of I pGq as a space of functions of π. In general, an endoscopic datum G1 for G represents a 4-tuple 1 pG , G 1, s1, ξ1q. According to Langlands, Shelstad and Kottwitz [LS1], [KS], one can attach a transfer factor ∆ to G1 . Recall that any ∆ comes with r 1 Ñ G1 over F some auxiliary data, namely a suitable central extension G 1 1 L 1 r r and an admissible L-embedding ξ : G Ñ G , and is then determined up to
2.1. LANGLANDS-SHELSTAD-KOTTWITZ TRANSFER
53
a complex multiplicative constant of absolute value 1. (See [KS, §2.2]. We shall review these notions later in the global context of §3.2.) The transfer factor is a function ∆pδ 1 , γ q, where δ 1 is a strongly G-regular stable conjur 1 pF q, and γ is a strongly regular orbit in GpF q under the gacy class in G 0 action of G pF q by conjugation. It serves as the kernel function for the transfer mapping, which sends functions f P HpGq to functions ¸ r1 1 G f 1 pδ 1 q f∆ pδ q ∆pδ1, γ qfGpγ q γ
of δ 1 . We recall that f 1 has an equivariance property r 1 p F q, f 1 pz 1 δ 1 q ηr1 pz 1 q1 f 1 pδ 1 q, z1 P C
r 1 is the kernel of the projection G r 1 Ñ G1 , and ηr1 is a character on where C 1 1 r r C pF q that depends on the choice of ξ . Suppose for example that G G0 is just a quasisplit group (with trivial r 1 equal to G and ξr1 equal to character ω), and that G1 G. If we set G r 1 L G, ∆ becomes a constant, the identity embedding of G 1 L G into L G which we can take to be 1. In this case, the transfer
f G pδ q
¸
γ
Ñδ
fG pγ q,
f
P H pG q,
of f is the stable orbital integral over a strongly regular, stable conjugacy class δ. It is a sum of invariant orbital integrals over the finite set of GpF qconjugacy classes γ in δ. We write S pGq for the space of functions of δ obtained in this way. More generally, if XG is a closed subgroup of GpF q, and χ is a character on XG , we write S pG, χq for the associated space of χ1 -equivariant functions fχG
pδ q
»
XG
f G pzδ qχpz qdz,
f
P H pG q ,
of δ. A χ-equivariant linear form S on HpGq is called stable if its value at f depends only on fχG . If this is so, we can identify S with the linear form (2.1.2)
SppfχG q S pf q,
f
P H pG q,
r1 , C r 1 pF q, ηr1 in on S pG, χq. These conventions can then be applied with G place of pG, XG , χq. The LSK (Langlands-Shelstad-Kottwitz) conjecture asserts that for any G and G1 , any associated transfer factor ∆, and any f P HpGq, the function r 1 , ηr1 q. The fundamental lemma is a variant of this, which f 1 belongs to S pG applies to the case that G is unramified. (Among other things [W6, 4.4], unramified means that F is p-adic, and that G0 is quasisplit and split over an unramified extension of F .) The fundamental lemma asserts that if f is the characteristic function of a product K θ, where K is a θ-stable, hyperspecial maximal compact subgroup of G0 pF q, then f 1 can be taken to be the image r 1 , ηr1 q of the characteristic function of a hyperspecial maximal compact in S pG
54
2. LOCAL TRANSFER
r 1 pF q. Both of these longstanding conjectures have now been subgroup of G resolved. We include a couple of brief remarks about the proofs, some of which are quite recent. If F is archimedean, the results are due to Shelstad. In the case G G0 , she established the transfer theorem in terms of the somewhat ad hoc transfer factors she introduced for the purpose [S3]. These earlier transfer factors for real groups were the precursors of the systematic transfer factors for both real and p-adic groups in [LS1]. In a second paper, Langlands and Shelstad showed that the two transfer factors were in fact the same for real groups, by an indirect argument that used Shelstad’s original proof of the transfer theorem [LS2, Theorem 2.6.A]. Shelstad has recently expanded her earlier results [S4]–[S6], establishing the transfer conjecture (again for G G0 ) directly in terms of the transfer factors of [LS1]. She has recently completed a proof of the general archimedean transfer [S7], using the explicit specialization to real groups of the twisted transfer factors of [KS]. (See also [Re].) For nonarchimedean F , the fundamental lemma has long been a serious obstacle. Its recent proof by Ngo [N] was a breakthrough, which has now opened the way for progress on several fronts. The general proof follows the special case of G U pnq treated earlier by Laumon and Ngo [LN], and is based on the new geometric ideas that were introduced by Goresky, Kottwitz and MacPherson [GKM1], [GKM2]. The methods in all of these papers exploit the algebraic geometry over fields of positive characteristic. However, by the results of Waldspurger [W3] on independence of characteristic (which have also been established by the methods of motivic integration [CHL], following [CH]), they apply also to our field F of characteristic 0. The paper of Ngo treats both the fundamental lemma for G G0 and the variant to which Waldspurger had reduced the general case [W6]. It therefore resolves the fundamental lemma for any G. As for the LSK conjecture, Waldspurger established some time ago that the case G G0 would follow from the fundamental lemma [W1]. His recent papers [W6], [W7] extend this implication to the general case. The general results of Waldspurger therefore yield the p-adic LSK conjecture in all cases. Our main focus in this volume will be on the three cases introduced in §1.2. In the first, we take G pG0 , θ, ω q to be the pair r pN q G r 0 pN q, θrpN q (with ω trivial) we were just discussing. In this case, G r pN q . G1 represents a twisted endoscopic datum G in the set ErpN q E G We have already fixed a homomorphism of L G into the dual group p 0 pN q GLpN, Cq, which of course determines an L-embedding of L G into G L G0 pN q. This amounts to a canonical choice of auxiliary data. In the second case, G and G1 are simply two objects G P Ersim pN q and G1 P E pGq. Here we have also fixed an L-embedding of L G1 into L G, so there is again no need of further auxiliary data. In the third case, we take G pG0 , θ, ω q to be the r pG r 0 , θrq (with ω trivial) attached to an even orthogonal group G in pair G
2.1. LANGLANDS-SHELSTAD-KOTTWITZ TRANSFER
55
r 1 in E pG r q. Once again, Ersim pN q, and G1 to be a twisted endoscopic datum G r 1 into L G r 0 , and no need for further auxiliary we have an L-embedding of L G data. This case will generally be easier to manage once we have taken care of the others, and will often be treated as an addendum. The first two cases on the other hand will have a central role in almost all of our arguments. In fact, we shall sometimes find it convenient to treat the corresponding pairs r pN q, G and pG, G1 q in tandem. G Since the groups we are considering here are quasisplit, we can fix canonical transfer factors. We shall use the Whittaker normalization of [KS, §5.3]. In general, a Whittaker datum consists of a rational Borel subgroup and a non-degenerate character on its group of rational points. In present circumstances, these objects can be chosen canonically, up to the choice of a fixed nontrivial additive character ψF of F . r pN q. We fix a θrpN q-stable Whittaker Consider the GLpN q-bitorsor G r 0 pN q by taking B pN q to be the standard Borel datum B pN q, χpN q for G r 0 pN q GLpN q, and χpN q to be the non-degenerate character subgroup of G on the unipotent radical NB pN q pF q of B pN, F q attached to ψF . That is,
χpN, xq ψF px1,2
xn1,n q,
x pxi,j q P NB pN q pF q.
Given a twisted endoscopic datum G P ErpN q, we take
∆r χpN q ∆r SpN q ε 21 , τF , ψF 1 to be the transfer factor assigned to B pN q, χpN q in [KS].
rN ∆
r S pN q is Here ∆ the transfer factor attached in [KS, p. 63] to the standard splitting S pN q r 0 pN q GLpN q, ψF is the additive character used to define χpN q, and of G τG is the representation of the Galois group ΓF on the space X pT q b Q, where T is the standard (diagonal) maximal torus in G. The local ε-factor ε 12 , τG , ψF is trivial unless G has aquasisplit group SOp2n, ηG q as a factor, in which case it equals ε 21 , ηG , ψF . (The general definition in [KS] also contains a factor ε 21 , τrN , ψF , where τrN is the representation of ΓF on θrpN q
the space X T pN q b Q attached to GrpN q, but since the associated maximal torus T pN q B pN q is split, this factor equals 1). A general group G P ErpN q is a product of (at most two) quasisplit orthogonal and symplectic groups, together with a number of general linear groups. The discussion of §1.2 leads to a standard splitting for each factor, and hence a splitting S for the product G. We use this splitting to form a Whittaker datum pB, χq for G. We then have a corresponding transfer factor 1 ε 12 , τG, ψF ∆ ∆χ ∆S ε 12 , τG1 , ψF for any G1 factor
P E pG q .
Similar considerations give rise to a canonical transfer
r ∆ rχ ∆
∆r S ε
1 rG1 , ψF 2, τ
1
ε
1 rG , ψF 2, τ
56
2. LOCAL TRANSFER
r of an even orthogonal group G P Ersim pN q, and for the twisted component G r 1 P E pG r q. a twisted endoscopic datum G r N , and with the knowledge that Given the normalized transfer factors ∆ the LSK-conjecture holds in general, we have a canonical transfer mapping r pN q to S pGq, for any G P ErpN q. The function frG actufr Ñ frG from H r N pGq-invariant functions in S pGq, since the ally lies in the subspace of Out r N is invariant under the action of this group on the first variable. function ∆ We also have a canonical transfer mapping f Ñ f 1 from HpGq to S pG1 q, for any G1 P E pGq, whose image has a similar symmetry property. Finally, r we have in the supplementary case of a twisted (even) orthogonal group G, 1 1 r r r r a canonical transfer mapping f Ñ f from HpGq to S pG q attached to any r 1 P E pG r q. G r p G q, I rpGq and SrpGq If G P Ersim pN q is simple, we follow §1.5 by defining H r N pGq-invariant functions in HpGq, I pGq and S pGq. as the subspaces of Out We extend this definition in the natural way to any G P ErpN q, or indeed to any connected, quasisplit group that is a product of orthogonal, symplectic and general linear factors, by imposing the appropriate symmetry condition at the orthogonal factors of G. For example, if
G1
G11 G12,
then
G1i
P ErsimpNiq, i 1, 2,
SrpG1 q SrpG11 q b SrpG12 q.
r pN q into SrpGq and H r p Gq The mappings fr Ñ frG and f Ñ f 1 then take H r N pGq is into SrpG1 q. Observe, however, that if G P ErpN q is not elliptic, Out larger than the symmetry group of the subspace SrpGq. In particular, the r pN q to SrpGq is not surjective in this case. transfer mapping fr Ñ frG from H Twisted transfer for GLpN q has a special role to play. It is what leads us to the essential objects for orthogonal and symplectic groups, in their guise as simple endoscopic data G P Ersim pN q, on which we will be able to base the local classification. To this end, we will need to know that the r pN q onto SrpGq. The property is part of a mapping fr Ñ frG does take H broader characterization of the image of the collective transfer mapping
fr
ÝÑ
à G
frG ,
r pN q, G P ErpN q, fr P H
which we will later also have to know. We sometimes denote general elements in ErpN q by M , since they can be regarded as Levi subgroups of elliptic endoscopic groups G P Erell pN q. Recall that any such M comes with an L-embedding ξM of M L M into L GLpN q, which we can treat as an equivalence class of N -dimensional representations of L M . We may as well take G also to be a general element in ErpN q. Suppose that its N -dimensional representation ξ ξG can be chosen so that it contains the image of (a representative) of ξM . In other
2.1. LANGLANDS-SHELSTAD-KOTTWITZ TRANSFER
57
words, ξM is the composition of an L-embedding L M L G with ξ. This x with a Levi subgroup of G, p which is dual to an L-embedding identifies M F -rational embedding λM : M ãÑ G as a Levi subgroup of G. We shall call λM a Levi embedding of M into G. It is uniquely determined up to composition α λM
αM r N pGq and αM P Aut r N pM q. We note that for any λM , by elements α P Aut the stable orbital integrals of a function f P HpGq over G-regular classes in M pF q give a function f M pfM qM in S pM q. Consider a family of functions
(2.1.3)
F
f
P HrpGq :
(
G P ErpN q ,
r pN q. We shall say that F is a parametrized by the endoscopic data for G compatible family if for any data G and M in ErpN q, and any Levi embedding r pGq and h P H r pM q in F λM of M into G, the associated functions f P H satisfy f M hM .
In other words, the function in SrpM q attached to M equals the restriction of the corresponding function for G. In particular, the family of stable funcr p Gq tions tf G u attached to F is determined by the subset of functions f P H parametrized by the elliptic elements G P Erell pN q, since any M has a Levi embedding into some elliptic G. We could therefore have formulated F as a family of functions parametrized by Erell pN q. Proposition 2.1.1. Suppose that F is any family of functions (2.1.3). Then r pN q such F is a compatible family if and only if there is a function fr P H that f G frG , G P ErpN q.
Proof. This is the twisted analogue (for GLpN q) of a property that plays a significant role in the stabilization of the trace formula. For the general untwisted case, the proof is implicit in [S3] and [A11], and will be r pN q , given explicitly in [A24]. (See [A11, p. 329].) For the twisted group G the proof follows similar lines, based on work in progress by Shelstad and Mezo on twisted characters for real groups and Waldspurger’s p-adic results [W4] on twisted endoscopy, which include the twisted analogue of a theorem from [W1]. However, since the lemma represents part of the stabilization of the twisted trace formula, which we will be adopting as a general hypothesis in the next chapter, we will not give a formal proof. We will be content simply to outline the main arguments. Suppose first that F is p-adic. This is the more difficult case. Its proof has three steps, which we discuss briefly in turn. It will be easier for us to
58
2. LOCAL TRANSFER
r pN q is at refer to the untwisted case in [A11], even though the proof for G least partly implicit in [W4, V–VI]. r pN q of the adjoint The first step is to establish twisted analogues for G relations [A11, (2.4), (2.5)] for geometric transfer factors. Let r reg,ell pN q Γreg,ell G r pN q Γ
be the set of G0 pN, F q-conjugacy orbits in GpN, F q that are strongly regular and elliptic. (We are using the natural twisted forms of these notions, for which the reader can consult [KS, p. 28, 74].) If G belongs to Erell pN q, let r N -reg,ell pGq ∆ r p Gq ∆ GpN q-reg,ell
r pN q-strongly regular be the set of stable conjugacy classes in GpF q that are G and elliptic. We then set E rE r Γ reg,ell pN q Γreg,ell GpN q
º
P p q
G Erell N
r N -reg,ell pGq{Out r N pG q , ∆
where the right hand quotient denotes the set of orbits under the finite group r N pGq. The twisted transfer factor for G r pN q is a function Out ∆pδ 1 , γ q ∆pδ, γ q
r reg,ell pN q and δ 1 P ∆ r N -reg,ell pN q, which depends only on the image δ of γ P Γ 1 E r of δ in Γreg,ell pN q. The adjoint relations take the form ¸
rE δ PΓ
reg,ell
and
pN q
¸ r reg,ell pN q γ PΓ
∆pγ, δ q∆pδ, γ1 q δ pγ, γ1 q
∆pδ, γ q∆pγ, δ1 q δ pδ, δ1 q,
P Γrreg,ellpN q and δ, δ1 P ΓrEreg,ellpN q, where ∆pγ, δ q ∆pδ, γ q is the adjoint transfer factor, and δ p, q is the Kronecker delta. Their proof
for elements γ, γ1
is similar to that of [A11, Lemma 2.2]. In the twisted case here, we use the equivariance relation @
D
∆pδ, γ q invpγ, γ1 q, κγ ∆pδ, γ1 q of [KS, Theorem 5.1.D], and the local form of the bijection established in [KS, Lemma 7.2.A]. The second step is to establish that the transfer mapping TE
à G
TG: a
ÝÑ
aE
à G
aG ,
a P Ircusp pN q,
2.1. LANGLANDS-SHELSTAD-KOTTWITZ TRANSFER
59
from the invariant space Ircusp pN q of cuspidal functions to its endoscopic counterpart (2.1.4)
E Ircusp pN q
à
P p q
G Erell N
Scusp pGq
Out r N pGq
,
r pN q is an isomorphism. (Recall that a function a in the space IrpN q I G r reg,ell pN q in is cuspidal if apγ q vanishes for every γ in the complement of Γ r reg pN q of all regular classes.) The proof of this fact is similar to the set Γ that of its untwisted analogue [A11, Proposition 3.5]. One uses the adjoint transfer factor ∆pγ, δ q to define a candidate
TE : aE
ÝÑ
a
for the inverse mapping. The adjoint relations then tell us that the mapping E pN q to TE is indeed the inverse of T E , once we know that it takes Ircusp Ircusp pN q. This last fact relies on deeper results of Waldspurger. One has to use his general methods of twisted descent [W6], for twisted transfer factors and the corresponding transfer mappings, in place of those of Langlands and Shelstad [LS2]. This reduces the problem to a local assertion for (untwisted) p-adic Lie algebras, which one verifies by inverting Waldspurger’s kernel formula [W1, (1.2)], as in the proof of [A11, Proposition 3.5]. (See [W4, V.1].) E The third step is to extend the isomorphism T to the full space r pN q . This is where we are taking the most for granted, namely IrpN q I G r pN q of Theorem 6.2 of [A11]. We shall the extension to the twisted group G be brief. Following [A11, §1], we form the graded vector space (2.1.5)
Irgr pN q
à
u tM
q Icusp pM
W pM q
,
u ranges over orbits of (semistandard) “Levi subsets” of G r pN q where tM under the Weyl group 0 N W G pN q SN , W 0 0 and q W N pM q Norm A , G r 0 pN q {M 0 W pM M
. (Recall that M G r pN q is defined [A4, §1] is the relative Weyl group for M r as a Levi component of a “parabolic subset” of GpN q, with split component r AM .) The twisted tempered characters for GpN q (which we will introduce formally at the beginning of the next section) then provide a canonical isomorphism from IrpN q onto Irgr pN q, as in the untwisted case [A11, §4]. If G belongs to Erell pN q, we can also form the graded vector space
(2.1.6)
Sgr pGq
à
tM u
Scusp pM q
W pM q
,
60
2. LOCAL TRANSFER
where tM u ranges over W0G -orbits of Levi subgroups of G, and W pM q is the relative Weyl group for M . The spectral construction from [A11, §5] then provides a canonical isomorphism from S pGq onto Sgr pGq. This assertion includes a nontrivial property of stability, and requires proof, unlike the r pN q above. The necessary justification is furnished by [A11, statement for G Theorem 6.1], the first main theorem of [A11]. It is the second main theorem r pN q we are taking for granted. [A11, Theorem 6.2] whose extension to G The extended assertion is that the two isomorphisms are compatible with transfer. In other words, the diagram
(2.1.7)
IrpN q
Ñ ÝÝÝÝ
Irgr pN q
S pG q
Ñ ÝÝÝÝ
Sgr pGq ,
T G
in which the left hand vertical arrow is the (twisted) transfer mapping, and the right hand vertical arrow is built out of restrictions of transfer mappings to cuspidal functions, is commutative. The assertion of the proposition for our p-adic field F is a consequence of these constructions. We introduce an invariant endoscopic space (2.1.8)
IrE pN q
$ & %
à
P p q
fG : f
G Erell N
as the subspace of stable functions in à
P p q
G Erell N
S pG q
, .
P F-
defined by the stable images of compatible families F. We also form the graded endoscopic space (2.1.9)
E Irgr pN q
à u tM
E q Icusp pM
W pM q
.
It is then not hard to construct a commutative diagram of isomorphisms IrpN q
(2.1.10)
E T
IrE pN q
Ñ ÝÝÝÝ Ñ ÝÝÝÝ
Irgr pN q
.
E pN q Irgr
The upper horizontal isomorphism is as above. The right hand vertical isomorphism is a consequence of the isomorphisms for spaces of cuspidal functions we have described. The lower horizontal isomorphism follows from the definition of a compatible family, and the existence of the corresponding isomorphism in (2.1.7). The three isomorphisms together then tell us
2.1. LANGLANDS-SHELSTAD-KOTTWITZ TRANSFER
61
that the left hand vertical arrow is also an isomorphism, which from the corresponding arrow in (2.1.7) we see is the endoscopic transfer mapping TE
à
P p q
G Erell N
T G.
The assertion of the proposition follows, for elements G P Erell pN q and hence for any G P ErpN q. Assume now that F is archimedean. In this case, we take for granted the extension of Shelstad’s theory of endoscopy, both geometric and spectral, to r pN q. (See [S7], [Me].) Then the archimedean forms of the twisted group G the spaces defined for p-adic F above all have spectral interpretations. They can each be regarded as a Paley-Wiener space of functions in an appropriate space of bounded, self-dual Langlands parameters φ : LF
ÝÑ
L
r 0 pN q GLpN, Cq ΓF . G
Such parameters can of course be identified with self-dual, unitary, N dimensional representations of WF . They can be classified by the analysis of §1.2, or rather the extension of this analysis to parameters that are not elliptic. It is then not hard to deduce the archimedean form of the last commutative diagram of isomorphisms, and hence the assertion of the proposition. Corollary 2.1.2. Suppose that G transfer mapping fr
ÝÑ
P ErsimpN q is simple.
frG ,
r pN q onto the subspace takes H
SrpGq S pGq
of S pGq.
Then the (twisted)
r pN q, fr P H Out r N pGq
Proof. We need to check that the transfer mapping T G in (2.1.7) takes
p q
p q
p q
Ir N onto Sr G . We can either show directly that any function in Sr G is
the image of a compatible family, or we can examine the lower horizontal mappings in (2.1.7) and (2.1.10). Taking the latter (less direct) course, we compose the inverse of the mapping in (2.1.10) with the projection of IrE pN q onto S pGq and the mapping in (2.1.7). This gives us a mapping (2.1.11)
E Irgr pN q
ÝÑ
Sgr pGq.
We must check that its image is the space of invariants in Sgr pGq under the r N p G q. natural action of Out E pN q from the If we restrict the mapping (2.1.11) to a summand of Irgr decompositions (2.1.9) and (2.1.4), which is then mapped to the corresponding summand of Sgr pGq in (2.1.6), we obtain an injection from a subspace
62
2. LOCAL TRANSFER
of Scusp pM q into a larger subspace. The problem is simply to characterize its image. It amounts to establishing a isomorphism of groups r OutM pM qW pM , M q W pM qOutN pG, M q,
q, P EellpM q, and Out r N pG, M q is the in W pM
M
, M q is the stabilizer of M where W pM r N pGq. This is a straightforward exercise, which we stabilizer of M in Out leave to the reader. For example, in the special case that M is the subgroup of diagonal matrices
T
tt :
t_
tu
in G, OutM pM q is trivial, and the product of groups on the right is indeed q SN . We note here that this equal to the stabilizer of M T in W pM property fails if G is composite, and therefore that the condition that G be simple is in fact necessary. Remarks. 1. One direction in the statement of the proposition is more or r pN q, the set tfrG u comes from a less obvious. The fact that for any fr P H compatible family F follows directly from the basic properties of transfer factors. It is the converse, the existence of some fr for any F, that is the main point. The argument we have given establishes both parts together. 2. The discussion from the proof of the corollary applies also to paramer pGq. It gives us another way to see that for any G P Ersim pN q, ters in the set Ψ r pGq to Ψ r pN q is injective. We have of course always been the mapping from Ψ r r pN q, a property that was already implicit in treating ΨpGq as a subset of Ψ the discussion of §1.2. 2.2. Characterization of the local classification The first of the three theorems stated in §1.5 applies to the local field F . It includes a local classification of representations of a quasisplit orthogonal or symplectic group G. However, the description is purely qualitative. In this section, we shall make the assertion precise. We shall state a theorem that characterizes the representations in terms of the LSK-transfer of functions. The classification is based on the transfer of self-dual representations of r pN q. GpN q GLpN q, or rather the extensions of such representations to G Having earlier normalized the LSK-transfer factors, we shall now normalize the extensions of these representations. We need to show that any irreducible r 0 pN, F q GLpN, F q has a canonical extension self-dual representation π of G r pN, F q. As was implicit for the transfer factors, we will use to the group G the theory of Whittaker models. Suppose first that π is tempered. Then π has a B pN q, χpN q -Whittaker functional ω, where B pN q, χpN q is the standard Whittaker datum fixed in the last section. By definition, ω is a nonzero linear form on the underlying
2.2. CHARACTERIZATION OF THE LOCAL CLASSIFICATION
space V8 of smooth vectors for π such that ω π pn q v
χpN, nqωpvq,
n P NB pN q pF q, v
63
P V8 .
It is unique up to a scalar multiple. We are assuming that π is self-dual, which is to say that the representation π θrpN q is equivalent to π. We can therefore choose a nontrivial intertwining operator Ir from π to π θrpN q, which is unique up to a nonzero scalar multiple. Since the Whittaker datum B pN q, χ pN q is θrpN q-stable, the linear form ω Ir on V is also a B pN q, χpN q -Whittaker model for π. It therefore equals cω, for some r Then π c P C . We set π rpN q π θrpN q equal to the operator c1 I. rpN q is r the unique intertwining operator from π to π θpN q such that ω
ω πrpN q.
r pN, F q generated by It provides a unitary extension π r of π to the group G r r pN, F q GpN, F q. In particular, it gives an extension of π to the bitorsor G that satisfies (2.1.1). Suppose next that π is replaced by a self-dual standard representation ρ. Recall that ρ is a (possibly reducible) representation, induced from the twist σ πM,λ of an irreducible tempered representation of a Levi subgroup M pF q by a regular, positive real valued character. We can identify M with a block diagonal subgroup
GLpm1 q GLpmk q
of GLpN q, and σ with a representation π1 px1 q| det x1 |λ1
b b πk pxk q| det xk |λ , k
x P M p F q,
for irreducible tempered representations π1 , . . . , πk and real numbers λ1 ¡ ¡ λk . It follows easily from the self-duality of ρ, and the form we have chosen for σ, that πi_
πk
1 i,
1 ¤ i ¤ k.
In particular, M is θrpN q-stable, and σ is isomorphic to σ θrpN q. By a minor variation of our discussion of the tempered representation π above, we see that σ has a canonical extension to the group M
pF q M pF q xθrpN qy.
Notice that the standard block upper triangular subgroup P in GLpN q of r pN q can be type pm1 , . . . , mk q is also θrpN q-stable. Its normalizer P in G regarded as a parabolic subgroup of G pN q. The pullback from M pF q to r pN, F q. P pF q of the extended representation σ can then be induced to G r We thus obtain an extension ρr of ρ to G pN, F q. We note that in the case of GLpN q here, there is a bijection φ Ñ ρ from r pN q of self-dual Langlands parameters φ for GLpN q and the set of the set Φ
64
2. LOCAL TRANSFER
r pN q, we self-dual standard representations ρ ρφ of GLpN, F q. Given φ P Φ shall write r pN q, frN pφq tr ρrpfrq , fr P H for the extension ρr ρrφ of ρφ we have just described. Observe that for ρ of the form above, we can replace the vector λ pλ1 , . . . , λk q in Rk by a complex multiple zλ, z P C. This gives us another self-dual standard representation ρz , with corresponding Langlands parameter φz . The function frN pφz q, defined for z near 1 in terms of the Whittaker model of M , extends to an entire function. Now if z is purely imaginary, ρz is tempered, and frN pφz q is also defined in terms of the Whittaker model of GLpN q. However, it follows easily from the general structure of induced Whittaker models [Shal], [CS] that the two functions of z P iR are equal. In other words, frN pφq can be obtained for general φ by analytic continuation from the case of tempered φ. Suppose finally that π is a general irreducible, self-dual representation of GLpN, F q. Then π is the Langlands quotient of a uniquely determined standard representation ρ. The bijective correspondence between irreducible and standard representations implies that ρ is also self-dual. The canonical r pN, F q then provides a canonical extension π r of π extension ρr of ρ to G r to G pN, F q. This is what we needed to construct. In the case that π is r pN, F q represents unitary, the restriction of this extension to the bitorsor G a canonical extension from the general class (2.1.1). r pN q. As in the global remarks Suppose now that ψ is a parameter in Ψ following the statement of Theorem 1.3.3, we define the local Langlands r pN q by setting parameter φψ P Φ
|u|
φψ puq ψ u,
0
1 2
u P LF .
0
, 1
|u| 2
We then have the standard representation ρψ ρφψ of GLpN, F q attached to φψ , and its Langlands quotient πψ πφψ . The irreducible representation πψ is unitary and self-dual. It therefore has a canonical extension π rψ to the r bitorsor GpN, F q. We write (2.2.1)
rψ pfrq , frN pψ q tr π
r pN q. fr P H
Our interest will be in the transfer of this linear form to a twisted endoscopic group. r pGq of Ψ r pN q attached to a simple Suppose that ψ belongs to the subset Ψ r datum G P Esim pN q. Then ψ can be identified with an L-homomorphism ψrG : LF
SU p2q ÝÑ
L
G.
On the other hand, we have just seen that ψ determines a linear form (2.2.1) r pN q H G r pN q . The next theorem uses this linear form to on the space H
2.2. CHARACTERIZATION OF THE LOCAL CLASSIFICATION
65
make the local correspondence of Theorem 1.5.1(a) precise. Specifically, it r ψ and the pairing x Ñ x , π y gives an explicit construction of the packet Π of Theorem 1.5.1(a) in terms of (2.2.1), and the endoscopic transfer of functions. r pG q. Theorem 2.2.1. (a) Suppose that G P Erell pN q, and that ψ belongs to Ψ Then there is a unique stable linear form
(2.2.2)
f
ÝÑ
f G p ψ q,
f
r pGq with the general property on H
frG pψ q frN pψ q,
(2.2.3)
together with a secondary property (2.2.4) in case
P HrpGq,
r pN q, fr P H
f G pψ q f S pψS qf O pψO q, G GS
GO , ψ ψS ψO ,
f
P HrpGq,
P ErsimpNεq, r p Gε q , ψε P Ψ Gε
and fG
f S f O,
fε
P SrpGεq,
ε O, S,
are composite. (b) Suppose that G P Ersim pN q. Then for every ψ r ψ over Π r unit pGq, together with a mapping set Π (2.2.5)
ÝÑ x , πy,
π
π
P Ψr pGq, there is a finite
P Πr ψ ,
r ψ to the group Spψ of irreducible characters on Sψ , with the following from Π property. If s is a semisimple element in the centralizer Sψ Sψ pGq and pG1, ψ1q is the preimage of pψ, sq under the local version of the correspondence (1.4.11) in §1.4, then
(2.2.6)
f 1 pψ 1 q
¸
P
rψ π Π
xsψ x, πy fGpπq,
where x is the image of s in Sψ .
f
P HrpGq,
Remarks. 1. If G P Ersim pN q is simple, the uniqueness of the linear form r pN q onto in (a) follows from (2.2.3), since the mapping fr Ñ frG takes H r r S pGq. If G R Esim pN q is composite, the uniqueness follows from the product formula (2.2.4). r ψ and the pairing 2. It is clear that (2.2.6) characterizes the packet Π xx, πy in (b). It therefore does provide a canonical formulation of the local correspondence of Theorem 1.5.1. 3. If G is composite, the symbol ψ on the right hand of (2.2.3) is understood to be the image of the given parameter under (the not necessarily
66
2. LOCAL TRANSFER
r pGq to Ψ r pN q. In this case, the identity (2.2.3) is not injective) mapping of Ψ needed to characterize either the linear form in (a) or the local correspondence in (b). However, it will still have an important role in our proofs. 4. Theorem 2.2.1 will be proved at the same time as the theorems stated in Chapter 1, together also with further theorems we will state in due course. The argument, which will take up much of the rest of the volume, will rely on long term induction hypothesis based on the integer N . r bdd pN q of generic 5. If F is archimedean and ψ φ lies in the subset Φ r pGq, the assertions of the theorem are included in the general parameters in Ψ results of Shelstad [S3], [S4]–[S7], and their twisted analogues for GLpN q [Me], [S8] in preparation.
Theorem 2.2.1 of course includes parameters ψ that are not generic, in the sense that they are nontrivial on the second factor of LF SU p2q. One ultimately wants to reduce their study to the case that ψ φ is generic. With this in mind, we shall describe how the twisted character on the right hand side of (2.2.3) can be expanded in terms of standard twisted characters. r pN, F q. Let us write PrpN q for the set of “standard representations” of G Elements in this set are induced objects rM,λ q, ρr IPr pπ
λ P paPr q ,
N r is a standard “parabolic subset” [A4, §1] of G r pN q, and π rM where Pr M is an extension (2.1.1) to M pF q of an irreducible tempered representation 0 of M 0 pF q. (The sets Pr and Pr pN q are of course different. The first rM π symbol P is an upper case p, while the second is supposed to be an upper r r pN q of a θ-stable, case ρ.) By definition, Pr is the normalizer in G standard 0 0 0 0 r r r parabolic subgroup P M N of G pN q GLpN q. Its Levi subset M has a split component AM AM 0 , and an associated real vector space aM a r 0 and corresponding dual chamber aM 0 with an open chamber aPr r P pa r q pa r0 q . As in (2.1.1), there is a simply transitive action P
P
rM u: π
ÝÑ
π rM,u ,
u P U p1 q,
0 , and therefore a correof the group U p1q on the set of extensions π rM of π rM 0 and λ. We rM sponding action ρr Ñ ρru on the set of ρr P PrpN q attached to π note that any U p1q-orbit in PrpN q contains a simply transitive pZ{2Zq-orbit. rM generates an irreducible representation of It consists of those ρr such that π the group M pF q generated by M pF q. Within this pZ{2Zq-orbit, we have the element ρr defined by the Whittaker normalization above. r πρr. It is an exAny ρr P PrpN q has a canonical Langlands quotient π r r 0 pN q r0 P Π G tension (2.1.1) to GpN, F q of an irreducible representation π (except for the fact that π r0 does not satisfy the unitary condition of (2.1.1)). r attached to π r0 in this way is a transiAs in (2.1.1), the set of extensions π tive U p1q-orbit. The mapping ρr Ñ πρr is then a U p1q-bijection from PrpN q
2.2. CHARACTERIZATION OF THE LOCAL CLASSIFICATION
67
r pN q of extensions to G r pN, F q of irreducible onto the corresponding set Π r 0 pN, F q. We write tPr pN qu and tΠ r pN qu for the set of representations of G r pN q respectively. U p1q-orbits in PrpN q and Π A standard parabolic subset Pr comes with a canonical embedding of its dual chamber par q into the closure paB q of the dual chamber of the P
standard Borel subgroup B. For any standard representation ρr P PrpN q with r pN q as above, we shall write r πρr in Π Langlands quotient π Λρ
Λπ ,
π
πr0, ρ ρr0,
for the corresponding image of the point λ. In general, one writes Λ1
Λ1 , Λ P paB q ,
¤ Λ,
if Λ Λ1 is a nonnegative integral combination of simple roots of pB, AB q. r pN qu. This determines a partial order on each of the sets tPrpN qu and tΠ We thus have two families
r pN q, ρr P Pr pN q, fr P H
r p N q, π r fr P H
frN pρrq tr ρrpfrq ,
and
rq tr π rpfrq , frN pπ
P Πr pN q,
r 0 pN, F q-invariant linear forms on H r pN q. Taken up to the action of U p1q, of G they represent two bases of the same vector space. More precisely, there are uniquely determined complex numbers rq, npπ r, ρrq : ρr P Pr pN q, π r mpρr, π
with and such that (2.2.7)
rv q u mpρr, π rqv 1 , mpρru , π rv , ρru q v npπ r, ρrqu1 , n pπ ¸
frN pρrq π r
and (2.2.8)
rq frN pπ
PtΠr pN qu ¸
Pt p qu
ρr Pr N
P Πr pN q
(
,
u, v
P U p1 q,
u, v
P U p1q,
rq frN pπ r q, mpρr, π
ρr P PrpN q,
r, ρrq fN pρrq, n pπ
π r
P Πr pN q.
This is a special case of a general result, which can be formulated in the same manner for any triplet G pG0 , θ, ω q. (See [A7, Lemma 5.1], for example.) The first expansion (2.2.7) follows from the decomposition of the standard representation ρr0 of G0 pN, F q attached to ρr into irreducible representations. The second expansion (2.2.8) follows by an inversion of the first, in which the coefficient function mpρr, π rq is treated as a unipotent matrix. rqu. We recall that Let us be more specific about the matrix tmpρr, π mpρr, π rq 1 if π r πρr. In general, if mpρr, π rq 0 for a given ρr, the linear
68
2. LOCAL TRANSFER
form Λρ satisfies Λρ ¤ Λπ , with equality holding only when π r πρr. This is the property that is responsible for the matrix being unipotent. In addition, mpρr, π rq vanishes unless ρr and π r have the same central character η : F
ÝÑ
C ,
and more significantly, the same infinitesimal character µ : Z pN q
ÝÑ
C.
Here, Z pN q denotes the center of the universal enveloping algebra of the r 0 pN, F q if F is archimedean, and the Bernstein complexified Lie algebra of G 0 r pN, F q if F is p-adic. It is an abelian C-algebra, with a two-sided center of G r pN q that is compatible with the associated two-sided action of action on H r 0 pN q. We can therefore regard tmpρr, π rqu as the Hecke algebra HpN q H a block diagonal matrix, whose blocks are parametrized by θ-stable pairs pµ, ηq. Since there are only finitely many irreducible or standard representar 0 pN, F q with a given infinitesimal character µ, the blocks represent tions of G unipotent matrices of finite rank. We obtain equivalent expansions if we fix representatives in the two sets of U p1q-orbits. Let us do so by choosing the Whittaker extensions introduced r pN qu are then each bijective with the family above. The sets tPrpN qu and tΠ r pN q. For any φ P Φ r pN q, we are thus taking ρr of Langlands parameters Φ to be the associated standard representation ρrφ , and π r to be the associated r irreducible Langlands quotient π rφ . For any ψ P ΨpN q, we take π r to be rψ . This gives a set of representatives for the the associated representation π r pN qu indexed by Ψ r pN q. The expansion (2.2.8) for elements in subset of tΠ this subset can then be written (2.2.9) where
frN pψ q
¸
P p q
r N φ Φ
rpψ, φq frN pφq, n
ψ
P Ψr pN q,
rpψ, φq npπ rψ , ρrφ q. n
Before we proceed to the initial study of Theorem 2.2.1, we shall describe a second well known feature of general Langlands parameters φ P ΦpN q for GLpN q, their infinitesimal characters. These are by definition the infinitesimal characters µφ of the standard representations ρφ of GLpN, F q (or the corresponding Langlands quotients πφ , or indeed, any of the constituents of ρφ ). Our immediate motivation is the fact that the terms on the right hand side of (2.2.9) all have the same infinitesimal character. Our remarks will r pGq, which we will need for the analysis lead to information about the set Φ of (2.2.9). They will also be useful later on. Suppose first that F is archimedean, and that φ : WF
ÝÑ
GLpN, Cq
2.2. CHARACTERIZATION OF THE LOCAL CLASSIFICATION
69
is a general Langlands parameter for GLpN q over F . The infinitesimal character of φ is given by a semisimple adjoint orbit in the Lie algebra of GLpN, Cq. To describe it, we first choose a representative of φ whose restriction to the subgroup C of WF takes values in the group of diagonal matrices. Following Langlands’ general notation [L11, p. 128], we write _ _ (2.2.10) λ_ φpz q z xµφ ,λ y z xνφ ,λ y , z P C , λ_ P ZN ,
for complex diagonal matrices µφ and νφ with µφ νφ integral. The infinitesimal character of φ is then equal to (the orbit of) µφ , which we can regard as a multiset of N complex numbers. r ell pN q of ΦpN q. Suppose for example that φ belongs to the subset Φ The condition that φ be self-dual is equivalent to the relation t µφ µφ . The ellipticity condition implies further that φ is a direct sum of distinct, self-dual two dimensional representations of WF (together with a self-dual one-dimensional character of F in case N is odd), from which one sees that νφ µφ . It follows that _ z P C , λ_ P ZN , λ_ φpz q pzz 1 qxµφ ,λ y ,
where µφ P 21 ZN is now a diagonal half integral matrix with t µφ µφ . We r 2 pGq, for a unique G P Erell pN q. The dual group of know that φ belongs to Φ G is given by pG pS G
GpO SppNS , Cq SOpNO , Cq,
where NO and NS are defined as the number of components of µφ , a priori half integers, which are respectively integral and nonintegral. This is an immediate consequence of the explicit description of Langlands parameters for orthogonal and symplectic groups, which we will review in §6.1. The p and the fact that its quadratic character group G is then determined by G ηG equals the determinant ηφ of φ. Conversely, suppose that φ P ΦpN q satisfies (2.2.10), for a diagonal matrix µφ P 12 ZN with t µφ µφ . This does not imply that φ lies in r ell pN q, even if the components of µφ are all distinct. However, we can Φ still attach a canonical element G P Erell pN q to µφ and ηφ , according to the prescription above. We can also choose a general endoscopic datum M P ErpN q such that φ is the image of a local Langlands parameter φM in r 2 pM q. It follows from the various definitions that M can be identified with Φ a Levi subgroup of G, and hence that φ is the image in ΦpN q of a canonical r pGq. If G P Esim pN q is simple, for example, Φ r pGq embeds into element in Φ r pGq of ΦpN q, so that φ itself represents a canonical element in the subset Φ Φ p N q. Suppose next that F is p-adic. The infinitesimal character of a Langlands parameter φ : LF WF SU p2q ÝÑ GLpN, Cq
70
2. LOCAL TRANSFER
for GLpN q over F is often called the cuspidal support of πφ or of ρφ . It can be regarded as a multiset µφ
r º
∆k ,
k 1
built from “segments” ∆k with r ¸
|∆k | N,
k 1
in the teminology of [Z]. Let us be more precise. We first recall [Z, §3] that a segment ∆ of norm
|∆| m msmu is a set
∆ rσαi , σαj s tσαi , σαi1 , . . . , σαj u,
where σ P Πscusp GLpms q is a (not necessarily unitary) supercuspidal representation of GLpms , F q, while mu 1 i j is an integer that is a ` difference of two given half integers i ¥ j, and σα P Πscusp GLpms q is defined by
pσα`qpxq σpxq | det x|`,
x P GLpms , F q.
(By a half integer, we again mean a number whose product with 2 is an integer!) The associated induced representation of GLpm, F q then has a unique subrepresentation π∆ . (See [Z, §3]. By implicitly ordering ∆ by decreasing half integers, as above, we have adopted the convention of Langlands rather than that of Zelevinsky. In other words, the special, essentially square integrable representation π∆ is a subrepresentation of the induced representation rather than a quotient.) Suppose that a is a multiset consisting of segments t∆k u of total norm N . We can then form the associated induced representation ρa
IP pπ∆ b b π∆ q 1
r
of GLpN, F q, and its corresponding Langlands quotient πa . The segments ∆ are bijective with simple Langlands parameters φ∆ P Φsim pmq. The associated mapping φ∆ Ñ π∆ is the Langlands correspondence for the relative discrete series of GLpmq. The multisets a are bijective with general Langlands parameters φa P ΦpN q. The associated two mappings φa Ñ πa and φa Ñ ρa in this case represent the general Langlands correspondence for irreducible and standard representations of GLpN q. Our description of the infinitesimal character µφ of a p-adic Langlands parameter φ P ΦpN q will now be clear. We write φ φa , for a multiset a whose elements are segments t∆k u. Then µφ is the multiset obtained by taking the disjoint union over k of the sets ∆k . For another description,
2.2. CHARACTERIZATION OF THE LOCAL CLASSIFICATION
71
which is a little closer to that of the archimedean case, we write χφ as at the beginning of §1.3 for the restriction of φ to the subgroup $ 1 & 2 w, w % 0
| |
0
1 |w| 2
: w
, .
P WF -
of WF SLp2, Cq. As an N -dimensional representation of WF , χφ P ΦpN q can then be regarded as a Langlands parameter that is trivial on the second factor of LF . The infinitesimal character µφ is then equal to the multiset µa that corresponds to χφ . As we observe from either description, µφ is a multiset of supercuspidal representations σα` , while a is a coarser multiset of segments ∆k . r ell pN q of ΦpN q. Then Suppose for example that φ belongs to the subset Φ φ is a direct sum of distinct, irreducible, self-dual representations of LF , which in turn correspond to distinct, self-dual segments. Since the dual of a general segment ∆ rσαi , σαj s equals ∆_ rσ _ αj , σ _ αi s, a self-dual segment may be written in the form (2.2.11)
∆ rσαi , σαi s tσαi , σαi1 , . . . , σαi u,
for a self-dual (unitary) supercuspidal representation σ P Πscusp pmq, and a nonnegative half integer i. This segment is symplectic if σ is symplectic and i is an integer, or if σ is orthogonal and i is not an integer. It is orthogonal otherwise, that is, if σ is orthogonal and i is an integer, or if σ is symplectic and i is not an integer. (Recall that the self-dual representation σ was designated symplectic or orthogonal in §1.4 according to whether its corresponding Langlands parameter factors through the associated complex subgroup of GLpm, Cq.) The dual group of G is again a product pG pS G
GpO SppNS , Cq SOpNO , Cq,
given in this case by the partition of the segments of φ into those of orthogonal and symplectic type. The group G is then determined by the dual group p and the character ηG ηφ . G Conversely, suppose that φ P ΦpN q is a parameter such that µφ is a disjoint union of self-dual sets (2.2.11). This does not imply that φ lies in r ell pN q. However, as in the archimedean case, we can still attach a canonical Φ element G P Erell pN q to µφ and ηφ , with the property that φ is the image r pGq. In the p-adic case, ηφ is just in ΦpN q of a canonical parameter in Φ the determinant of µφ , and is therefore superfluous here. However, we will continue to carry it in order to accommodate the Archimedean case. With these notions in mind, we consider again the expansion (2.2.9). r pN q on the right has an infinitesimal character µφ and Each parameter φ P Φ r pN q on the left is likewise equipped determinant ηφ . The parameter ψ P Ψ with data µψ µφψ and ηψ ηφψ . These parameters also come with the linear functionals Λ φ Λ πφ Λ ρφ
72
2. LOCAL TRANSFER
and Λψ
Λφ
ψ
.
Given ψ, we define r pN, ψ q φ P Φ r pN q : pµφ , ηφ q pµψ , ηψ q, Λφ Φ
¤ Λψ
(
.
The expansion (2.2.9) then remains valid if the sum is restricted to the r pN, ψ q of Φ r pN q. subset Φ Lemma 2.2.2. Suppose that the assertion (a) of Theorem 2.2.1 holds in the special case that ψ φ is generic. It is then valid for any ψ. Proof. The assertion is assumed to hold if ψ φ lies in the subset r bdd pN q of generic parameters in Ψ r pN q. Suppose that φ belongs to the Φ r pN q. In other larger set of generic parameters in the general family Ψ r pN q is a general self-dual Langlands parameter. We have words, φ P Φ observed that the associated (twisted) standard character frN pφq frN pρrφ q,
r pN q, fr P H
can be obtained by analytic continuation from the tempered case. Since the left hand side of (2.2.3) is also an analytic function in the relevant complex variable, our assumption implies that the obvious variant of assertion (a) of the theorem holds if ψ is replaced by any φ P ΦpN q. Suppose first that G is simple. In this case, we have only the definition r pG q, (2.2.3) to contend with. We must show that for the given element ψ P Ψ r the linear form fN pψ q in (2.2.3) depends only on the image of the mapping r pN q. fr Ñ frG . We shall use the expansion (2.2.9) for ψ as an element in Ψ Given the supposition of the lemma, and its extension above to general r pGq, we need only show that if the coefficient n rpψ, φq in parameters φ P Φ r r pGq of (2.2.9) is nonzero for some φ P ΦpN q, then φ lies in the subset Φ r pN q. We first note that ψ belongs to the complement of Ψ r ell pN q in Ψ r pN q Φ r if and only if it is the image of a parameter ψM P ΨpM q attached to a proper of G r pN q. In this case, the terms in (2.2.9) are each standard Levi subset M . The required assertion then follows induced from analogous terms for M inductively from its analogue for the proper Levi subgroup M of G attached . We may therefore assume that ψ lies in the intersection of Ψ r pGq with to M r r r Ψell pN q, which is just the subset Ψ2 pGq of ΨpGq. Another induction argument, combined with the knowledge we now have of the elliptic endoscopic data for G, reduces the problem further to the case that ψ is simple. r sim pGq to be simple. We are thus taking both G P Ersim pN q and ψ P Ψ Then ψ µ b ν, N m n, for irreducible unitary representations µ and ν of LF and SU p2q of respective dimensions m and n. The infinitesimal character µψ of ψ is by definition the infinitesimal character µφψ of the irreducible representation πψ πφψ
2.2. CHARACTERIZATION OF THE LOCAL CLASSIFICATION
73
in (2.2.1). It equals the tensor product of the infinitesimal characters of µ and ν, defined in the obvious way, and is naturally compatible with the tensor product of the representations themselves. (The overlapping notation in µ and µψ is unfortunate, but should not cause confusion.) If we compare the remarks for real and p-adic infinitesimal characters above with the local version of the discussion following (1.4.7), we see that the datum in Erell pN q attached to µψ µφψ and ηψ ηφψ equals G. But we have agreed that the r pN, ψ q of Φ r pN q. For any sum in (2.2.9) can be taken over φ in the subset Φ such φ, the datum attached to µφ µψ and ηφ ηψ is therefore also equal r pGq of Φ r pN q. This is what we to G. In particular, φ belongs to the subset Φ had to show. r pG q, We have established that for simple G P Ersim pN q, and any ψ P Ψ the set r pG, ψ q Φ r pN, ψ q Φ
r pGq of Φ r pN q. The definition (2.2.3) is thus valid is contained in the subset Φ for G. It takes the form
(2.2.12)
f G pψ q
¸
P p
r G,ψ φ Φ
q
rpψ, φq f G pφq, n
f
P HrpGq,
which we obtain from (2.2.9). r pGq are composite, as in (2.2.4). Suppose now that G P Erell pN q and ψ P Ψ In this case, the linear form (2.2.2) is defined by the product (2.2.4) and what we have established for the simple data GS and GO . The problem in this case is to show that (2.2.3) is valid. r ell pN q of It is again enough to assume that ψ belongs to the subset Ψ r r r ΨpN q. In other words, ψ lies in the subset Ψ2 pGq of ΨpGq. We are now dealing with decompositions G GS GO , ψ ψS ψO and N NS NO that are nontrivial. However, we shall treat them in the same way, by applyr pN, ψ q of Φ r pN q. ing the expansion (2.2.9) with φ summed over the subset Φ r pN, ψ q has the same infinitesimal character and determinant as Any φ P Φ r ell pN q. It follows from the earlier remarks for the original parameter ψ P Ψ real and p-adic infinitesimal characters that φ is the image of a canonical r pGq. In particular, we can identify φ with an element in parameter in Φ r pGq, even though the mapping from Φ r pGq to Φ r pN q is not injective. With Φ this interpretation, we obtain a canonical decomposition (2.2.13)
φ φS
` φO ,
φS
P Φr pNS , ψS q,
φO
P Φr pNO , ψO q,
r pN, ψ q. for any φ P Φ It is convenient to write IrpN, ψ q for the finite dimensional space of comr pN, ψ q. This space represents plex valued functions frN pφq on the finite set Φ r r pN q, since the twisted a quotient of the invariant Hecke space I pN q of G trace Paley-Wiener theorem for GLpN q implies that the restriction mapping from IrpN q to IrpN, ψ q is surjective. We can identify it with the space
74
2. LOCAL TRANSFER
of functions on the set r pN, ψ qu, PrpN, ψ q tρrφ : φ P Φ
r pN, ψ q. It is also canonically isomorphic since frN pρrφ q frN pφq for any φ P Φ to the space of functions on the set r pN, ψ q tπ r pN, ψ qu, rφ : φ P Φ Π
where we recall that like ρrφ , π rφ represents the canonical Whittaker extension r pN, F q of the underlying representation of GLpN, F q. Indeed, the gento G eral expansions (2.2.7) and (2.2.8) remain valid with the sets PrpN, ψ q and r pN, ψ q in place of tPr pM qu and tΠ r pN qu. For any function frN P I rpN, ψ q, Π r pN, ψ q is defined by this modification the corresponding function on Π frN pπ rφ1 q
¸
P p
r N,ψ φ Φ
q
n pπ rφ1 , ρrφ q frN pρrφ q,
φ1
P Φr pN, ψq,
of (2.2.8). We define a linear transformation IrpN, ψ q
(2.2.14)
ÝÑ IrpNS , ψS q b IrpNO , ψO q
P IrpN, ψq to the function frN ,N pφS φO q frN pφq,
by mapping frN
S
φ φS
O
` φO ,
in the tensor product. The mapping is actually an isomorphism, by virtue r pN, ψ q. It is of the canonical decomposition (2.2.13) of any parameter φ P Φ closely related to the twisted transfer mapping fr Ñ frG . More precisely, it is part of a commutative diagram of isomorphisms IrpN, ψ q
p
∨
Ir G, ψ
q
>
IrpNS , ψS q b IrpNO , ψO q
>
Ir GS , ψS
p
∨
q b IrpGO , ψO q,
in which IrpG, ψ q is the quotient of IrpGq defined by restriction of functions r pG, ψ q Φ r pN, ψ q of Φ r pGq. The left hand vertical arrow is the to the subset Φ corresponding lift of the transfer mapping frN Ñ frG , while the right hand vertical arrow is its analogue for ψS ψO . This diagram is a consequence of the supposition of the lemma, or rather its extension to non-tempered generic parameters φ described at the beginning of the proof. We shall show that the isomorphism dual to (2.2.14) maps the product rψS π π rψO , regarded as a linear form on the right hand tensor product, to rpN, ψ q. At the same time, we shall show that the the linear form π rψ on I coefficients in (2.2.9) satisfy the natural formula (2.2.15)
n rpψ, φq n rpψS , φS q n rpψO , φO q,
r pN, ψ q. φPΦ
2.2. CHARACTERIZATION OF THE LOCAL CLASSIFICATION
75
It follows from the definition of (2.2.14) that the dual isomorphism takes r pN, ψ q to the the linear form ρrS ρrO ρrφS ρrφO attached to any φ P Φ linear form ρr ρrφ . Viewed directly in terms of standard representations, this correspondence is a composition
ρO q ρ ÝÑ ρr, where Pψ is the standard parabolic subgroup of GLpN q of type pNS , NO q. (2.2.16)
ρrS
ρrO ÝÑ
ρS
ρO ÝÑ
IPψ pρS
The expansion (2.2.7) for ρr is obtained from the decomposition of ρr (as a standard representation of G pN, F q) into irreducible constituents. Its analogues for ρrS and ρrO are of course obtained in the same way. The reprψ is the Langlands quotient of ρrψ ρrφψ , or more precisely, the resentation π r pN, F q of the Langlands quotient πψ of ρψ that is compatible extension to G rψS π rψO is the Langlands with the Whittaker extension ρrψ of ρψ , while π quotient of ρrψS ρrψO . Using a local form of the global notation (1.3.6), we could write π rψS ` π rψO for the image of π rψS π rψO under the composition (2.2.16). But since the representations πψS and πψO are unitary, the induced representation IPψ pπψS πψO q is irreducible [Be]. It follows that π rψS ` π rψO , a priori a quotient of ρrψ , is actually the Langlands quotient π rψ . This is what we wanted to show. Moreover, if we apply the dual of (2.2.14) to the terms in the product of the expansions (2.2.9) for ψS and ψO , we see that rψS frN pπ
`
π rψO q
¸
P p
r N,ψ φ Φ
q
n rpψS , φS q n rpψO , φO q frN pφq,
in the obvious notation. The expansion (2.2.9) itself can be written rψ q frN pπ
¸
P p
r N,ψ φ Φ
q
n rpψ, φq frN pφq.
Since the left hand sides are equal, the required decomposition (2.2.15) of the coefficients follows as well. We can now complete our proof of the second half of Lemma 2.2.2. r pN q. According to the definition (2.2.12) Suppose that fr is any function in H G r of f pψ q, we can write frG pψ q frG pψS
ψO q rpψS , φS q n rpψO , φO q frG pφS φO q, n
¸
φS ,φO
for a double sum over φS equals ¸
n rpψS , φS q n rpψO , φO q frNS ,NO pφS
φS ,φO
¸
P p
P Φr pNS , ψS q and φO P Φr pNO , ψO q.
r N,ψ φ Φ
q
rpψ, φq frN pφq, n
φO q
This in turn
76
2. LOCAL TRANSFER
by the definition of frNS ,NO , the commutative diagram above, and the coefficient formula (2.2.15). The last sum is then equal frN pψ q, according to (2.2.9). We have established that frG pψ q equals frN pψ q, which is the required identity (2.2.3). Remarks. 1. My original proof was incorrectly based on the analogue of (2.2.9) for the connected group GLpN q. I thank the referee for pointing this out, and for suggestions that motivated the revised proof above. 2. The analogue of the decomposition (2.2.15) holds more generally for the coefficients np π r, ρrq,
π r
P Πr pN, ψq, ρ P PrpN, ψq,
in (2.2.8). This is because each induced representation IPψ pπS
πO q,
rS π
P Πr pNS , ψS q,
rO π
P Πr pNO , ψO q,
is irreducible, an observation I owe to the referee. The point is that there is no interaction between the infinitesimal characters of πS and πO . To be precise, we can attach an “affine Z-module” to the infinitesimal character µπ of any π P ΠpN q by writing Zrµπ s tσα` : σ
P µπ , ` P Zu
in the earlier notation for p-adic F , and Zrµπ s ts
` : s P µπ , ` P Zu
if F is archimedean and µπ is treated as a multiset of order N . The sets ZrµπS s and ZrµπO s are then disjoint, from which it follows by standard methods that the induced representation is irreducible. This presumably leads to a generalization of (2.2.3) that is tied to (2.2.8) (and a general representation π rφ ) rather than (2.2.9) (with the representation π rψ ). 3. The generic case of Theorem 2.2.1(a), which was our hypothesis for the lemma, will be established later by global means (Lemmas 5.4.2 and 6.6.3). It is possible that the general case of Theorem 2.2.1(a), the actual assertion of the lemma, could also be proved globally as a nongeneric supplement of Lemma 5.4.2. (See Lemmas 7.3.1 and 7.3.2, which represent nongeneric supplements of the other results in §5.4.) However, the local proof we have given here is perhaps more instructive. The following lemma gives another reduction of Theorem 2.2.1. This one is much easier, and has to some extent been implicit in our earlier discussion. Lemma 2.2.3. Suppose that Theorem 2.2.1 is valid if N is replaced by r pGq, for any integer N N . Then it also holds for any parameter ψ P Ψ r G P Esim pN q, such that the group Sψ has infinite center.
r pGq is as given. The centralizer in L G of Proof. Suppose that ψ P Ψ any nontrivial central torus in Sψ is then the L-group L M of a proper Levi subgroup M of G. We fix M , and choose an L-homomorphism ψM from
2.2. CHARACTERIZATION OF THE LOCAL CLASSIFICATION
77
r pGq equals ψ. The centralizer Sψ in LF SU p2q to L M whose image in Φ M x of the image of ψM is then equal to the original centralizer Sψ in G p of M the image of ψ. The Levi subgroup M is a product of general linear groups with a group G P Ersim pN q, for some N N . (See (2.3.4), for example.) It follows from the supposition of the lemma that the natural analogue of Theorem 2.2.1 holds for M . This gives us a stable linear form hM pψM q on SrpM q, a r ψ over Π r unit pM q, and a mapping πM ÝÑ x , πM y from Π r ψ to packet Π M M SpψM . We define the corresponding objects for G by setting
f G pψ q f M pψM q,
P HrpGq, ( r ψ π IP pπM q : πM P Π rπ , Π f
M
and
P
P P pM q,
x , πy x , IP pπM qy x , πM y. The induced representations IP pπM q are presumably irreducible, which of course would mean that π equals IP pπM q. But in any case, the required character identity (2.2.6) then follows from the familiar descent formula ¸
p q
π IP πM
fG pπ q tr IP pπM , f q
fM pπM q,
f
P HrpGq.
Theorem 2.2.1 characterizes the local classification in terms of two kinds of local endoscopy, twisted endoscopy for GLpN q and ordinary endoscopy for the group G P Ersim pN q. Recall that we have also been thinking of a third case, that of twisted endoscopy for an even orthogonal group. This case is not needed for the statement of the classification. However, it has an indispensable role in the proof, and it also provides supplementary information about characters. We shall state what is needed as a theorem, to be established along with everything else. Even orthogonal groups G P Ersim pN q of course represent the case that r N pGq is nontrivial. They are responsible for our general nothe group Out r r ψ (rather than ΨpGq and Πψ ), since it is in this case that tation ΨpGq and Π r ψ can be a nontrivial Out r N pGq-orbit of irreducible reprean element in Π sentations π (rather than a singleton). Twisted endoscopy is defined by the r N pGq. It applies to the subset nontrivial automorphism of order 2 in Out r r r pGq, or more accurately, those ΨpGq of OutN pGq-fixed parameters ψ in Ψ r pGq that can be represented by an Out r N pGq-fixed L-homomorphism ψPΨ L from LF SU p2q to G. For any such ψ, we would expect the elements r ψ to be singletons. In other words, they should consist of irreducible in Π r pF q generated by representations π of GpF q that extend to the group G r GpF q. For it is clear that the corresponding linear characters (2.2.5) on Sψ extend to characters on the group Srψ generated by Srψ . However, we shall establish this only in case ψ is generic.
78
2. LOCAL TRANSFER
Theorem 2.2.4. Suppose that N is even, that G P Ersim pN q is orthogonal in r p SOpN, Cq, and that ψ lies in the subset ΨpG r q of θ-stable the sense that G r pG q . elements in Ψ r (a) Suppose that sr is a semisimple element in the G-twisted centralizer r Sψ , and that
pGr1, ψr1q, Gr1 P E pGrq, ψr1 P ΨpGr1q, is the preimage of pψ, srq under the analogue of the correspondence (1.4.11).
Then we have an identity (2.2.17)
fr1 pψr1 q
¸
P
rψ π Π
xsψ xr, πry frGr pπrq,
r q, fr P HpG
r pF q, and r is the image of sr in Srψ , π r is any extension of π to G where x x, πry is a corresponding extension of the linear character (2.2.5) to Srψ such that the product xsψ xr, πry frGr pπrq, xr P Srψ ,
in (2.2.17) depends only on π (as an element in Πunit pGq). r φ is an Out r N pGq-stable representa(b) If φ ψ is generic, each π P Π r pF q. (In tion of GpF q, which consequently does have an extension π r to G r ψ does not have an extension, we agree simply to set π general, if π P Π r equal to 0). r pF q with stable Remarks. 1. The theorem relates twisted characters on G r 1 pF q. Recall that G r 1 is decharacters on the twisted endoscopic group G 1 1 r ,N r q whose sum equals N , termined by a pair of odd positive integers pN 1 2 1 1 and a pair of quadratic characters pηr1 , ηr2 q on ΓF whose product equals the quadratic character η that defines G as a quasisplit group. The left hand r 1 q, at a product of side of (2.2.17) is the value of fr1 , the transfer of fr to S pG stable linear forms
ψ1
ψr11 ψr21 ,
ψri1
P Ψr pNri1q,
each defined by Theorem 2.2.1(a). On the right hand side, the sum is taken r ψ that as irreducible representations of GpF q have extenover those π P Π r pF q. If π does have an extension, there is in general no canonical sions to G choice (in contrast to the case of twisted GLpN q). However, the theorem asserts that the summand in (2.2.17) is independent of the choice, which is all we need. 2. The extension x , π ry of x , π y is uniquely determined by the extension r of π. Given the condition that the product of x , π ry with frGr pπ rq depends π only on π as a representation of GpF q (which allows for the possibility that r ψ occurs several times with the same linear character x, π y), this π P Π easily established from (2.2.17). Indeed, if (2.2.17) is valid for two different extensions x , π r1 y and x , π r1 y1 of (2.2.5) to Srψ , for a given π1 P Πψ with
2.3. NORMALIZED INTERTWINING OPERATORS
79
r pF q, we can identify the corresponding summands on the extension π r1 to G r q, which r1 q vanishes for any fr P HpG right hand side. This implies that frGr pπ is contrary to the basic properties of irreducible characters. r ψ , since ψ is 3. We could have denoted the packet of ψ by Πψ rather Π r N pGq-stable. We shall often do so in the future. Out
2.3. Normalized intertwining operators Intertwining operators play an essential role in the proof of our theorems. Global intertwining operators are the main terms in the discrete part of the trace formula. It is important to understand their local factors in order to interpret the stabilization of this global object. Local intertwining operators also play a role in the local classification. They lead to a partial construction r ψ and pairings x, π y of Theorem 2.2.1, which reduces their of the packets Π r 2 pG q. study to the case that π belongs to Ψ We shall devote the remaining three sections of the chapter to this topic. One of the main problems is to normalize local intertwining operators. This has been done quite generally by Shahidi [Sha4], for inducing representations with Whittaker models. However, our inducing representations will frequently not have Whittaker models, even when they are tempered. We will apply global methods to their study, both in the initial stage here and in arguments from later chapters. The problem is not simply to normalize intertwining operators, but to do so in a way that is compatible with endoscopic transfer. To focus on the different questions that arise, it will be best to separate the problem into three distinct steps. The first is to normalize intertwining operators between representations induced from different parabolic subgroups. We begin by introducing a slightly nonstandard family of normalizing factors. Recall that if φ is an N -dimensional representation of the local Langlands group, we can form the local L-function Lps, φq and ε-factor εps, φ, ψF q. The ε-factor follows the notation [T2, (3.6.4)] of Langlands, and depends on a choice of nontrivial additive character ψF of F . For p-adic F , it has the general form
nps 12 q
εps, φ, ψF q εpφ, ψF q qF for a nonzero complex number εpφ, ψF q ε
1 2 , φ, ψF
δ pφ, ψF q εp0, φ, ψF q ε
1 2 , φ, ψF
,
,
and an integer n npφ, ψF q. The integer n is the simpler of the two constants, since it can be expressed explicitly in terms of the Artin conductor of φ and the conductor of ψF [T2, (3.6.4), (3.6.5)]. The quotient (2.3.1)
1
pqF q
n 2
is therefore a more elementary object than the ε-factor. Suppose now that G represents an element in Ersim pN q, and that M is a fixed Levi subgroup of G. Intertwining operators will be attached to
80
2. LOCAL TRANSFER
parameters and representations for M . For the time being, therefore, we shall denote such objects by symbols φ, ψ, π, etc., that have hitherto been reserved for G. r pM q is a Langlands parameter for In particular, suppose that φ P Φ M . The normalizing factors of φ include special values of local L-functions, which are only defined for parameters in general position. To take care of this, we let λ be a point in general position in the vector space aM,C
The associated twist
X pM qF b C.
φλ pwq φpwq|w|λ ,
w
P LF ,
of φ is then in general position. Suppose that P and P 1 belong to P pM q, the set of parabolic subgroups of G over F with Levi component M . We write ρP 1 |P for the adjoint representation of L M on the quotient
{
p nP 1 p nP 1
X pnP ,
where p nP 1 denotes the Lie algebra of the unipotent radical of Pp1 . The composition ρ_ P 1 |P
(2.3.2)
φλ
of φλ with the contragredient of ρP 1 |P is of course a finite dimensional representation of LF . We define a corresponding local normalizing factor rP 1 |P pφλ q rP 1 |P pφλ , ψF q
as the quotient
Lp0, ρ_ P 1 |P
(2.3.3)
φλq δpρ_P 1|P φλ, ψF q1Lp1, ρ_P 1|P φλq1.
Both M and φ have parallel decompositions (2.3.4)
M
GLpN11 q GLpNr1 1 q G,
and
G
P ErsimpNq,
r pG q, φ φ11 φ1r1 φ , φ P Ψ for positive integers N11 , . . . , Nr1 1 and N such that
2N11
2Nr1 1
N
N.
(Keep in mind that each general linear factor of M has a diagonal embedding g Ñ g g _ into G.) We shall ultimately argue by induction on N . This means that if M is proper in G, we will be able to assume that the local Langlands correspondence of Theorem 1.5.1 holds for the factor G of M . r pM q belongs to a In particular, we will be able to assume that any π P Π r φ , and thereby write unique L-packet Π (2.3.5)
rP 1 |P pπλ q rP 1 |P pφλ q.
In case F is archimedean, the notation here matches that of [A7], provided that ψF is taken to be the standard additive character [T2, (3.2.4), (3.2.5)]. For in this case, the general ε-factors are independent of s. The general
2.3. NORMALIZED INTERTWINING OPERATORS
81
quotient (2.3.1) is then equal to 1, and can be removed from the definition (2.3.3). The formula (2.3.5) requires a word of explanation. It is not a definition, since the right hand side is defined (2.3.3) in terms of local Artin L-functions and δ-factors, while the left hand side is the corresponding quotient
_ 1 _ 1 Lp0, πλ , ρ_ P 1 |P q δ pπλ , ρP 1 |P q Lp1, πλ , ρP 1 |P q
of representation theoretic objects. This in turn is defined as the quotient
_ 1 _ 1 Lp0, πφ,λ , ρ_ P 1 |P q δ pπφ,λ , ρP 1 |P q Lp1, πφ,λ , ρP 1 |P q
attached to the product
0 pN q GLpN 1 q GLpN 1 1 q GLpN q M 1 r
0 pN, F q given by of general linear groups, where πφ is the representation of M φ, and where the finite dimensional representation ρP 1 |P is identified with 0 pN q. The Rankin-Selberg its transfer from L M to the dual group of M constituents of each side of (2.3.5) are equal, thanks to the local classification for GLpN q. The symmetric and exterior square constituents are more subtle. However, it is known that their L-functions and δ-factors are equal [He2], even though at the time of writing, this has not been established for their ε-factors. The identity (2.3.5) is therefore valid. We need a more general formulation, which applies to nongeneric parameters ψ. Suppose first ψ is an n-dimensional representation of the product of LF with SU p2q. Then ψ extends to a representation of the product of LF with the complex group SLp2, Cq. Its pullback
φψ puq ψ u,
|u| 0
1 2
0
|u|
1 2
,
u P LF ,
becomes an N -dimensional representation of LF . We define Lps, ψ q Lps, φψ q,
εps, ψ, ψF q εps, φψ , ψF q, and δ pψ, ψF q δ pφψ , ψF q. In following standard notation here, we have had to use the symbol ψ in two different roles. To minimize confusion, we shall always include the subscript F whenever ψ denotes an additive character. r pM q is a general parameter for the Levi subgroup Suppose now that ψ P Ψ 1 M . For P, P P P pM q as above, the meromorphic function (2.3.6)
rP 1 |P pψλ q rP 1 |P pφψλ q rP 1 |P pφψ,λ q
of λ P aM,C depends implicitly on the additive character ψF . It will serve as our general normalizing factor.
82
2. LOCAL TRANSFER
The basic unnormalized intertwining operators are defined as for example in [A7, (1.1)]. We take π to be an irreducible unitary representation of M pF q, with twist πλ pmq π pmqeλpHM pmqq ,
m P M pF q ,
by λ P aM,C . We then have the familiar operator JP 1 |P pπλ q : HP pπ q
ÝÑ
H P 1 pπ q
that intertwines the induced representations IP pπλ q and IP 1 pπλ q. It is defined as a meromorphic function of λ by analytic continuation of an integral, taken over the space (2.3.7)
NP 1 pF q X NP pF qzNP 1 pF q,
which converges absolutely for Repλq in an affine chamber in aM . (See [A7, §1].) We do have to specify the underlying invariant measure on (2.3.7). The standard splitting on the quasisplit group G determines a Chevalley basis, and hence an invariant F -valued differential form of highest degree on the quotient (2.3.7). Its absolute value, together with the Haar measure on F that is self-dual with respect to ψF , then determines an invariant measure on (2.3.7). This is the measure that we take in the definition of JP 1 |P pπλ q. It follows easily from [T2, (3.6.5)] that the dependence of JP 1 |P pπλ q on ψF is parallel to that of the normalizing factor rP 1 |P pψλ q. We leave the reader to check that the measure is compatible with the convention in [A7] for real groups. In other words, if F is archimedean and ψF is the standard additive character, the canonical measure defined on p. 31 of [A7] in terms of a certain bilinear form coincides with the measure on (2.3.7) chosen here. We will therefore be able to apply the results of [A7] without modification. Our version of the normalized intertwining operators will require an assumption of the kind mentioned above. We assume that M is proper in G, and that the local Theorem 1.5.1 holds if G is replaced by M . There is of course nothing new to prove for the general linear factors of M . The assumption refers specifically to the factor G of M . It will later be treated as an induction hypothesis, applied to the integer N N . We can then r ψ of (OutN pG q-orbits) of representaassume the existence of the packet Π r tions of M pF q for every ψ P ΨpM q. We define the normalized intertwining r pM q and π P Π r ψ by operator attached to any ψ P Ψ (2.3.8)
RP 1 |P pπλ , ψλ q rP 1 |P pψλ q1 JP 1 |P pπλ q.
It is independent of the additive character ψF . We note that (2.3.8) differs in some respects from the standard normalized intertwining operator, conjectured in general in [L5, p. 281–282], and established for archimedean F in [A7]. For example, the δ-factor in (2.3.3) differs from the ε-factor of [L5]. This is essentially because the remarks of
2.3. NORMALIZED INTERTWINING OPERATORS
83
[L5] apply to elements w in the relative Weyl group
W pM q W G pM q NormpAM , Gq{M,
or more precisely, representatives of such elements in GpF q, rather than to pairs pP, P 1 q of parabolic subgroups. (We have already noted that the δfactor reduces to 1 in the archimedean case of [A7].) Another point is that π here represents an orbit of irreducible representations of M pF q under the group r N pM q Out r P pM q Out r N pG q. Out However, any use we make of (2.3.8) will be a context that is independent of the representative of the orbit. Finally, the normalizing factor (2.3.6) is defined in terms of the L-function and δ-factor of ψ, rather than π (which is to say, the Langlands parameter φ such that π lies in Πφ ). If ψ is not generic, the two possible normalizing factors can differ. The nonstandard form (2.3.6) turns out to be appropriate for the comparison of trace formulas. Proposition 2.3.1. Assume that the local and global theorems are valid if N is replaced by any integer N N . Then the operators (2.3.8) satisfy the multiplicative property (2.3.9)
RP 2 |P pπλ , ψλ q RP 2 |P 1 pπλ , ψλ qRP 1 |P pπλ , ψλ q,
for any P , P 1 and P 2 in P pM q, as well as the adjoint condition RP 1 |P pπλ , ψλ q
(2.3.10)
RP |P 1 pπλ, ψλq.
In particular, RP 1 |P pπλ , ψλ q is unitary and hence analytic if λ is purely imaginary, and the operator RP 1 |P pπ, ψ q RP 1 |P pπ0 , ψ0 q
is therefore defined. Proof. Suppose first that φ ψ is generic, which we recall means that rφ Π r ψ is an L-packet, it is trivial on the extra factor SU p2q. In this case, Π and (2.3.8) is the normalized intertwining operator RP 1 |P pπλ q rP 1 |P pπλ q1 JP 1 |P pπλ q,
π
P Πr ψ ,
studied for real groups in [A7]. Since the required assertions were established for real groups in [A7], we can assume that F is p-adic here. Moreover, by standard reductions [A7, p. 29], it suffices to consider the case that π represents an element in the set Π2 pM q of representations of M pF q that are square integrable modulo the center. In fact, we shall see that it suffices to consider the case that the general linear components of π are supercuspidal. Imposing these conditions on π, we write (2.3.11) π
π11 πr1 1 π,
πi
Πscusp
GLpNi1 q , π
P Πr 2pGq,
for its decomposition relative to (2.3.4). To handle this basic case, we shall use global means. Having reserved the symbol F in this chapter for our local field, we write F9 for a global field.
84
2. LOCAL TRANSFER
u S tuu, If u is any valuation of F9 , we write S8 puq S8 Y tuu and S8 8 where S8 is the set of archimedean valuations of F9 . We shall apply the following two lemmas to the components π and πi1 of π.
Lemma 2.3.2. Given the local objects F, G P Ersim pN q and π P Π2 pG q, we can find corresponding global objects F9 , G9 P Ersim pN q and π9 P Π2 pG9 q, together with a valuation u of F9 , with the following properties. (i) pF, G , π q pF9u , G9 ,u , π9 ,u q (ii) For any v R S8 puq, π9 ,v has a vector fixed by a special maximal compact subgroup of G9 pF9v q.
We shall establish a stronger version of Lemma 2.3.2 in §6.2, as an application of the simple invariant trace formula. In the meantime, we shall take the lemma for granted. Lemma 2.3.3 (Henniart, Shahidi, Vign´eras). We can choose the objects F9 and u of the last lemma so that the other local factors πi P Πscusp GLpNi1 q of π also have global analogues. More precisely, for each i we can find a cuspidal automorphic representation π9 i of GLpNi1 q over F9 such that πi π9 i,u , and such that for any v R S8 puq, π9 i,v has a vector fixed by a maximal compact subgroup of GLpNi1 , F9v q. This lemma is a consequence of the fact that πi has a Whittaker model. In the form stated here, it is Proposition 5.1 of [Sha4], which applies to a generic representation of any quasisplit group over F . We note, however, that π need not be generic, which is why we are using Lemma 2.3.2 in place of the analogue of Lemma 2.3.3 for G . Applying the lemmas to the components of the representation (2.3.11), we obtain an automorphic representation π9
π11 πr1 1 π 9
9
9
of the group M9
M11 Mr1 1 G, 9
9
9
M9 i1
GLpNi1q,
such that π9 u π. We can identify M9 with a Levi subgroup of a group G9 over F9 , which represents a simple endoscopic datum in Ersim pN q over F9 , and has the property that G9 u G. We then have the global induced representation IP pπ9 q,
P
P P p M q, 9
and the global intertwining operator, defined by analytic continuation in λ of the product â MP 1 |P pπ9 λ q JPv1 |Pv pπ9 v,λ q, from IP pπ9 λ q to IP 1 pπ9 λ q.
v
2.3. NORMALIZED INTERTWINING OPERATORS
85
The operator MP 1 |P pπ9 λ q here is a minor modification of the global operator that is at the heart of Langlands’ theory of Eisenstein series. In this setting Langlands’ functional equation takes the form (2.3.12)
MP 2 |P pπ9 λ q MP 2 |P 1 pπ9 λ qMP 1 |P pπ9 λ q.
(See for example [A3, §1].) We shall combine it with what can be obtained from the intertwining operators at places v u. We can write (2.3.13)
MP 1 |P pπ9 λ q rP 1 |P pπ9 λ q RP 1 |P pπ9 λ q,
for the normalized global intertwining operator (2.3.14)
RP 1 |P pπ9 λ q
â v
RP 1 |P pπ9 v,λ q,
and the global normalizing factor defined by analytic continuation in λ of a product ¹ rP 1 |P pπ9 λ q rP 1 |P pπ9 v,λ q. v
The factor of v in this last product is our canonical normalizing factor for the generic parameter φ9 v . It is defined by the analogue for π9 v,λ of (2.3.3) and (2.3.5). The existence of the corresponding L-function and δ-factor is guaranteed by the implicit assumption that Theorem 1.5.1 holds for the localization G9 ,v of G9 . We note that the measures and additive characters used to define the local factors in (2.3.14) can be chosen to be compatible with corresponding global objects. This follows from the well known fact [T1, (3.3)] that if each ψF9v is obtained from a fixed, nontrivial additive 9 is self-dual character ψF9 on A9 {F9 , the associated product measure on A 9 9 (relative to ψF9 ) and assigns volume 1 to the quotient A{F . We claim that (2.3.15)
RP 2 |P pπ9 v,λ q RP 2 |P 1 pπ9 v,λ qRP 1 |P pπ9 v,λ q,
for any valuation v u of F9 . For archimedean v, this is the concrete form of [A7, Theorem 2.1(R2 )], established as Proposition 3.1 in the Appendix of [A7]. For p-adic v, we use the property (ii) of π9 ,v in Lemma 2.3.2. Combined with the theory of p-adic spherical functions [Ma] and Whittaker models [C], it implies that π9 ,v has a Whittaker model. Since for any i, the generic representation π9 i,v of GLpNi , F9v q also has a Whittaker model, the same is true of the representation π9 v of M9 pF9v q. We can therefore apply Shahidi’s results [Sha4] to the induced representation IP pπ9 v,λ q. They imply that Langlands’ conjectural normalizing factors have the desired properties. In particular, the formula (2.3.15) does hold for v, as claimed. (We will discuss Shahidi’s results further in §2.5.) The global normalizing factors have an expression
_ q1 Lp1, π9 λ , ρ_1 q1 9 λ, ρ 1 rP 1 |P pπ9 λ q Lp0, π9 λ , ρ_ P 1 |P q δ pπ P |P P |P
86
2. LOCAL TRANSFER
in terms of global L-functions and δ-factors
_ qε 9 λ, ρ 1 δ pπ9 λ , ρ_ P |P P 1 |P q εp0, π
_
1 2 , πλ , ρP 1 P
|
1
.
We claim that they satisfy the identity
rP 2 |P pπ9 λ q rP 2 |P 1 pπ9 λ qrP 1 |P pπ9 λ q.
(2.3.16)
By standard reductions, it suffices to consider (2.3.16) when the Levi subgroup M9 of G9 is maximal. (See [A7, §2] for example.) In this case there are only two groups P and P in P pM9 q, and it suffices to take P 1 P and P 2 P , since rP |P pπ9 λ q 1. But ρP |P is just the direct sum of the standard
(tensor product) representation of GLpN11 , Cq L G9 with either the symmetric square or skew symmetric square representation of GLpN11 , Cq. We are assuming that the global theorems hold for G9 . This allows us to express Lps, π9 λ , ρ_ q as the product of a Rankin-Selberg L-function with either a P¯ |P symmetric square or skew symmetric square L-function. In particular, this L-function has analytic continuation and functional equation
_ qLp1 s, π9 λ , ρ ¯ q. 9 λ, ρ ¯ Lps, π9 λ , ρ_ P |P P¯ |P q εps, π P |P
It follows that
1 rP¯ |P pπ9 λ q Lp1, π9 λ , ρP¯ |P qLp1, π9 λ , ρ_ P¯ |P q ε
_ . |
1 9 λ, ρ ¯ 2, π P P
Similarly, we have
1 rP |P¯ pπ9 λ q Lp1, π9 λ , ρP |P¯ qLp1, π9 λ , ρ_ P |P¯ q ε
_ . |
1 9 λ, ρ 2, π P P¯
Since ρ_ ρP¯|P , the product of the four L-functions equals 1. Another P |P¯ application of the functional equation tells us that the product of the two ε-factors also equals 1. The product of the two normalizing factors thus equals 1. In other words, the identity (2.3.16) holds in the special case at hand, and hence for any M , P 1 and P , as claimed. Consider the product expression for the operator MP 1 |P pπ9 λ q given by (2.3.13) and (2.3.14). We need only combine the multiplicative property (2.3.12) of this operator with its analogues (2.3.16) and (2.3.15) for the scalar factor in (2.3.13) and the factors with v u in (2.3.14). It follows that the same property holds for the remaining factor, that corresponding to v u in (2.3.13). This is the required specialization RP 2 |P pπλ q RP 2 |P 1 pπλ qRP 1 |P pπλ q
of (2.3.9) to the case that φ ψ is generic, and π is of the form (2.3.11). It is easy to relax the conditions on π. From the remarks leading to [A7, (2.2)], we see that the last product formula remains valid for representations (2.3.11) whose general linear components πi are induced super cuspidal. Since any representation in Π2 GLpNi q is a subrepresentation of such a πi , and the corresponding normalizing factors can be related by [A7, r 2 pM q. Similar reProposition 6.2], the product formula holds for any π P Π r pM q. marks then allow us to extend the formula to any representation π P Π
2.3. NORMALIZED INTERTWINING OPERATORS
87
(See [A7, §2], for example.) The required formula (2.3.9) is thus valid if r pM q. ψ φ is any parameter in Φ Suppose now that the local parameter ψ for M is arbitrary. The normalizing factor in (2.3.8) is attached to ψ rather than π. This is the distinction noted prior to the statement of the proposition. It is pertinent because the r ψ to which π belongs will not in general be an L-packet. In other set Π words, the local Langlands parameter φ of π may be different from the local Langlands parameter φψ of ψ. By our assumption on G , we are free to regard both φψ and φ as local Langlands parameters for a product of general linear groups. The normalizing factor (2.3.6) to which the proposition applies is taken relative to φψ . The extension of the required identity (2.3.9) from tempered representations to arbitrary Langlands quotients, which was established for any F on p. 30 of [A7], is implicitly based on the parameter φ. We must show that if (2.3.9) holds for the one set of normalizing factors, it also holds for the other. As Langlands parameters attached to a product of general linear groups, φψ and φ correspond to irreducible representations πψ and πφ of the corresponding products of groups over F . The two representations are related by the algorithm of Zelevinsky, used in the proof of Lemma 2.2.2, (and its analogue [AH] for F R.) In other words, πψ and πφ are block equivalent, in the sense of D. Vogan. (See [A7, p. 42].) It then follows from [A7, Proposition 5.2] that the quotient rP 1 |P pφλ , ψλ q rP 1 |P pφλ q1 rP 1 |P pψλ q
of the two normalizing factors satisfies the analogue
rP 2 |P pφλ , ψλ q rP 2 |P 1 pφλ , ψλ qrP 1 |P pφλ , ψλ q
of (2.3.9). This implies that the two formulations of the identity (2.3.9), the one here and the original version in [A7], are equivalent. The identity therefore holds as stated. We still have the adjoint condition (2.3.10) to verify. It is a straightforward matter to establish a general identity JP 1 |P pπλ q
JP |P 1 pπ λq, for any irreducible representation π of M pF q with adjoint π .
One first takes λ to lie in the domain of absolute convergence of the integral that defines JP 1 |P pπλ q. One obtains the formula in this case by a suitable change of variables in the quasi-invariant measure on P 1 pF qzGpF q determined by the inner product on HP 1 pπ q. The formula for arbitrary λ then follows by r ψ , and is hence analytic continuation. In (2.3.10), π lies in the packet Π unitary by our assumption that Theorem 1.5.1 holds for G . (We of course know that the unitary condition holds also for the remaining, general linear factors of M .) In other words, π π . It is also not difficult to see that rP 1 |P pψλ q rP |P 1 pψλ q,
88
2. LOCAL TRANSFER
r pM q as in (2.3.10). In the special case that ψ φ is generic, for ψ P Ψ this follows from the explicit formulas in [T2] for the terms in (2.3.3). For r pM q, we write the parameter φψ P Φ r pM q as a a general parameter ψ P Ψ r temp pM q is the restriction of ψ to the subgroup LF of twist φµ , where φ P Φ LF SU p2q, and µ is a point in aM . Since the Langlands parameter φµ is x-conjugate to φµ , we obtain M
rP 1 |P pψλ q rP 1 |P pφµ
q rP |P 1 pφµλq rP |P 1 pφµλq rP |P 1 pψλq, λ
as claimed. The adjoint condition (2.3.10) then follows from the definition (2.3.8), since π is assumed to be unitary. The identity (2.3.9) reduces to when P 2 that
P.
RP |P 1 pπλ q RP 1 |P pπλ q1 . If we take λ to be purely imaginary in (2.3.10), we then see
RP 1 |P pπλ , ψλ q
RP |P 1 pπλ, ψλq RP |P 1 pπλ, ψλq RP 1|P pπλ, ψλq1.
In other words RP 1 |P pπλ q is unitary, as noted in the last assertion of the proposition. The proof of Proposition 2.3.1 is complete. We have constructed normalized intertwining operators RP 1 |P pπ, ψ q between induced representations IP pπ q and IP 1 pπ q. This is just the first step. It remains to convert these objects to self-intertwining operators of IP pπ q, which will be attached to elements w P W pM q that stabilize π. Suppose that P P P pM q, π P Πunit pM q and w P W pM q are fixed. We can certainly choose a representative w r of w in the normalizer N pM q of M in GpF q. This gives us another representation
pwπqpmq πpwr1mwrq,
m P M p F q,
of M pF q on the underlying space Vπ of π, which depends only on the image r in the quotient of N pM q by the center of M pF q. It also gives us an of w intertwining isomorphism r π q : H P 1 pπ q `pw,
ÝÑ
from IP 1 pπ q to IP pwπ q by left translation (2.3.17) where
r π qφ 1 p x q φ 1 pw r1 xq, `pw,
P1
HP pwπ q φ1
P H P 1 p π q , x P G p F q,
w1P wr1P wr1.
If wπ is equivalent to π, we can in addition choose an intertwining isomorrpwq of Vπ from wπ to π. Its pointwise action on HP pwπ q intertwines phism π
2.3. NORMALIZED INTERTWINING OPERATORS
IP pwπ q with IP pπ q. The composition
rpwq `pw, π r π q RP 1 |P pπ, ψ q,
π
89
P Πr ψ ,
of intertwining maps H P pπ q
ÝÑ
HP 1 p π q
ÝÑ
HP pwπ q
ÝÑ
H P pπ q
will then be a self intertwining operator of IP pπ q. It is in the choices of w r rpwq that we will need to be aware of the implications for endocopic and π transfer. We will deal with the first at the end of this section, and the second in the beginning of the next. In each of these remaining two steps, we will also have to adjust the relevant intertwining map by its own scalar normalizing factor. r of w. The group There is a natural way to choose the representative w G comes with a splitting S
T, B, txα u .
We assume that M and P are standard, in the sense that they contain T and B respectively. Let wT be the representative of w in the Weyl group WF pG, T q that stabilizes the simple roots of pB X M, T q. We then take rw rS to be the representative of wT in GpF q attached to the splitting S, w as on p. 228 of [LS1]. Thus, w rw rα1 w rαr ,
where wT wα1 wαr is a reduced decomposition of wT relative to the simple roots of pG, T q, and rα w
exppXαq exppXαq exppXαq,
in the notation of [LS1]. r introduces another difficulty. For it is well However, our representative w known that the mapping w Ñ w r from W pM q to N pM q is not multiplicative in w. The obstruction is the co-cycle of [LS1, Lemma 2.1.A], which plays a critical role in the construction of transfer factors. To compensate for it, we must introduce the factor ε 21 , that was taken out of the original definition (2.3.1). For global reasons, we will use the representation theoretic ε-factor, 0 pN, F q of general linear given by the representation πψ of the product M r ψ attached to groups. We are assuming now that π belongs to the packet Π r pM q. Since M is proper, we can assume as before that the a given ψ P Ψ packet exists, and that π is unitary. The choice of P allows us to identify xq of M x. (We are regarding M x here as a W pM q with the Weyl group W pM xq complex group with an L-action of the Galois group Γ ΓF , so that W pM xq{M x.q We is the group of Γ-invariant elements in the quotient NormGp pM define the ε-factor (2.3.18)
εP pw, ψ q ε
_
1 2 , πψ , ρw1 P P , ψF
|
,
w
P W pM q.
90
2. LOCAL TRANSFER
We know from the general results of Shahidi that we must also include a supplementary term. It is the λ-factor (2.3.19)
λpwq λpw, ψF q
¹
λa pψF q,
w
P W pM q,
a
of [KeS, (4.1)] and [Sha4, (3.1)], in which a ranges over the reduced roots of pB, AB q such that wT a 0, and AB is the split component of B (or T ) over F . This term is built out of the complex numbers λpE {F, ψF q attached to finite extensions E {F by Langlands [L3] to account for the behaviour of ε-factors under induction. In the case at hand, λa pψF q λpFa {F, ψF q,
where Fa is either a quadratic extension of F or F itself. This field is given by Ga,sc ResFa {F SLp2q , where Ga,sc is the simply connected cover of the derived group of Ga , the Levi subgroup of G of semisimple rank one attached to a. It is known that neither εP pw, ψ q or λpwq are multiplicative in w. However, the next lemma tells us that their combined obstruction matches that r π q. of `P pw, Lemma 2.3.4. The product (2.3.20)
`pw, π, ψ q λpwq1 εP pw, ψ q`pw, r π q,
satisfies the condition
`pw1 w, π, ψ q `pw1 , wπ, wψ q`pw, π, ψ q,
w
P W pM q ,
w1 , w
P W pM q. r1 w r Proof. We consider general elements w1 and w in W pM q. Since w 1 1 and w w are two representatives in GpF q of w w, and since they preserve
the same splitting of M , they satisfy
1 w z pw 1 , w q , rw r1 w w
for a point z pw1 , wq in the center of M pF q. It follows that
1 w, πq ηπ z pw1 , wq `pwr1 , wπq`pw, `pw r π q,
where ηπ is the central character of π. Suppose for a moment that M equals the minimal Levi subgroup T . The elements w1 and w then belong to the rational Weyl group W pT q WF pT q of T . We write ΣB pw1 , wq for the set of roots α P ΣB of pB, T q such that wα R ΣB and w1 wα P ΣB . The corresponding sum λ pw 1 , w q
¸ α
α_ ,
α P ΣB pw1 , wq,
is a co-character in X pT q, which is fixed by the Galois group Γ ΓF , since both w1 and w are defined over F . It then follows from [LS1, Lemma 2.1.A] that 1 z pw1 , wq p1qλpw ,wq ,
2.3. NORMALIZED INTERTWINING OPERATORS
91
where the right hand side is understood as a point in T pF q X pT q b F
Γ .
Suppose now that M is arbitrary, subject only to our requirement that P P P pM q contains B. For any w P W pM q, we have constructed the rational Weyl element wT P W pT q. The definitions tell us that 1 z pw1 , wq p1qλpw ,wq , w 1 , w P W pM q, where
λ pw 1 , w q
¸
α_ ,
α P ΣB pwT1 , wT q.
α
In other words, z pw1 , wq z pwT1 , wT q and λpw1 , wq λpwT1 , wT q. We claim that λpw1 , wq belongs to the subgroup X pAM q of co-characters of the split
component AM of the center of M . To see this, consider an element u in the rational Weyl group W0M of pM, T q. If α is any root in ΣB pwT1 , wT q, uα obviously belongs to ΣB . Moreover, we see that wT puαq R ΣB and wT1 wT puαq P ΣB , since wT and wT1 wT each conjugate u to another element in W0M . In other words, uα also lies in ΣB pwT1 , wT q. The Weyl group W0M thus acts by permutation on ΣB pwT1 , wT q, and consequently fixes the sum λpw1 , wq. The claim then follows from the fact that λpw1 , wq is also fixed by Γ. In particular, z pw1 , wq lies in the group AM pF q X pAM q b F . The split torus AM is the product of the centers of the general linear factors of M in (2.3.4). We can therefore write ηπ z p w 1 , w q
η ψ z pw 1 , w q ,
where ηψ is the central character of the representation πψ , since π and πψ are equal on these general linear factors. We have established an obstruction _ 1 1 w, πq `pw, `pw r π q1 `pw r1 , wπ q1 ηψ p1qλ pw ,wq
r π q to be multiplicative in w. Before we compare it with for the operator `pw, the corresponding obstructions for the other two factors in (2.3.20), let us first take note of its L-group interpretation. pM of AM can be identified with the “split The complex dual torus A x, namely the largest connected quotient of M x on part” of the cocenter of M L which the action of M by conjugation is trivial. This provides the second of the two canonical isomorphisms pM q X pM xqF . X pAM q X pA
x in the set X pM xqF to a Moreover, we can extend any character on M L character on M that is trivial on the semidirect factor WF of L M . If r pM q is a Langlands parameter with central character ηφ on AM pF q, it φPΦ is then easy to see that _ (2.3.21) ηφ pxλ q pλ_ φqpuq λ_ φpuq , x P F , λ_ P X pAM q,
92
2. LOCAL TRANSFER
where λ_ is identified with a character on L M , and u is any element in WF whose image in the abelian quotient F WFab equals x. Let λ _ pw 1 , w q
¸
λ_ β
β
be the natural decomposition of the character λ_ over roots β in the subset
λ_pw1, wq into a sum (
p r pw 1 , w q β Σ P
pr of the reduced roots Σ P
P Σp rP : wβ 0, w1wβ ¡ 0 of pPp, AM x q. The summand of β thus equals ¸ λ_ α_ , α P Σβ , β α
where Σβ is the subset of roots α of pG, T q whose coroot α_ restricts to a _ 1 _ positive multiple of β on AM x . The components λβ of λ pw , w q each belong _ 1 to the group AM x , by virtue of the argument we applied above to λ pw , w q. r π q can be written as a product It follows that the obstruction for `pw, ¹ _ 1 (2.3.22) ηψ p1qλ pw ,wq pλ_β φψ qp1q β
p r pw 1 , w q . over β P Σ P p r , let ρβ For any β P Σ P the subspace p nβ
ρ_β be the adjoint representation of
à
LM
on
α P Σβ ,
p nα_ ,
α
of p nP attached to β. Th ε-term in (2.3.20) then has a decomposition εP pw, ψ q
¹
ε
1 2 , πψ , ρβ , ψF
,
β
P Σp rP ,
0.
wβ
β
This is similar to the decomposition we have come to expect of normalizing factors. For example, it tells us that the obstruction for εP pw, ψ q to be multiplicative in w, which in general we write in the inverse form epw1 , wq εP pw1 , wψ q εP pw, ψ q εP pw1 w, ψ q1 ,
is trivial if the length of w1 w equals the sum of the lengths of w1 and w. For arbitrary elements w1 and w in W pM q, we observe from this decomposition that e pw 1 , w q
¹
ε
1 2 , πψ , ρβ , ψP
ε
1 2 , πψ , ρ β , ψF
,
β
P Σp rP pw1, wq.
β
For each β, the product of the two representation theoretic ε-factors here equals the corresponding product of Artin ε-factors. It then follows from the general property [T2, (3.6.8)] of Artin ε-factors that
ε
1 2 , πψ , ρβ , ψF
ε
1 2 , πψ , ρ β , ψF
pdet ρβ φψ qp1q,
2.3. NORMALIZED INTERTWINING OPERATORS
93
where the right hand side is understood as the value of the determinant of ρβ φψ at any u P WF whose image in F WFab equals p1q. We can therefore write the obstruction for εP pw, ψ q as a product epw1 , wq
(2.3.23)
¹
pdet ρβ φψ qp1q
β
pw 1 , w q .
over β P Suppose for example that G is split. Then the supplementary term p r , we can regard ρβ as a λpwq in (2.3.20) is trivial. Moreover, for any β P Σ P x L M on p representation of the complex connected group M nβ . Its weights _ p on T are the characters α parametrized by α P Σβ , each of which occurs with multiplicity 1. It follows that the restriction of ρβ to Tp has determinant x equal to the character λ_ β . Since every element in M is conjugate to an ΣrP
x element in Tp, the characters λ_ β and det ρβ on M are equal. It follows that ¹
pλ_β φψ qp1q
β
¹
pdet ρβ φψ qp1q,
β
P Σp rP pw1, wq,
β
so that the obstructions for the two primary terms in (2.3.20) match. The lemma is therefore valid in case G is split. Assume now that G P Ersim pN q is a general element in Ersim pN q. Our remaining concern is the nonsplit group G SOp2n, ηE {F q, where η ηG ηE {F is the quadratic character on F attached to a fixed quadratic extension E {F . This is the case in which the supplementary term λpwq in (2.3.20) is nontrivial. We recall that λpwq is given by a product (2.3.19) of λ-factors λpFa {F, ψF q, taken over the reduced roots a of pB, AB q with wT a 0. Each a determines an orbit tαu of roots of pG, T q under the Galois group Γ ΓF . The field Fa is the extension of F that corresponds to the stabilizer of a in Γ. It equals either F , in which case the associated factor λpF {F, ψF q is trivial, or the quadratic extension E, in which case the factor satisfies λpE {F, ψF q2
ηE{F p1q.
(See [KeS, (2.7)].) It follows that the obstruction for λpwq to be a character in w equals the product λpw1 wq λpwq1 λpw1 q1
(2.3.24)
¹
ηE {F p1qµβ
β
pw1, wq, where µβ is the number of roots a with Fa
over β P E such that the associated orbit tαu is contained in Σβ . We need to combine this with the other two obstructions (2.3.23) and (2.3.22). x For the general group G P Ersim pN q, the characters λ_ β and det ρβ on M ΣrP
p P remain equal. However, we must also consider their attached to any β P Σ x ΓE {F . As an element extensions to the nonconnected group L M M _ λ in X pAM q, following the general identity (2.3.21), λ_ β has a canonical L extension that is trivial on the factor ΓE {F of M . The character det ρβ is
94
2. LOCAL TRANSFER
the determinant of a representation ρβ that is defined a priori on L M . It is generally not trivial on ΓE {F . p r . The quotient ΓE {F of WF acts by permutation on Fix the root β P Σ P the coroot spaces p nα_ in p nP , in a way that is compatible with the natural action of ΓE {F on the roots α P Σβ . To describe the extension of det ρβ to L M , it suffices to consider the case that P is a maximal parabolic subgroup of the group G, which we are implicitly assuming equals SOp2n, ηE {F q. The Levi component M of P then takes the form M
GLpmq SOp2n 2m, ηE{F q,
m n.
The set of roots α can be written as Σβ
ptq ti ptj q1 : 1 ¤ i ¤ m 1, m ¤ j αij
¤n
(
,
where t represents a diagonal matrix pti q in the standard maximal torus T of G. The group ΓE {F acts simply transitively on any pair pi
pαin, αin q,
i ¤ m 1,
and acts trivially on each of the remaining roots of Σβ . Consequently, the determinant of the restriction of ρβ to the factor ΓE {F of L M equals a product of copies of the nontrivial character of ΓE {F , taken over the set of pairs pi . It follows that det ρβ p1 σ q p1qm1 ,
for the nontrivial element σ P ΓE {F . This will allow us to compare the L M . Moreover, it is clear characters λ_ β and det ρβ on the factor ΓE {F of from the definitions that m 1 equals the integer µβ assigned to β in (2.3.24). In comparing the two primary obstructions (2.3.22) and (2.3.23), we need to choose an element u in the Weil group WF whose image in F WFab equals p1q. We write φψ puq mψ puq σ puq
x, and σ puq lies in the quotient ΓE {F of WF . We where mψ puq belongs to M are assuming here that G is not split, so that E is defined as a quadratic extension of F . We then observe that the element σ puq P t1u equals ηE {F p1q, a property that follows from the fact that ηE {F p1q equals 1 if and only if the element p1q P F lies in the image of the norm mapping p r pw1 , wq, we then have from E to F . For any β P Σ P
pdet ρβ φψ qp1q det ρβ φψ puq det ρβ mψ puq det ρβ 1 σpuq λ_β mψ puq ηE{F p1qµ λ_β φψ puq ηE{F p1qµ pλ_β φψ qp1q ηE{F p1qµ . β
β
β
2.3. NORMALIZED INTERTWINING OPERATORS
95
It remains only to take the product over β of the numbers in each side of the identity we have obtained. We conclude that the obstruction (2.3.23) equals the product of the original obstruction (2.3.22) with the supplementary obstruction (2.3.24). It then follows from the definitions that the threefold product (2.3.20) is indeed multiplicative in w, as required. Remarks. 1. The operator `pw, π, φq makes sense for an arbitrary pair pG, M q, under the appropriate hypothesis on the local Langlands correspondence for M . The proof of the lemma should carry over to the general case, although I have not checked the details. I thank the referee for pointing out that λpwq is generally not a character in w. 2. It is interesting to see the local root numbers (2.3.18) appearing as explicit factors of the operators (2.3.20). They remind us of the global root numbers (1.5.7) used to construct the sign character (1.5.6). As orthogonal root numbers, however, they are less deep, and better understood [D2] than the symplectic root numbers (1.5.7) and their local analogues. We can think of Lemma 2.3.4 as a normalization of the “miniature” r π q. It is reminiscent of the main property (2.3.9) intertwining operator `pw, of Proposition 2.3.1, if somewhat simpler. We combine the two normalized operators by writing (2.3.25)
RP pw, π, ψ q `pw, π, ψ qRw1 P |P pπ, ψ q,
w
P W pM q.
This operator then has a decomposition (2.3.26)
r πλ q, RP pw, πλ , ψλ q rP pw, ψλ q1 JP pw,
π
P Πr ψ ,
for the unnormalized intertwining operator r πλ q `pw, r πλ qJw1 P |P pπλ q JP pw,
from HP pπλ q to HP pwπλ q, and a corresponding normalizing factor rP pw, ψλ q λpwqεP pw, ψλ q1 rw1 P |P pψλ q
that reduces to the product of λpwq with the familiar quotient
_ 1 _ 1 (2.3.27) Lp0, πψ,λ , ρ_ w1 P |P qεp0, πψ,λ , ρw1 P |P , ψF q Lp1, πψ,λ , ρw1 P |P q . We are using the ψ-variant of (2.3.5) here, while continuing to suppress the implicit dependence on the additive character ψF from the notation. It follows from Proposition 2.3.1 and Lemma 2.3.4 that the normalized intertwining operator RP pw, π, ψ q : HP pπ q satisfies the familiar co-cycle relation
ÝÑ
HP pwπ q
(2.3.28) RP pw1 w, π, ψ q RP pw1 , wπ, wψ qRP pw, π, ψ q,
w1 , w
P W pM q.
This is essentially what has been conjectured in general by Langlands [L5, Appendix II].
96
2. LOCAL TRANSFER
2.4. Statement of the local intertwining relation The purpose of this section is to state a theorem that relates local intertwining operators with endoscopic transfer. The relation will have global implications for our later analysis of trace formulas. It will also be closely tied to Theorem 2.2.1, both as a supplement and as a part of the proof. We are assuming that G belongs to Ersim pN q, and that M is a proper Levi subgroup of G. There is still some unfinished business from the last section. It is to convert the intertwining operator RP pw, π, ψ q from IP pπ q to IP pwπ q to a self-intertwining operator for IP pπ q. This is the last step in the normalization process, and the one that is most closely tied to endoscopy. r pM q, and that π belongs to the packet Π r ψ . We Recall that ψ lies in Ψ will usually take w to be in the subgroup (2.4.1)
Wψ pπ q W pπ q X Wψ pM q tw
P W pM q :
wπ
π,
wψ
ψu
xq that stabilize the equivalence classes of of elements w in W pM q W pM both π and ψ. We will then be able to introduce an intertwining operator rpwq from wπ to π. π r ψ . Suppose that w lies in We are assuming that π lies in the packet Π Wψ pπ q, and that π rpwq is an intertwining operator from wπ to π. We can r as an extension of π to the M pF q-bitorsor think of π w pF q M pF q w, M r
pF q in the sense of (2.1.1), or even an extension of π to the group M w w pF q, if π rpwq is chosen so that its order equals that of w. generated by M The product r, ψ q π rpwq RP pw, π, ψ q RP pw, π
(2.4.2)
is then a self-intertwining operator of IP pπ q, which of course depends on the rpwq. Suppose that the operators π rpwq on Vπ could be chosen to choice of π be multiplicative in w. It would then follow from (2.3.28) and the definition of RP pw, π, ψ q that the mapping w
ÝÑ
RP pw, π r , ψ q,
w
P W ψ p π q,
is a homomorphism from Wψ pπ q to the space of intertwining operators of IP pπ q. However, there is no canonical choice for π rpwq in general. While this does not necessarily preclude an ad hoc construction, it represents a problem that left unattended would soon lead to trouble. The solution is provided by an application of Theorem 2.2.4 to M . We first recall the various finite groups that can be attached to the r pM q. However, its parameter ψ. We are regarding ψ as an element in Ψ L composition with the standard embedding of M into L G is an element in r pGq, which we shall also denote by ψ. From these two objects, we obtain Ψ the two complex reductive groups Sψ pM q Sψ pGq Sψ ,
2.4. STATEMENT OF THE LOCAL INTERTWINING RELATION
97
and their two finite quotients
Sψ pM q Sψ pGq Sψ .
Recall that Sψ is the group of connected components in the quotient p qΓ . It is clear that Sψ pM q is the subgroup of elements in S ψ Sψ {Z pG Sψ that leave the complex torus AM x
xqΓ Z pM
F
0
pointwise fixed, and it is easily seen that Sψ pM q is the image of Sψ pM q in Sψ . There are other interesting finite groups as well. We write Nψ pG, M q for the normalizer of the complex torus AM x in Sψ and Nψ pG, M q for the group of components in the quotient p qΓ . N ψ pG, M q Nψ pG, M q{Z pG
Then Sψ pM q is a normal subgroup of Nψ pG, M q. Its quotient is isomorphic to the Weyl group Wψ pG, M q W pSψ , AM x q, which is to say, the group of automorphisms of AM x induced from S ψ . We write Wψ0 pG, M q to be the normal subgroup of automorphisms in Wψ pG, M q that are induced from the connected component S 0ψ , and Rψ pG, M q Wψ pG, M q{Wψ0 pG, M q
for its quotient. On the other hand, there is a canonical injection from Wψ0 pG, M q into Nψ pG, M q. This map is well defined, since the centralizer of 0 0 AM x in Sψ is connected, and is therefore contained in Sψ pM q. In other words, 0 Wψ pG, M q can also be regarded as a normal subgroup of Nψ pG, M q. The quotient Sψ pG, M q of Nψ pG, M q by Wψ0 pG, M q is a subgroup of Sψ . We can summarize the relations among these finite groups in a commutative diagram 1 1 ∨
Wψ0 pG, M q
(2.4.3)
1
1
>
Sψ pM q
} > Sψ pM q
∨
>
Nψ pG, M q
∨
>
∧
Sψ pG, M q
Wψ pG, M q
>
1
>
1
∧
∨
>
∨
Wψ0 pG, M q
∨
>
Rψ pG, M q
∨
∨
1
1
98
2. LOCAL TRANSFER
of short exact sequences. The dotted arrows stand for splittings determined by the parabolic subgroup Pψ
pPp X Sψ0 q{Z pGpqΓ
F
0
of S ψ , or rather the chamber of P ψ in the Lie algebra of AM x. Suppose that u belongs to Nψ pG, M q. We write wu for the image of u in Wψ pG, M q given by the horizontal short exact sequence at the center x of (2.4.3). Since wu stabilizes AM x , it normalizes M , and can therefore be xq. The choice of P then allows us to identify treated as an element in W pM wu with an element in W pM q. Recall that M is a product of groups (2.3.4), all of which all have standard splittings. We set u M
w pM, w ru q, M u
where we recall that w ru is the automorphism in the inner class of wu that u is a twisted group, preserves the corresponding splitting of M . Then M u q of Ψ r pM q. As earlier, the and ψ belongs to the corresponding subset ΨpM centralizer quotient u q Srψ,u Sψ pM
is an Sψ pM q-bitorsor. We shall write u r for the element in this set defined by u. ru -stable element in the packet Assume that π represents a w r r rpwu q from Πψ Πψ pM q. We can then choose an intertwining operator π wu π to π, from which we are led to the intertwining operator RP pwu , π r, ψ q of IP pπ q by (2.4.2). As we have said, there is generally no canonical choice for π rpwu q. However, M is a product (2.3.4) of groups to which either Theorem 2.2.1 or Theorem 2.2.4 can be applied. In particular, the products
xur, πry πrpwuq and
xur, πry RP pwu, πr, ψq
(2.4.4)
rpwu q. (The automorcan be defined, and are independent of the choice of π phism w ru of M can also include a permutation of factors in (2.3.4), but the extension of Theorem 2.2.1 to this slightly more general situation is clear. We have taken the liberty of expressing the assertion (2.2.6) of this theorem in the same form as its counterpart (2.2.17) in Theorem 2.2.4, with the extension x , y attached to the Whittaker extension of the general linear components of π being trivial.) We therefore obtain a canonical linear form
(2.4.5)
fG pψ, uq
¸
P
rψ π Π
xur, πry tr RP pwu, πr, ψqIP pπ, f q
in f P HpGq. The convention here is similar to that of Theorem 2.2.4, in ru -stable. that the summand on the right is zero if π happens not to be w
2.4. STATEMENT OF THE LOCAL INTERTWINING RELATION
99
We will later have to consider the role of u in the vertical split exact sequence at the center of (2.4.3). We will then write xu and wu0 for the images of u in the respective groups Sψ pG, M q and Wψ0 pG, M q. Suppose now that s is any semisimple element in the complex reductive group Sψ . We write fG1 pψ, sq f 1 pψ 1 q
(2.4.6)
r pGq attached to the preimage pG1 , ψ 1 q of pψ, sq for the linear form in f P H under the local form of the correspondence (1.4.11). Any such s projects to an element in the quotient Sψ Sψ pGq, which may or may not lie in the subgroup Sψ pG, M q.
Theorem 2.4.1 (Local intertwining relation for G). For any u in the local normalizer Nψ pG, M q, the identity (2.4.7)
fG1 pψ, sψ sq fG pψ, uq,
f
P HrpGq,
holds for any semisimple element s P Sψ that projects onto the image xu of u in Sψ pG, M q.
r 2 pM q of Ψ r pM q. The most important case is when ψ lies in the subset Ψ Then the group Tψ AM x
is a maximal torus in Sψ , and Sψ pG, M q equals the full group Sψ . In this case, we write
Nψ pG, M q, Wψ Wψ pG, M q, Wψ0 Wψ0 pG, M q, Nψ
and Rψ
Rψ pG, M q.
The last group Rψ can be regarded as the R-group of the parameter ψ, since it is closely related to the reducibility of the induced representations IP pπ q,
P P pM q, π P Πr ψ pM q. r pGq in this case, and we We shall often treat ψ strictly as an element in Ψ P
shall write
Sψ1
Sψ pM q,
r N pG q . since M is uniquely determined up to the action of the group Aut
Lemma 2.4.2. Suppose that for any M , Theorem 2.4.1 is valid whenever r 2 pM q. Then it holds for any ψ in the general set Ψ r pM q . ψ belongs to Ψ
100
2. LOCAL TRANSFER
The proof of the lemma will be straightforward, but it does depend on some further observations. We shall postpone it until the end of this section. In the meantime, we shall discuss an important implication of the local intertwining relation postulated by the theorem. We first observe that the groups in (2.4.3) can be described in concrete r pGq, we write terms. For any ψ P Ψ ψ
`1ψ1 ` ` `r ψr ,
following the local form of the notation (1.4.1). As in the global case ² of §1.4, the subscripts k P Kψ range over disjoint indexing sets Iψ Iψ pGq Iψ pGq, Jψ and Jψ_ . We recall that if i
P Iψ , ψi P Ψr simpGiq is a simple parameter for an endoscopic datum Gi P Ersim pNi q, while if j P Jψ , ψj belongs to r sim pNi q in Ψsim pNi q. The subset I pGq consists of the complement of Ψ ψ p and G p i are of the same type (both orthogonal or both those i such that G p and G p i are of symplectic), while Iψ pGq consists of those i such that G opposite type. We then have
(2.4.8) Sψ
¹
P p q
Op`i , Cq
i Iψ G
as in (1.4.8), where ξψ :
¹
gi
¹
G i Iψ
P p q
Spp`i , Cq
¹
P
j Jψ
i
i
The identity component of this group obviously equals
¹
P p q
pq pqψ denotes the kernel of the character ¹ ÝÑ pdet giqN , gi P Op`i, Cq, i P Iψ pGq.
i
Sψ0
GLp`j , Cq
SOp`i , Cq
i Iψ G
¹
G i Iψ
P p q
Spp`i , Cq
¹
P
GLp`j , Cq .
j Jψ
Finally S ψ is the quotient of Sψ by a central subgroup, of order 2 if N is even and order 1 if N is odd, which may or may not lie in Sψ0 . Consider for simplicity the basic case that ψ is the image of a parameter r in Ψ2 pM q. We can then identify the complex torus Tψ AM x with the product of standard maximal tori in the simple factors of Sψ0 . It is clear that π0 pSψ q is the group of connected components in the left hand factor p q of Sψ , and that its image in π0pS ψ q is the group Sψ . The subgroup Sψ1 is given by the contribution of the subfactors Op`i , Cq with `i odd, while the quotient Rψ is given by the subfactors Op`i , Cq with `i even. The group Nψ is an extension of Sψ by the product Wψ0 of standard Weyl groups of the simple factors of Sψ0 . It is worth looking at one example in greater detail. Consider the lower horizontal short exact sequence (2.4.9)
1
ÝÑ
Sψ1
ÝÑ
Sψ
ÝÑ
Rψ
ÝÑ
1
2.4. STATEMENT OF THE LOCAL INTERTWINING RELATION
101
p is orthogonal. We write from (2.4.3), in case N is even and G
Iψ pGq
I
for simplicity, and identify the group of connected components in the product ¹
P
O p`i , C q
i I
from (2.4.8) with the group Σ of functions
ÝÑ Z{2Z. The point is to account for the indices i P I in terms of the parity of both associated integers `i and Ni . For example, the group π0 pSψ q corresponds σ: I
to the subgroup
Σ1
!
σ
PΣ:
±1 i
)
σ piq 1 ,
where the product is over the set Io of indices i with Ni odd. The group Sψ π0 pS ψ q corresponds to the quotient Σ
1 Σ1 { xσy,
where σ is the element in Σ1 (of order 1 or 2) such that σ piq 1 whenever i belongs to the set I ,o of indices with `i odd. (The set I ,o X Io of i with both `i and Ni odd is even, so σ does lie in Σ1 .) The group Sψ1 corresponds
1
1
to the subgroup pΣ q1 of σ P Σ such that σ piq 1 whenever i belongs to the subset I ,e of indices with `i even. The R-group Rψ then corresponds to 1 the associated quotient of Σ , which is canonically isomorphic to the group R1 of functions ρ : I ,e ÝÑ Z{2Z, with the supplementary requirement that ±1
i ,o
ρpiq 1,
in the exceptional case that I X Io is empty. The short exact sequence (2.4.9) can thus be identified with the more concrete exact sequence (2.4.10)
1
ÝÑ pΣ1q1 ÝÑ
Σ
1
ÝÑ
R1
ÝÑ
1.
p symplectic) are similar, but The other two cases (with N odd or G simpler. In all three cases, the short exact sequence (2.4.9) (and its analogue p is orthogonal there is generally (2.4.10)) splits. However, if N is even and G no canonical splitting. The sign character
(2.4.11)
εpρq
±2 i
ρpiq,
ρ P R1 ,
where the product is over the intersection I ,e X Io , is a distinguishing feature of this case. It represents the obstruction to a canonical splitting of (2.4.10). Observe also that as an element in Rψ , ρ can act by outer automorphism on the even orthogonal factor G P Erell pN q, as well as the
102
2. LOCAL TRANSFER
general linear factors. It is not hard to infer from our discussion that the action on G is nontrivial if and only if εpρq 1. The character (2.4.11) thus governs the ambiguity of both factors in the product (2.4.4). In fact, it is fair to say that this character is what ultimately forces us to deal with twisted endoscopy for even orthogonal groups, as stated in Theorem 2.2.4, if only to resolve the ambiguity in the definition (2.4.2). We now continue with our discussion of the local intertwining relation. The last lemma tells us that its general proof can be reduced to the case that r 2 pM q. In dealing with this basic case, we shall make a slight change ψPΨ of notation. Following an earlier suggestion, we will generally take ψ to be r pGq. We can then choose a pair pM, ψM q, simply a parameter in the set Ψ r 2 pM, ψ q of parameters in Ψ r 2 pM q whose image where ψM lies in the set Ψ equals ψ. As we have noted, the Levi subgroup M is uniquely determined r N pGq-orbit of G-conjugacy classes by ψ. as an Out We are assuming that M is proper in G, which is to say that ψ does not r 2 pGq. We can therefore assume as usual that we have constructed belong to Ψ r ψ of (Out r N pM q-orbits of) representations πM of M pF q. We the packet Π M shall combine this with the local intertwining relation to construct the packet r N pGq-orbits) of representations π of GpF q. The construction r ψ of pOut Π provides an important reduction of the local classification. It also gives us r ψ , in the more concrete terms of an independent way to view the packet Π induced representations and intertwining operators. Its proof amounts to a reordering of some of the discussion of this section. We shall try to present it so as to be suggestive of more general situations. Proposition 2.4.3. Assume that for any proper Levi subgroup M of G, r 2 pM q. Then if ψ is any parameter Theorem 2.4.1 holds for parameters in Ψ r r r ψ and pairing xx, π y of in the complement of Ψ2 pGq in ΨpGq, the packet Π Theorem 2.2.1 exist, and satisfy (2.2.6). r ψ provides a representation Proof. The packet Π M
(2.4.12)
ΠψM
à
pξM ,πM q
pξ M b π M q
of the group Sψ1 M pF q. We let πM range here over a set of representatives r N pM q-orbits that comprise the packet Π r ψ and ξM range over of the Out M the associated (1-dimensional) representations
ξM pxM q xxM , πM y of Sψ1 . We are of course free to identify ΠψM with a representation of
r ψ to the subalgebra the group algebra C pSψ1 q b HpM q. Its restriction Π M
r pM q is then independent of the choice of representatives. The C pSψ1 q b H overlapping notation is deliberate.
2.4. STATEMENT OF THE LOCAL INTERTWINING RELATION
The group Nψ acts on M pF q by u: m
ÝÑ
ru mw ru1 , w
103
m P M p F q, u P N ψ ,
where the image wu of u in Wψ is regarded as an element in W pM q. The associated semidirect product M pF q N ψ
is then an extension of the group Wψ by the direct product M pF q Sψ1
Sψ1 M pF q.
We claim that there is a canonical extension of the representation of ΠψM to M pF q Nψ . To see this, we have first to expand slightly on the discussion of the short exact sequence (2.4.10) above. The action of Nψ on M pF q obviously stabilizes the orthogonal or symplectic factor G of M . Let Nψ, be the p is subgroup of elements that act on G by inner automorphism. If G symplectic or N is odd, for example, Nψ, equals Nψ . In the other case p is orthogonal and N is even, Nψ, is the kernel of the pullback to that G Nψ of the sign character (2.4.11) on Rψ . In all cases, there is a canonical splitting Nψ, Sψ1 ` Wψ, , where Wψ, is the image of Nψ, in Wψ . It follows from the definitions that any constituent πM of (2.4.12) has a canonical extension to M pF q Wψ, . The representation ΠψM therefore has a canonical extension to the subgroup M pF q Nψ, (of index 1 or 2) of M pF q Nψ . The claim is that ΠψM can be extended canonically to the larger group M pF q Nψ . Suppose that πM is Wψ -stable. Applying Theorems 2.2.4 and 2.2.1 separately to the components of πM relative to the decomposition (2.3.4), as in the definition (2.4.4), we see that ξM b πM has a canonical extension to M pF q Nψ . In general, Nψ (and Wψ ) act by permutation on the set of πM attached to a given ξM . If o is a nontrivial orbit (necessarily of order 2), we observe that as a representation of M pF q Nψ, , the sum à
P
πM o
pξM b πM q
is the restriction of the representation of M pF q Nψ induced from any of the summands. We conclude that the full sum (2.4.12) does have a canonical r ψ , despite extension to M pF q Nψ , as claimed. We shall denote it by Π M the further ambiguity in the notation. Let (2.4.13)
Πψ
IP pΠψ q M
à
pξM ,πM q
ξM
b IP pπM q
be the representation of Sψ1 GpF q induced parabolically from the representation ΠψM of Sψ1 M pF q. Our canonical extension of ΠψM to M pF q Nψ
104
2. LOCAL TRANSFER
can then be combined with the intertwining operators RP pw, πM , ψM q from the last section. A modest generalization of (2.4.2), in which the irreducible representation π of M pF q is replaced by the (reducible) restriction of ΠψM w pF q is replaced by our canonical r of π to M to M pF q, and the extension π u r extension ΠψM of ΠψM to M pF q Nψ , allows us to attach an intertwining operator (2.4.14)
r ψ , ψM q Π r ψ puq RP pwu , Πψ , ψM q RP pu, Π M M M
to any u P Nψ . This operator then commutes with the restriction of Πψ to GpF q. The product (2.4.15)
r ψ , ψM qΠψ pg q, Πψ pu, g q RP pu, Π M
u P Nψ , g
P G pF q ,
therefore gives a canonical extension of the representation Πψ to the product N ψ G p F q. We consider the character
r ψ pu, f q , tr Π
u P Nψ , f
P HrpGq,
r ψ of Πψ to the product of Nψ with the symmetric Hecke of the restriction Π r pGq. Suppose that o is an Nψ -orbit of representations πM . If algebra H r ψ puq in the orbit contains one element, the restriction of the operator Π M (2.4.14) to the subspace of pξM b πM q in (2.4.12) equals rM pwu q, ξrM puq b π
where ξrM puq xu r, π rM y. It follows from (2.4.2) that the restriction of the intertwining operator (2.4.14) to the corresponding subspace in (2.4.13) equals rM , ψM q. ξrM puqRP pwu , π
The contribution of this subspace to the character is therefore
xur, πrM y tr RP pwu, πrM , ψM qIP pπM , f q .
On the other hand, if o contains two elements, the operator ΠψM puq interchanges the subspaces of the two representations pξM b πM q. The restriction of (2.4.14) therefore interchanges the corresponding two subspaces of (2.4.13), so the contribution of these subspaces to the character vanishes. It follows from the definition (2.4.5) (with our notation now dictating a sum r ψ in place of the sum over π P Πψ in (2.4.5)) that over πM P Π M (2.4.16)
r ψ pu, f q tr Π
fGpψ, uq,
u P Nψ , f
P HrpGq.
We will be concerned with the restriction of Πψ to the subgroup Sψ GpF q of Nψ GpF q, relative to the splitting of the left hand vertical exact sequence in (2.4.3). The local intertwining relation suggests that the representation is trivial on the complementary subgroup Wψ0 . We will indeed establish this fact by global means later, so there will be no loss of
2.4. STATEMENT OF THE LOCAL INTERTWINING RELATION
105
information in the restriction. As a representation of Sψ GpF q, Πψ has a decomposition à Πψ p ξ b π q,
pξ,πq
for irreducible representations ξ of Sψ and π of GpF q. The sum is of course rψ finite, but the constituents can have multiplicities. We define the packet Π to be the disjoint union of the set of π, each identified with it associated r N pGq-orbit, that occur. Any π P Π r ψ then comes with a corresponding Out character ξ pxq xx, π y, x P Sψ , of the abelian group Sψ . It follows that (2.4.17)
r ψ px, f q tr Π
¸
P
rψ π Π
xx, πyfGpπq,
x P Sψ , f
P HrpGq.
r ψ and pairing xx, π y. To show that We have constructed the packet Π they satisfy (2.2.6), we apply our assumption that Theorem 2.4.1 is valid. Combined with (2.4.16) and (2.4.17), it tells us that
fG1 pψ, sψ sq fG pψ, uq
tr Πr ψ pu, f q tr Πr ψ px, f q ¸ xx, πy fGpπq,
P
rψ π Π
for elements s and u that project to the same point x in Sψ . A similar iden1 tity holds for fG1 pψ, sq, provided that we replace x by the point s ψ x sψ x. Since fG1 pψ, sq f 1 pψ 1 q, we obtain ¸ xsψ x, πy fGpπq, f P HrpGq. f 1 pψ 1 q
P
rψ π Π
This is the required identity (2.2.6). Remarks. 1. It is only at the end of the proof that we appeal to our hypothesis that Theorem 2.4.1 holds for G. In particular, the actual construction (2.4.17) of the packet and pairing of Theorem 2.2.1 depends only on the application of Theorems 2.2.1 and 2.2.4 to the smaller group G . 2. The structure of the proof appears to be quite general. We shall add several comments on what might be expected if G is a general connected group. (i) There are certainly examples of quasisplit groups G for which the finite groups Sψ1 , Sψ and Rψ are nonabelian. This will make the associated representations ΠψM and Πψ more interesting. (In general, Πψ would continue to denote both a packet and a representation.) It seems likely
106
2. LOCAL TRANSFER
that there will again be a canonical extension of the representation ΠψM of Sψ1 M pF q to the semidirect product M pF q Nψ . However, there will also be new phenomena. (ii) The stabilizer Wψ pπM q of a representation πM P ΠψM could be proper in Wψ . However, we would expect, at least if ψ φ is generic (and in contrast to the hypothetical possibility treated in the proof of the proposition) that Wψ pπM q is also equal to the stabilizer of the corresponding irreducible representation ξM of Sψ1 attached to πM . It is conceivable that the representation πM will not have an extension to M pF q Wψ pπM q. The obstruction would presumably reduce to a complex valued 2-cocycle on the image Rψ pπM q of Wψ pπM q in Rψ , of the kind considered in [A10, §3]. However, we could expect the same obstruction (or rather its inverse) to account for the failure of ξM to have an extension to the preimage Nψ pπM q of Wψ pπM q in Nψ . In fact, there ought to be a canonical extension of the representation ξM b πM of Sψ1 M pF q to the semidirect product M pF q Nψ pπM q. One could then induce this extended representation to the full group M pF q Nψ . The direct sum of the representations so obtained, taken over the Wψ -orbits tπM u in the packet ΠψM , would be the canonical extension of the representation ΠψM from Sψ1 M pF q to M pF q Nψ . With this object in hand, one would then be able to construct the representation Πψ of Sψ GpF q from rM , ψ q, induced representations IP pπM q and intertwining operators RP pw, π as in the proof of the proposition. (iii) If G is not quasisplit, the situation is more complicated. In this 1 case, one would need to replace the groups Sψ1 and Sψ by extensions Sψ,sc and Sψ,sc , defined as in [A19, §3]. With this proviso, the arguments for quasisplit groups would presumably carry over. We shall discuss inner twists of orthogonal and symplectic groups in Chapter 9. We return to our quasisplit classical group G P Ersim pN q. We have still to establish the reduction asserted in Lemma 2.4.2. However, we shall first state a variant of the local intertwining relation, which applies to a twisted r orthogonal group G. r pG r 0 , θrq, where G G r 0 is an even orthogonal group Recall that G in Ersim pN q. The statement will be almost the same as that of Theorem 2.4.1. In particular, M is a proper Levi subgroup of G, and P P P pM q is a parabolic subgroup of G. (We do not assume that P is normalized by any r If π lies in the packet Π r ψ , IP pπ q denotes the corresponding element in G.) r pF q generated by G r pF q. representation induced from P pF q to the group G The normalized intertwining operator (2.4.2) is then the earlier operator attached to G and M , but with one difference. The element w must be r M q induced from G. r This taken here from the rational Weyl set W pG, implies that the four groups in the lower right hand block in (2.4.3) have r in place of G. For any u in the set also to be formulated as sets, with G r u and x ru , M ru as above, and we Nψ pG, M q, we construct the objects wu , w
2.4. STATEMENT OF THE LOCAL INTERTWINING RELATION
107
r q as in (2.4.5). We also define the define the linear form frGr pψ, uq on HpG 1 r r linear form f r pψ, srq on HpGq as in (2.4.6), for any semisimple element sr in G
the centralizer Srψ
Sψ pGrq.
r For any u P Nψ pG, r M q, Theorem 2.4.4 (Local intertwining relation for G). the identity
(2.4.18)
frG1r pψ, sψ srq frGr pψ, uq,
r q, fr P HpG
ru of u holds for any semisimple element sr P Srψ that projects to the image x r in Sψ pG, M q.
This theorem joins our growing body of unproved assertions that will be established later by interlocking induction arguments. In the meantime, we assume as necessary that it holds if G is replaced by an even orthogonal group G P Ersim pN q, for any N N .
r pM q, Proof of Lemma 2.4.2. We are given a local parameter ψ P Ψ where M is a Levi subgroup of the simple group G P Ersim pN q. We are also given a point u P Nψ pG, M q, and a point s P Sψ whose image in the quotient Sψ equals the image xu of u in the subgroup Sψ pG, M q of Sψ . We have to show that the linear form
fG pψ, uq
¸
P
rψ π Π
xur, πry tr RP pwu, πr, ψqIP pπ, f q
equals its endoscopic counterpart fG1 pψ, sψ sq. The parameter ψ is the image of a “square integrable” parameter r 2 pM1 q, for a Levi subgroup M1 of M . The point u is a coset in ψ1 P Ψ 0 the complex group Sψ that normalizes the torus AM x in Sψ . It has a representative that also normalizes the maximal torus Tψ in Sψ0 . In other words, we can choose a representative u1 of u in the group Nψ1 , which one sees is determined up to translation by the subgroup Wψ01 pM, M1 q of Nψ1 in the chain Wψ01 pM, M1 q Wψ01 pG, M1 q Nψ1 pG, M1 q Nψ1 .
The point s can of course be identified with an element s1 P Sψ1 pGq, since the groups Sψ1 pGq and Sψ pGq are equal. Moreover, s1 projects to the image xu1 of u1 in the group Sψ1 pGq Sψ1 pG, M1 q, since xu1 equals xu . The information we are given from the lemma therefore tells us that the linear form (2.4.19)
fG pψ1 , u1 q
¸
P
rψ π1 Π 1
xur1, πr1y tr RP pwu , πr1, ψ1qIP pπ1, f q 1
1
1
equals its endoscopic counterpart fG1 pψ1 , sψ1 s1 q. Since sψ1 s1 equals sψ s, the linear forms fG1 pψ1 , sψ1 s1 q and fG1 pψ, sψ sq are equal. We have therefore to verify that fG pψ1 , u1 q equals fG pψ, uq.
108
2. LOCAL TRANSFER
The problem is simply to interpret the terms in the expression (2.4.19) for fG pψ1 , u1 q. Assuming an appropriate choice of M1 , we can arrange that the parabolic subgroup P1 P P pM1 q is contained in P . The Weyl element in (2.4.19) then satisfies wu1
wuM wu, 1
where wuM1 P W M pM1 q is a Weyl element for M , and the point wu P W pM q is identified with the Weyl element in W pM1 q that stabilizes the parabolic subgroup R1 P1 X M of M . The induced representation in (2.4.19) satisfies
M pπ 1 q , IP1 pπ1 q IP IR 1
by induction in stages, while the intertwining operator has a corresponding decomposition M RP1 pwu1 , π r1 , ψ1 q RR pwuM1 , πr1, ψ1qRP pwu, πr, ψq. 1
We assume implicitly that the analogues of Theorems 2.2.1 and 2.2.4, and of Theorems 2.4.1 and 2.4.4, hold for the proper Levi subgroup M in place u q, the sum of G. They tell us that for any function h P HpM ¸
P
xur1, πr1y tr RRM pwuM , πr1, ψ1qIRM pπ1, hq 1
1
h1M pψ1 , sψ1 s1 q
¸
rψ π1 Π 1
1
equals the linear form
P
rψ π Π
xur, πry hM pπrq.
Finally, from general principles, we know that the trace in (2.4.19) fibres as a product over the Hilbert space
M H P 1 pπ 1 q H P H R pπ1q . 1
It follows that the expression for fG pψ1 , u1 q reduces to the earlier expression for fG pψ, uq, as required. We have now established preliminary reductions of our local theorems. Lemma 2.4.2 reduces the local intertwining relation of Theorem 2.4.1 to r 2 pM q. Proposition 2.4.3 interprets the the case of parameters in the set Ψ local, intertwining relation itself as an explicit construction of some of the r ψ of Theorem 2.2.1, and hence reduces this theorem to parameters packets Π r 2 pGq. Similar reductions, which we shall leave to the reader, in the set Ψ apply to the assertions of Theorems 2.2.4 and 2.4.4. For the basic cases that remain, we will need global methods. We shall establish them, along with the supplementary local assertions of Theorem 1.5.1(b), in Chapters 6 and 7.
2.5. RELATIONS WITH WHITTAKER MODELS
109
2.5. Relations with Whittaker models The theory of Whittaker models has been an important part of representation theory for many years. For example, it is at the heart of the theorem of multiplicity 1 for GLpN q [Shal], and its generalization Theorem 1.3.2 by Jacquet and Shalika. In the hands of Shahidi, Whittaker models have yielded a broader understanding of intertwining operators. We shall review a couple of his main results, for comparison with our earlier discussion. Suppose for the moment that G is an arbitrary quasisplit, connected reductive group over the local field F . Suppose also that pB, T, tXk uq is a ΓF -stable splitting of G over F , where
¤ n, are fixed root vectors for the simple roots of pB, T q. If ψF is a fixed nontrivial Xk
1¤k
Xα , k
additive character on F , the function χpuq ψF pu1
un q,
u P NB pF q,
is a nondegenerate character on the unipotent radical NB pF q of B pF q. As usual, tuk u are the coordinates of logpuq relative to the simple root vectors tXk u, or more correctly, any basis of root vectors of nB pF q that includes tXk u. Then χ represents a Whittaker datum pB, χq for GpF q. We are interested in irreducible tempered representations π P Πtemp pGq of GpF q that have a pB, χq-Whittaker model. As in the special case of G GLpN q from §2.2, a pB, χq-Whittaker functional for π is a nonzero linear form ω on the underlying space of smooth vector V8 for π such that ω π puqv
χpuq ωpvq,
u P NB pF q, v
P V8.
We recall that the vector space spanned by the pB, χq-Whittaker functionals for π has dimension at most 1 [Shal], and that π is said to be generic if the space is actually nonzero. For any such ω, the function
W px, v q ω π pxqv ,
satisfies
W x, π py qv
x P G p F q, v
ωpxy, vq,
y
P V,
P G pF q,
and therefore represents an intertwining operator from π to the representation INB pχq of GpF q induced from the character χ of NB pF q. The Whittaker model of ω is the corresponding space of functions W pπ, χq W pxq W px, v q : v
P V8
(
of x, equipped with the representation of GpF q by right translation. Conversely, suppose that W pπ, χq is a subrepresentation of INB pχq that is equivalent to π. If v ÝÑ W px, v q, v P V, x P GpF q, is a nontrivial intertwining operator from π to W pπ, χq, the linear form ω pv q W p1, v q,
v
P V8,
110
2. LOCAL TRANSFER
is a pB, χq-Whittaker functional for π. Whittaker models and Whittaker functionals are thus essentially the same. Our focus will be on the functionals. Suppose that M is standard Levi subgroup of G over F . Then M is the Levi component of a parabolic subgroup P M NP of G that contains B. The Whittaker datum pB, χq for G restricts to a Whittaker datum pBM , χM q for M pF q, which can be expressed as above in terms of the splitting of M attached to that of G. Assume now that π and ω are attached to M instead of G. That is, π is an irreducible representation of M pF q, with a pBM , χM qWhittaker functional ω. There is then a canonical Whittaker functional Ωχ,ω pπ q for the induced representation IP pπ q. It is defined in terms of the “Whittaker integral” (2.5.1)
Wχ,ω px, h, πλ q
»
ω hπ,λ pw1 n xq χpn q1 dn ,
p q
N F
whose ingredients we recall. The vector h lies in the space HP,8 pπ q of smooth functions in HP pπ q. We are following the convention that HP pπ q is a Hilbert space of functions from K to V , which does not change if π is replaced with its twist πλ by a point λ P aM,C . We have therefore to write
hπ,λ pxq π MP pxq h KP pxq epλ
ρP
qpHP pxqq ,
x P G p F q,
in the notation of [A7, p. 26], to obtain a vector in the usual space on which IP pπλ q acts. The group N NP is the unipotent radical of the standard parabolic subgroup P M N that is “adjoint” to P , in the sense that M
w M w1,
w
w`w`M ,
where w` and w`M are the longest elements in the restricted Weyl groups of G and M respectively. The Whittaker integral (2.5.1) converges absolutely for Repλq in a certain chamber, and has analytic continuation as an entire function of λ. (See [CS, Proposition 2.1] and [Sha1, Proposition 3.1].) Its value Wχ,ω px, φ, π q is therefore a well defined intertwining operator from IP pπ q to INB pχq. We are interested in the corresponding pB, χq-Whittaker functional Ωχ,ω pπ q : h
ÝÑ
Ωχ,ω ph, π q Wχ,ω p1, h, π q,
h P HP,8 pπ q,
for IP pπ q. The constructions from the earlier sections of this chapter pertain to the special case of G P Ersim pN q. To compare them with the results of Shahidi, we take π P Πtemp pM q to be tempered. We assume that Theorem 1.5.1 holds for the Levi subgroup M , which we take to be proper. Then π belongs to r φ of a parameter φ P Φ r bdd pM q. We can denote the normalizing the packet Π factor from the end of §2.3 by rP pw, πλ q rP pw, φλ q,
w
P W pM q,
2.5. RELATIONS WITH WHITTAKER MODELS
111
since φ is uniquely determined by π. For the same reason, we write RP pw, πλ q RP pw, πλ , φλ q for the normalized intertwining operator in (2.3.26). Suppose in addition that π is generic, and is equipped with a pBM , χM q-Whittaker functional ω. rpwq from wπ There is then a canonical choice for the intertwining operator π r of w in GpF q is defined by to π. To see this, recall that the representative w r preserves the splitting of G. It follows from [Sp, Proposition 11.2.11] that w the splitting of M , and therefore stabilizes the Whittaker functional ω. We can therefore choose π rpwq uniquely so that (2.5.2)
ω
ω πrpwq.
The definition (2.4.2) thus gives us a canonical self-intertwining operator RP pw, π rq RP pw, π r, φq
of IP pπ q, in the case π P Πtemp pM q is both generic and equivalent to wπ. In this section, we have taken G to be an arbitrary quasisplit group. With the assumption that π is generic, Shahidi was nonetheless able to construct the local L and ε-functions that appear in the normalizing factors rP pw, πλ q. This was a local ingredient of the paper [Sha4] that became known as the Langlands-Shahidi method, in which Langlands’s original results [L4] were extended, and the analytic continuation and functional equation for a broad class of automorphic L-functions were established. In particular, the local definitions of §2.3 and §2.4 all carry over to the general group G, if π P Πtemp pM q is a representation that is generic. We therefore have the normalizing factor rP pw, πλ q, and the normalized intertwining operator r πλ q, RP pw, πλ q rP pw, πλ q1 JP pw,
w
P W pM q,
in (2.3.26), to go with the basic unnormalized intertwining operator JP pw, r πλ q. By the general results of Harish-Chandra [Ha4] and the properties of rP pw, πλ q established by Shahidi, these functions all have analytic continuation as meromorphic functions of λ P aM,C . Moreover, if w belongs to the stabilizer W pπ q of π in W pM q, we obtain the canonical self-intertwining operator of IP pπ q from (2.5.2).
RP pw, π rq π rpwq RP pw, π q
Theorem 2.5.1 (Shahidi). Suppose that G is a quasisplit group over F with Levi subgroup M , and that π P Πtemp pM q is generic. (a) The normalized intertwining operators Rpw, π q RP pw, π q,
w
P W pM q,
Rpw1 w, π q Rpw1 , wπ q Rpw, π q,
w1 , w
are unitary, and satisfy the relation. (2.5.3)
P W pM q.
112
2. LOCAL TRANSFER
(b) Suppose that w belongs to the subgroup W pπ q of W pM q. Then the rq satisfies the relation canonical self intertwining operator RP pw, π (2.5.4)
r q, Ωχ,ω pπ q Ωχ,ω pπ q RP pw, π
if ω is a pBM , χM q-Whittaker functional for π.
The property (a) is [Sha4, Theorem 7.9]. Its proof is contained in several papers, which include [Sha1] and [Sha2] as well as [Sha4]. Since the argument is by induction on the length of w, the assertion has to be proved in slightly greater generality. Namely, w must be taken from the more general set W pM, M 1 q of Weyl elements that conjugate M to a second standard Levi subgroup M 1 , while w1 is taken from a second set W pM 1 , M 2 q. The operators in (2.5.3) are otherwise defined exactly as above, and the required identity takes the same form. Shahidi’s starting point is the meromorphic scalar valued function Cχ,ω pw, πλ q, defined by the relation Ωχ,ω pπλ q Cχ,ω pw, πλ q Ωχ,ω pwπλ q J pw, r πλ q
given by the unnormalized intertwining operator r πλ q JP pw, r πλ q. J pw,
The existence of this function follows from the multiplicity 1 of Whittaker functionals, and the fact that J pw, r πλ q is meromorphic in λ. Since
1 w, πλ q rpw1 , w, πλ q J pwr1 , wπλ q J pw, J pw r πλ q,
for a meromorphic scalar valued function rpw1 , w, πλ q, one sees that rP pw1 , w, πλ q Cχ,ω pw1 w, πλ q1 Cχ,ω pw1 , wπλ q Cχ,ω pw, πλ q.
To split the 2-cycle rpw1 , w, πλ q in terms of suitable normalizing factors, it then suffices to compute the local coefficients Cχ,ω pw, πλ q. Therein lies the problem. Shahidi’s construction of these objects eventually leads to the construction of L and ε-functions, which he uses to express Cχ,ω pw, πλ q and to define normalizing factors rP pw, πλ q such that the corresponding normalized operators satisfy (2.5.3). The general version of (2.5.4) must be written differently. It takes the form (2.5.5)
Ωχ,ω pπ q Ωχ,ω pwπ q Rpw, π q,
for a general element w P W pM, M 1 q. In the special case that M 1 M and w belongs to the subgroup W pπ q of W pM q W pM, M q, we have the rpwq from wπ to π that stabilizes ω. It follows from intertwining operator π the definitions that the right hand side of (2.5.5) equals rpwq1 RP pw, π rq Ωχ,ω pπ q RP pw, π r q. Ωχ,ω pwπ q π
Therefore (2.5.5) does reduce to (2.5.4) in this case.
2.5. RELATIONS WITH WHITTAKER MODELS
113
The proof of (2.5.5) is only implicit in [Sha4]. The right hand side of (2.5.5) equals the value at λ 0 of the operator r πλ q Ωχ,ω pwπλ q rP pw, πλ q1 J pw,
rP pw, πλq1 Cχ,ω pw, πλq1 Ωχ,ω pπλq. The problem is to show that the product rP pw, πλ q Cχ,ω pw, πλ q
is analytic at λ 0, with value at λ 0 equal to 1. We shall have to leave the reader to extract this fact from the formulas of [Sha4] and [Sha2], following the calculation from the special case in [KeS, §4]. We shall review the calculation in the paper [A27] in preparation. We will have a special interest in the following five cases: (i) F arbitrary, G GLpN q, M arbitrary; (ii) F C, G P Ersim pN q, M arbitrary; (iii) F arbitrary, G P Ersim pN q, M GLprN {2sq; (iv) F arbitrary, G P Ersim pN q, M T minimal; (v) F arbitrary, G P Ersim pN q, Mder Spp2q.
In cases (i)–(iv), it is known that any π P Πtemp pM q is generic. In the last case (v), G is of the form Spp2nq, while M is an product of several copies of GLp1q with the group SLp2q Spp2q. From the general relation between representations of SLp2q and GLp2q described in the introduction of [LL], we see that any π P Πtemp SLp2q is generic, for some choice of Whittaker datum. The same is therefore true for any π P Πtemp pM q. This case will not be needed in any of our future arguments, unlike the other four. We have included it here for general perspective, and because it will arise in a natural setting later in Lemma 6.4.1. It follows that in all five cases, the canonical self intertwining operator RP pw, π rq is defined for any w P W pπ q, and satisfies (2.5.4). Corollary 2.5.2. Suppose that pF, G, M q is as in one of the cases (i)– (v), that π P Πtemp pM q, and that w P W pπ q. Then if pΠ, V q is the unique irreducible pB, χq-generic subrepresentation of IP pπ q, the operator RP pw, π rq satisfies RP pw, π rqφ φ, φ P V8 . In particular, if IP pπ q is irreducible, we have rq 1. RP pw, π
Proof. The restriction of RP pw, π rq to the irreducible subspace V of rq satisfies (2.5.4). HP pπ q is a nonzero scalar. Since π is generic, RP pw, π Since the Whittaker functional Ωχ,ω pπ q on HP,8 pπ q is supported on the subspace V, the scalar in question equals 1.
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2. LOCAL TRANSFER
We note that in the cases (i) and (ii), the induced representation IP pπ q is always irreducible. In (iii), suppose that π corresponds to the Langlands parameter φ P Φ2 pM q, and that w P W pπ q is nontrivial. The centralizer Sφ is then isomorphic to either Spp2, Cq or Op2, Cq. We will apply the corollary later to the case that the centralizer is Spp2, Cq, in the proof of Lemma rq is a scalar. In (iv), 5.4.6, after first showing that the operator RP pw, π our interest will be in the case that π is trivial on the maximal compact subgroup of T pF q. The representation IP pπ q need not be irreducible in this case. However, the generic subrepresentation Π is easily identified. For it follows from (2.5.1) (together with an approximation argument in case F is archimedean) that Π is the irreducible constituent of IP pπ q that contains the trivial representation of K. It is only in the last case (v) that the L-packets for M can be nontrivial. In the same paper, Shahidi conjectured that any tempered L-packet for the quasisplit group G has a pB, χq-generic constituent [Sha4, Conjecture 9.4], motivated by the proof of the property for archimedean F that followed from the results of [Kos] and [V1]. Shelstad [S6] has recently established a strong form of this conjecture for real groups. Namely, if the transfer factors are normalized as in [KS, (5.3)], the pB, χq-generic representation in a tempered packet is the unique representation for which the linear character (2.2.5) is trivial. For nonarchimedean F and G P Ersim pN q, Konno [Kon] showed that the conjecture would follow from local twisted endoscopy for G, which is to say, the assertion of Theorem 2.2.1. In §8.3, we will show that the strong form of the conjecture (apart from the uniqueness condition) is also valid in this case, after proving Theorem 2.2.1 in Chapter 6. In retrospect, we could have assumed Shahidi’s conjecture inductively for M , and then appealed to his conditional proof of Proposition 2.3.1 in [Sha4, §9]. I have retained the proof of the proposition in §2.3 for its different perspective, and its role in the transition to the self intertwining operators (2.4.4). We need to expand on the case (i) of GLpN q. It will be important to r pN q of the identity of Corollary establish an analogue for the component G 2.5.2. Of equal importance, but harder, is the task of proving an identity r p N q. for general parameters ψ P Ψ Suppose that M is a standard Levi subgroup of GLpN q, and that
SU p2q ÝÑ LM is a general parameter in ΨpM q. Then ψ corresponds to an irreducible unitary representation πψ of M pF q, which is the Langlands quotient of a pM q for the Weyl set of outer standard representation ρψ . We write W r pN q, and W ψ pM q for automorphisms of M induced from the component G pM q. Any element in W ψ pM q then stabilizes πψ and the stabilizer of ψ in W ρψ as well as ψ. We assume that the set Wψ pM q is nonempty. This implies that the image of ψ in ΨpN q, which we again denote by ψ, lies in the subset r pN q. Ψ ψ : LF
2.5. RELATIONS WITH WHITTAKER MODELS
115
ψ pM q. Then Suppose that w belongs to W
w
θrpN q w0,
where θrpN q is the standard outer automorphism of GLpN q defined in §1.3, and w0 belongs to a set W pM, M 1 q. Bearing in mind that θrpN q acts as an involution on the set of standard parabolic subgroups of GLpN q, we note that M 1 is the standard Levi subgroup that is paired with M . The representative r θrpN q w w r0
r pN, F q preserves the standard Whittaker datum pBM , χM q for M . of w in G We would like to define twisted intertwining operators by some variant of (2.4.2) and (2.3.25). The operator
RP pw, πψ q `pw, πψ q Rw1 P |P pπψ q in (2.3.25) has to be slightly modified. For we cannot define the operator `pw, r πψ q Hw1 P pπψ q in the product
ÝÑ
HP pwπψ q
r πψ q `pw, πψ q εP pw, ψ q `pw,
by a simple left translation by w r1 , since the operation would not take values in HP pwπψ q. We instead write (2.5.6)
r πψ q φ pxq φ w r1 x θrpN q , `rpw,
φ P H w 1 P pπ ψ q,
in order that the right hand side make sense, as well as `rpw, πψ q εP pw, ψ q `rpw, r πψ q. The product
rP pw, πψ q `rpw, πψ q R 1 R w P |P pπψ q
is then an operator rP pw, πψ q : HP pπψ q R
ÝÑ
HP pwπψ q.
If ω is a pBM , χM q-Whittaker functional for ρψ , we define the intertwining operator ρrψ pwq from wρψ to ρψ uniquely by analytic continuation, and the property ω ρrψ pwq ω. This lifts to an intertwining operator π rψ pwq from wπψ to πψ . The product rP pw, π rP pw, πψ q R rψ q π rψ pwq R
is then a canonical operator on HP pπψ q. We note that its construction is a variant of both the definition at the beginning of §2.2, and the definition of the operator in Theorem 2.5.1(b).
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2. LOCAL TRANSFER
rP pw, π The twisted operator R rψ q does not intertwine IP pπψ q with itself. Indeed, it follows from (2.5.6) and the other definitions that rP pw, π rψ q : IP pπψ q R
ÝÑ IP pπψ q θrpN q. rP pω, π rψ q intertwines IP pπψ q with the representation In other words, R θrpN q1 IP pπψ q IP pπψ q θrpN q.
Recall that we have already introduced an operator with the same intertwining property. In §2.2, we defined an operator with
r pN q, IrP pπψ , N q Π
IrP pπψ , N q : IP pπψ q
ÝÑ
Π I P pπ ψ q,
IP pπψ q θrpN q,
in terms of a Whittaker functional for IP pπψ q. How are the two objects related? Theorem 2.5.3. The intertwining operator attached to ψ ψ pM q satisfies wPW (2.5.7)
P ΨpM q
and
rP pw, π rP pπψ , N q. R rψ q I
In case φ ψ is generic, one can apply the formula (2.5.5) to the element w0 P W pM, M 1 q. The required identity (2.5.7) then follows without much difficulty from the various definitions. The general case is more difficult, and as far as I know, has not been investigated. It requires further techniques, based on some version of minimal K-types. Rather than introduce them here, we shall leave the general proof of Theorem 2.5.3 for a separate paper [A26]. There is another way to interpret the identity (2.5.7). We begin by writing Srψ pN q as in §1.4 for the centralizer in Then Srψ pN q is a bitorsor under the centralizer
r pN q of the image of ψ. G Srψ0 pN q of the image of ψ p
r 0 pN q. The analogue of the diagram (2.4.3) for the G0 G r 0 pN q torsor in G r pN q certainly makes sense. If we define its objects in the natural way GG in terms of the bitorsor Sψ Srψ pN q, p
we see that
ψ pM q, Nψ pG, M q Wψ pG, M q W
and that this last set is a bitorsor under the group Wψ0 pG, M q Wψ pM q.
All the other sets in the diagram are trivial. We shall use the diagram r pN q of the local intertwining relation of to formulate the analogue for G Theorem 2.4.1. The formalities of the process will be helpful in the next section in understanding the spectral terms of the twisted trace formula
2.5. RELATIONS WITH WHITTAKER MODELS
117
r pN q will be an essential for GLpN q. The actual intertwining relation for G ingredient of the global comparison in Chapter 4. The twisted intertwining relation can be most easily stated if we inflate r pN, F q. With this interpretation, IP pπψ q to an induced representation of G IP pπψ q acts on the larger Hilbert space r pπψ q HP pπψ q ` H r P pπ ψ q, H P
r P pπψ q is the space of functions supported on the component G r pN, F q. where H ψ pM q, we then obtain a linear transformation For any w P W
by setting
rψ q : HP pπψ q RP pw, π
ÝÑ
r P pπ ψ q H
rP pw, π rψ qφ pxq R rψ qφ xθrpN q1 , RP pw, π
r pN, F q. In other words, RP pw, π rψ q is defined for φ P HP pπψ q and x P G exactly as in (2.4.2) and (2.3.25), namely, with the translation operation r πψ q defined by excluding the right translation by θrpN q from (2.5.6). If `pw, r pN q, its integration against IP pπψ q gives a fr belongs to the Hecke space H linear transformation r P pπ ψ q IP pπψ , frq : H
ÝÑ
HP pπψ q
that we can compose with RP pw, π rψ q. Following (2.4.5), we set (2.5.8)
rψ q IP pπψ , frq , frN pψ, uq tr RP pw, π
r pN q, fr P H
ψ pM q. for any point u w in the set Nψ pG, M q W 0 r pN, F q on HP pπψ q also has an extension The representation IP pπψ q of G r to the bitorsor GpN, F q. This is provided by the operator
IrP pπψ , N q IrP πψ , θrpN q ,
which then yields the linear form
frN pψ q tr IrP pπψ , frq , of (2.2.1). Observe that if
r pN q, fr P H
IP pπψ , N qφ pxq IrP pπψ , N qφ xθrpN q1 ,
r pN, F q, then for φ P HP pπψ q and x P G
frN pψ q tr IP pπψ , frq IP pπψ , N q
It follows from Theorem 2.5.3 that (2.5.9)
frN pψ q frN pψ, uq,
tr IP pπψ , N q IP pπψ , frq . ψ pM q. uPW
Suppose that s is a semisimple element in the torsor Srψ pN q. The pair pψ, sq then has a preimage under the general correspondence (1.4.11), which r pGq that we continue consists of a datum G P ErpN q, and a parameter in Ψ
118
2. LOCAL TRANSFER
to denote by ψ. If the assertion (a) of Theorem 2.2.1 is valid, we have the linear form r pN q, frG pψ q frN pψ q, fr P H
on SrpGq. Following (2.4.6), we then set (2.5.10)
G frN pψ, sq frGpψq,
r pN q. fr P H
r pN q). Assume that the Corollary 2.5.4 (Local intertwining relation for G assertion (a) of Theorem 2.2.1 is valid for any pair
pG, ψq,
G P ErpN q, ψ
P Ψr pGq.
P ΨpM q is as in Theorem 2.5.3, the identity G (2.5.11) frN pψ, sψ sq frN pψ, uq, fr P HrpN q, ψ pM q and any semisimple element s P S rψ pN q. holds for any u P W G pψ, sq equals fr pψ, uq, Proof. It follows from (2.5.10) and (2.5.9) that frN N Then if ψ
for any s and u. The two linear forms are therefore independent of the points s and u. This is something we could have expected from general principles, given that the set Srψ pN q is connected. Its proof, together with that of the equality of the two linear forms, is thus an immediate consequence of Theorem 2.5.3. The required identity (2.5.11) obviously follows if we replace s by sψ s.
We return briefly to the case that G P Ersim pN q is a quasisplit orthogonal or symplectic group. A central theme of Chapter 2 has been that the general r, ψ q in (2.4.2) has no canonical definition. self-intertwining operator RP pw, π Only when the operator is balanced with the pairing xx ru , π ry in (2.4.4) do we obtain a canonical object in general. We have seen in this section that the definition (2.4.2) can be made canonical in the special case that the parameter φ ψ and the representation π are generic. Before we move on, we remind ourselves of another case that leads to a canonical definition. To r 2 pM q, with π P Π r pM q describe it, we take ψ to be a general parameter in Ψ r ψ. being a general element in the corresponding packet Π We are assuming that w lies in the stabilizer Wψ pπ q of ψ and π. As an element in W pM q, w stabilizes the orthogonal or symplectic part G of M . Suppose that it acts on G by inner automorphism. This is always so p is orthogonal, in which case it is equivalent to the unless N is even and G condition that the value corresponding to w of the explicit sign character (2.4.11) be 1. The representative w r of w then commutes with G . We rpwq from wπ to π by asking can therefore define the intertwining operator π that it stabilize the relevant Whittaker functional on each of the general linear factors of M . The product (2.4.2) is then a canonical self intertwining r, ψ q of IP pπ q. It owes its existence to the fact that the operator RP pw, π q in (2.4.5) is extension x , π ry of the character x , π y from Sφ pM q to Sφ pM canonical.
2.5. RELATIONS WITH WHITTAKER MODELS
119
Suppose for example that w lies in the subgroup Wψ0 of elements in Wψ induced by points in the connected group Sψ0 pGq. The corresponding value of (2.4.11) is then 1, for trivial reasons, and we obtain a canonical r, ψ q. The local intertwining relation for G self intertwining operator RP pw, π r, ψ q equals 1. We shall establish this fact later, in the suggests that RP pw, π course of proving the intertwining relation. There is one other application of the theory of Whittaker models that we will need. It is a kind of converse of the theorem of Konno, for nonarchimedean F , which we will need at one point in the proof of the local intertwining relation. Since the results in [Kon] rely on the exponential map, we do have to assume that the residual characteristic charpF q of F is not equal to 2. Lemma 2.5.5. Suppose that F is nonarchimedean, and that the objects r pG q, φ M P Φ r 2 pM, φq, M G and pBM , χM q over F G P Ersim pN q, φ P Φ have the following three properties. (i) p charpF q 2. r φ with x , πM y 1 is the unique pBM , χM q(ii) The element πM P Π M rφ . generic representation in the packet Π M (iii) For any pair ps, uq as in the statement of Theorem 2.4.1, there is a constant eps, uq such that fG1 pφ, sq eps, uq fG pφ, uq,
f
P HrpGq.
Then eps, uq 1 for every ps, uq. In other words, the local intertwining relation is valid for φ and φM . Results of this nature are probably known. The proof in any case is not difficult, given Theorem 2.5.1(b) and the papers [Kon], [MW1] and [Rod2]. Since we need to move on, we shall leave the details for [A27]. We shall use Lemma 2.5.5 only in very special cases, which will come up in the proofs of Lemma 6.4.1 and Lemma 6.6.2.
CHAPTER 3
Global Stabilization 3.1. The discrete part of the trace formula We can now begin to set in place the means for proving the three main theorems. The methods are based on the trace formula. In recent years, it has been possible to compare many terms in the trace formula directly with their stable analogues for endoscopic groups. The resulting cancellation has led to a corresponding identity for the remaining terms. These are the essential spectral terms, the ones which carry the desired information about automorphic representations. They comprise what is called the discrete part of the trace formula. We shall review the discrete part of the trace formula in this section. The theorems we are trying to prove concern quasisplit orthogonal and symplectic groups. In the long term, we are of course interested in more general groups so it would be sensible to think as broadly as possible in discussing the general theory. We shall do so whenever practical, with the understanding that the results will ultimately be specialized to orthogonal and symplectic groups, and the corresponding global parameters defined in §1.4. We assume until further notice that the field F is global. Suppose for the moment that G is a connected, reductive algebraic group over F . The discrete part of the trace formula for G is a linear form on the Hecke algebra HpGq. We recall that HpGq is the space of smooth, compactly supported, complex-valued functions on GpAq that are K-finite relative to the left and right actions of a suitable maximal compact subgroup K GpAq. It will be convenient to have this linear form depend on two other quantities, which we will use to account for two minor technical complications. The first point concerns the nominal failure for G to have any discrete spectrum. This is governed by the split component AG of the center of G, or equivalently, the quotient of GpAq by GpAq1 . The subgroup GpAq1 of GpAq equals the kernel of the homomorphism HG : GpAq
from GpAq onto the real vector space aG
ÝÑ
aG
HomZ X pGqF , R ,
which is defined by the familiar prescription exHG pxq,χy
|χpxq|,
x P G pA q , χ P X pG q F . 121
122
3. GLOBAL STABILIZATION
It has a group theoretic complement AG,8
AG pRq0,
GQ
Q
RF {QpGq,
in GpAq, where RF {Q pq denotes the restriction of scalars, and p q0 stands for the connected component of 1. In particular, the mapping HG restricts to a group theoretic isomorphism from AG,8 onto aG . It gives rise to an isometric isomorphism L2 GpF qAG,8 zGpAq
ÝÑ
L2 GpF qzGpAq1 .
We are therefore free to work withthe discrete spectrum of the left hand space in place of L2disc GpF qzGpAq1 . To allow flexibility for induction arguments, it is useful to treat a slightly more general situation. We choose a closed subgroup XG of the full center Z GpAq of GpAq such that the product XG Z GpF q is closed and cocompact in Z GpAq . (In general, we write Z pH q for the center of any group H.) The quotient GpF qXG zGpAq GpF qzGpAq{XG
then has finite invariant volume, as in the special case that XG also fix a character χ on the quotient Z Gp F q
AG,8. We
X XGzXG Z GpF q zZ GpF q XG,
and write
L2disc GpF qzGpAq, χ L2 GpF qzGpAq, χ for the space of χ-equivariant functions on GpF qzGpAq that are squareintegrable modulo XG , and decompose discretely under the action of GpAq. The central character datum pXG , χq for G gives rise to corresponding data for Levi subgroups M of G. Given M , we write AM,,G8 for the kernel in AM,8 of the composition AM,8
H ÝÝÑ
The product XM
M
aM
ÝÑ
aG .
pAM,,G8qXG
is then an extension of XG , to which we can pull back the character χ. We obtain a triplet pM, XM , χq that satisfies the same conditions as pG, XG , χq. In particular, we obtain a representation of M pAq on L2disc M pF qzM pAq, χ that decomposes discretely. If P NP M belongs to the set P pM q of parabolic subgroups of G with Levi component M , we write IP pχq IPG pχq
for the corresponding parabolically induced representation. It acts on the Hilbert space HP pχq of left NP pAq-invariant functions φ on GpAq such that the function φpmk q on M pAq K belongs to the space
L2disc M pF qzM pAq, χ
b L2pK q.
3.1. THE DISCRETE PART OF THE TRACE FORMULA
123
This discussion can of course be applied to the special case above, namely where XG AG,8 , χ 1, and XM AM,8 . The second point has to do with analytic estimates. W. M¨ uller has established [Mu] that the restriction of any operator IP pχ, f q
»
p q
G A
f pxqIP pχ, xqdx,
f
P H p G q,
to the discrete spectrum is of trace class. Rather than work with the estimates of M¨ uller, however, we shall rely on the consequences in [A5] of what was later called the multiplier convergence estimate [A14, §3]. These allow us to work with small subrepresentations of IP pxq, defined by restricted Archimedean infinitesimal characters. We fix a minimal Levi subgroup M0 of G, which we can assume is in good position relative to K. We can then construct the real vector space h ihK
` h0 in terms of the group M0,Q RF {Q pM0 q, as in [A2, §3]. Thus, h0 is the Lie algebra of a maximal real split torus in the real group M0,Q pRq M0 pF8 q, and hK is a Cartan subalgebra of the Lie algebra of the compact real group K X M0,Q pRq K X M0 pF8 q. The complexification hC of h is a Cartan subalgebra of the Lie algebra of the complex group GQ pCq, whose real form h is invariant under the complex Weyl group W . The role of the real space h hG is simply to control infinitesimal characters. We represent the Archimedean infinitesimal character of an irreducible representation π of GpAq by a complex valued linear form µπ µπ,R iµπ,I on h. The imaginary part µπ,I , regarded as a W -orbit in h , is the object of interest. It has a well defined norm }µπ,I }, relative to a fixed, W -invariant Hermitian metric on hC . There is then a decomposition IP pχq
à
¥
t 0
IP,t pχq
of IP pχq, where IP,t pχq is the subrepresentation of IP pχq composed of those irreducible constituents π with }µπ,I } t. The endoscopic comparison of trace formulas reduces ultimately to a reciprocity law among the representations IP,t pχq attached to different groups. The discrete part of the trace formula will be a linear form that depends on χ and t. We may as well build the dependence on χ into the test function f . Let HpG, χq be the equivariant Hecke algebra of functions f on GpAq that satisfy f pxz q f pxqχpz q1 , z P XG , and are compactly supported modulo XG . The operator IP,t pχ, f q
»
p q{
G A XG
f pxqIP,t pχ, xqdx
124
3. GLOBAL STABILIZATION
on HP pχq is then defined for f in HpG, χq. In this setting, the discrete part of the trace formula is the linear form G Idisc,t pf q Idisc,t pf q ,
defined by (3.1.1) ¸ |W0M ||W0G|1
P
M L
¸
t ¥ 0, f
P HpG, χq,
P p qreg
w W M
| detpw 1qa |1tr MP,tpw, χqIP,tpχ, f q . G M
The outer sum in (3.1.1) is over the finite set L LpM0 q of Levi subgroups of G that contain M0 . The inner sum is over the set of regular elements W pM qreg
w
P W pM q :
detpw 1qaG
M
0
(
in the relative Weyl group
W pM q W G pM q NormpAM , Gq{M
for G and M . Here aG M is the canonical complement of aG in aM , so W pM qreg is the set of w whose fixed subspace in aM equals the minimal space aG . Following standard practice, we have written W0M W M pM0 q and W0G W G pM0 q for the Weyl groups with respect to M0 . The remaining ingredient of (3.1.1) comes from the standard global intertwining operator MP pw, χq that is at the heart of Langlands’ theory of Eisenstein series. We can define MP pw, χq as the value at λ 0 of a meromorphic composition
w1P, λ P paGM qC, of operator valued functions on HP pχq. The factor `pwq stands for the mapping from HP 1 pχq to HP pχq defined by left translation by any representative r1 of w1 in GpF q, while P 1 w1 P is the parabolic subgroup w w r 1 P w r in P pM q. The other factor is the operator MP 1 |P pχλ q : HP pχq ÝÑ HP 1 pχq, P, P 1 P P pM q, (3.1.2)
MP pw, χλ q `pwq MP 1 |P pχλ q,
whose value
MP 1 |P pχλ qφ pxq,
P1
φ P HP pχq, x P GpAq,
is defined for the real part of λ in a certain cone in paG M q by the familiar intertwining integral »
p qXNP pAqzNP 1 pAq
NP 1 A
φpnxqepλ
ρP
qpHP pnxqq dn epλ
qp
p qq .
ρP 1 H P 1 x
We recall that HP is the mapping from GpAq to aM defined by HP pnmk q HM pmq,
n P NP pAq, m P M pAq, k
P K,
that ρP is the usual linear form on aM defined by half the sum of the roots (with multiplicity) of pP, AM q, and that χλ puq χpuq eλpHM puqq ,
u P XM ,
3.1. THE DISCRETE PART OF THE TRACE FORMULA
125
is the twist of χ by λ. The operator MP pw, χλ q leaves the subspace HP,t pχq of HP pχq invariant, and intertwines the two induced representations IP,t pχλ q and IP,t pχwλ q on this subspace. It has analytic continuation as a meromorphic func G tion of λ P paG M qC , whose values at ipaM q are analytic and unitary. Thus, MP pw, χλ q restricts to a unitary operator MP,t pw, χq on the Hilbert space HP,t pχq, which intertwines the representation IP,t pχq. This operator is the last of the terms in (3.1.1). It is also the most interesting. Its analysis includes the local intertwining relation stated in the last chapter, and will be an important aspect of future chapters. More generally, suppose that G pG0 , θ, ω q
(3.1.3)
is an arbitrary triplet over F . Then G0 is a connected reductive group over F , θ is a semisimple automorphism of G over F , and in the global context here, ω is a character on G0 pAq that is trivial on the subgroup G0 pF q. As in the local case of §2.1, we also write G G0 θ more narrowly for the associated G0 -bitorsor over F , with distinguished point θ
1 θ.
The Hecke module HpG, χq is then a space of functions on GpAq defined exactly as in the case G G0 above. We note that there are two morphisms from G G to G0 . They are defined in the obvious way as formal algebraic operations on the bitorsor by #
(3.1.4)
y11 y2
θ1px1 1x2q, 1 y1 y21 x1 x 2 ,
for two points yi xi θ in G. Our general convention of letting G stand for both a G0 -torsor and the underlying triplet should not lead to confusion. It is similar to our convention for endoscopy, in which a symbol G1 represents both an endoscopic group and the underlying endoscopic datum. The discrete part of the trace formula for G again takes the form (3.1.1), provided that the terms are properly interpreted. We recall how these terms can be understood in the general twisted case. For a start, the central character datum pXG , χq has to be adapted to G (rather than G0 q. We write aG
aθG 0
H
P aG
0
: AdpθqH
(
H aG
0
for the subspace of θ-fixed vectors in the real vector space aG0 , and AG
pAθG q0 AG 0
0
126
3. GLOBAL STABILIZATION
for the maximal F -split torus in the centralizer of G in G0 . We assume implicitly that ω is trivial on the subgroup AG,8
AG pRq0, Q
GQ
RF {QpGq,
of G0 pAq, since the twisted trace formula for G otherwise becomes trivial. If A is any ring that contains F , we write Z G pAq
Z G 0 pAq θ Z G 0 pAq for the centralizer of GpAq in G0 pAq. We then take XG Z G p A q
to be any closed subgroup satisfying the conditions above, with the further requirement that it lie in the kernel of ω, and χ to be any character on the quotient of XG by its intersection with Z GpF q X XG . The standard example is of course the pair
pXG, χq pAG,8, 1q.
For any such choice, we can form the equivariant Hecke module HpG, χq of functions in GpAq, relative to pXG , χq and a suitably fixed maximal compact subgroup K G0 pAq. The general version of (3.1.1) will be a linear form in a function f in HpG, χq. The general version of (3.1.1) is again based on a fixed minimal Levi subgroup M0 G0 . Suppose that M belongs to the set L LpM0 q of Levi subgroups of G0 that contain M0 . The composition aM
ÝÑ
aG0
ÝÑ
aG0 ,θ ,
together with the natural isomorphism between aG and the space of θcovariants ( aG0 ,θ aG0 { X AdpθqX : X P aG0 in aG0 , gives a canonical linear projection from aM onto aG . We write aG M for its null space in aM . We then have the regular set G Wreg pM q Wreg pM q w
P W pM q :
detpw 1qaG
M
0
(
,
in the Weyl set W pM q W G pM q NormpAM , Gq{M of outer automorphisms of M induced by the conjugation action of G on G0 . The set Wreg pM q is generally quite distinct from its untwisted analogue 0 pM q W G0 pM q, defined for G0 above. These conventions account for Wreg reg the general indices of summation M and w, and the corresponding coefficients | detpw 1qaG |1, M
in the twisted interpretation of (3.1.1).
3.1. THE DISCRETE PART OF THE TRACE FORMULA
127
We observe that pXG , χq again provides a central character datum pXM , χq for any M P L. The first component is the extension XM
pAM,,G8qXG
of XG , and the second component is the character on XM obtained by pulling back χ. The only difference from the case G G0 above is that AM,,G8 is now the kernel of the projection of AM,8 onto the subspace aG attached to G (rather than G0 ). The remaining term is the trace in (3.1.1). Suppose that P belongs to the set P pM q of parabolic subgroups of G0 with Levi component M . We 0 pχq for the Hilbert space of the induced representation defined for write HP,t G0 above. We reserve the symbol HP,t pχq for its analogue for G, namely the space of complex-valued functions φ on GpAq such that for any y P GpAq, the function φpxy q, x P G 0 p A q, 0 pχq. The operators in (3.1.1) attached to f P HpG, χq and belongs to HP,t w P W pM q make sense in this more general context. However, they have now to be interpreted as linear transformations (3.1.5) and (3.1.6)
IP,t pχ, f q : HP,t pχq
ÝÑ
0 HP,t pχq
0 MP,t pw, χq : HP,t pχq
ÝÑ
HP,t pχq.
The first transformation equals
G IP,t pχ, f q IP,t pχ, f q
for the induced mapping
»
p q{
G A XG
IPG pχ, y qφ pxq φpxy qω pθ1 xy q,
G f py qIP,t pχ, yqdy,
φ P HP,t pχq, x P G0 pAq,
0 pχq. (There is no need to include ω in the notation, as it from HP,t pχq to HP,t is implicit in superscript G.) The second transformation is given by (3.1.2), with the translation `pwq defined now as a map from HP0 1 pχq to HP pχq by (3.1.4). The product MP,t pw, χqIP,t pχ, f q is then an operator on HP,t pχq, for which the trace in (3.1.1) is defined. With this notation, all of the terms in the expression (3.1.1) makes sense. The discrete part of the trace formula for G pG0 , θ, ω q is the linear form G Idisc,t pf q Idisc,t pf q,
t ¥ 0, f
P HpG, χq,
defined by the general form of this expression. It is the main spectral component of a general formula usually known as the twisted trace formula. The twisted trace formula was established in [LW] and [A5]. Our notation here differs from that of [A5] in three minor respects, which might be worth pointing out explicitly.
128
3. GLOBAL STABILIZATION
In [A5], G represents a component of a (nonconnected) reductive algebraic group, and ω is trivial. This is slightly less general that the triplet pG0, θ, ωq here. However, the earlier arguments all extend without difficulty. The given datum of [A5] is also less precise, in that it specifies the underlying automorphism of G0 only up to inner automorphism. The point of view here is more suitable to stablization, which we will discuss in the next section. 0 and H Secondly, the induced Hilbert spaces HP,t P,t do not take this form in [A5]. They reflect the fact here that the G0 -torsor G need not be attached to a component of a reductive algebraic group. They also streamline the notation in all cases. Finally, we reiterate that the elements M P L here are Levi subgroups of G0 . The complementary geometric terms in the twisted trace formula require another notion, that of a Levi subset M of G, which we can recall from the local context of §2.2. To keep the two kinds of objects straight, we 0 wrote L0 LG pM0 q in [A5] for the set of Levi subgroups of G0 containing M0 and L LG pM0 q for the collection of Levi subsets of G, noting at the same time that there was an injective mapping M Ñ M 0 from L to L0 . In the present volume, the discrete part Idisc,t is the only term we will consider. Since it is based only on the Levi subgroups of G0 , we are free here to denote these objects by M , and to write L for the set of M that contain M0 . It is with this notation that the expression from Idisc,t pf q takes exactly the same form (3.1.1) in general as it does in the original case of G G0 . The starting point for the trace formula is the kernel of an integral operator. In the general case at hand, the operator is a composition IG pχ, f q MG pθ, χq,
f
P HpG, χq,
of what amount to specializations to M G0 of the transformations (3.1.5) and (3.1.6), but with the χ-equivariant Hilbert space L2 G0 pF qzG0 pAq, χ in place of
0 2 0 0 HG 0 ,t pχq Ldisc,t G pF qzG pAq, χ .
The reader can check that the kernel of this operator equals K px, y q
¸
P p q
f px1 γy qω py q,
x, y
P G0pF qzG0pAq.
γ G F
As a general rule, we have not tried to be explicit about measures. In the global setting here, we note simply that the operator IP,t pχ, f q in the formula (3.1.1) for Idisc,t pf q depends on a choice of G0 pAq-invariant measure on GpAq{XG . This amounts to a choice of Haar measure on G0 pAq{XG . Observe that a Haar measure on the group XG determines a Haar measure on AG,8 such that
vol XG Z GpF q zZ GpAq
vol AG,8Z GpF q zZ GpAq
,
3.1. THE DISCRETE PART OF THE TRACE FORMULA
129
for any Haar measure on Z GpAq . The linear form Idisc,t pf q thus depends on a choice of G0 pAq-invariant measure on GpAq{AG,8 , or equivalently, a Haar measure on G0 pAq{AG,8 . Finally, we observe that the linear form Idisc,t pf q is “admissible” in f . In other words, it can be written as a linear combination of twisted characters
tr π pf q ,
π0
P Π0,
where Π0 is a finite set of irreducible unitary representations of G0 pAq, and π is an extension of π 0 to GpAq, in the sense of the global analogue of (2.1.1). The set Π0 Π0t pf q depends on f , but only through its K-type. To be more precise, let us agree that a Hecke type for G means a pair pτ8, κ8q, where κ8 is an open compact subgroup of G0pA8q and τ8 is a finite set of irreducible representations of a maximal compact subgroup K8 of GpA8 q. We have written A8
a P A : av
0,
and
if v
(
R S8
¹
P
v S8
Fv
(
F8
A8 a P A : av 0, if v P S8 here for the respective subrings of archimedean and finite adeles in A. We also write ( AS a P A : av 0, if v P S and ( 8 A8 S a P A : av 0, if v R S , if S is any set of valuations of F that contains the set S8 of archimedean S valuations. We can then decompose κ8 as a product κ8 S K , where S S8 is 0 8 S finite, κ8 S is an open compact subgroup of G pAS q, and K is a product over v R S of hyperspecial maximal compact subgroups Kv G0 pFv q. Assume that the product κ K8 κ8 is contained in the maximal compact subgroup K (and is in particular a subgroup of finite index in K). We shall say that pτ8 , κ8 q is a Hecke type for f if f is bi-invariant under translation by κ8 , and transforms under left and right translation by K8 according to representations in the set τ8 . Any f P HpG, χq of course has a Hecke type. The assertion above is that the finite set Π0 depends only on a choice of Hecke type for f . This follows from the definition (3.1.1) of Idisc,t pf q and the theory of Eisenstein series, specifically Langlands’ decomposition of L2 G0 pF qzG0 pAq in terms of residues of cuspidal Eisenstein series [L5, Chapter 7]. (See [A5, Lemma 4.1]. Once κ8 has been fixed, the archimedean infinitesimal character of π is controlled by the K8 -type τ8 and the norm t }µπ,I } of its imaginary part.) Notice that for the extension π of π 0 , µπ,I µπ0 ,I is a linear form on the quotient h h0θ
h0{tH AdpθqH : H P h0u of the real vector space h0 hG attached to G0 . 0
130
3. GLOBAL STABILIZATION
3.2. Stabilization In this section we shall formally state the condition on which our theoG rems depend. We have just described a linear form Idisc,t pf q on HpG, χq. The condition is that it can be stabilized. There has been considerable progress on this problem in recent years, and a general solution in case G G0 is now in place. One can hope that its extension to twisted groups might soon be within reach. For our part here, we require a solution only for the two r pN q and G G, r initially introduced in §1.2. twisted groups G G We continue to suppose for the time being that G represents a general triplet pG0 , θ, ω q over the global field F . We need to discuss what it means to G stabilize the linear form Idisc,t pf q. Let us first review a few of the background notions from the beginning of [KS], which underlie both the local discussion of §2.1 and the global discussion we are about to undertake here. General endoscopic transfer has to be formulated in terms of several supplementary data. These include a connected quasisplit group G over F , equipped with an inner class of inner twistings ψ: G0 Ñ G of the connected group G0 . Given G , one fixes an F -automorphism θ of G that preserves some F -splitting, and is of the form θ
Intpgθ q ψ θ ψ1,
for some element gθ P Gsc . (As usual, Gsc denotes the simply connected cover of derived group of G .) The automorphism θ of G0 also induces p 0 of G0 that preserves some ΓF an automorphism θp of the dual group G splitting, and is determined up to conjugation. One fixes a 1-cocycle aω from p 0 q of G p 0 that is the Langlands the Weil group WF of F to the center Z pG 0 dual of the automorphic character ω on G pAq. This then gives rise to an L-automorphism L
θ
Lθω : g w ÝÑ θppgqaω pwq1 w,
of the L-group
L G0
gw
P L G0 ,
p 0 -bitorsor of G0 . We use it to define the dual G pG pω G
Gp0 Lθω ,
on which the L-group L G0 acts by conjugation. These matters are discussed (with slightly different notation) in the early stages (2.1) of [KS]. So is the general notion of endoscopic datum G1 for G. As in the local case mentioned briefly in §2.1, a general endoscopic datum G1 represents a 4-tuple pG1 , G 1 , s1 , ξ 1 q. We recall that the first component, denoted also by G1 , is a connected quasisplit group over F . The other p 1 , a semisimple element components are a split extension G 1 of WF by G 1 1 1 L 0 p and an L-embedding ξ of G into G . These four components s in G, are required to satisfy the conditions (2.1.1)–(2.1.4) of [KS], the most basic p 1 q equals the connected centralizer of being the assertion (2.1.4b) that ξ 1 pG p 0 . Similar definitions apply to the local completions Fv of F . In s1 in G particular, an endoscopic datum G1 for G over F localizes to an endoscopic
3.2. STABILIZATION
131
datum G1v for Gv over Fv . The general notion of isomorphism between endoscopic data is defined on p. 18 of [KS]. As in our earlier special cases, we write OutG pG1 q AutG pG1 q{IntG pG1 q for the group of outer automorphisms of G1 over F obtained from automorphisms of G1 as an endoscopic datum. We also write E pGq for the set of isomorphism classes of endoscopic data G1 that are locally relevant to G, by which we mean that for every v, G1 pFv q G1v pFv q contains elements that are norms from GpFv q ([KS, p. 29]). Endoscopic data that are not locally relevant play no role in transfer, and can be ignored. For a given endoscopic datum G1 , the group G 1 need not be L-isomorphic L to G1 . Even if it is, there often is not a canonical isomorphism. To remedy r 1 , ξr1 q to G1 , as in the local this defect, one attaches an auxiliary datum pG r 1 represents a central extension of G1 discussion of §2.1. We recall that G 1 r over F that is induced, in the sense that it is a product of by a torus C r 1 . For tori of the form RE {F pGm q, and ξr1 is an L-embedding of G 1 into L G r 1 to be a z-extension of G1 (See [KS, (2.2)].) example, one can always take G
r 1 q, which The L-embedding ξr1 gives rise to a cohomology class in H 1 pF, C r 1 pAq{C r 1 pF q, by the global in turn yields an automorphic character ηr1 on C Langlands correspondence for tori. (With a suitable choice of ξr1 , we can assume that ηr1 is unitary.) r 1 , ξr1 q) The global transfer factor for G and G1 (and the auxiliary datum pG 1 1 is a canonical function ∆pδ , γ q of two ad`elic variables δ and γ. These lie in the ad`elized varieties of strongly G-regular, stable conjugacy classes in r 1 and G respectively. It again serves as the kernel of a transfer mapping, G which takes functions f P HpGq to functions p
f 1 pδ 1 q
¸
∆pδ 1 , γ qfG pγ q
γ
of δ 1 . The global transfer factor can be written as a (noncanonical) product ∆ pδ 1 , γ q
(3.2.1)
¹
∆v pδv1 , γv q
v
of local transfer factors. The transfer mapping therefore takes a decomposable function ¹ f fv v
in HpGq to the decomposable function f 1 pδ 1 q
¹
fv1 pδv1 q
v
of δ 1 . The LSK transfer conjecture, now a theorem, tells us that for any r 1 , ηr1 q. The fundamental lemma tells us that v, fv1 belongs to the space S pG v v r 1 , ηr1 q of the characteristic function for almost all v, f 1 is the image in S pG v
v
v
132
3. GLOBAL STABILIZATION
r 1 pFv q. It follows that f 1 of a hyperspecial maximal compact subgroup of G belongs to the restricted tensor product r 1 , ηr1 q S pG
â v
r 1 , ηr1 q S pG v v
of local stable Hecke algebras. The general situation is actually slightly more complicated than we have indicated. It comes with a cohomology class z 1 P H 1 F, Z pGsc qθ with values in the group of θ -covariants Z pGsc qθ
Z pGscq{
zθ pz q1 : z
P Z pGscq
(
in the center of Gsc [KS, Lemma 3.1.A]. For much of [KS], the authors assume that this class (or rather its local analogue) is trivial. They explain how to take care of the more general situation in (5.4), at the expense of r 1 , ηr1 q by a twisted stable Hecke algebra relative to an inner replacing S pG r 1 over F . However, it seems likely that by choosing automorphism θr1 of G pGr1, ξr1q suitably, for example so that Gr1 is a z-extension of G1 with connected center, one could arrange that θr1 be trivial. It ought to be easy to check this point. I have not done so, since the most general case will not be our main r 1 , ξr1 q focus here. Instead, let us simply assume as a condition on G that pG can be chosen for each G1 so that θr1 is trivial, and therefore that any transfer r 1 , ηr1 q. mapping does indeed take values in the untwisted Hecke algebra S pG r 1 , ξr1 q gives rise to a There is also another point. The auxiliary datum pG surjective, affine linear mapping r h1
ÝÑ h h0θ
r 1 and G. This is dual to a between the real vector spaces attached to G mapping µπ,R iµπ,I ÝÑ µπ1 ,R iµπ1 ,I of archimedean infinitesimal characters of corresponding irreducible tem1 . (The representation π1 of Gr1 pF8 q is pered representations π8 and π8 8 1 on Cr1 pF8 q, while π8 is understood required to have central character ηr8 to be an extension of a representation to GpF8 q of the form (2.1.1). They can be any corresponding representations in Shelstad’s endoscopic transfer of archimedean L-packets.) We can always modify the L-homomorphism r 1 pAq. For simplicity, ξr1 by tensoring it with an automorphic character on G we assume that this has been done so that the affine mapping from r h1 to h 1 descends to a linear isomorphism from r h to h. We can then assume that
}µπ,I } }µπ1,I }1, for the dual of a suitable Hermitian norm }}1 on hC . (The alternative would be to take } } to be an affine norm, in the sense that it is a translate in hC of a Hermitian norm, and } }1 to be an affine norm on r h1C that is related to } } in a way that depends on pGr1, ξr1q.)
3.2. STABILIZATION
133
In the last section, we worked with a central character datum pXG , χq for G. To relate it to global transfer, we recall that for any G1 , there is a canonical injection of the group of covariants Z pG 0 qθ
Z pG0q{ zθpzq1 : z P Z pG0q
(
of Z pG0 q into Z pG1 q. (See [KS, p. 53].) We assume here that χ is trivial on the kernel of the restriction of this mapping to XG . Then χ pulls back to 1 a character on the image of XG in Z G pAq , and hence also to a character (which we denote again by χ) on its preimage r1 X
XG˜1
r 1 pAq . Recall also that the automorphic character ηr1 on C r 1 pAq atin Z G r 1 p Aq r 1 , ξr1 q is actually defined on the preimage of Z G0 pAq in Z G tached to pG r 1 . The prod[KS, p. 53 and p. 112]. It therefore restricts to a character on X uct r1 χηr1 χ r 1 . The pair pX r 1, χ r1 q will serve as our then represents a third character on X r 1 . It follows from the discussion on p. 112 of central character datum for G [KS] that the global transfer of functions f Ñ f 1 can be defined if f belongs r1 , χ r1 q. By to HpG, χq, and descends to a mapping from HpG, χq to S pG 1 1 r definition, S pG , χ r q is the space of strongly regular, stable orbital integrals of r1 , χ r1 q. We can therefore functions in the equivariant global Hecke algebra HpG 1 1 1 r ,χ p1 on r q with a linear form S identify any stable linear form S on HpG r1 , χ S pG r1 q by the global analogue of the local prescription (2.1.2). For any G1 , we write
(3.2.2) where
p 1 qΓ Z pG
Z pGp1qΓ{ Z pGp1qΓ X Z pGpqΓ Z pGp1qΓZ pGpq{Z pGpq, p q Z pG p 0 qθ Z pG
p
p 0 . The general endoscopic is the subgroup of θp -fixed points in the center of G 1 1 Γ p q is finite. We write Eell pGq as usual for the datum G is called elliptic if Z pG subset of elliptic isomorphism classes in E pGq. The elliptic classes represent the global endoscopic data that are used to stabilize the trace formula. In their stabilization of the regular elliptic terms in the twisted trace formula, Kottwitz and Shelstad introduced global coefficients
ιpG, G1 q,
G1
P EellpGq.
These objects are defined by an explicit formula [KS, p. 115], which generalizes the formula of Kottwitz [K3, Theorem 8.3.1] for the original coefficients introduced by Langlands [L10] in case G G0 . They are also essentially the same as the provisional coefficients defined in [A9, (3.3)]. These were introduced to describe a conjectural stabilization of the discrete part of the trace formula [A9, Hypothesis 3.1].
134
3. GLOBAL STABILIZATION
As an aside, we note that the discrete part of the trace formula is the spectral analogue of what was called the elliptic part of the trace formula, in the case G G0 studied in [A15]. The elliptic part of the trace formula is a sum of invariant orbital integrals, geometric terms that include semisimple orbital integrals. The semisimple elliptic part was stabilized for G G0 in [K5]. It is the regular (semisimple) elliptic part that was stabilized for any G in [KS], generalizing the first results of Langlands [L10] for G G0 . We note that all of these partial stabilizations were originally dependent on the LSK-transfer conjecture and the fundamental lemma. They are now unconditional. What is meant by a stabilization of some part of the trace formula? For the discrete part (as in general), it is a decomposition (3.2.3)
Idisc,t pf q
¸
G1 Eell G
P p q
1 pf 1 q, ιpG, G1 qSpdisc,t
f
P HpG, χq,
of the given linear form in terms of stable linear forms G1 1 r1 , χ r1 q ÝÑ C Sdisc,t : H pG Sdisc,t attached to elliptic endoscopic data. The correspondence 1 f ÝÑ f 1 f G
r1 , χ r1 q we have just deis the global transfer mapping from HpG, χq to S pG 1 scribed. The stable linear form Sdisc,t is universal, in that it depends only
r 1 (and not G). The coefficient ιpG, G1 q depends on G as well as G1 , on G but not on which part of the trace formula is being stabilized. The set of summation Eell pGq is generally infinite in the global case here. However, the sum can be taken over a finite subset of Eell pGq, which depends only on a choice of finite set S S8 of valuations outside of which f is unramified. We note that if G G0 is quasisplit, the stable linear form G on GpAq is defined inductively by (3.2.3). The assertion in Sdisc,t Sdisc,t this case is simply that the difference
Idisc,t pf q
¸
G 1 G
1 pf 1 q, ιpG, G1 qSpdisc,t
f
P HpG, χq,
is stable. For any other G, there are no further definitions that can be made. The assertion then becomes an identity, which ties the original linear form Idisc,t pf q to the seemingly unrelated forms f
ÝÑ
1 pf 1 q . Spdisc,t
In general, the summand of G1 in (3.2.3) is supposed to be independent of the r 1 , ξr1 q, while the linear form S 1 r1 choice of pG disc,t on G pAq is to be independent
of ξr1 . But the transfer mapping f Ñ f 1 does depend on ξr1 . This apparent contradiction can be resolved directly by induction, and an appeal to [A5, Lemma 4.3].
3.2. STABILIZATION
135
If G G0 is a connected group, the decomposition (3.2.3) was established in [A16], subject to a general condition on the fundamental lemma. As we have noted, the standard fundamental lemma has now been established. The condition in [A16] was actually a generalization of the fundamental lemma, which was conjectured [A12, Conjecture 5.1] for weighted orbital integrals. It was used in the stabilization of the general geometric terms, its role being roughly similar to that of the standard fundamental lemma in the analysis of the elliptic geometric terms. The weighted fundamental lemma has now been established by Chaudouard and Laumon [CL1], [CL2], using the global methods introduced by Ngo [N], and a reduction by Waldspurger [W7], [W8] to fields of positive characteristic. This removes the condition that had qualified the results for G G0 in [A16]. In particular, the decomposition (3.2.3) is now valid if G is any connected group. If G is a general triplet (3.1.3), the decomposition (3.2.3) remains conjectural. The problem is to extend the results of the papers [A14]–[A16] to the twisted case, or in other words, to stabilize the twisted trace formula. The linear form Idisc,t pf q whose stabilization (3.2.3) we seek is given by the explicit formula (3.1.1). However, there is no direct way to stabilize the terms in this formula without first assuming the theorems we are trying to prove in this volume, (or rather their general analogues for G). One has instead to stabilize all the complementary terms in the twisted trace formula. Once this is done, the decomposition will follow from the twisted r 1 (which was estrace formula itself, and the stable trace formula for each G tablished for connected quasisplit groups in [A16]). Some of the techniques will certainly carry over to the twisted case without much change. Others will call for refinement, and no doubt new ideas. Still, there is reason to be hopeful that the general stabilization can be established in the not too far distant future. In any case, it is not our intention in this volume to discuss the complementary terms in the (twisted) trace formula. We shall instead simply take the decomposition (3.2.3) as a hypothesis in the cases under consideration. We note that the setting for the papers [A14]–[A16] was actually slightly different from that of a connected reductive group G G0 . It pertains to what we called a K-group, a somewhat artificial object consisting of a finite disjoint union of connected groups over F with extra structure. The point is that one can arrange for G to be one of these components. The identity (3.2.3) for G G0 follows from the results of [A16] (with minor adjustments for the slightly different kind of central character datum pXG , χq) by extending a given function f on GpAq to be zero on the complementary adelic components of the K-group. Before we leave the general case, we recall the general formula
(3.2.4)
p 1 qΓ |1 |OutG pG1 q|1 ιpG, G1 q |π0 pκG q|1 k pG, G1 q|Z pG
136
3. GLOBAL STABILIZATION
for the global coefficients. Here, π0 pκG q is the group of connected components in the complex group κG and
Z pGpqΓ X Z pGp0qΓ
0
,
p 0 q 1 ker1 F, Z pG p 1 q , k pG, G1 q ker1 F, Z pG
where ker1 pF, q denotes the subgroup of locally trivial classes in the global p 1 qΓ given by (3.2.2) is finite, cohomology group H 1 pF, q. The group Z pG 1 since G is assumed to be elliptic. Notice that π0 pκG q equals the group of p acting on the complex torus fixed points of the automorphism θ, p 0 qΓ Z pG
0
{ Z pGpqΓ
0
.
On the other hand, the dual automorphism θ acts as a linear isomorphism Adpθq on the real vector space aG0 , for which aG is the subspace of pointwise fixed vectors. The order of the group of fixed points of θp equals the absolute value of the determinant of Adpθq 1 on the subspace aG G0 of aG0 . (See [SpS, II.17].) It follows that (3.2.5)
π0 κG 1
p q
det
Adpθq 1
aG0
1 .
G
This factor in (3.2.4) does not occur in the corresponding formula on p. 115 of [KS]. (It was taken from the formula [A9, (3.5)], which was used in a provisional stabilization of general spectral terms.) The discrepancy can be traced to the choice of Haar measure on GpAq{GpF qAG,8 on p. 74 of [KS]. With the conventions of [KS], the determinant of pw 1q in (3.1.1) would 0 G have to be taken on the space aG M rather than aM . We return to our special cases of study. For the two principal cases, r pN q G r pN q0 , θrpN q, 1 , or G G0 represents either G equals the triplet G an element in Ersim pN q. It is pretty clear how to specialize the general notions above. In the first case, we take the inner twist to be the identity r pN q0 onto G r pN q GLpN q, and we set θrpN q θrpN q. morphism from G p We also set θrpN q
LθrpN q θrpN q, regarded as an automorphism of eip r pN q0 GLpN, Cq or L G r pN q0 GLpN, Cq ΓF . In both cases, an ther G 1 endoscopic datum G P Eell pGq (or rather its isomorphism class) takes the
simplified form from Chapter 1. That is, G 1 is just the L-group L G1 and ξ 1 is the canonical L-embedding of L G1 into L G0 that accompanies G1 as an endoscopic datum. Since there is no need for the supplementary datum pGr1, ξr1q, we simply set Gr1 G1 and ξr1 1. There is also no need of central r 1 1 in all cases. character data. We can therefore take XG X We also have the supplementary case, in which G is the triplet r 1q attached to an even orthogonal group G r pG r 0 , θ, r 0 P Ersim pN q. SimG r There ilar remarks apply here. In particular, we take ψ 1 and θr θ.
3.3. CONTRIBUTION OF A PARAMETER ψ
137
r 1 and ξr1 is again no call for auxiliary datum, which leaves us free to have G r 1 , Gr1 , sr1 , ξr1 q for G. r stand for components in an endoscopic datum pG 1 The general formula (3.2.4) for ιpG, G q contains a quotient k pG, G1 q of orders of groups of locally trivial cohomology classes. In the three cases of p 0 q or Z pG p 1 q factors through an interest, the action of Γ ΓF on either Z pG abelian quotient of Γ. It follows that these cohomology groups are trivial and that k pG, G1 q 1. The general formula therefore reduces to
1 p1 Z G
p qΓ1OutGpG1q1. r pN q, we can write For example, in the twisted general linear group G p qΓ 1 Out r N pGq1 , r pN q, G 1 Z pG r ιpN, Gq ι G 2 for any G P Eell pN q, since κGr pN q Z{2Z in this case. The global transfer factor ∆pδ 1 , γ q is canonical, but the local factors in (3.2.6)
ιpG, G1 q π0 pκG q
(3.2.1) are generally not. However, in the quasisplit case where we are now working, the local transfer factors can be specified uniquely. In fact, as we saw in the explicit discussion of our three special cases in §2.1, there are two ways to normalize the local transfer factors. One depends on a choice of splitting over Fv , while the other depends on a choice of Whittaker datum over Fv . They differ by the ε-factor ε
1 2 , rv , ψFv
of a virtual representation rv of Γv [KS, §5.3]. It is a consequence of the global definitions ([LS1, §6.2–6.3], [KS, §7.3]) that the product formula (3.2.1) is valid for quasisplit G if the local transfer factors are all normalized in terms of a common splitting over F . If on the other hand, they are all normalized in terms of a common global Whittaker datum, as has been the understanding for our three special cases here, the product changes by the global ε-factor ε 12 , r of a virtual representation r of Γ. But it is a consequence of the general construction in §5.3 of [KS] that the virtual representation r is orthogonal. It then follows from [FQ] that the global ε-factor ε 12 , r equals 1. The product formula (3.2.1) is therefore valid for the Whittaker normalizations of the local transfer factors we have adopted. We shall state formally the hypothesis on which the results of this volume rest. Keep in mind that (3.2.3) has now been established in case G G0 , and in particular, if G is any element in Ersim pN q. The question therefore concerns our other two basic cases. r pN q or G, r in the notation Hypothesis 3.2.1. Suppose that G equals either G G introduced initially in §1.2. Then the stabilization (3.2.3) of Idisc,t pf q holds for any t ¥ 0 and f P HpGq.
3.3. Contribution of a parameter ψ The decomposition (3.2.3) is what will drive the proofs of our theorems. In the end, our results will come from the interplay of formulas obtained
138
3. GLOBAL STABILIZATION
r pN q and G P Ersim pN q. We have by specializing (3.2.3) to the cases G G r pN q back into the discussion. In first to bring the global parameters ψ P Ψ this section, we shall describe the contribution of ψ to the terms in the specializations of (3.2.3). We may as well continue to work more broadly while this is still feasible. We do not have global parameters ψ for general G, but we are free to replace them by Hecke eigenfamilies c for G. Some of this discussion will be rather formal. The reader might prefer to pass directly to Corollary 3.3.2, which contains its application to the parameters ψ. For simplicity, suppose first that G is a connected reductive group over the global field F . In §1.3, we introduced the set Caut pGq of (equivalence classes of) families c of semisimple classes in L G. We may as well work with the subset Caut pG, χq of classes in Caut pGq that are compatible with the central character datum pXG , χq. Any class c P Caut pGq determines an unramified character ζv on Z GpFv q for almost all v, namely the central character of the unramified representation of GpFv q attached to cv . Recall that χ is a character on the closed subgroup XG of Z GpAq . We define Caut pG, χq somewhat artificially as theset of c in Caut pGq such that χ extends to a character ζ on Z GpF q zZ GpAq whose restriction to Z GpFv q equals ζv for almost any v. We need to see how the set Caut pG, χq is related to the terms in the explicit expression (3.1.1) for the discrete part
Idisc,t pf q,
P HpG, χq, of the trace formula. Consider the operator IP,t pχ, f q in this expression. It f
is isomorphic to a direct sum of induced representations of the form π
IP pπM q,
}µπ,I } t,
in which πM is taken from the set of irreducible subrepresentations of L2disc M pF qzM pAq, χ . For any such π, the class cpπ q belongs to Caut pG, χq. If c is an arbitrary class in Caut pG, χq, we write à
IP,t,c pχ, f q
tπ: cpπqcu
IP,π pχ, f q,
where π IP pπM q as above, and IP,π pχq is the subrepresentation of IP,t pχq corresponding to π. We also write MP,t,c pw, χq for the restriction of the operator MP,t pw, χq in (3.1.1) to the invariant subspace HP,t,c pχq on which IP,t,c pχq acts. Then tr MP,t pw, χqIP,t pχ, f q
¸
P
It follows that (3.3.1)
Idisc,t pf q
p
c Caut G,χ
q
¸
P
p
c Caut G,χ
q
tr MP,t,c pw, χqIP,t,c pχ, f q . Idisc,t,c pf q,
3.3. CONTRIBUTION OF A PARAMETER ψ
139
where Idisc,t,c pf q is the c-variant of Idisc,t pf q, obtained by replacing the trace in (3.1.1) by the summand of c in its decomposition above. Suppose now that G represents an arbitrary triplet pG0 , θ, ω q. To extend the decomposition to this case, we need only agree on the meaning of the associated set Caut pG, χq. We define it to be the subset of classes c P C pG0 , χq that are compatible with θ and ω, in the sense that for almost all valuations v, the associated conjugacy classes cv satisfy θpv pcv q cv cpωv q.
With this interpretation of Caut pG, χq, the decomposition (3.3.1) remains valid as stated. We note that in general, the sum in (3.3.1) can be taken over a finite set that depends on f only through a choice of Hecke type. Recall that HpG, χq is the χ1 -equivariant Hecke module of GpAq, relative to a suitably chosen maximal compact subgroup K
¹
Kv
v
of G0 pAq. It is a direct limit
S HpG, χq lim ÝÑ HpG, K , χq, S
where Hp q is the space of functions in HpG, χq that are biinvariant under the product G, K S , χ
KS
¹
R
Kv ,
Kv
G0pFv q,
v S
of hyperspecial maximal compact subgroups. The set Caut pG, χq is a direct limit S Caut pG, χq lim ÝÑ CautpG, χq, S
where pG, χq is the relevant variant of the set defined in §1.3, composed of families of semisimple conjugacy classes S Caut
cS
tcv : v R S u.
In both limits, S S8 represents a large finite set of valuations outside of which G is unramified. The two are related through the unramified Hecke algebra S S Hun Hun pG0q Cc8 K S zG0pAS q{K S on G0 pAS q. On the one hand, an element cS complex-valued character (3.3.2)
h
ÝÑ phpcS q,
S . On the other, there is an action on Hun
P
S pG, χq determines a Caut
S h P Hun ,
S ÝÑ fh, f P HpG, K S , χq, h P Hun , S on HpG, K S , χq that is characterized by multipliers. Namely, if π is of Hun an extension to GpAq of an irreducible unitary representation π 0 of G0 pAq,
f
140
3. GLOBAL STABILIZATION
which is unramified outside of S and satisfies the global analogue of (2.1.1), then tr π pfh q p h cS pπ q tr π pf q . It is easy to describe the decomposition (3.3.1) in terms of eigenvalues S . Any function f P HpG, χq belongs to HpG, K S , χq, for some S. We of Hun fix S and f , and consider the linear form Idisc,t pfh q,
S h P Hun ,
S . It follows from the expression (3.1.1) for I on Hun disc,t pf q that this linear form is a finite sum of eigenforms. More precisely, we can write
Idisc,t pf q
¸
Idisc,t,cS pf q,
cS
where Idisc,t,cS is a linear form on HpG, K S , χq such that (3.3.3)
Idisc,t,cS pfh q p hpcS qIdisc,t,cS pf q.
S pG, χq, which again depends on f only The sum is over a finite subset of Caut S through a choice of Hecke type (necessarily of the form pτ8 , κ8 S K q in this case). If c belongs to Caut pG, χq, we can then write
(3.3.4)
Idisc,t,c pf q
¸
cS
Ñc
Idisc,t,cS pf q,
S pG, χq. This is the summand of the sum being over the preimage of c in Caut c on the right hand side of (3.3.1). Suppose that G1 P Eell pGq is an elliptic endoscopic datum, with G again r 1 , ξr1 q is an auxiliary datum for G1 , we being a general triplet (3.1.3). If pG r 1 q of C pG r 1 q. The properties of pG r 1 , ξr1 q are such that have the subset Caut pG 1 1 1 r ,χ an element c P C pG r q transfers to a family c of conjugacy classes for G. However, because we do not know the principle of functoriality for G and r 1 , we cannot say that c lies in Caut pG, χq. To sidestep this formal difficulty, G let us simply write S CA pG, χq lim ÝÑ CA pG, χq S
for the set of all equivalence classes of families cS of semisimple conjugacy classes in L G that are compatible with χ, and such that for each v, cv projects a Frobenius class in WFv at v. It follows from the various construcr1 , χ r1 q to CA pG, χq. tions that there is a canonical mapping c1 Ñ c from CA pG We can of course take CA pG, χq to be the domain of summation in (3.3.1) if we set Idisc,t,c pf q 0 for c in the complement of Caut pG, χq in CA pG, χq. We would like to convert the decomposition (3.2.3) for Idisc,t pf q into a parallel decomposition for Idisc,t,c pf q. We need to be clear about our assumptions, since (3.2.3) remains conjectural in the twisted case. However, the decomposition is valid in case G G0 , and in particular if G G is quasisplit. This means that the stable linear forms on the right hand side of (3.2.3) are defined unconditionally for any G. As an assumption on
3.3. CONTRIBUTION OF A PARAMETER ψ
141
G, (3.2.3) can therefore be regarded as a conjectural identity, both sides of which are well defined. Lemma 3.3.1. Assume that G satisfies (3.2.3). (a) Suppose that G equals G0 and is quasisplit. Then there is a decomposition (3.3.5)
¸
Sdisc,t pf q
P p
c CA G,χ
q
Sdisc,t,c pf q,
f
P HpG, χq,
G for stable linear forms Sdisc,t,c Sdisc,t,c that satisfy the analogues of (3.3.3) and (3.3.4), and vanish for all c outside a finite subset of CA pG, χq that depends on f only through a choice of Hecke type. (b) Suppose that G is arbitrary. Then for any c P CA pG, χq, there is a decomposition
(3.3.6)
Idisc,t,c pf q
¸
G1 PEell pGq
ιpG, G1 q
¸
c1 Ñ c
1 Spdisc,t,c 1 pf 1 q,
f
P HpG, χq,
r1 , χ r1 q that map to c. where c1 is summed over the set of classes in CA pG
1 Proof. We prove that the stable linear form Sdisc,t,c exists for any 1 1 G P Eell pGq. Assume inductively that if G G, this form satisfies the analogue for G1 of the assertion (a). Suppose first that G G0 is quasisplit, as in (a). If f P HpG, K S , χq and cS P CAS pG, χq, we set Sdisc,t,cS pf q Idisc,t,cS pf q
¸
G1 G
¸
ιpG, G1 q
1 1 Spdisc,t,c 1,S pf q,
c1,S ÑcS r1 , χ r1 q that map to cS . If h the inner sum being over elements c1,S in CAS pG S for G, the fundamental lemma belongs to the unramified Hecke algebra Hun for spherical functions [Hal] implies that
pfhq1 fh1 1 .
We can then write
Sdisc,t,cS pfh q
Idisc,t,c pfhq S
Idisc,t,c pfhq S
¸ G1 G
¸
G1 G
ιpG, G1 q ιpG, G1 q
phpcS qIdisc,t,c pf q phpcS q S
p q
pq
¸ G1
¸ c1,S
Ñ cS
¸
c1,S
1 1 Spdisc,t,c 1,S pfh q 1 1 Spdisc,t,c 1,S pfh1 q
Ñ cS ιpG, G1 q
p h cS Sdisc,t,cS f ,
¸ c1,S
1 1 Spdisc,t,c 1,S pf q
by the stable analogue for G1 of (3.3.3), and the identity p h1 c1,S
p q phpcS q.
142
3. GLOBAL STABILIZATION
This is the analogue for G of (3.3.3) To complete the proof of (a), we sum over cS in CAS pG, χq. We obtain ¸
cS
¸
Sdisc,t,cS pf q
Idisc,t,cS pf q
cS
Idisc,tpf q Idisc,tpf q Sdisc,tpf q,
¸ G1 G
¸
G1 G
¸ ¸ cS G 1 G
ιpG, G1 q
¸ ιpG, G1 q Sp1
¸ c1,S
disc,t,c
c1
Ñ cS 1 pf 1 q
1 1 Spdisc,t,c 1,S pf q
1 pf 1 q ιpG, G1 qSpdisc,t
by the analogues for G1 of (3.3.4) and (3.3.5) (which we assume inductively), and the fact that the double sum over c1,S and cS reduces to a simple sum r1 , χ over c1 P CA pG r1 q. The triple sum over cS , G1 and pc1 qS above can be taken over a finite set that depends on f only through a choice of Hecke type. This follows from the corresponding property for Idisc,t pf q, the finiteness of the sum over G1 , the uniformity properties of the transfer mapping f Ñ f 1 that follow from the two theorems in [A11, §6], and our induction hypothesis above for G1 . We set ¸
Sdisc,t,c pf q
cS
Ñc
Sdisc,t,cS pf q.
This definition gives the analogue for G of (3.3.4). It thus completes our proof of the assertion (a) for G, as well as the accompanying induction argument. Suppose that G is arbitrary, as in (b), and that f again belongs to HpG, K S , χq. To establish (3.3.6), we consider the expression (3.3.7)
¸
G1 Eell G
P p q
ιpG, G1 q
¸
c1,S
ÑcS
1 1 Spdisc,t,c 1,S pf q
attached to any cS P CAS pG, χq. If f is replaced by its transform fh under S , the expression is multiplied by the factor p a general element h P Hun hpcS q. Recall that the linear form Idisc,t,cS pf q transforms the same way under the S , and that its sum over cS in C S pG, χq equals I action of Hun disc,t pf q. But A S the sum over c of (3.3.7) equals ¸ G1
1 pf 1 q Idisc,t pf q ιpG, G1 qSpdisc,t
as well, by (3.2.3) and the analogues of (3.3.4) and (3.3.5) for G1 . Thus, both Idisc,t,cS pf q and (3.3.7) represent the cS -component of the linear form Idisc,t pf q, relative to its decomposition into eigenfunctions under the action S . Therefore (3.3.7) equals I of Hun disc,t,cS pf q. The required decomposition
3.3. CONTRIBUTION OF A PARAMETER ψ
143
(3.3.6) is then given by the sum of each side of this identity over those cS that map to the given class c P CA pG, χq.
Remark. The spherical fundamental lemma [Hal] was needed to justify the first steps in the proof above. As we noted in the introduction, it is closely related to the the two theorems of [A11] that we used in the proof of Proposition 2.1.1 (and will use again in Corollary 6.7.4). In fact, it is not hard to see that the specialization of these theorems to unramified spherical functions, combined with the spherical fundamental lemma for SLp2q, imply the spherical fundamental lemma for G. This is perhaps not surprising, given that the results of [Hal] and [A11] are proved by a similar global argument, based on the simple trace formula and the fundamental lemma for units. (The paper [Hal] predates [A11], and was itself motivated by an earlier result [Clo2] of Clozel.) It is perhaps worth noting that the results of [Hal] and [A11] differ in a way that is suggestive (though very much simpler) of a difference between two general ways of trying to classify representations of p-adic groups, by types and by characters. More generally, suppose that CrA pG, χq is the family of equivalence classes relative to some given equivalence relation on CA pG, χq. If c now belongs to CrA pG, χq, we write Idisc,t,c pf q for the sum of the corresponding linear forms attached to elements of CA pG, χq in this equivalence class. We obtain the obvious variant ¸ Idisc,t pf q Idisc,t,c pf q
P p
c CrA G,χ
q
of the decomposition (3.3.1). Similarly, we have the stable variant Sdisc,t pf q
¸
c P CrA pG,χq
Sdisc,t,c pf q
of (3.3.5), in case G G0 is quasisplit. If G1 belongs to Eell pGq, we can pull r1 , χ r1 q. We write the equivalence relation back to CA pG (3.3.8) and (3.3.9)
1 Idisc,t,c pf 1 q
1 Sdisc,t,c pf 1 q
¸
c1 Ñc
¸ c1
1 Idisc,t,c 1 pf 1 q
1 Sdisc,t,c 1 pf 1 q
Ñc r1 , χ r1 q, the sum being over the elements H pG
for any c P CrA pG, χq and f 1 P r1 , χ r1 q that map to c. This is compatible with the convention above in CA pG for G, in case c lies in the image of the map and is consequently identified r1 , χ r1 q. It is also compatible with the summation over c1 with a class in CrA pG in the decomposition (3.3.6). We can therefore rewrite (3.3.6) slightly more simply as (3.3.10)
Idisc,t,c pf q
¸
G1 Eell G
P p q
1 ιpG, G1 qSpdisc,t,c pf 1 q,
144
3. GLOBAL STABILIZATION
for any c in CrA pG, χq. This is the form of the general decomposition we will specialize. Our discussion so far has been quite formal and considerably more r pN q and general than we need. It simplifies in the two cases of G G G P Ersim pN q that are our primary concern. We recall that in both of these r 1 G1 and ηr1 1. cases, we take χ 1, G r pN q. It follows from Theorems 1.3.2 and Suppose first that G equals G r pN q Ψ G r pN q to the 1.3.3 that the mapping ψ Ñ cpψ q from the set Ψ r pN q of CA G r pN q is a bijection. These theorems tell us subset CrpN q C G in addition that the mapping cS Ñ c is injective on the preimage of CrpN q r pN q , and hence that the sums over cS above are in the set CrA pN q CA G r pN q also gives rise to an all superfluous in this case. Any element ψ P Ψ archimedean infinitesimal character, the norm of whose imaginary part we denote by tpψ q. With this notation, we write (3.3.11)
Idisc,ψ pfrq Idisc,tpψq,cpψq pfrq,
r pN q. fr P H
Suppose next that G represents an element in Ersim pN q. Following the general notation (3.3.8) and (3.3.9), we set (3.3.12)
Idisc,ψ pf q Idisc,tpψq,cpψq pf q
and (3.3.13)
Sdisc,ψ pf q Sdisc,tpψq,cpψq pf q,
r pN q and f P HpGq. We will obviously have a special interest for any ψ P Ψ r pGq of parameters in Ψ r pN q attached in the case that ψ lies in the subset Ψ to G. We note, however, that the existence of this subset will require a running induction hypothesis based on the assertion of Theorem 1.4.1. The notation (3.3.12) and (3.3.13) holds more generally if G belongs to the larger set Erell pN q. We shall state the specialization of (3.3.10) as a corollary of Lemma 3.3.1. It is of course Hypothesis 3.2.1 that tells us that the condition of the r pN q. lemma holds if G G r pN q. Corollary 3.3.2. Suppose that ψ belongs to Ψ r pN q, we have (a) If G G
(3.3.14)
Idisc,ψ pfrq
¸
P p q
G Erell N
(b) If G P Ersim pN q, we have (3.3.15)
Idisc,ψ pf q
¸
G1 PEell pGq
p
q
p q
G r ι N, G Spdisc,ψ frG ,
1 ιpG, G1 qSpdisc,ψ pf 1 q,
r pN q. fr P H
f
P H pG q.
3.4. A PRELIMINARY COMPARISON
145
Suppose that G and ψ are as in (b). The linear form Idisc,ψ pf q on the left hand side of (3.3.15) then has a spectral expansion ¸
P
M L
|W0M ||W0G|1
¸
P p qreg
det w
p 1 qa
w W M
G M
1 tr MP,ψ w IP,ψ f ,
p q
pq
where MP,ψ pwqIP,ψ pf q MP,tpψq,cpψq pw, 1M qIP,tpψq,cpψq p1M , f q.
It is understood here that the last expression vanishes if cpψ q lies in the the complement of Craut pGq, and that 1M stands for the trivial character on AM,8 . The same formula obviously applies to the left hand side of (3.3.14)
r pN q and f fr. if we set G G We have come at length to the decompositions (3.3.14) and (3.3.15) r pN q will serve as the foundation for our proofs. They that for any ψ P Ψ are the culmination of the general discussion of the last three sections. The discussion has been broader than necessary, in the hope that it might offer some general perspective, and in addition, to serve as a foundation for future investigations. We turn now to the business at hand. We shall begin a study of the implications of these decompositions for the proofs of our theorems.
3.4. A preliminary comparison Our interest will be focused on the decompositions (3.3.14) and (3.3.15) r pN q. One of our goals will be to establish attached to a parameter ψ P Ψ transparent expansions of the two sides of each identity that we can compare. This will be the topic of Chapter 4. In the remaining two sections of this chapter, we shall establish a complementary result that is more modest. r pN q. The Suppose first that G equals the twisted general linear group G classification of Theorems 1.3.2 and 1.3.3 then characterizes the automorphic spectrum in terms of self-dual families c. For we have observed that the r pN q onto the subset mapping ψ Ñ cpψq in this case is a bijection from Ψ r pN q . We recall that CrpN q represents the set of self-dual CrpN q of CA G representations in the automorphic spectrum of GLpN q. In particular, it r pN q. represents all the representations that occur in the formula (3.1.1) for G The summand of c in (3.3.1) therefore vanishes in this case unless c cpψ q r pN q. The decomposition (3.3.1) thus reduces to for some ψ P Ψ (3.4.1)
Idisc,t pfrq
¸
tψPΨr pN q:tpψqtu
Idisc,ψ pfrq,
r pN q . fr P H
We would like to establish a similar formula if G is any element in
p q
p q
Ersim N . Following the general convention in §3.3, we write CrA G for the set of fibres of the mapping C A p Gq
ÝÑ
r pN q . CrA pN q CA G
146
3. GLOBAL STABILIZATION
In this case, we will generally be working with the symmetric subalger pGq of HpGq. Recall from §1.5 that H r pGq equals HpGq unless bra H p G SOp2m, Cq, in which case it consists of functions that at each v are symmetric under the automorphism of G we have fixed. The fibres in CA pGq of points in CrA pGq have a similar description. However, so that there be no misunderstanding, we note that even if f belongs to the symmetric Hecke r pGq, the function Idisc,t,c pf q of c P CA pGq is not constant on the algebra H fibres. Indeed, a fibre can be infinite, while this function has finite support. We recall that if c is taken to be a class in CrA pGq instead of CA pGq, Idisc,t,c pf q is the sum of the function over the fibre of c in CA pGq. The following proposition will reduce the study of automorphic spectra r pN q. of our groups G P Ersim pN q to the subsets attached to parameters ψ P Ψ Its proof over the next two sections will be a model for the more elaborate global comparisons that will occupy us later. Proposition 3.4.1. Suppose that G P Ersim pN q, f c P CrA pGq. Then Idisc,t,c pf q 0 Sdisc,t,c pf q, unless pt, cq tpψq, cpψq , for some ψ
P HrpGq, t ¥
0 and
P Ψr pN q.
We shall begin the proof later in this section, and complete it in the next. Before doing so, we shall establish two important corollaries of the proposition, and comment on some related matters. The first corollary follows immediately from the decompositions (3.3.1) and (3.3.5) (stated in terms of CrA pGq rather than CA pGq), and the definitions (3.3.12) and (3.3.13). Corollary 3.4.2. For any t ¥ 0 and f (3.4.2)
Idisc,t pf q
P HrpGq, we have ¸ G Idisc,ψ pf q
tψPΨr pN q: tpψqtu
and G Sdisc,t pf q
(3.4.3)
¸
tψPΨr pN q: tpψqtu
G Sdisc,ψ pf q .
The other corollary bears directly on our main global theorem. For a group G P Ersim pN q, we define an invariant subspace
L2disc,t,c GpF qzGpAq ,
t ¥ 0, c P CrA pGq,
of L2disc GpF qzGpAq in the obvious way. It is the direct sum à
tπ: }µπ,I }t, cpπqcu
mpπ qπ,
3.4. A PRELIMINARY COMPARISON
147
of the irreducible representations π of GpAq attached to t and c that occur in L2disc GpF qzGpAq with positive multiplicity mpπ q. Then L2disc GpF qzGpAq
r pN q, we set If ψ belongs to Ψ
L2disc,ψ GpF qzGpAq
à t,c
L2disc,t,c GpF qzGpAq .
L2disc,tpψq,cpψq GpF qzGpAq . Corollary 3.4.3. If t ¥ 0 and c P CA pGq, then L2disc,t,c GpF qzGpAq 0,
unless for some ψ (3.4.4)
pt, cq tpψq, cpψq ,
P Ψr pN q. In particular, we have a decomposition à L2disc,ψ GpF qzGpAq . L2disc GpF qzGpAq P p q
r N ψ Ψ
Proof of Corollary 3.4.3. Assume that pt, cq is not of the form tpψ q, cpψ q . The proposition then tells us Idisc,t,c pf q vanishes. The formula for Idisc,t,c pf q given by the c-analogue of (3.1.1) is a sum over Levi subgroups M of G. If M is proper in G, it is a product of general linear groups with a group G P Ersim pN q, for some integer N N . We can assume inductively that the corollary holds if N is replaced by N . Since the same property holds for the general linear factors of M , the analogue of the corollary holds for M itself. The operator IP,t,c p1M , f q in the summand of M is induced from the pt, cq-component of the automorphic discrete spectrum of M , or more correctly, the component of the discrete spectrum given by some partition of pt, cq among the factors of M . This operator then vanishes, by our assumption on pt, ψ q, and so therefore does the summand of M . The remaining term in the formula for Idisc,t,c pf q is the summand tr IG,t,c p1G , f q
G. We have established that it vanishes. But IG,t,c pf q IG,t,c p1G , f q is the operator on the space L2disc,t,c GpF qzGpAq by right convolution of f . corresponding to M
Its trace equals the sum ¸
mpπ qtr π pf q ,
}µπ,I } t, cpπq c,
π
of the irreducible subrepresentations of this space. The function f belongs r pGq. However, since the only to the (locally) symmetric Hecke algebra H multiplicities mpπ q are all positive, it is clear that the sum cannot vanish r pGq unless it also vanishes for all f P HpGq. It follows that for all f P H
148
3. GLOBAL STABILIZATION
mpπ q 0 for each π. In other words, the sum is over an empty set, and the assertions of the corollary hold. It will be convenient to write (3.4.5)
G Rdisc,ψ pf q IG,tpψq,cpψqpf q,
f
P H pG q ,
r pN q. Then RG for any G P Ersim pN q and ψ P Ψ stands for the regular disc,ψ 2 representation of GpAq on Ldisc,ψ GpF qzGpAq . The second corollary of (the yet unproven) Proposition 3.4.1 reduces the study of the automorphic G discrete spectrum of G to that of the subrepresentations Rdisc,ψ . Given the statement of Theorem 1.5.2, we will obviously need to reduce the problem r pGq of Ψ r pN q. This question further to parameters ψ that lie in the subset Ψ turns out to be surprisingly difficult. We will be able to give a partial answer below, after some elementary remarks on central characters. However, its full resolution will not come before the end of our general induction argument in Chapter 8. Recall that any G P Ersim pN q comes with a character ηG of ΓF of order 1 or 2. If this character is nontrivial, it determines G uniquely. The same is true if ηG 1 and N is odd. However, if ηG 1 and N is even, there p _ equals is a second element G_ P Ersim pN q with ηG_ 1. Its dual group G p p SOpN, Cq if G SppN, Cq and SppN, Cq if G SOpN, Cq. In other words, G_ is the split group SOpN q if G equals SOpN 1q, while G_ equals SOpN 1q if G is the split group SOpN q. r pN q also comes with a character ηψ on ΓF of Any parameter ψ P Ψ order 1 or 2. It is the id`ele class character detpπψ q, where πψ is the self dual automorphic representation of GLpN q attached to ψ. If ηψ ηG , the Tchebotarev density theorem tells us that we can find a valuation v, at which ψ, G, ηψ and ηG are unramified, such that the local characters ηψ,v ηψv and ηG,v ηGv are distinct. It follows from the definitions that the conjugacy class cpψv q in GLpN, Cq does not meet the image of L Gv . This implies that
(3.4.6)
G Rdisc,ψ
0,
ηψ
ηG ,
G P Ersim pN q.
r pGq of On the other hand, it follows from the construction of the subset Ψ r pN q in §1.4, which we will presently formalize with a running induction Ψ r pGq. This gives a partial charhypothesis, that ηψ ηG if ψ belongs to Ψ r r pN q, for our simple endoscopic acterization of the complement of ΨpGq in Ψ datum G P Ersim pN q. Our partial answer to the question posed above is that G r pGq in Ψ r pN q, except posRdisc,ψ vanishes for any ψ in the complement of Ψ sibly in the case that ηψ 1, N is even and G is split (which is to say that ηG ηψ 1). It is this last case that contains major unresolved difficulties. r pN q. Suppose again that ηψ ηG , for some G P Ersim pN q, and that ψ P Ψ If v is a valuation such that ηψ,v ηG,v as above, the conjugacy class cpψv q
3.4. A PRELIMINARY COMPARISON
in GLpN, Cq does not meet the image of subgroup M of G. This implies that IP,tpψq,cpψq pf q 0,
LM
v
149
in GLpN, Cq, for any Levi
P P pM q, f P HrpGq. It follows from the analogue of (3.1.1) for ψ that Idisc,ψ pf q 0. P
Applying this assertion to the left hand side of (3.3.15), we see inductively that a G similar assertion holds for Sdisc,ψ pf q. In other words, G G (3.4.7) Idisc,ψ pf q 0 Sdisc,ψ pf q ,
f
P HrpGq,
ηψ
ηG ,
G P Ersim pN q.
There is one other remark to be made before we start the proof of the proposition. It concerns a minor point of possible confusion. Our convention of letting the same symbol denote both a group and the endoscopic datum it represents has usually been harmless. For we agreed in the general case 1 (3.2.3) that the linear form Sdisc,t depends on G1 only as a group (or more
r1 , χ r1 q), and not on the other components G 1 , s1 and correctly, on the pair pG 1 1 (as ξ of the endoscopic datum. However, the c-component (3.3.9) of Sdisc,t 1 1 well as its analogue (3.3.8) for Idisc,t ) depends by definition on G as an
endoscopic datum. In the case G P Ersim pN q we are considering here, this G means that the ψ-component (3.3.13) of the stable linear form Sdisc,t Sdisc,t G ) depends nominally on (as well as its analogue (3.3.12) for Idisc,t Idisc,t G as an endoscopic datum. The question arises when N is odd. In this p SOpN, Cq, while the case, G SppN 1q is the unique group with G endoscopic datum is further parametrized by the quadratic character ηG . However, this distinction is still innocuous. For the ψ-component (3.3.13) vanishes if ηG ηψ , and in the case that ηG ηψ , it remains unchanged if ψ is replaced with a translate ψ b η by a quadratic character η, and G is adjusted accordingly. Proof of Proposition 3.4.1. (First step): We assume inductively that the proposition holds if N is replaced by any integer N N . We will let G P Ersim pN q vary, so we need to fix a general pair
pt, cq,
t ¥ 0, c P CrA pN q,
that is independent of G. We assume that pt, cq is not of the form tpψ q, cpψ q r pN q. We must establish that the linear forms Idisc,t,c pf q and for any ψ P Ψ r p G q. Sdisc,t,c pf q both vanish for any G P Ersim pN q and f P H We fix G for the moment, and begin as in the proof of Corollary 3.4.3. Our induction hypothesis implies that the corollary also holds if N is replaced by N N . It follows that the pt, cq-component of the discrete spectrum of any proper Levi subgroup M of G vanishes. The operator IP,t,c pf q in the summand of M in the c-analogue of (3.1.1) therefore vanishes, and so then does the summand itself. Since the only remaining summand corresponds to M G, we see that (3.4.8)
Idisc,t,c pf q tr IG,t,c pf q ,
150
3. GLOBAL STABILIZATION
for any f P HpGq. We can apply similar arguments to the decomposition (3.3.10) given by the c-analogue of (3.2.3). Consider an index of summation G1 P Eell pGq in this decomposition with G1 G. Then G1 is a proper product G1
G11 G12,
G1i
P EsimpNi1q, Ni1 N. 1 The corresponding linear form Spdisc,t,c pf 1q in (3.3.10) is defined in turn by
a sum (3.3.9). It follows from our induction hypothesis and our condition on pt, cq that each of the summands in (3.3.9) vanish. The linear form in (3.3.10) thus vanishes, and so therefore does the summand of G1 . The remaining term in the decomposition (3.3.10) of Idisc,t,c pf q is the stable linear form Sdisc,t,c pf q. We have shown that Idisc,t,c pf q Sdisc,t,c pf q.
(3.4.9)
r pN q in place of G. Next, we apply (3.3.10) again, but this time with G r pN q. The datum G plays We have then to replace f by a function fr in H 1 the role of G in (3.3.10), and is summed over the general indexing set r r Eell pN q Eell GpN q . If G lies in the complement of Ersim pN q in Erell pN q, it equals a proper product
G GS
GO ,
Gε
P ErsimpNεq,
Nε
N.
Arguing as above, we see from our induction hypothesis, the condition on pt, cq, and the definition (3.3.9) that G Spdisc,t,c pfrGq 0
in this case. The sum on the right hand side of (3.3.10) can therefore be taken over G in the subset Ersim pN q of Erell pN q. The left hand side of the identity is the linear form Idisc,t,c pfrq. It vanishes by the remarks preceding (3.4.1) and our condition on pt, cq. The identity becomes (3.4.10)
¸
P
p q
G Ersim N
p
q
p q 0.
G r ι N, G Spdisc,t,c frG
This completes the first step. To state what we have established so far, we shall introduce a global form of the local object (2.1.3) from §2.1. We shall formulate it as a family of functions (3.4.11)
F
f
P HrpGq :
G P Erell pN q
(
parametrized by global endoscopic data that are elliptic, with the condition that f 0 for all but finitely many G. Let us say that F is a decomposable compatible family if for each v, there is a local compatible family of functions Fv
fv
P HrpGv q :
Gv
P Erv pN q
(
3.4. A PRELIMINARY COMPARISON
151
on the local endoscopic groups Gv pFv q such that for any G completion Gv P Erv pN q, the corresponding function satisfies f
¹
P ErellpN q with
fv .
v
We shall then say simply that F is a compatible family if there is a finite set of decomposable compatible families Fi
fi
P HrpGq :
(
G P Erell pN q ,
such that the functions attached to any G satisfy f
¸
fi .
This is a natural analogue of the local definition. We observe from Proposition 2.1.1 that F is a compatible family if and only if there is a function fr r pN q such that in H f G frG , G P Erell pN q.
In other words, the function f P F attached to any G P Erell pN q has the same image in SrpGq as fr. The notion of Hecke type can obviously be formulated here. By a Hecke type for Erell pN q, we shall mean a pair (3.4.12)
S,
pτ8, κ8q
(
,
(
where S S8 is a finite set of valuations, and pτ8 , κ8 q is a finite set of Hecke types, one for each G P Erell pN q that is unramified outside of S, such that S 8 κ8 κ8 κ8 SK , S GpAS q, for a hyperspecial maximal compact subgroup K S of GpAS q. Then (3.4.12) is a Hecke type for F if the function f attached to G vanishes whenever G ramifies outside of S, and has Hecke type pτ8 , κ8 q if G is unramified outside of S. Any compatible family F obviously has a Hecke type. r pN q Suppose that F is any compatible family (3.4.11), and that fr P H G G r is chosen so that f f for any G. We can then write ¸
GPErsim pN q
p
q
pq
r ι N, G tr IG,t,c f
¸
p
q
pq
p
q
p q
p
q
p q
r ι N, G Sdisc,t,c f
G
¸
r ι N, G Spdisc,t,c f G
G
¸
r ι N, G Spdisc,t,c frG ,
G
by (3.4.8) and (3.4.9). According to (3.4.10), this last sum vanishes. We conclude that (3.4.13)
¸
P
p q
G Ersim N
p
q
p q 0,
r ι N, G tr IG,t,c f
f
P F,
152
3. GLOBAL STABILIZATION
if F is any compatible family (3.4.11). Since r ιpN, Gq ¡ 0, we can write the summand of G as a linear combination ¸
cG pπ qtr π pf q ,
π
P ΠunitpGq,
π
of irreducible unitary characters on GpAq with nonnegative coefficients cG pπ q. We can then write (3.4.13) in the form (3.4.14)
¸
¸
GPErell pN q π PΠunit pGq
cG pπ qtr π pf q
0,
f
P F,
for nonnegative coefficients cG pπ q, which actually vanish if G lies in the complement of Ersim pN q. The identity (3.4.14) is valid for any compatible family F, and the double sum can be taken over a finite set that depends on F only through the choice of a Hecke type. In the next section we use the identity (3.4.14) to show that all of the coefficients cG pπ q vanish. In fact, we will establish general vanishing properties, which will be a foundation for more subtle comparisons later on. The coefficients are defined by the decomposition of the pt, cq-component IG,t,c of the representation of GpAq on the discrete spectrum. If they vanish, we see that ¸ cG pπ qtr π pf q 0, tr IG,t,c pf q r ιpN, Gq1 for any G P (3.4.9) that
π
p q
Ersim N and any f
P H p G q.
It then follows from (3.4.8) and
Idisc,t,c pf q Sdisc,t,c pf q 0, as required. In other words, we will have a proof of Proposition 3.4.1 once we have shown that the coefficients cG pπ q vanish. 3.5. On the vanishing of coefficients We have to complete the proof of Proposition 3.4.1. In the last section, we reduced the problem to proving that the coefficients cG pπ q in the identity (3.4.14) all vanish. We shall now deduce that in any such identity, which we recall is associated to the global field F and the positive integer N , the coefficients automatically vanish. Proposition 3.5.1. Suppose that there are nonnegative coefficients cG pπ q,
G P Erell pN q, π
P Πp G q,
such that for every global compatible family F (3.4.11), the function
pG, πq ÝÑ
cG pπ qfG pπ q,
f
P F,
is supported on a finite set that depends only on a choice of Hecke type for F, and such that the double sum (3.5.1)
¸¸ G
π
cG pπ qfG pπ q,
f
P F,
3.5. ON THE VANISHING OF COEFFICIENTS
153
vanishes. Then cG pπ q 0, for every G and π. The proof of Proposition 3.5.1 is primarily local. In order to focus on the essential ideas, we shall revert briefly to the local notation of Chapter 2. Thus, in contrast to the convention that has prevailed to this point in Chapter 3, and until further notice later in the section, we take F to be a r p G q, H r pGq, and so on, are then to be understood local field. The sets ErpN q, Π as local objects over F . We shall take advantage of this interlude to recall some notions from local harmonic analysis, both for the proof and for later use. We shall review the theory of the representation theoretic R-group [A10, §1–3], which is founded on work of Harish-Chandra, Knapp, Stein and Zuckerman on intertwining operators (see [Ha4], [Kn], [KnS], [KnZ1] and [Si2]). For these remarks, we may as well allow G to be an arbitrary connected reductive group over F. As in the global notation of §3.1, we let L LG
LGpM0q,
be the finite set of Levi subgroups of G that contain a fixed minimal Levi subgroup M0 , and we write W0G for the Weyl group W pM0 q of M0 . The set of GpF q-conjugacy classes of Levi subgroups of G is bijective with the set of W0G -orbits in L. We then write T pGq for the set of W0G -orbits of triplets (3.5.2)
τ
τr pM, σ, rq,
M
where Rpσ q is the R-group of σ in G. short exact sequence
ÝÑ Rpσq ÝÑ 1, where W pσ q is the stabilizer of σ in the Weyl group W pM q of M , and W 0 pσ q 1
ÝÑ
W 0 pσ q
ÝÑ
P L, σ P Π2pM q, r P Rpσq, We recall that Rpσ q is given by a
W pσ q
is the Weyl group of a root system defined by the vanishing of Plancherel densities. Any choice of positive chamber for the root system determines a splitting of the sequence, and thereby allows us to identity Rpσ q with a subgroup of W pσ q. It is known how to construct general normalized intertwining operators RP pw, σ q : IP pσ q
ÝÑ
IP pσ q,
w
P W pσ q ,
from the basic intertwining integrals, which satisfy analogues of (2.3.9) and (2.3.10). (The focus of the discussion of §2.2.3 and §2.3.4 was more specific, namely to establish particular normalizations that would be compatible with endoscopy.) The group W 0 pσ q becomes the subgroup of elements w P W pσ q such that RP pw, σ q is a scalar. What is not known in general is whether these normalizations can be chosen to be multiplicative in w. The question really pertains to the R-group Rpσ q [A10, p. 91, Remark 1], and is equivalent to whether the irreducible representation σ has an extension to a semidirect product M pF q Rpσ q.
154
3. GLOBAL STABILIZATION
One sidesteps the problem by introducing a finite central extension 1
ÝÑ
Zσ
ÝÑ Rrpσq ÝÑ Rpσq ÝÑ
1
of Rpσ q. It then becomes possible to attach normalized intertwining operarpσ q such that tors RP pr, σ q to elements r in R RP pzr, σ q χσ pz q1 RP pr, σ q,
z
P Zσ , r P Rrpσq,
for a fixed character χσ of Zσ , and so that the mapping r
ÝÑ
RP pr, σ q,
r
P Rrpσq,
rpσ q to the space of intertwining operators of is a homomorphism from R IP pσ q. We write TrpGq for the set of W0G -orbits of triplets of the form rpσ q of Rpσ q. We then set (3.5.2), but with r in the extension R
(3.5.3)
fG pτ q fG pτr q tr RP pr, σ qIP pσ, f q ,
f
P H p G q,
for any element τ τr in TrpGq. rpσ q do not have to be abelian. However, we can The groups Rpσ q and R r r pσ q still work with the set Π Rpσ q, χσ of irreducible representations ξ of R whose Zσ -central character equals χσ . For any such ξ, the linear form (3.5.4)
fG pρξ q |Rpσ q|1
¸
P pq
tr ξ prq fG pτr q,
f
r R σ
P H pG q ,
is the character of a subrepresentation of IP pσ q, which turns out to be irreducible if σ is tempered. We write Ttemp pGq and Trtemp pGq for the subsets of triplets τ in the respective sets T pGq and TrpGq for which the second component σ of (3.5.2) belongs to Π2,temp pM q, the subset of tempered representations in Π2 pM q. For any such σ, we write Πσ pGq for the set of irreducible constituents of the induced tempered representations IP pσ q of GpF q. It is well known that the set Πtemp pGq of all irreducible tempered representations of GpF q is a disjoint union, over W0G -orbits of pairs pM, σ q, of the sets Πσ pGq. (See for example [A10, Proposition 1.1].) Moreover, the basis theorem of HarishChandra [Ha4] tells us that if σ belongs to Π2,temp pM q, and tru is a set of rpσ q of the conjugacy classes in Rpσ q, the corresponding representatives in R set of linear forms r ÝÑ fG pτr q, f P H p G q, is a basis of the space spanned by the characters of representations in Πσ pGq. It follows that the mapping
ÝÑ πξ ρξ , ξ P Π Rrpσq, χσ , rpσ q, χσ to Πσ pGq. The full set Πtemp pGq can thereis a bijection from Π R ξ
fore be identified with the family of W0G -orbits of triplets π
πξ pM, σ, ξq,
M
P L, σ P Π2,temppM q, ξ P Π Rrpσq, χσ
.
3.5. ON THE VANISHING OF COEFFICIENTS
155
(See [A10, §3], where the set Ttemp pGq was denoted by T pGq. We need to reserve the symbol T pGq here for nontempered triplets.) One thus obtains a rough classification of Πtemp pGq from local harmonic analysis. It is independent of the finer endoscopic classification, which for simple endoscopic data G P Ersim pN q is contained in the assertions of the local theorems. We will eventually relate the two classifications in Chapter 6, as part of the proof of the local theorems. This will obviously entail letting r φ , with φM P Φ r 2 pM, φq being a preimage σ represent an element πM P Π M r pGq. The resulting relations will then confirm of a generic parameter φ P Φ 0 that W pσ q Wφ , W pσ q Wφ0 , Rpσ q Rφ , and therefore that the two kinds of R-groups are the same. In the general nontempered case, we have to distinguish between irreducible representations π P ΠpGq and standard representations ρ P P pGq. The general relations between the two kinds of representations are parallel r pN q and ρr P Pr pN q discussed in §2.2. Let to those between extensions π rPΠ us review them. There is a bijection π
ÝÑ
ρπ ,
π
P ΠpG q ,
between ΠpGq and P pGq such that π is the Langlands quotient of ρ. The character of any π has a decomposition fG pπ q
¸
P p q
npπ, ρq fG pρq,
f
ρ P G
P H pG q ,
into standard characters. The coefficients npπ, ρq are uniquely determined integers, with npπ, ρπ q 1,
which have finite support in ρ for any π. Both π and ρ determine real linear forms Λπ and Λρ . These objects lie in the dual closed chamber pa0 q in a0 aM0 attached to a preassigned minimal parabolic subgroup P0 P P pM0 q of G, and are a measure of the failure of the representations to be tempered. In particular, Λπ Λρπ is the point in the relatively open cone
paP q pa0 q
,
P
P0
that represents the nonunitary part of ρπ , as a representation induced from P pF q. If npπ, ρq 0, then Λρ ¤ Λπ , in the usual sense that Λπ Λρ is a nonnegative integral combination of simple roots of pP0 , A0 q, with equality Λρ Λπ holding if and only if ρ equals ρπ . (See [A7, Proposition 5.1], which is based on results in [BoW] and underlying ideas of Vogan.) Standard characters can in turn be decomposed into the virtual characters parametrized by TrpGq. For it follows from the definition of a standard representation, and the general classification of Πtemp pGq described above, that the set P pGq of standard representations of GpF q can be identified with
156
3. GLOBAL STABILIZATION
the family of W0G -orbits of triplets (3.5.5)
ρ ρξ
pM, σ, ξq,
M
P L, σ P Π2pM q, ξ P Π Rrpσq, χσ
We define
.
xρ, τ y |Rpσq|1tr ξprq , for ρ ρξ as in (3.5.5) and τ τr as in the analogue for TrpGq of (3.5.2). We also set xρ, τ y 0 for elements ρ P P pGq and τ P TrpGq that are com-
plementary, in the sense that their corresponding triplets do not share the same W0G -orbit of pairs pM, σ q. It then follows that (3.5.6)
fG pπ q
¸
P p q
npπ, τ qfG pτ q,
τ T G
where
npπ, τ q
¸
P p q
npπ, ρqxρ, τ y,
π
P Πp G q, f P H pG q, τ
P TrpGq.
ρ P G
If τ belongs to TrpGq, we set Λτ Λρ , for any element ρ P P pGq with xρ, τ y 0. Then Λτ is a linear form in pa0 q , with the property that Λτ ¤ Λπ if npπ, τ q 0. Elliptic elements in T pGq have a special role. We recall that Tell pGq is the subset of (W0G -orbits of) triplets τr in (3.5.2) such that r is regular, in the sense that aG tH P aM : rH H u. In general, we will write Greg
Gstr-reg
for the open connected subset of strongly regular points in G, and Greg,ell pF q Gstr-reg,ell pF q
for the subset of elements in Greg pF q that are elliptic over F . Thus, Greg,ell pF q is the subset of elements in GpF q whose centralizer is a torus T such that T pF q{AG pF q is compact. (As in Chapter 2, we will generally use the simpler notation on the left, since we will not need to consider regular points that are not strongly regular.) Elements in Tell pGq have the important property that they are uniquely determined by the restriction of their virtual characters to Greg,ell pF q. To be more precise, let tτ u be a set of representatives of the set Tell pGq in its preimage Trell pGq in TrpGq. Then the corresponding set of distributions τ
ÝÑ
fG pτ q,
f
P Cc8
Greg,ell pF q ,
on Greg,ell pF q is a basis for the space spanned by the restriction to Greg,ell pF q of characters of representations in either of the sets ΠpGq or P pGq. (See [Ka] and [A11]. This is also a consequence of the orthogonality relations in [A11, §6].) It follows from the definitions that T pGq is a disjoint union, over W0G orbits of Levi subgroups M , of images in T pGq of the corresponding elliptic
3.5. ON THE VANISHING OF COEFFICIENTS
157
sets Tell pM q. For any τ P T pGq, we write Mτ for a Levi subgroup such that τ lies in the image of Tell pMτ q. Suppose that π P ΠpGq. If the associated standard representation ρπ P P pGq corresponds to the triplet pMπ , σπ , ξπ q under (3.5.5), we define an element τπ in either T pGq or TrpGq by the triplet pMπ , σπ , 1q. The two elements ρπ and τπ are such that both npπ, ρπ q and xρπ , τπ y are nonzero, and they satisfy Λτπ Λρπ Λπ .
Conversely, suppose that ρ P P pGq and τ P T pGq are elements such that both npπ, ρq and xρ, τ y are nonzero (τ being identified here with a representative in TrpGq). Then Λτ Λρ ¤ Λπ . If these linear forms are all equal, Mτ contains Mπ (with Mτ and Mπ taken to be suitable representatives of the associated W0G -orbits in L). Moreover, with the assumption that Λτ Λπ , the group Mτ equals Mπ if and only if τ τπ and ρ ρπ . We observe that in this case, we have npπ, τπ q npπ, ρπ qxρπ , τπ y 1 |Rpσπ q|1 dimpξπ q ¡ 0.
We now return to the case that G represents a twisted endoscopic datum for GLpN q. The statement of Proposition 3.5.1 makes sense over the local field F . The set F has of course to be understood as a local compatible family (2.1.3), with the implication that the global set Erell pN q is replaced by the general local set ErpN q. A Hecke type for F will be family of objects, parametrized by the finite set of G P ErpN q, consisting of open compact subgroups K0 GpF q if F is p-adic, and finite sets τR of irreducible representations of maximal compact subgroups KR GpRq if F R. Lemma 3.5.2. Proposition 3.5.1 is valid, stated with ErpN q in place of Erell pN q, over the local field F . Proof. We are given that the sum ¸¸
(3.5.7)
G
cG pπ qfG pπ q,
π
over G P ErpN q and π P ΠpGq, vanishes for any compatible family F over the local field F . The function f of course stands for the element of F indexed by G. We shall let F vary over the compatible families attached to a fixed local Hecke type. By the general discussion above, we can write the sum (3.5.7) as ¸
¸
¸
Pp q P p q P p q
G Er N π Π G τ T G
¸¸ G
¸
¸
P
P p q
π M LG τ Tell M
cG pπ qnpπ, τ qfG pτ q
|W0Gpτ q||W0G|1cGpπqnpπ, τ qfM pτ q,
158
3. GLOBAL STABILIZATION
where W0G pτ q is the stabilizer of pM, τ q in W0G . The coefficient npπ, τ q in the last expression is understood as a W0G -invariant function of τ P Trell pM q, which is to say, a function whose product with fM pτ q depends only on the image of τ in T pGq. The multiple sum can be taken over a finite set of indices pG, π, M, τ q, which depends only on the given local Hecke type. A Levi subgroup M of G can be treated in its own right as an element in ErpN q. Conversely, for any M P ErpN q, there will generally be several G P ErpN q in which M embeds as a Levi subgroup. The best way to account r pN q for these groups is as elements in the twisted analogue ErM pN q EM G of the set EM 1 pGq defined (for any connected reductive group G and any endoscopic datum M 1 for G) on p. 227 of [A12]. (See [Wh, Appendix, p. 434–436].) We leave the reader to formulate the definition of ErM pN q as a r pN q, which contain M as a Levi set of twisted endoscopic data pG, s, ξ q for G subgroup, and are taken up to translation of s by elements in the product
p p qΓ 0 Z G r pN q . Z pG
0
p qΓ is a departure from (The inclusion of the connected component Z pG the convention of [A12], and is designed to make the set ErM pN q finite.) We can then replace the last double sum over G P ErpN q and M P LG by a double sum over M P ErpN q and G P ErM pN q, provided that we adjust the coefficients to compensate for the fact that several elements in ErM pN q could represent the same isomorphism class in ErpN q. It follows that (3.5.7) equals ¸
¸
¸
¸
Pp q P p q P p q P p q
M Er N τ Tell M G ErM N π Π G
cG pπ qnpπ, τ qγG pτ qfM pτ q,
for positive constants γG pτ q. For any M P ErpN q, the function fM pτ q in the last sum depends on G, since it comes from the function f P F attached to G. The family fM pτ q,
M
P ErpN q,
G P ErM pN q, τ
P TrellpM q,
of such functions is indexed by F. As F varies, the corresponding set of families is a natural vector space. It is not hard to see from the definition of a compatible family (2.1.3), together with the trace Paley-Wiener theorem [CD] [DM], that this vector space is a direct sum over M P ErpN q of the subspaces of (families of) functions that are supported on Trell pM q. It follows that the sum ¸
¸
¸
P p q P p q P p q
τ Tell M G ErM N π Π G
cG pπ q npπ, τ q γG pτ q fM pτ q
vanishes for any M , and any compatible family F. We fix M P ErpN q, and consider the function fM pτ q attached to G M . r N pM q-invariant funcBy the definition (2.1.3), fM lies in the subspace of Out tions in I pM q. We will assume that fM also lies in the subspace Icusp pM q
3.5. ON THE VANISHING OF COEFFICIENTS
159
of cuspidal functions in I pM q, but this will be the only other constraint. It then follows from the trace Paley-Wiener theorem for M , and the linear independence of the set of distributions on Mstr-reg,ell pF q attached to Tell pM q, r N pM q-invariant, χ1 -equivariant functhat we can take fM pτ q to be any Out σ tion in the natural Paley-Wiener space on Trell pM q. Since we are working with a preassigned Hecke type, we are actually appealing here to an implicit consequence of the trace Paley-Weiner theorem. Namely, if Γ is a finite set of irreducible representations of the maximal compact subgroup KM of M pF q, and I pM qΓ and HpM qΓ are the corresponding subspaces of I pM q and HpM q, there is a suitable section I pM qΓ
ÝÑ HpM qΓ.
(See [A5, Lemma A.1], for example.) The only other constraint imposed by the definition of F is that as G varies, the images f M in SrpM q of the r pGq are all equal. It follows from this that we can take functions f P H the functions fM pτ q parametrized by G to be equal. In particular, we can r N pM q-orbit choose the common function fM pτ q so that it isolates any Out in Tell pM q. It follows that the sum (3.5.8)
cM pτrq
¸
¸
¸
P p q P p q P
G ErM N π Π G τ τr
cG pπ q npπ, τ q γG pτ q
r N pM q-orbit τr in Trell pM q. vanishes for any M P ErpN q and any Out It is instructive to introduce an equivalence relation on the set of pairs
pG, πq,
For any pG, π q, and any pair
pM, τ q,
G P ErpN q, π τ
P ΠpG q .
P TrellpM q,
we set npπ, τ q equal to the coefficient we have defined if G represents a datum in Erell pN q, and 0 otherwise. With this understanding, we write pG, πq pG1, π1q if there is a pair pM, τ q such that both npπ, τ q and npπ1, τ q are nonzero. We can then define τ -equivalence to be the equivalence relation generated by this basic relation. If P is any τ -equivalence class, we write P for the adjoint equivalence class in the set of pairs pM, τ q. It consists of those pM, τ q such that npπ, τ q is nonzero for some pG, π q in P . One sees without difficulty that the classes P and P are both finite. These two equivalence relations are natural variants of the familiar relations of block equivalence, which are defined on the two sets ΠpGq and P pGq attached to any connected reductive group G over F . We shall use them in the analysis of (3.5.8). Assume that the assertion of the lemma is false. We can then fix a τ -equivalence class P such that the subset P1
pG, πq P P :
(
cG pπ q 0
160
3. GLOBAL STABILIZATION
is not empty. We define λ1 and
µ1
max }Λπ } : pG, πq P P 1
( (
dimpMπ q : pG, π q P P 1 , }Λπ } λ1 .
min
(As in the general setting of §3.2, the Hermitian norm } } on the space a0 attached to G is defined in terms of a suitable, preassigned norm on the r pN q.) Let pG1 , π 1 q be a fixed pair in P 1 such that space r a0 attached to G
}Λπ1 }, dimpMπ1 q pλ1, µ1q.
The pair
pM 1, τ 1q pMπ1 , τπ1 q
belongs to P , since we can treat τ1
pM 1, π1, 1q
as an element in Trell pM 1 q. It also has the property that
}Λτ 1 }, dimpMτ 1 q pλ1, µ1q. Moreover, G1 is represented by a datum in ErM 1 pN q. We shall use the fact that cM 1 pτr1 q 0 to derive a contradiction. Suppose that pG, π q is any pair in P 1 such that G is represented by a r N pM 1 q-orbit τr1 such datum in ErM 1 pN q, and that τ is any element in the Out that npπ, τ q 0. The latter condition implies that Λτ ¤ Λπ , that Mτ Mπ in case Λτ Λπ , and that τ τπ in case both Λτ Λπ and Mτ Mπ hold. But it is clear that
}Λτ }, dimpMτ q }Λτ 1 }, dimpMτ 1 q }Λπ1 }, dimpMπ1 q , all three pairs being equal to the extremal pair pλ1 , µ1 q. It then follows from the definitions that the two relations Λτ Λπ and Mτ Mπ do indeed hold, so that τ τπ . The coefficient npπ, τ q is therefore strictly positive. The summand of G, π and τ in the expression (3.5.8) for cM 1 pτr1 q is consequently positive. The first two sums in (3.5.8) reduce to a sum over pairs pG, π q in P , according to the definition of τ -equivalence. We have just seen that a summand of pG, π q and τ is either positive or zero. Since the summand of pG1 , π 1 q and τ 1 is positive by assumption, the coefficient cM 1 pτr1 q itself is positive. This gives the desired contradiction, and completes the proof of Lemma 3.5.2.
We now go back to the global setting. For the rest of the chapter (and indeed until the end of Chapter 5) we assume that the field F is global. We need to establish Proposition 3.5.1. We shall do so by applying the local proof above to the completions Fv of F .
3.5. ON THE VANISHING OF COEFFICIENTS
161
Proof of Proposition 3.5.1. Let S S8 be a finite set of valuations of the global field F . We shall write ErS pN q for the finite set of products GS
¹
Gv ,
P
Gv
v S
P Erv pN q,
r v pN q of twisted endoscopic data for GLpN q where Erv pN q denotes the set E G over Fv , as earlier. We can then form the set !
ΠpGS q πS
â
P
πv : πv
P ΠpG v q
)
v S
of irreducible representations of GS . Let us also write Erell pN, S q for the finite set of global endoscopic data G P Erell pN q that are unramified outside of S, and ΠpG, S q for the set of irreducible representations of GpAq that are unramified outside of S. For a given pair
pGS , πS q, we then define
cGS pπS q
¸
pG,πq
GS
cG pπ q,
P ErS pN q,
πS
P ΠpG S q,
G P Erell pN, S q, π
P ΠpG, S q.
The sum here is taken over the preimage of pGS , πS q under the localization mapping pG, πq ÝÑ pGS , πS q, and can be restricted to a finite set, in view of the given condition on the Hecke type. We are also given that each coefficient cG pπ q is nonnegative, so the same is true of cGS pπS q. It therefore suffices to show that cGS pπS q vanishes for any S and any pGS , πS q. Suppose that FS
¹
P
v S
Fv
!
fS
¹
fv : fv
P p q
v
r G v , Gv H
P p q Erv N
)
is a product of local compatible families Fv . Then FS determines a decomposable global compatible family F
f
P H p Gq :
G P Erell pN q
(
in the natural way. That is, f vanishes unless G lies in Erell pN, S q, in which r pGS q with the characteristic case f is the product of the function fS P H function of a hyperspecial maximal compact subgroup K S of GpAS q. The expression (3.5.1) becomes a double sum (3.5.9)
¸¸ GS πS
cGS pπS qfS,GS pπS q,
fS
P FS ,
which can be taken over a finite set that depends only on a choice of Hecke type for FS . We are told that (3.5.9) vanishes. We must use this to deduce that each cGS pπS q vanishes.
162
3. GLOBAL STABILIZATION
We have simply to duplicate the proof of Lemma 3.5.2, amplifying the notation in the appropriate way (generally without comment). We find that (3.5.9) equals (3.5.10)
¸ ¸¸¸
MS
cGS pπS qnpπS , τS qγGS pτS qfS,MS pτS q,
τS GS πS
for sums over MS P ErS pN q, τS P Tell pMS q, GS P ErMS pN q and πS P ΠpGS q, and for positive constants γGS pτS q. It is understood here that ErMS pN q stands for the product of the local sets ErMv pN q (rather than the global object defined on [A12, p. 241]). We can then take each fS,MS pτS q to be r N pMS q-invariant Paley-Wiener function that is any general, cuspidal, Out independent of GS . Varying fS , we conclude that for any MS P ErS pN q, and r N pMS q, the sum any orbit τrS of Out (3.5.11)
cMS pτrS q
¸
P
¸
p q
¸
P p q P
GS ErMS N πS Π GS τS τrS
cGS pπS qnpπS , τS qγGS pτS q
vanishes. Following the proof of the lemma further, we fix a τS -equivalence class PS
pGS , πS q :
GS
P ErS pN q,
τS
P ΠpGS q
(
.
To apply this part of the local argument, we enumerate the elements v in S as v1 , . . . , vn , and introduce a lexicographic order on the associated local objects }Λπv } and dimpMπv q. We take the “alphabet” A to be the set of pairs x pλ, µq, for λ a nonnegative real number and µ a nonnegative integer. It has a linear order defined by x pλ, µq ¡ x1
pλ1, µ1q,
if λ ¡ λ1 , or if λ λ1 and µ µ1 . We take the “dictionary” DS to be the set AS An , with the associated lexicographic order. There is then a mapping
pGS , πS q ÝÑ wpGS , πS q
¹
}Λπ }, dimpMπ q v
P
v
v S
from PS to DS , as well as a mapping
pMS , τS q ÝÑ wpMS , τS q
from the adjoint class PS to DS .
We need to show that the set PS1
¹
}Λτ }, dimpMτ q v
P
v
v S
pGS , πS q P PS :
cGS pπS q 0
(
is empty. Suppose that it is not. We can then take the largest element w1
pG1S , πS1 q P PS1 , in the image of PS1 in DS . The choice of pG1S , πS1 q in the preimage also gives us an associated pair
wpG1S , πS1 q,
pMS1 , τS1 q pMπ1 , τπ1 q S
S
3.5. ON THE VANISHING OF COEFFICIENTS
163
in PS . Suppose that pGS , πS q is any pair in PS1 , and that τS is any element in the orbit τrS1 of τS1 such that npπS , τS q 0. It follows from the lexicographic order that wpMS , τS q ¤ wpGS , πS q, with equality if and only if τS τπS . By definition, we have wpMS , τS q wpMS1 , τS1 q wpG1S , πS1 q w1 .
It then follows from the definition of w1 that τS τπS , and hence that npπS , τS q is strictly positive. The summand of GS , πS and τS in (3.5.11) is therefore positive. The first two sums in (3.5.11) reduce to a double sum over pairs pGS , πS q in PS . Any summand in (3.5.11) is consequently either positive or zero. Since the summand of pG1S , πS1 q and τS1 is positive, the sum itself is positive, contradicting the fact that cMS1 pτrS1 q vanishes. This completes the proof of Proposition 3.5.1. The proof we have just completed is largely formal, though this may not be apparent in the general framework we have had to adopt. It is essentially a consequence of the decomposition of irreducible representations into standard representations. We will also need a more technical generalization of the proposition. We shall state it as a corollary, since its proof is the same. Corollary 3.5.3. Suppose that we have coefficients cG pπ q that are as in the proposition, except that for a fixed valuation v and a simple datum G1 P Ersim pN q, the given sum (3.5.1) equals a separate expression ¸
(3.5.12)
P p
τv T G1,v
q
d1 pτv , f1v qfv,G1 pτv q,
in which the function f1 corresponding to G1 in the compatible family F is taken to be a product f1
fv f1v ,
and where the coefficient
fv
P HrpG1,v q,
d1 pτv , f1v q,
τv
f1v
P Hr GpAv q ,
P T pG1,v q,
r N pG1,v q-invariant function of τv , which is supported on a finite set is an Out that depends only on a choice of Hecke type for fv , and which equals 0 for any τv of the form pMv , σv , 1q. Then the coefficients cG pπ q and d1 pτv , f1v q all vanish.
Proof. Suppose that the coefficients cG pπ q all vanish. Then the sum (3.5.1) vanishes for any F, and so therefore does the expression (3.5.12). r pG1,v q to SrpG1,v q is Since G1 is simple, the mapping fv Ñ fvG1 from H surjective by Corollary 2.1.2. We can therefore choose the compatible family F so that the function fv,G1 pτv q takes preassigned values on any finite set of r N pG1,v q-orbits τv . It follows that the coefficients d1 pτv , f v q also vanish. Out 1 We have thus only to show that each cG pπ q vanishes.
164
3. GLOBAL STABILIZATION
As in the proof of the proposition, we fix a finite set of valuations S S8 , which we can assume contains v. We then see that it is enough to show that the coefficients cGS pπS q all vanish. Following the next step in the proof of the proposition, we see that the original sum (3.5.1) takes the form (3.5.10), an expression that is therefore equal to (3.5.12). We can again choose the function fS,MS pτS q in (3.5.10) as we wish, and in particular, so that it isolates r N pMS q-orbit τrS . If the v component of any representative of this a given Out orbit is of the form pMv , σv , 1q, the expression (3.5.12) vanishes, by the given condition on the coefficient d1 pπv , f1v q. It follows that the coefficient cMS pτrS q defined by (3.5.11) vanishes in this case. This is all we need to complete the argument in the proof of the proposition. For it was the vanishing of cMS1 pτrS1 q, in the case of the chosen pair
pMS1 , τS1 q pMπ1 , τπ1 q, S
S
from which we deduced that the coefficients cMS pπS q all vanish. But τS1 is of the form τS1 τπS1 pMπS1 , σπS1 , 1q, by our definition of maximality. The assertions of the corollary follow. With the proof of Proposition 3.5.1, we have established that the coefficients cG pπ q in the earlier identity (3.4.14) vanish. This completes the proof of Proposition 3.4.1, the problem we started with. In particular, the expansions of Corollaries 3.4.2 and 3.4.3 are valid. They tell us that for any G P Ersim pN q, the discrete spectrum and the discrete part of the trace r pN q of formula are both delimited to the original global parameters ψ P Ψ Chapter 1. We can therefore concentrate on the contributions of these parameters. This leads to a finer analysis, which will occupy the rest of the volume, and in which Proposition 3.5.1 and its corollary will have a critical role.
CHAPTER 4
The Standard Model 4.1. Statement of the stable multiplicity formula We assume that G is as in the theorems stated in Chapter 1. It is therefore a connected, quasisplit orthogonal or symplectic group, which is to say that it represents an element in the set
r pN q , Ersim pN q Esim G
N
¥ 1, GrpN q GLpN q θ,r
of isomorphism classes of simple, twisted endoscopic data for GLpN q over F . We will later need to broaden the discussion somewhat, in order to include r pN q. It will also be useful to add more general at least the case G G comments from time to time for added perspective, and to lay foundations for future study. However, we shall always be explicit whenever G represents something other than the basic case of an element in Ersim pN q. We also assume that the underlying field F is global. We have seen that the only families c P Caut pN q that contribute to the discrete part of the trace r pN q. We expect that ψ must formula for G come from parameters ψ P Ψ r r actually belong to the subset ΨpGq of ΨpN q. But as we have noted, this fact is deep, and will be established only after a sustained analysis of other r pN q for questions. We must therefore work with a general parameter ψ P Ψ the time being. r pGq, we are making the imObserve that in even mentioning the set Ψ plicit assumption that Theorem 1.4.1 is valid. This was the seed theorem, which when applied to the simple generic constituents φi of ψ led us in §1.4 r pGq. In this sense, Chapter 4 resembles Chapter 2 in reto a definition of Ψ lying on some cases of our stated theorems for its definitions and arguments. We recall that Chapter 3 was more direct, since it was independent of any part of the local and global theorems. At the beginning of §4.3, we shall take on some formal induction hypotheses, which replace these informal implicit assumptions, and which will carry the argument into later chapters. In §3.3, we described the spectral expansion (4.1.1) ¸ ¸ Idisc,ψ pf q |W0M ||W0G|1 | detpw 1qaG |1 tr MP,ψ pwqIP,ψ pf q M
M
w
165
166
4. THE STANDARD MODEL
r pN q at a test function f for the contribution of our parameter ψ P Ψ We also established an endoscopic expansion
(4.1.2)
Idisc,ψ pf q
¸
G1 Eell G
P p q
P HrpGq.
1 ιpG, G1 q Spdisc,ψ pf 1 q
for this contribution. Our long term goal is to extract as much information as possible from the identity of these two expansions. In this section, we 1 shall state a formula for the linear form Spdisc,ψ pf 1q on the right hand side of (4.1.2). It will suffice to describe the case that G1 G. We have therefore to state a putative formula for the stable linear form G Sdisc,ψ pf q Sdisc,ψ pf q ,
f
P HrpGq.
The formula depends on local information yet to be established. Specifically, it is based on the stable linear form postulated in Theorem 2.2.1(a), or rather a global product (4.1.3)
f G pψ q
± v
fvG pψv q,
f
± fv , v
of these objects. Theorem 1.4.2 asserts that this product makes sense if ψ r pGq of Ψ r pN q. However, the formula also has a global lies in the subset Ψ component, which will add to the complexity of its eventual proof. The stable linear form Sdisc,ψ pf q is uniquely determined by the expansion (4.1.2). The stable linear form f G pψ q is uniquely determined by the conditions of Theorem 2.2.1(a). Our formula asserts that the first linear form equals a multiple of the second by an explicitly given constant, which r pGq. It is this constant that contains the global vanishes unless ψ lies in Ψ information. We will call it the stable multiplicity of ψ. The global constant has three factors that will be familiar from the definitions of §1.5. They are the integer mψ , the inverse of the order of the finite group Sψ , and a special value (4.1.4)
εG pψ q εψ psψ q εG ψ ps ψ q
of the sign character εG ψ . The point sψ P Sψ was defined in §1.4 as the image of the nontrivial central element of SLp2, Cq. Since mψ vanishes r pGq, in which case the objects Sψ and εG make sense, unless ψ belongs to Ψ ψ the product of the three factors is well defined and vanishes unless ψ lies r pGq. There will also be a fourth factor. This is the number σ pS 0 q in Ψ ψ constructed from ψ in [A9]. We shall review its definition. The factor σ pS 0ψ q is part of a general construction that applies to any complex (not necessarily connected) reductive group S. To allow for induction, we take S more generally to be any union of connected components in some complex reductive group. We write S 0 for the connected component of 1, and Z pS q CentpS, S 0 q
4.1. STATEMENT OF THE STABLE MULTIPLICITY FORMULA
167
for the centralizer of S in S 0 . Let T be a fixed maximal torus in S 0 . We can then form the Weyl set W pS q NormpT, S q{T
of automorphisms of T induced from S. The Weyl group W 0 W pS 0 q of S 0 is of course an obvious special case. We write Wreg pS q for the subset of elements w P W pS q that are regular, in the sense that the fixed point set of w in T is finite. This property holds for w if and only if the determinant detpw 1q is nonzero, where pw 1q is regarded as a linear transformation on the real vector space aT
Hom X pT q, R .
Finally, we define the sign s0 pwq sgn0 pwq 1
of an element w P W to be the parity of the number of positive roots of pS 0, T q mapped by w to negative roots. Given these various objects, we attach a real number (4.1.5)
ipS q |W pS q|1
¸
P
p q
s0 pwq| detpw 1q|1
w Wreg S
to S. The number ipS q bears a formal resemblance to the spectral expansion (4.1.1) of Idisc,ψ pf q. The rest of the construction amounts to the definition of a number epS q that bears a similar resemblance to the endoscopic expansion (4.1.2). This second number is defined inductively in terms of factors σ pS1 q attached to connected complex reductive groups S1 . Let us write Sss for the set of semisimple elements in S. For any s P Sss , we set (4.1.6)
Ss
Centps, S 0q,
the centralizer of s in the connected group S 0 . Then Ss is also a complex reductive group, with identity component Ss0
pSsq0 Centps, S 0q0.
(The notation here differs from the convention used in some places, in which the symbol Ss is reserved for the identity component of the centralizer. We must also take care not to confuse elements s P Sss with the sign character in (4.1.5), which we will always denote with the superscript 0.) If Σ is any subset of S that is invariant under conjugation by S 0 , we shall write E pΣq for the set of equivalence classes in Σss
Σ X Sss,
with the equivalence relation defined by setting s1
s
168
4. THE STANDARD MODEL
if
s1 s0 zsps0 q1 , The essential case is the subset (4.1.7)
Sell
s0
P S 0, z P Z pSs0q0. (
s P Sss : |Z pSs0q| 8
of S. For among other things, the equivalence relation in Sell that defines the quotient Eell pS q E pSell q 0 is simply S -conjugacy. The resemblance here with the (more complicated) definition of endoscopic data [KS] is not accidental. The following proposition is a restatement of Theorem 8.1 of [A9]. Proposition 4.1.1. There are unique constants σ pS1 q, defined whenever S1 is a connected complex reductive group, such that for any S, the number (4.1.8)
e pS q
¸
P p q
|π0pSsq|1σpSs0q
s Eell S
equals ipS q, and such that
σ pS1 q σ pS1 {Z1 q|Z1 |1 ,
(4.1.9)
for any central subgroup Z1 of S1 . To see the uniqueness of the constants σ pS1 q, one notes that (4.1.9) implies that σ pS1 q equals 0 if the center Z pS1 q is infinite. If Z pS1 q is finite, we define σ pS1 q inductively from (4.1.8) by setting σ pS1 q|Z pS1 q| ipS1 q
¸
0 |π0pS1,sq|1σpS1,s q,
s
where s is summed over the complement of Eell Z pS1 q in Eell pS1 q. We refer the reader to [A9, §8] for the proof of the general identities (4.1.8) and (4.1.9). Remark. If S is a single connected component, it is a bitorsor under the actions of S 0 by left and right translation. The constructions actually make sense if S is taken to be any bitorsor, relative to a complex connected reductive group S 0 . Proposition 4.1.1 remains valid in this setting, since the group multiplication on S plays no role in the proof. More generally one could take S to any finite union of S 0 -torsors. r pN q. Recall that Sψ stands for the We return to our parameter ψ P Ψ p centralizer in G of the image of (some representative of) ψ. As in the local discussion from §2.4, we shall often work directly with the quotient
(4.1.10)
Sψ
Sψ {Z pGpqΓ,
Γ ΓF .
It is of course the group Sψ
π0pS ψ q S s{S 0s
4.1. STATEMENT OF THE STABLE MULTIPLICITY FORMULA
169
of connected components of S ψ that governs the main assertion of our basic global Theorem 1.5.2. The fourth global factor is the number σ pS 0ψ q, defined for the connected reductive group S1 S 0ψ by the proposition. We can now state the formula for Sdisc,ψ pf q. Theorem 4.1.2 (Stable multiplicity formula). Given ψ G P Ersim pN q, we have (4.1.11)
P Ψr pN q
and
Sdisc,ψ pf q mψ |Sψ |1 σ pS 0ψ q εG pψ q f G pψ q,
r pGq. In particular, Sdisc,ψ pf q vanishes unless ψ belongs to the for any f P H r pGq of Ψ r pN q. subset Ψ
This represents one of the main results of the volume. It will have to be proved at the same time as the theorems stated in Chapter 1, and their local refinements from Chapter 2. However, we will be able to apply it inductively to endoscopic groups G1 P Eell pGq that are proper. We write the specialization of the formula to G1 in slightly different terms. r pGq can be identified with Aut r N pGq-orbits of Recall that elements ψ P Ψ L-homomorphisms ψrG : Lψ SLp2, Cq
ÝÑ LG. Following a suggestion from §1.4, we write ΨpGq for the corresponding set p of Aut r N pGq. There is then a surjective of orbits under the subgroup G r mapping ψG Ñ ψ from ΨpGq to ΨpGq, for which the order of the fibre of ψ r pGq and ψG maps to ψ, we write equals the integer mψ . If f lies in H 1 G f G pψG q m ψ f pψ q. p 1 -orbits of L-homomorphisms The corresponding set ΨpG1 q for G1 consists of G ψ 1 : Lψ SLp2, Cq ÝÑ L G1 .
The composition of any such ψ 1 with the endoscopic L-embedding ξ 1 of L G1 r pN q. This into L G is an element ψG in ΨpGq, which in turn maps into Ψ 1 r gives a mapping from ΨpG q to ΨpN q, whose image is contained in the subset r pGq of Ψ r pN q. We write Ψ ΨpG1 , ψ q ΨpG1 , G, ψ q
r pN q, a set that is empty for the fibre in ΨpG1 q of our given element ψ P Ψ r p G q. unless ψ belongs to Ψ We want to specialize the formula of Theorem 4.1.2 to the linear form 1 Spdisc,ψ pf 1q in (4.1.2). It is a consequence of the definitions (3.3.9) and (3.3.13) that ¸ 1 1 1 pf 1 q, Spdisc,ψ pf 1q Spdisc,ψ ψ1
170
4. THE STANDARD MODEL
r pG1 q that map to ψ. Since where ψ 1 is summed over those elements in Ψ r pG1 q, the specialization of the this sum is over the image of ΨpG1 , ψ q in Ψ 1 theorem to G yields the following corollary.
Corollary 4.1.3. Given G P Ersim pN q, G1 r pGq, we have an expansion f PH
1 Spdisc,ψ pf 1 q
(4.1.12) where
¸ ψ 1 Ψ G1 ,ψ
P p
q
P
Eell pGq, ψ
P Ψr pN q
and
|Sψ1 |1σpS 0ψ1 q ε1pψ1q f 1pψ1q,
1 1 ε1 p ψ 1 q εG p ψ 1 q εG ψ 1 psψ 1 q.
Remark. Suppose that the endoscopic group G1 is proper, in the sense that it is distinct from G. In other words, it is a proper product G1
G11 G12,
G1i
P ErsimpNi1q,
N11
N21
N,
of simple endoscopic groups. We note that the informal assumptions under which the corollary makes sense depend only on the positive integer N . They could be replaced by a formal induction hypothesis that the theorems we have stated so far, including Theorem 4.1.2, are valid if N is replaced by a smaller integer N . Corollary 4.1.3 would then hold for G1 . The group S ψ given by (4.1.10) is of course insensitive to whether we r pGq. With either interpretation of ψ, the treat ψ as an element in ΨpGq or Ψ group will be at the heart of our analysis of the linear form Idisc,ψ pf q. As ψ varies, the properties of S ψ characterize an important chain of subsets of ΨpGq. We write (4.1.13) where
Ψsim pGq Ψ2 pGq Ψell pGq Ψdisc pGq ΨpGq,
Ψsim pGq tψ
P ΨpGq : |S ψ | 1u, Ψ2 pGq tψ P ΨpGq : S ψ is finiteu, Ψell pGq tψ P ΨpGq : S ψ,s is finite for some s P S ψ,ss u,
and Ψdisc pGq tψ
P ΨpGq : Z pS ψ q is finiteu.
r N pGq (of These sets are obviously stable under the action of the group Out order 1 or 2). The associated sets of orbits give a corresponding chain of r pGq, which can be regarded as a refinement of the earlier chain subsets of Ψ (1.4.7) from §1.4. r pGq) have a The subsets (4.1.13) of ΨpGq (or rather their analogues for Ψ direct bearing on how we interpret the corresponding linear forms Idisc,ψ pf q. r disc pGq should be the subset of parameters ψ P Ψ r pGq such For example, Ψ
4.2. ON THE GLOBAL INTERTWINING RELATION
171
r ell pGq are characthat Idisc,ψ pf q is nonzero. Elements ψ in the smaller set Ψ 1 terized by the property that for some G P Eell pGq, the subset
Ψ2 pG1 , ψ q Ψ2 pG1 q X ΨpG1 , ψ q
of ΨpG1 q is nonempty. In other words, ψ should contribute to the discrete r 2 pGq of Ψ r ell pGq consists of those ψ for which spectrum of G1 . The subset Ψ 1 we can take G G. This is of course to say that ψ should contribute to the discrete spectrum of G, or equivalently, that the term with M G in the r sim pGq consists expansion (4.1.1) of Idisc,ψ pf q is nonzero. The smallest set Ψ 1 r simply of the parameters ψ P Ψ2 pGq such that for any G P Eell pGq distinct from G, the set Ψ2 pG1 , ψ q (or for that matter the set ΨpG1 , ψ q) is empty. These last remarks are intended as motivation for what follows. They offer some guidance for our efforts to compare the two expansions (4.1.1) and (4.1.2). Notice that the properties that define the subsets in (4.1.13) are given entirely in terms of the group S ψ . They make sense if S ψ is replaced by any complex reductive group, or even a disjoint union S of bitorsors under a connected group S 0 . For example, one could take S to be bitorsor under a complex reductive (but not necessarily connected) group S , a set that can obviously be regarded as a finite disjoint union of bitorsors under the connected group S 0 pS q0 . It is necessary to formulate the definitions in this setting if one wants to replace G by an arbitrary triplet (3.1.2). 4.2. On the global intertwining relation We continue to work with a fixed group G P Ersim pN q and a parameter r pN q, both taken over the global field F . Our long term aim will be ψ PΨ to compare the spectral expansion (4.1.1) of Idisc,ψ pf q with its endoscopic expansion (4.1.2). In the last section, we stated a formula for the linear form
1 pf 1 q, Spdisc,ψ
f
P HrpGq,
in the endoscopic expansion. In this section, we shall establish an expansion for the analogous term (4.2.1)
tr MP,ψ pwq IP,ψ pf q ,
f
P HrpGq,
in the spectral expansion. We shall also reformulate the local intertwining relation stated in Chapter 2 as a global relation. This will ultimately serve as a link between the two formulas. The trace (4.2.1) depends on fixed elements M P L, P P P pM q and w P W pM qreg . The Levi subgroup M is a product of several general linear groups with an orthogonal or symplectic group of the same form as G, as in the local case (2.3.4). We can therefore construct the sets r 2 pM q Ψ r pM q Ψ
of parameters as products of sets of the sort we have already defined. We r M pGq for the image of Ψ r 2 pM q in Ψ r pGq under the natural mapping. write Ψ
172
4. THE STANDARD MODEL
We can also form the set Ψ2 pM q. For later reference, we write Ψ2 pM, ψ q Ψ2 pM, G, ψ q r pN q in Ψ2 pM q, a set that is empty for the fibre of our given element ψ P Ψ r M pGq. We may as well also write Ψ r 2 pM, ψ q for the unless ψ belongs to Ψ r fibre of ψ in Ψ2 pM q. r pGq (or for that matter, any of corresponding sets Once again, the set Ψ attached to M ) depends on the implicit assumption that Theorem 1.4.1 is valid. The rules here are the same as in the last section. Each object we introduce is predicated on the validity of any cases of the stated theorems required for its existence. As we have noted, we will replace this implicit assumption with formal induction hypotheses at the beginning of §4.3. Our informal assumptions will include Theorem 1.5.2, since among other things, we will need to introduce a global analogue of the expression (2.4.5). In particular, we can assume that the relative discrete spectrum of M der 2 pM q. The trace (4.2.1) composes into subspaces indexed by parameters in Ψ comes from the contribution of ψ to a representation induced from the disr M pGq. We shall crete spectrum of M . It thus vanishes if ψ does not lie in Ψ r therefore assume that ψ belongs to ΨM pGq, and in particular, that it lies in r pGq of Ψ r pN q. the subset Ψ Our central algebraic object will continue to be the complex reductive group S ψ . We digress to consider some of its global implications, including our formulation of the global intertwining relation. The group S ψ gives rise to several finite groups, which are global analogues of the finite groups introduced in §2.4. For example, the basic component group Sψ π0 pS ψ q has a normal subgroup Sψ1 . This is isomorphic
r 2 pM, ψ q of to the component group SψM attached to any ψM in the subset Ψ r r 2 pM q elements in Ψ2 pM q that map to ψ. Our assumption that ψM lies in Ψ means that the quotient
(4.2.2)
Tψ
p qΓ xqΓ {Z pG p qΓ A x { A x X Z p G Z pM M M
,
where AM x
xqΓ Z pM
0
,
is a maximal torus in S ψ . The associated finite groups Wψ0 and Wψ of automorphisms can then be identified with the full Weyl groups of S 0ψ and
4.2. ON THE GLOBAL INTERTWINING RELATION
173
S ψ . They take their places in the global version 1 1 ∨
∨
Wψ0
Wψ0
∨
(4.2.3)
1
>
Sψ1
>
Sψ1
>
}
1
>
∨
Nψ
>
Wψ
∧
>
1
>
1
∧
∨
∨
Sψ
>
∨
Rψ ∨
1 1 of the commutative diagram (2.4.3) (or rather the special case of (2.4.3) in r 2 pM q). which ψ is represented by an element in Ψ It is interesting to compare the general diagram (4.2.3) with its specialization to parameters in the subsets (4.1.13). The set Ψell pGq in (4.1.13), for example, is closely related to the R-group Rψ in (4.2.3). One sees easily r ell pGq if and only if there is an element w P Rψ whose that ψ belongs to Ψ fixed point set in T ψ (relative to the embedding of Rψ into Wψ defined by a choice of Borel subgroup B ψ S ψ ) is finite. This condition in turn implies that S 0ψ T ψ . Such elements will be an important part of our future study. Consider a point u from the group Nψ
N ψ pG, T ψ q{T ψ
at the center of the diagram. Following the local notation from §2.4, we shall write wu and xu for the images of x in Wψ and Sψ respectively. We can identify the first point wu with an element in the group W pM q. This gives a global twisted group u M
w pM, w ru q. M u
We have also been following the later convention from §2.4, in which ψM r 2 pM q that maps to ψ. This parameter actually lies denotes an element in Ψ r 2 pM q. We again write u r (somewhat superfluously) in the subset Ψ2 pMu q of Ψ for the image of u in the associated SψM -bitorsor SrψM ,u
u q. S ψ pM M
This notation will be useful to describe the normalization of the global intertwining operator in (4.2.1). We are assuming that the analogue of Theorem 1.4.2 is valid for M . This implies that the localization ψM,v of ψM at any v belongs to the correspondr pMv , ψv q. The localization ψv of ψ therefore belongs to Ψ r p G v q. ing set Ψ
174
4. THE STANDARD MODEL
It follows easily that S ψ embeds into S ψv , and that there is a morphism of the diagram (4.2.3) into its local counterpart (2.4.3). In particular, any element u in Nψ has a local image uv in the group Nψv pGv , Mv q, for any valuation v. We can therefore define the local linear form fv,G pψv , uv q,
fv
P HrpGv q,
by (2.4.5) (with ψMv , πMv and uv taking the place here of the local symbols ψ, π and u in (2.4.5)). We obtain a global linear form fG pψ, uq, by setting
fG pψ, uq
(4.2.4)
± v
f
P HrpGq,
fv,G pψv , uv q,
f
± fv , v
for a product that by the discussion of §2.5 can be taken over a finite set. It is clear from §2.4 that the definition has an equivalent global formulation
p4.2.4q1
fG pψ, uq
¸
P
rψ πM Π M
xur, πrM y tr RP pwu, πrM , ψM qIP pπM , f q ,
rM , ψM q is a normalized global intertwining operator. This where RP pwu , π operator is defined as a product over v of its local analogues, constructed in Chapter 2 by (2.3.26), (2.4.2) and (2.4.4). In particular, it depends implicu pAq. The pairing xu rM of πM to the torsor M r, π rM y itly on an extension π represents the corresponding extension of the character x , πM y on the group SψM Sψ1 to the bitorsor SrψM ,u , which is again defined as a product of its local analogues. Consider also a semisimple element s in the original group S ψ . It has a local image sv in the group S ψv , for any valuation v. We can therefore define the local linear form
1 pψv , sv q, fv,G
fv
P HrpGv q,
by (2.4.6) (with ψv and sv in place of ψ and s). We obtain a global linear form r pG q, fG1 pψ, sq, f PH by setting (4.2.5)
fG1 pψ, sq
± v
1 pψv , sv q, fv,G
f
± fv , v
for a product that can again be taken over a finite set. The global formulation of this definition is of course
p4.2.5q1 where pG1 , ψ 1 q
fG1 pψ, sq f 1 pψ 1 q,
is the preimage of pψ, sq under the global correspondence (1.4.11). The local intertwining relation stated as Theorem 2.4.1 applies to the local factors in (4.2.4) and (4.2.5). We can state the corresponding global identity for the products as a corollary of this theorem.
4.2. ON THE GLOBAL INTERTWINING RELATION
175
Corollary 4.2.1 (Global intertwining relation for G). For any u in the global normalizer Nψ , the identity (4.2.6)
fG1 pψ, sψ sq fG pψ, uq,
f
P HrpGq,
holds for any semisimple element s P S ψ that projects onto the image xu of u in Sψ . We remind ourselves that Theorem 2.4.1 has yet to be established. Its proof in fact will be one of our major concerns. The same therefore goes for the corollary. We have formulated it here to give our discussion some sense of direction, and in particular, to see how we will eventually relate the summands of (4.1.1) and (4.1.2). We note that the diagram (4.2.3) has an obvious analogue if S ψ is replaced by any complex reductive group S. More generally, it makes sense if we take S to be a bitorsor under a complex reductive group S . (We write S here because we need to reserve the symbol S 0 pS q0 for the connected component of 1 in S .) The corresponding objects N, W , S and R in the diagram are then bitorsors under their analogues N , W , S and R for the group S . On the other hand, the objects W 0 and S 1 are the equal to the associated groups for S . In particular, S 1 is the projection onto the quotient S π0 pS q of a subgroup S 1 of S with S0
S1 S.
For example, we can take S
Srψ Sψ pGrq{Z pGprqΓ,
r is a twisted orthogonal group over F , and ψ belongs to the subset where G r r pGq. It would be easy to formulate a global intertwining relation ΨpGq of Ψ r as a corollary of Theorem 2.4.4 that is parallel to Corollary 4.2.1. We for G, will not do so, since we do not need it. However, we will formulate a global r We shall state it here for application supplement of Theorem 1.5.2 for G. to the fixed group M above, even though its proof will only come later. r pG r 0 , θrq, where G G r 0 is an even orthogonal We are supposing that G r 2 pG q. group in Ersim pN q over the global field F . Consider a parameter ψ P Ψ 2 r For any function φ P Ldisc,ψ GpF qzGpAq , and any y P GpAq, the function
(4.2.7)
G pyqφ pxq φpθr1xyq, Rdisc,ψ
r
x P GpAq,
also belongs to L2disc,ψ GpF qzGpAq . We thus obtain a canonical extension
G r pAq generated by G r p A q. of the representation Rdisc,ψ of GpAq to the group G Theorem 1.5.2, together with Theorem 2.2.1, describes the character of the original representation in terms of the transfer of characters from GLpN q. The theorem we are about to state, in combination with Theorem 2.2.4, plays a similar role in it extension.
176
4. THE STANDARD MODEL
Suppose that ψ lies in the subset r 2 pGq. Then the set of Ψ
r q ΨpG rq X Ψ r 2 pG q Ψ2 pG
Srψ π0 pSrψ q is a (nonempty) Sψ -torsor. Any localization ψv of ψ lies in the subset r v q of Ψ r pGv q, and determines a mapping x r Ñ x rv from Srψ to Srψv . Ψ pG r ψ , equipped with Suppose that π is a representation in the global packet Π r pAq (or equivalently, an intertwining operator an extension to the group G r π pθq of π of order 2). We can then choose extensions of the local components r pFv q whose tensor product is compatible with the πv of π to the groups G extension of π. Theorem 2.2.4 tells us that there is a corresponding product
xxr, πry ± xxrv , πrv y, xr P Sψ , v of extensions of linear characters x , πv y, which determines an extension of the linear character x , π y on Sψ to the group Srψ generated by Srψ . Assume that x , π y equals the sign character εψ On the one hand, εψ has a canonical extension to Srψ . This follows from the definition (1.5.6) and the fact that
p SOpN, Cq on its Lie algebra the adjoint action of the complex group G has a canonical extension to the group OpN, Cq. On the other, Theorem G . Moreover, we 1.5.2 asserts that π occurs in the decomposition of Rdisc,ψ G r pAq . The theorem has a canonical extension to G have just seen that Rdisc,ψ stated below asserts that the restriction of this extension to the subspace of π coincides with the extension of εψ under the local correspondence of extensions postulated by Theorem 2.2.4.
r with G P Ersim pN q, is a twisted orthogonal Theorem 4.2.2. Suppose that G, r 2 pG q . group over F , and that ψ belongs to Ψ r q in Ψ r 2 pGq, we have (a) If ψ lies in the complement of Ψ2 pG G tr Rdisc,ψ pfrq
r
0, r q, Ψ2 pG
r q. fr P HpG
(b) Assume that ψ lies in and that π is a representation in r the global packet Πψ such that the linear character x , π y on Sψ is equal to 1 G εψ ε ψ . Then the restriction to π of the canonical extension of Rdisc,ψ to r pAq corresponds to the canonical extension of εψ to Sr under the product G ψ of local correspondences of extensions given by Theorem 2.2.4.
We will now take up the expansion of the global trace (4.2.1). As an aid to the reader, we shall try to formulate it so that it is roughly parallel to its endoscopic counterpart, Corollary 4.1.3. In particular, we shall state it as a corollary of Theorem 4.2.2 and Theorem 1.5.2, just as its endoscopic analogue was stated as a corollary of Theorem 4.1.2. We recall that Theorem 4.1.2 was the stable multiplicity formula, while Theorem 1.5.2 is the ordinary
4.2. ON THE GLOBAL INTERTWINING RELATION
177
multiplicity formula (with its twisted supplement Theorem 4.2.2). We will want to apply the two corollaries in the next two sections, long before any of these three theorems have been proved. We will then have to apply them under a common induction hypothesis. Our proof of the expansion of (4.2.1) in the remainder of this section will be more complicated than that of its endoscopic counterpart. In addition to the two theorems on which it depends as a corollary, the proof also requires r pN q. However, an analogue of Theorem 4.2.2 for the original GLpN q-torsor G this is elementary, unlike Theorem 4.2.2, in the sense that we will be able to establish it on the spot. r sim pN q Ψsim G r pN q . The Consider a parameter ψ in the global set Ψ analogue
N pyqφ pxq φ θrpN q1xy , Rdisc,ψ
P GrpN, Aq, x P GLpN, Aq, of (4.2.7), defined for φ P L2disc,ψ GLpN, F qzGLpN, Aq , is a canonical exN r pN, Aq . It of GLpN, Aq to the group G tension of the representation Rdisc,ψ (4.2.8)
y
corresponds to a canonical extension of the automorphic representation πψ of GLpN, Aq attached to ψ. On the other hand, for any completion ψv of ψ, we also have the extension of the representation πψv of GLpN, Fv q described in §2.1.
r sim pN q. Then the extension of the autoLemma 4.2.3. Assume that ψ P Ψ morphic representation πψ of GLpN, Aq defined by (4.2.8) equals the tensor rψv pN q of representations πψv defined in terms product of local extensions π of Whittaker functionals in §2.1. N Proof. The definition (4.2.8) makes sense if Rdisc,ψ is replaced by the N right regular representation R of GpN, Aq on any invariant space of functions on the quotient of GLpN, Aq by GLpN, F q. In particular, the operator
rpN q RN θrpN q R
intertwines RN with RN θrpN q. It suffices to show that the intertwining rdisc,ψ pN q of RN operator R disc,ψ corresponds to the intertwining operator rψ pN q π
â
â
of πψ
v
v
π rψv pN q
πψv .
We write ψ µ b ν and N mn, where µ is a unitary cuspidal automorphic representation of GLpmq, and ν ν n is the irreducible representation of SLp2, Cq of dimension n. Let IP pσµ,λ q be the induced representation of GLpN, Aq defined as in the statement of Theorem 1.3.3, but with a general vector λ P Cn in place
178
4. THE STANDARD MODEL
of the fixed vector ρP
n 1 n 2 2 , 2 ,
, pn1q
2
of exponents. It acts on a Hilbert space HP pσµ q that is independent of λ. Given that µ is self-dual, we see that there are isomorphisms IP pσµ,λ q IP pσµ_ ,λ_ q IP pσµ,λ_ q_ ,
in which
λ_ pλn , . . . , λ1 q. It follows that there is a canonical intertwining isomorphism IrP pσµ,λ , N q : IP pσµ,λ q
ÝÑ
IP pσµ,λ_ q_
IP pσµ,λ_ q θrpN q
that is compatible with the corresponding two standard global Whittaker functionals (namely, the global analogues of (2.5.1) for λ and λ_ ) on the space HP pσµ q. Suppose now that λ is in general position. Then for any φ in the subspace HP0 pσµ q of κ-finite vectors in HP pσµ q, the Eisenstein series E pφ, λq : x
ÝÑ E px, φ, λq is a well defined automorphic form on GLpN, Aq. Let RN pµ, λq be the rer pN q, to striction of RN , regarded as a representation of the Hecke algebra H the space
The mapping
Apµ, λq tE pφ, λq : φ P HP0 pσµ qu
E pλq : φ ÝÑ E pφ, λq is then an intertwining isomorphism between the irreducible representations r pN q. Our third ingredient is the intertwinIP pσµ,λ q and RN pµ, λq of H rpN q above. It takes Apµ, λq onto the space Apµ_ , λ_ q ing mapping R Apµ, λ_ q, as one sees readily from an examination of the relevant constant terms, and therefore represents an intertwining isomorphism rpN q : RN pµ, λq R
ÝÑ
RN pµ, λ_ q_
RN pµ, λ_q θrpN q.
The main point is to show that (4.2.9)
rpN qE pφ, λq E I rP pσµ,λ , N qφ, λ_ . R
The putative identity (4.2.9) asserts that the diagram IP pσµ,λ q
E pλq
pσ ,N q ÝIÝÝÝÝÝÝ Ñ rP
µ,λ
IP pσµ,λ_ q θrpN q
p q
E λ_
p q
r N R
RN pµ, λq ÝÝÝÝÝÝÝÑ RN pµ, λ_ q θrpN q , is commutative. Since the four arrows each represent intertwining isomorphisms between irreducible representations, we can write rpN qE pλq, E pλ_ qIrP pσµ,λ , N q cλ R
for a multiplicative complex number cλ
P C. We must prove that cλ 1.
4.2. ON THE GLOBAL INTERTWINING RELATION
179
The group GLpN q has a standard global Whittaker datum pB, χq. It is defined as in the local discussion of §2.5, except that χ is a left NB pF qinvariant, nondegenerate character on NB pAq. The Eisenstein series in (4.2.9) has a canonical automorphic χ-Whittaker functional
pωE qpφ, λq
»
p qz p q
NB F NB A
E pn, φ, λq χpnq dn.
A change of variables in this integral, combined with the definitions of χ and θrpN q in terms of the standard splitting of GLpN q, tells us that rpN qE pφ, λq ω R
ω E pφ, λq .
Let us write Ωpφ, λq for the χ-Whittaker functional for IP pσµ,λ q with respect to which we have normalized the isomorphism IrP pσµ,λ , N q. It is induced from the automorphic χMP -Whittaker functional on the space of cuspidal functions on MP pAq, and is defined by the global analogues of the integral (2.5.1). We then have Ω IrP pσµ,λ , N qφ, λ_
Ωpφ, λq,
φ P HP0 pσµ q,
by definition. Moreover, one can show from the basic definition of E pφ, λq as an absolutely convergent series on an open set of λ that ω E pλ qφ
Ωpφ, λq,
φ P HP0 pσµ q.
(See [Sha1, p. 351].) Combining these identities, we see that ω E pλ_ qIrP pσµ,λ , N qφ
Ω IrP pσµ,λ, N qφ, λ_ Ωpφ, λq ω E pλqφ ω RrpN qE pλqφ .
It follows that cλ 1, as required, and therefore that the identity (4.2.9) is valid. The embedding of the global Langlands quotient πψ into the automorphic discrete spectrum is provided by residues of Eisenstein series. More precisely, it is the sum of iterated residues of E px, φ, λq at the point λ ρP ρ_ P given by the general scheme in [L5, Chapter 7], and its specialization to GLpN q in [MW2], that yields the intertwining isomorphism from πψ to N Rdisc,ψ . We recall that the residue scheme is noncanonical. In particular, it could be replaced by its “adjoint”, in which the iterated residues are taken rpN q and π rψ pN q. in λ_ rather than λ. We are trying to relate the operators R But π rψ pN q is the Langlands quotient of the value at λ ρP of the operator IrP pσµ,λ , N q. The lemma then follows from an application of the residue operation to the left hand side of (4.2.9), and its adjoint to the right hand side. We now consider the global trace (4.2.1). We shall expand it into an expression that is roughly parallel to the formula (4.1.12) of Corollary 4.1.3, and which, as we noted above, is a corollary of Theorems 4.2.2 and 1.5.2.
180
4. THE STANDARD MODEL
P Ψr 2pM q determines a subspace L2disc,ψ M pF qzM pAq L2disc AM pRq0 M pF qzM pAq of the discrete spectrum. This subspace in turn has an M pAq-equivariant Any parameter ψM M
decomposition
L2disc,ψM M pF qzM pAq
à πM
mψM pπM qπM ,
for irreducible representations πM of M pAq such that cpπM q maps to cpψM q, and nonnegative multiplicities mψM pπM q. The induced representation in (4.2.1) therefore has a decomposition IP,ψ pf q
àà ψM πM
mψM pπM q IP pπM , f q,
r 2 pM, ψ q of Ψ r 2 pM q. where the sum over ψM can be restricted to the subset Ψ Let us write MP pw, πM q for the transfer of the intertwining operator MP,ψ pwq in (4.2.1) to the space on which IP pπM , f q acts. We can then express the global trace (4.2.1) as a double sum ¸ ¸
ψM πM
mψM pπM q tr MP pw, πM q IP pπM , f q ,
with the understanding that the trace vanishes if wπM is not equivalent to πM . The original intertwining operator MP,ψ pwq in (4.2.1) acts on the Hilbert space HP,ψ of the induced representation IP,ψ . It is defined by analytic continuation of Langlands’ original integral over Nw1 P pAq X NP pAqzNw1 P pAq.
Since the definition also includes a left translation by w1 , the transfer r πM q MP pw, πM q JP pw,
of MP,ψ pwq to the Hilbert space of IP pπM , f q is the global analogue of the r πλ q (with πM in place of πλ ) in (2.3.26). The unnormalized operator JP pw, global form of the entire expression (2.3.26) is the product RP pw, πM , ψM q rP pw, ψM q1 MP pw, πM q,
where rP pw, ψM q is the product of the global λ-factor λpwq with the global normalizing factor (4.2.10)
_ 1 Lp1, πψ , ρ_1 Lp0, πψM , ρ_ M w1 P |P q εp0, πψM , ρw1 P |P q w P |P
1
defined by analytic continuation of a product of local normalizing factors (2.3.27). Since the global λ-factor equals 1, the global normalizing factor is in fact just equal to (4.2.10). In the discussion of Chapter 2, the next step was to transform the local object (2.3.26), which occurs on the right hand side of (2.4.2), into the selfintertwining operator on the left hand side of (2.4.2). In the global setting
4.2. ON THE GLOBAL INTERTWINING RELATION
181
here, RP pw, πM , ψM q is already a self-intertwining operator, under the condition w P Wψ pπM q of (2.4.2). In fact, a global version of the intertwining operator π pw rq of (2.4.2) is built into the definition of RP pw, πM , ψM q. It comes from operators (4.2.7) and (4.2.8), applied to the relevant factors of r and x 1 at which they become trivial. M , and taken at values y w For the general linear factors of M , we must use Lemma 4.2.3 to verify that the analogue of (4.2.8) equals the tensor product of the local Whittaker extensions that are implicit in the global self-intertwining operator in (4.2.4)1 . We can therefore write MP pw, πM q rP pw, ψM q RP pw, π rw , ψM q,
w
P W ψ pπ M q ,
in the global form of the notation (2.4.2). It remains to apply Theorems 1.5.2 and 4.2.2 to the other factor G of M . We have expressed the global trace (4.2.1) as a double sum over ψM and πM . We first apply Theorem 4.2.2(a) to the G -component of the global multiplicity mψM pπ q that occurs in any summand. We see that the summand vanishes unless ψM belongs to the subset w , ψ q ΨpM w q X Ψ r 2 pM, ψ q Ψ2 pM
r 2 pM, ψ q. We can also restrict the second sum to of w-fixed elements in Ψ r ψ . It follows that (4.2.1) equals elements πM in the packet Π M ¸
P p
¸
P
q
rψ w ,ψ πM Π ψM Ψ2 M M
rM , ψM q IP pπM , f q . mψM pπM q rP pw, ψM q tr RP pw, π
Next, we apply Theorem 1.5.2 to G . This gives us the multiplicity formula mψM pπM q mψM |SψM |1
¸
P
εψM pxM q xxM , πM y,
xM SψM
by Fourier inversion on the abelian group SψM . The last step will be to apply Theorem 4.2.2(b) to G . w , ψ q means that w lies in the subThe condition that ψM lies in Ψ2 pM group Wψ of W pM q. (Like the other groups in the diagram (4.2.3), Wψ is in fact defined in terms of an implicit representative ψM of ψ.) Let us write Nψ pwq
SrψM ,u ,
wu
w,
for the fibre of w in Nψ , relative to the horizontal exact sequence at the center of (4.2.3). We can then write ¸
P
xM SψM
εψM pxM q xxM , πM y
¸
P p q
rqxu r, π rM y, εψM pu
u Nψ w
where the summand on the right is an extension to the group SrψM ,u of the character on SψM defined by the summand on the left. The extension must be trivial if the character on the left is trivial, but can otherwise be arbitrary, since both sums then automatically vanish. We take εψM pu rq to be the canonical extension of εψM pxM q. If the linear character xxM , πM y
182
4. THE STANDARD MODEL
1 on SψM equals ε r, π rM y will then have to be the canonical ψM , the pairing xu 1 extension of εψM . Applying Theorem 4.2.2(b) to M in this case, we conclude that RP pw, π rM , ψM q is defined in terms of an implicit extension of πM to w pAq that is compatible with the product of local extensions associated M 1 by Theorem 2.2.4 to xu r, π rM y. In case xxM , πM y is distinct from ε ψM , which is to say that πM does not occur in the discrete spectrum, we have only to rM , ψM q by any product of local intertwining operators (2.4.2) define RP pw, π r, π rM y we have chosen. that is compatible with the extension of xu Having agreed upon these conventions, we take the sum over u P Nψ pwq r ψ . It then follows from the definition (4.2.4) outside the sum over πM P Π M that the resulting sum
¸
P
rψ πM Π M
xur, πrM y tr RP pw, πrM , ψM q IP pπM , f q
equals fG pψ, uq. Combining this with the other terms discussed above, we obtain the expansion of (4.2.1) that represents our common corollary of Theorems 4.2.2 and 1.5.2. Corollary 4.2.4. Given G we have an expansion (4.2.11)
¸
w ,ψ q ψM PΨ2 pM
P ErsimpN q, M P L, ψ P Ψr M pGq and f P HrpGq,
|Sψ |1 M
¸
P p q
rP pw, ψM q εψM pu rq fG pψ, uq
u Nψ w
for the global trace
tr MP,ψ pwq IP,ψ pf q ,
w
P W pM q.
Remarks. 1. Suppose that the Levi subgroup M is proper in G. In other words, it is a product of general linear groups with a simple endoscopic group G P Ersim pN q that is a proper subgroup of G. As in Corollary 4.1.3, the informal assumptions under which this corollary makes sense depend only on the positive integer N . They could be replaced by a formal induction hypothesis that the theorems stated so far, including Theorems 1.5.2 and 4.2.2, are valid if N is replaced by a smaller integer N . Corollary 4.2.4 would then hold for M . 2. Corollary 4.2.4 bears a formal resemblance to Corollary 4.1.3 on several accounts. This is no accident. We shall see in the next two sections that their roles in our comparison of the two expansions (4.1.1) and (4.1.2) are entirely parallel. The global intertwining relation (4.2.6) and the formula (4.2.11) for the global trace (4.2.1) are obviously designed to be used together. They have important implications for the two expansions (4.1.1) and (4.1.2), which we shall analyse over the next few sections. However, we should first describe twisted versions of the formulas for GLpN q, since they will also be needed.
4.2. ON THE GLOBAL INTERTWINING RELATION
183
r pN q, rather than an eleSuppose then that G equals the component G ment in Ersim pN q. Then G represents the triplet
r pN q0 , θrpN q, 1 G
GLpN q, θrpN q, 1 ,
as in the general discussion of Chapter 3. We take ψ to be a parameter in the set r p Gq Ψ r pN q Ψ G r pN q , Ψ and f to be a function in the space
r pG q H r pN q H G r pN q . H
We then have the spectral expansion (4.1.1) (which has been established in general), and the endoscopic expansion (4.1.2) (provided by Hypothesis 3.2.1). One of our aims has been to arrange matters so that the twisted case r pN q does not require separate notation. It is for this reason that we GG write p Sψ Sψ pGq Cent Impψ q, G and
Sψ {Z pGp0qΓ Sψ {Z pGp0q, p These two sets are bitorsors under the p is the G p 0 -torsor G p 0 θ. where G Sψ
corresponding complex reductive groups Sψ
and
Sψ pG0q CentpImpψq, Gp0
Sψ
Sψ {Z pGp0qΓ Sψ {Z pGp0q.
Similarly, the quotient Sψ
π0pS ψ q
Sψ
π0pS ψ q.
is a bitorsor under the finite group
Observe that Sψ is a product of general linear groups, and that
p 0 q Z pG p 0 qΓ equals the group C , embedded diagonally in S . In parZ pG ψ ticular, the groups Sψ and S ψ are both connected. Therefore S ψ equals S 0ψ , so the sets Sψ and Sψ are both trivial in this case. Despite the fact that Sψ is trivial, the constructions of §4.1 are still interesting when applied to the bitorsor S S ψ . Proposition 4.1.1 gives a noteworthy relationship among the associated numbers ipS q ιpS ψ q and σ pSs0 q σ pS 0ψ,s q. The properties formulated in terms of S ψ at the end of §4.1 still make sense, and provide a chain of subsets (4.1.13) of the set r pGq. In this case, the subsets Ψsim pGq and Ψ2 pGq are equal. ΨpGq Ψ The formula of Corollary 4.1.3 also makes sense as stated, that is, with G1 representing any datum in the set Eell pGq Erell pN q. It is an immediate
184
4. THE STANDARD MODEL
consequence of Theorem 4.1.2 (with G being as originally understood, an element in Ersim pN q.) The earlier discussion of this section also extends to the twisted case. The diagram (4.2.3) becomes simpler, since the sets in the lower horizontal exact sequence are now all trivial. It follows that the sets Nψ and Wψ are equal, and are both bi-torsors under the finite group Wψ0 . The families Ψ2 pM, ψ q, ΨpM, ψ q, ΨpG1 , ψ q, etc., all have obvious meaning. It is understood here that M is as in the twisted form of (4.1.1), a Levi subgroup of G0 GLpN q, which we assume is proper. Furthermore, the notation (4.2.4) and (4.2.5) continues to make sense. In the case of (4.2.4), we note r ψ Πψ attached to any ψM P Ψ2 pM, ψ q contains one repthe packet Π M M resentation πM , and that the corresponding global factor xu r, π rM y is trivial. The other definition (4.2.5) is formulated in terms of the correspondence
pG1, ψ1q ÝÑ pψ, sq,
G1
P E pGq, s P S ψ,ss,
which is given by the obvious construction. The statements of Corollaries 4.2.1 and 4.2.4 also extend without change in notation. In the formula (4.2.11) of Corollary 4.2.4, the factors |SψM | and εψM pxM q are both equal to 1, while rP pw, ψM q represents a global normalizing factor for G0 GLpN q. The derivation of this formula is similar. (The formula cannot be described as a corollary in this case, since Theorem 4.2.2 no longer has a role.) The proof of the twisted global intertwining relation (4.2.6), or rather its reduction to the local relation Corollary 2.5.4, also remains the same. We recall that Corollary 2.5.4 depends on the definitions provided by Theorem 2.2.1 (which of course still has to be proved). The discussion of this section thus holds uniformly if G is either an r pN q itself. We shall apply it in this element in Ersim pN q or if G equals G generality over the next few sections. 4.3. Spectral terms We are now at the stage where we can begin the proof of our results in earnest. There are essentially nine theorems. They are the original three theorems of classification, Theorems 1.5.1, 1.5.2 and 1.5.3 (which for their statements implicitly include Theorems 1.4.1 and 1.4.2), their local refinements Theorems 2.2.1, 2.2.4, 2.4.1 and 2.4.4 from Chapter 2, and the global refinements Theorems 4.1.2 and 4.2.2. We shall prove all of the theorems together. The argument will be a multilayered induction, which will take up much of the rest of the volume. We need to be clear about our induction assumptions. For a start, they are to be distinguished from our structural condition (Hypothesis 3.2.1) on the stabilization of the twisted trace formula. Our primary induction hypothesis has already been applied informally (and often implicitly) a number of times. It concerns the rank N .
4.3. SPECTRAL TERMS
185
Recall that the nine theorems all apply to a group G in the set r pN q , where Ersim pN q Esim G r pN q GLpN q θrpN q, G
for some positive integer N . We fix N . We then assume inductively that the theorems all hold if G is replaced by any group G P Ersim pN q (and hence also for any G in one of the larger sets Erell pN q or ErpN q), for any positive integer N with N N . We will later have occasion to take on more delicate induction hypotheses. We assume for the rest of this chapter that F is global. For any r pN q, we have been writing RG G P Ersim pN q and ψ P Ψ disc,ψ for the repre-
r pGq on the ψ-component sentation of the symmetric global Hecke algebra H 2 Ldisc,ψ GpF qzGpAq of the discrete spectrum. It follows from Corollary 3.4.3 (one of the results we have established, rather than just stated) that the r pGq on the full discrete spectrum is a direct sum over representation of H r pN q of the subrepresentations RG ψ P Ψ disc,ψ . Theorem 1.5.2 (one of our G stated, but yet unproven theorems) describes the decomposition of Rdisc,ψ into irreducible representations, at least up to an understanding of the local r ψ . It asserts that RG constituents of the packet Π disc,ψ is zero if ψ does not r 2 pGq of Ψ r pN q, and that if ψ does lie in Ψ r 2 pGq, then belong to the subset Ψ G Rdisc,ψ
where mψ pπ q
#
à
P
rψ π Π
mψ , if 0
mψ pπ q π,
x , πy εψ 1 εψ ,
otherwise.
This is of course one of the assertions we can apply inductively (having already done so implicitly in the last section). Its validity for G, however, will be among the most difficult things we have to prove. r pGq of Ψ r pN q, but Suppose for example that ψ belongs to the subset Ψ r lies in the complement of Ψ2 pGq. Then ψ factors through a parameter ψM P Ψ2 pM q, for a proper Levi subgroup M of G. By our induction assumption, applied to the factors of M as a product of groups, the image r 2 pM, ψ q contributes to the automorphic discrete spectrum of M of ψM in Ψ (taken modulo the center). The theory of Eisenstein series then tells us that ψ contributes to the continuous spectrum of G. Theorem 1.5.2 includes the assertion that ψ does not contribute to the discrete spectrum of G. In other words, the automorphic spectrum of G has no embedded eigenvalues, in the sense of unramified Hecke eigenvalues in Caut pGq. This question is obviously an important part of the theorem. It will be a focal point of the long term study we are about to undertake. For the next few sections, we shall be working from the slightly broader perspective reached at the end of the last section. That is, we assume that G
186
4. THE STANDARD MODEL
r pN q itself. The basic expansions is either an element in Ersim pN q or equal to G (4.1.1) and (4.1.2), which are our starting points, were originally formulated in this generality. The rest of the discussion of §4.1 and §4.2 (apart from Theorem 4.1.2, which is expressly limited to the case that G is a connected quasisplit group) also applies. We note that the operator G Rdisc,ψ p f q,
f
P HrpGq,
r pN q. It is defined by the extension (4.2.8). Equivamakes sense if G G lently, it is simply the composition of the two factors in (4.2.1), under the condition that M P G0 . In particular, its trace gives the corresponding contribution to the term with M G0 in (4.1.1). The other contribution to this term comes from the sole element w θ in Wreg pM q. We observe that
| detpw 1qa |1 | detpθ 1qa |1 |κG|1, G M
G G0
by (3.2.5), recalling at the same time that
|κG|1 |π0pκGq|1 12 ,
r pN q at hand. The operator RG in the case G G disc,ψ pf q is straightforward in this case, since we know from Theorem 1.3.2 that GLpN q has no embedded r sim pGq. eigenvalues. In particular, the operator equals 0 unless ψ belongs to Ψ We include it in the discussion in order to have uniform statements of the results. r pN q and f P H r pGq. We are looking at the expansion (4.1.1), We fix ψ P Ψ r pN q itself. The with G now allowed to be either an element in Ersim pN q or G 0 term with M G equals the product
G G pf q . pf q |κG|1tr Rdisc,ψ rdisc,ψ
It follows that the difference G Idisc,ψ pf q rdisc,ψ pf q
(4.3.1)
equals the sum of those terms in (4.1.1) with M G. The summand of M depends only on the W0G -orbit of M . We can therefore sum over the set tM u of W0G-orbits in L distinct from G, provided that we multiply by the number |W0G||W0M |1|W pM q|1 of elements in a given orbit. The difference (4.3.1) therefore equals ¸
tM utGu
|W pM q|1
¸
P
p q
w Wreg M
| detpw 1qa |1 tr MP,ψ pwq IP,ψ pf q . G M
We can apply Corollary 4.2.4, which is founded on our underlying induction hypothesis, to the factor
tr MP,ψ pwq IP,ψ pf q .
4.3. SPECTRAL TERMS
The difference (4.3.1) becomes a fourfold sum, over tM u w , ψ q and u P Nψ pwq, of the product of w P Wreg pM q, ψM P Ψ2 pM
187
t G u,
|W pM q|1| detpw 1qa |1 G M
with
|Sψ |1rP pw, ψM q εψ purq fGpψ, uq. M
M
Despite its notation, the second factor is independent of the representative ψM of ψ. We will therefore be able to remove the sum over the set of ψM , so long as we multiply the summand by its order. The result will be clearer if we first rearrange the double sum over w and ψM . w , ψ q as a double sum We start by writing the sum over ψM P Ψ2 pM over ψG in ΨpG, ψ q and ψM in Ψ2 pMw , ψG q, the subset of parameters in w , ψ q that map to the G-orbit p Ψ2 pM ψG . We then interchange the sums over w and ψG . This allows us to combine the resulting sums over w and ψM into a double sum over the set
pψM , wq P Ψ2pM, ψGq Wreg pM q : Intpwq ψM ψM Now, the projection pψM , wq Ñ w gives a canonical fibration Vψ ÝÑ tWψ,reg u Vψ
(
.
of Vψ over the set of orbits of Wψ by conjugation on Wψ,reg . We claim p that the group W pM q acts transitively on the fibres. Since ψG is a G-orbit, 0 and since AM x is a maximal torus in SψM , for any ψM P Ψ2 pM, ψ q, any two
p that stabilizes parameters in Ψ2 pM, ψG q are conjugate by an element in G AM x . Therefore W pM q acts transitively on Ψ2 pM, ψG q. The claim follows from the fact that the stabilizer in W pM q of any parameter in Ψ2 pM, ψG q is isomorphic to Wψ . The summand is constant on the fibres. We can therefore replace the double sum over Vψ with a constant multiple of a simple sum over tWψ,reg u. The scaling constant is slightly simpler if we take the simple sum over the set Wψ,reg , rather than its quotient tWψ,reg u. With this change, we have then only to multiply the summand by the integer
|Ψ2pM, ψGq| |W pM q||Wψ |1. For any ψG , there is at most one orbit tM u such that the set Ψ2 pM, ψG q is not empty. Different elements ψG P ΨpG, ψ q can give different orbits tM u, but the corresponding summands remain equal. We can therefore remove the exterior sum over ψG , provided that we multiply the summand by the second integer |ΨpG, ψq| mψ . We conclude that the difference (4.3.1) equals (4.3.2) ¸ ¸ mψ |Wψ |1 | detpw 1qaG |1|SψM |1 rP pw, ψM q εψM purq fGpψ, uq, w
M
u
188
4. THE STANDARD MODEL
where w and u are summed over Wψ,reg and Nψ pwq respectively, and pM, ψM q w , ψ q. Of course, if there is no is any pair with M G0 and ψM P Ψ2 pM such pM, ψM q, the expression is understood to be 0. This is the case if ψ r 2 pGq or in the complement of the subset Ψ r pGq of Ψ r pN q. lies in either Ψ r pN q, which is now part of our analysis, is generally The case that G G the simpler of the two. However, it does have one minor complication. This is because aG , the p 1q-eigenspace of the operator θ on aG0 , is a proper subspace of aG0 . (In fact, it is clear that dim aG0 1 and dim aG 0.) We obtain a decomposition
| detpw 1qa | | detpw 1qa | | detpw 1qa |,
for any w to write (4.3.3)
P Wψ .
G0 M
G M
G G0
Since the restriction of w to aG0 equals θ, we use (3.2.5)
| detpw 1qa | | detpθ 1qa | |κG| G G0
G G0
for the second factor on the right. The first factor may be written as
| detpw 1qa | | detpw 1q|, where the operator pw 1q on the right is to be regarded as an endomorphism G0 M
0
of the real vector space aT ψ , as in §4.1. Indeed, aG M is isomorphic to the dual of aT ψ , since as a maximal torus in the complex connected group S 0ψ
S ψ ,
p0 Γ the group T ψ is isomorphic to AM x {Z pG q . We substitute the resulting product into our expression (4.3.2) for the difference (4.3.1). We also add a couple of housekeeping changes to the notation in (4.3.2). The group SψM is isomorphic to the group Sψ1 in the diagram (4.2.3). It then follows from the relevant two short exact sequences in the diagram that
|Wψ |1|Sψ |1 |Wψ |1|Sψ1 |1 |Nψ |1 |Sψ |1|Wψ0 |1. If u is any element in the coset Nψ pwq, we set rq ε1ψ puq εψ pu M
M
and
rψG pwq rP pw, ψM q,
since these functions depend only on ψ. With these changes, we have now left ourselves with no reference to M in our expression for (4.3.1). This is not a problem, since the orbit of M is still implicit in the torsor S ψ . However, we will need a marker to rule out the case that M G0 . We take care of this by setting 1 tw P Wψ,reg : w 1u, Wψ,reg since it is clear that W1
ψ,reg
#
H,
if S ψ is finite,
Wψ,reg , otherwise.
4.3. SPECTRAL TERMS
189
Collecting the various terms, we see that the difference (4.3.1) equals (4.3.4)
Cψ |Wψ0 |1
¸
¸
1 u Nψ w w Wψ,reg
P p q
P
where (4.3.5)
Cψ
| detpw 1q|1rψGpwq ε1ψ puq fGpψ, uq,
mψ |κG|1|Sψ |1.
r pGq of Ψ, r then mψ Keep in mind that if ψ does not belong to the subset Ψ and Cψ both vanish, and so therefore does the expression. The expression consequently makes sense, even though Sψ is not defined in this case. There is a critical lemma, which gives a different way to express the product of the two signs rψG pwq and ε1ψ puq. To state it, we need to recall the original sign function s0 pwq s0ψ pwq on Wψ that came up in (4.1.5). If only for reasons of symmetry, we shall write
s0ψ puq s0ψ pwu q,
for its pullback to Nψ . Then s0ψ puq equals p1q raised to a power given by the number of roots of pB ψ , T ψ q mapped by wu to negative roots. This object depends on ψ only through the group Wψ . It it not to be confused with the deeper sign character εψ pxq εG ψ pxq on Sψ , defined by (1.5.6) in terms of symplectic root numbers attached to ψ, or its twisted analogue ε1ψ puq for M . It is of course also unrelated to the fixed point sψ in Sψ . Lemma 4.3.1. The sign characters satisfy the relation 0 rψG pwu q ε1ψ puq εG ψ pxu q sψ puq,
u P Nψ .
We shall give the proof of this lemma in §4.6. Assuming it for now, we make the appropriate substitution in the last expression (4.3.4) for the difference (4.3.1). The next step is to rearrange the double sum in (4.3.4) by using the vertical exact sequence of Nψ in the diagram (4.2.3) in place of the horizontal exact sequence. Accordingly, we write N ψ px q,
x P Sψ ,
for the fibre of x under the mapping from Nψ to Sψ . We also write Wψ pxq for the bijective image of Nψ pxq in Wψ . We then let Nψ,reg pxq and N1ψ,reg pxq denote the preimages in Nψ pxq of the respective subsets Wψ,reg pxq Wψ pxq X Wψ,reg
and
1 pxq Wψ pxq X W 1 Wψ,reg ψ,reg
of Wψ pxq. With this notation, we obtain the following result. Lemma 4.3.2. The difference G Idisc,ψ pf q rdisc,ψ pf q
190
4. THE STANDARD MODEL
equals (4.3.6)
Cψ
¸
P
0 1 εG ψ pxq |Wψ |
x Sψ
¸ u N1ψ,reg x
P
pq
s0ψ puq | detpwu 1q|1 fG pψ, uq,
P Ψr pN q and f P HrpGq. The existence of the pairing xxM , πM y for M G0 , which is implicit in
for any ψ
(4.2.4), is part of our basic induction hypothesis. Suppose that we happen to know also that it is defined when M G0 . Since ψ is assumed to belong r M pGq in (4.2.4), this is only relevant to the case that ψ P Ψ r 2 pGq. It to Ψ is essentially a local hypothesis, namely that for any ψG P Ψ2 pG, ψ q, the localization ψG,v belongs to Ψ pGv , ψv q, and the local pairing xxG,v , πG,v y is defined. In particular, it is weaker than asking that the global multiplicity formula of Theorem 1.5.2 hold. However, it does allow us to form the difference 0 G rdisc,ψ
G pf q |κG|1mψ |Sψ |1 pf q rdisc,ψ
¸
P
x Sψ
εG ψ pxq
¸
P
rψ π Π
xx, πy fGpπq,
G pf q and its expected value. between the (normalized) trace rdisc,ψ It follows easily from the definitions (4.2.4) and (4.3.5) that
(4.3.7)
0 G rdisc,ψ
G p f q Cψ pf q rdisc,ψ
¸
P
εG ψ pxq fG pψ, xq,
x Sψ
r 2 pGq. In particular, the existence of the pairing xxM , πM y for in case ψ P Ψ 0 r 2 pGq. In M G amounts to the existence of the function fG pψ, xq for ψ P Ψ this case, the second term in the difference (4.3.7) is equal to the expresssion given by Lemma 4.3.2, but with the index of summation Nψ,reg pxq in place of N1ψ,reg pxq. This is of course because Nψ,reg pxq equals the point txu when ψ r 2 pGq. (The original indexing set N1 belongs to Ψ ψ,reg pxq is empty in this case.)
r 2 pGq, on the other hand, N1 If ψ does not lie in Ψ ψ,reg pxq equals Nψ,reg pxq. In this case, we simply set
G pf q rdisc,ψ p f q, G since the expected value of the trace rdisc,ψ pf q is then equal to 0. It follows G that the formula of Lemma 4.3.2 remains valid with 0 rψ,disc pf q in place of G 1 rψ,disc pf q, and Nψ,reg pxq in place of Nψ,reg pxq. Suppose that in addition to being well defined, the function fG pψ, uq
(4.3.8)
0 G rdisc,ψ
depends only on the image x of u in Sψ . This is implied by the global intertwining identity, but is obviously weaker. The function can then be taken outside the sum over u in Nψ,reg pxq. Lemma 4.3.2, or rather its minor extension above, can then be formulated as follows. Corollary 4.3.3. Suppose that the function fG pψ, xq fG pψ, uq,
x P Sψ , u P Nψ pxq,
4.4. ENDOSCOPIC TERMS
191
is defined , and depends only on x. Then the difference G Idisc,ψ pf q 0 rdisc,ψ pf q
equals (4.3.9)
¸
Cψ
P
iψ pxq εG ψ pxq fG pψ, xq,
x Sψ
where
iψ pxq |Wψ0 |1
¸
P
pq
s0ψ pwq| detpw 1q|1 .
w Wψ,reg x
There is perhaps more notation in this section than is strictly necessary. As in other sections, it has been designed to suggest how various results fit together, and thus offer some guidance to the course of the overall argument. In the case here, it is meant to emphasize the intrinsic symmetry between formulas in this section and those of the next. 4.4. Endoscopic terms This section is parallel to the last one. It is aimed at the terms in the endoscopic expansion (4.1.2) of Idisc,ψ pf q. We shall derive a finer expansion of (4.1.2) that can be compared with the one we have established for (4.1.1). r pN q We are working in the setting adopted in the last section. Then G r pN q is fixed, and G is allowed to denote either an element in Ersim pN q or G r pN q is also fixed, as is the function of f P H r pG q. itself. The parameter ψ P Ψ The terms in (4.1.2) are parametrized by elliptic endoscopic data G1 P Eell pGq. The subset Esim pGq of Eell pGq is of particular interest, because its terms are not amenable to induction. In recognition of this, we set sG disc,ψ pf q
¸
G1 Esim G
P
p q
1 pf 1 q. ιpG, G1 q Spdisc,ψ
If G1 belongs to the complement of Esim pGq in Eell pGq, it is represented by a product of groups to which we can apply our induction hypothesis. In particular, we can apply the stable multiplicity identity of Corollary 4.1.3 1 to the corresponding term Sdisc,ψ pf 1q in (4.1.2). We can of course also apply the specialization at the end of §3.2 of the general formula (3.2.4) to the coefficient ιpG, G1 q in (4.1.2). We see that the difference Idisc,ψ pf q sG disc,ψ pf q
(4.4.1) equals (4.4.2)
¸ G1
ιpG, G1 q
¸ ψ1
1 1 pf 1 q , Spdisc,ψ
where G1 is summed over the complement of Esim pGq in Eell pGq and ψ 1 is summed over ΨpG1 , ψ q, while (4.4.3)
1 1 pf 1 q |Sψ1 |1 σpS 0 1 q ε1 pψ1 q f 1 pψ1 q Spdisc,ψ ψ
192
and (4.4.4)
4. THE STANDARD MODEL
p 1 qΓ |1 |OutG pG1 q|1 . ιpG, G1 q |κG |1 |Z pG
p 1 -orbits of L-embeddings of We recall that ΨpG1 , ψ q is the set of G L 0 Lψ SLp2, Cq to G that map to ψ, and that 1 ε1 p ψ 1 q εG ψ 1 ps ψ 1 q . r pGq, a subset of Ψ r pN q that is proper if G P Ersim pN q If ψ does not lie in Ψ r r and is equal to ΨpN q if G GpN q, each set ΨpG1 , ψ q is empty. The difference (4.4.1) thus vanishes in this case. We can therefore assume that ψ does lie r pG q . in Ψ The computations of the present section are again founded on the centralizer p 0 qΓ . S ψ Sψ {Z pG
p 0 -orbit) under our assumption that This object is well defined (as a G r ψ P ΨpGq. It is represented by the original complex reductive group (4.1.10) if G P Ersim pN q, and a bitorsor under the complex connected reductive group
S ψ
Sψ pG0q{Z pGp0qΓ,
r pN q. in case G G Any semisimple element s P S ψ,ss in S ψ gives rise to a pair
pG1, ψ1q, G1 P E pGq, ψ1 P ΨpG1, ψq, which maps to pψ, sq under the correspondence (1.4.11) of §1.4. The subset ( S ψ,ell s P S ψ,ss : |Z pS 0ψ,s q| 8 of S ψ,ss consists of those elements s such that the endoscopic datum G1 G1s belongs to Eell pGq. Our concern will be the subset ( S 1ψ,ell s P S ψ,ell : G1s R Esim pGq of S ψ,ell . In the case that G P Ersim pN q, for example, S 1ψ,ell is just the subset of elements s P S ψ,ell with s 1. Following notation from §4.1, we write Eψ,ell E pS ψ,ell q and
1 Eψ,ell
E pS 1ψ,ellq
for the set of orbits in S ψ,ell and S 1ψ,ell respectively under the action of S 0ψ by conjugation. These are of course contained in the finite set Eψ
E pS ψ q E pS ψ,ssq
defined in §4.1. Our aim is to replace the double sum over G1 and ψ 1 in 1 . This leads to some our expression for (4.4.1) by a simple sum over Eψ,ell rescaling constants, which we need to compute.
4.4. ENDOSCOPIC TERMS
Consider the set of pairs
1 pG q y Ydisc,ψ
pψG, sGq
193
(
,
where ψG is an actual L-homomorphism from Lψ SLp2, Cq to L G0 that maps to ψ, and sG belongs to S 1ψG ,ell . The centralizer set S 1ψG ,ell here is obviously the analogue of S 1ψ,ell , which is to say that it is defined directly for the L-homomorphism ψG rather than some unspecified representative of the equivalence class of L-homomorphisms ψrG attached to ψ in §1.4. There p 0 on Y 1 pG, ψ q, defined by conjugation is a natural left action of the group G ell on the two components of a pair y. We would like to replace the double sum over G1 and ψ 1 in (4.4.2) by a sum over the corresponding set p 0 zzY 1 G disc,ψ pGq
p 0 -orbits in Y 1 of G disc,ψ pGq. (We will denote general orbit spaces by a double bar, so we can reserve a single bar for the special case of cosets.) 1 pGq is the image of a unique pair Any pair y pψG , sG q in Ydisc,ψ 1 1 x pG , ψ q, under the correspondence (1.4.11). It is understood here that G1 is simply an elliptic (nonsimple) endoscopic datum for G, taken up to p by Z pG p 0 qΓ , and translation of the associated semisimple element s1 P G that ψ 1 is an actual L-homomorphism from Lψ SLp2, Cq to L G1 that maps to ψ. However, it is the isomorphism class of pG1 , ψ 1 q, as an element in Eell pGq ΨpG1 , ψ q, that indexes the double sum (4.2.2). This class maps to p 0 -orbit of pψG , sG q. The summand of pG1 , ψ 1 q actually dea subset of the G p 0 -orbit, a property that follows easily from the fact that pends only on the G r pGq of HpGq. We have therefore to count the f belongs to the subspace H p 0 -orbit of pψG , sG q. number of isomorphism classes pG1 , ψ 1 q that map to the G p 0 -orbit of y pψG , sG q, we first note that the stabilizer To describe the G 0 p is the group of ψG in G
SψG
Sψ pG0q Cent ImpψGq, Gp0 G
.
p 0 is the stabilizer It then follows that the stabilizer of y in G
Sy
Sψ
G ,sG
g
P Sψ
G
: gsG g 1
sG
(
p 0 qΓ -coset sG in S . The orbit of y in G p 0 is therefore bijective of the Z pG ψG
p 0 {S . with the quotient coset space G y p 0 -orbit of y, we note To describe the isomorphism classes pG1 , ψ 1 q in the G 1 p 0 -orbit. Its stabilizer that the class of G in Eell pGq can also be treated as a G p 0 equals the subgroup AutG pG1 q. (These assertions follow directly from in G the definitions in [KS, p. 18–19], and the fact that we are regarding G1 as a p 0 qΓ -orbit of endoscopic data.) The natural mapping from the orbit of y Z pG to the class of G1 can then be identified with the projection p 0 {S G y
ÝÑ
p 0 {AutG pG1 q. G
194
4. THE STANDARD MODEL
In particular, the stabilizer of ψ 1 in AutG pG1 q can be identified with Sy . However, the class of ψ 1 in ΨpG1 , ψ q is not its orbit under AutG pG1 q, but p 1 of AutG pG1 q. This rather its orbit (by conjugation) under the subgroup G p 0 -orbit of is the reason that there are several isomorphism classes in the G 1 1 p y. Now the G -orbit of ψ is the same as its orbit under the product p 1 Z pG p 0 qΓ , IntG pG1 q G
p 0 qΓ commutes with the image of ψ 1 . We recall that IntG pG1 q is a since Z pG normal subgroup of AutG pG1 q, whose quotient is the finite group OutG pG1 q. p 0 -orbit It follows from these remarks that the set of pairs pG1 , ψ 1 q in the G of y is bijective with the set
AutG pG1 q{Sy IntG pG1 q OutG pG1 q{ Sy IntG pG1 q{IntG pG1 q . Writing
Sy IntG pG1 q{IntG pG1 q Sy {Sy
X Gp1 Z pGp0qΓ,
we see that the number of such pairs is equal to the quotient (4.4.5)
|OutGpG1q| |Sy {Sy X Gp1 Z pGp0qΓ|1.
We have established that the double sum over G1 and ψ 1 in (4.4.2) can p 0 -orbits of pairs y pψG , sG q, provided that be replaced by a sum over G the summand is multiplied by (4.4.5). The next step is to replace the new sum by an iterated sum over ψG and sG . The outer sum will be over the set p 0 -orbits of L-homomorphisms ψG that map to the given element ΨpG, ψ q of G r pGq. Since the function lies in H r pGq, the corresponding summand is ψPΨ independent of ψG , and the outer sum collapes. In other words, we can identify ψG with ψ, provided that we multiply by the order (4.4.6)
mψ
|ΨpG, ψq|.
p 0 is by definition the group The stabilizer of ψG in G
SψG pG0 q Sψ pG0 q Sψ .
The remaining inner sum can therefore be taken over the finite set SψG pG0 qzzS 1G,ell
1 Sψ zzSψ,ell S ψ zzS 1ψ,ell
s in S 1ψ,ell, under action by conjugation of either of the groups p 0 qΓ . Sψ or S ψ Sψ {Z pG We prefer to take orbits of the connected component S 0ψ pS ψ q0 of S ψ rather than S ψ . The stabilizer in S ψ of an element s P S 1ψ,ell is the of orbits sG
centralizer
S ψ,s
Sψ,s{Z pGp0qΓ Centps, S ψ q
4.4. ENDOSCOPIC TERMS
195
of s in S ψ . The S ψ -orbit of s is therefore bijective with S ψ {S ψ,s . The S 0ψ -orbit of s is bijective with S 0ψ {S ψ,s , where we recall that S ψ,s is the centralizer of s in S 0ψ . We can therefore sum s over the set S 0ψ zzS 1ψ,ell
1 E pS 1ψ,ellq Eψ,ell of S 0ψ -orbits in S 1ψ,ell , rather than the set S ψ z S 1ψ,ell above, provided that we multiply the summand by the quotient
|S ψ,s{S ψ,s| |S ψ {S 0ψ |1.
(4.4.7)
The last few observations have been directed at the sum over G1 and ψ 1 in (4.4.2). We have now established that this double sum can be replaced 1 , provided that the summand is multiplied by a simple sum over the set Eψ,ell by the product of the factors (4.4.5), (4.4.6), and (4.4.7). The summand itself becomes the product of these three factors with the right hand sides of (4.4.4) and (4.4.3). We thus obtain an expanded formula for the difference (4.4.1) with which we started. Multiplying the various factors together, we 1 of the product of see that (4.4.1) can be written as the sum over s in Eψ,ell two expressions
|Sψ,s{Sψ,s X Gp1Z pGp0qΓ|1 |Sψ1 |1 |Z pGp1q|1 |S ψ,s{S ψ,s|
and
|κG|1 mψ |S ψ {S 0ψ |1 σpS 0ψ1 q ε1pψ1q f 1pψ1q,
in which pG1 , ψ 1 q maps to the pair y
pψ, sq pψG, sGq.
We consider each of the two expressions in turn. To simplify the first expression, we note that S 0ψ,s
pS ψ,sq0 Centps, S 0ψ q0,
and hence that
|Sψ1 | |π0pS ψ1 q| S ψ,s X Gp1{pS ψ,sq0 Z pGp1q,
p 1 denotes the quotient where G
p 0 q{Z pG p0 q G p 1 {Z pG p 1 q. p 1 Z pG G
Consequently,
|Sψ,s{Sψ,s X Gp1Z pGp0q|1|Sψ1 |1 |S ψ,s{ S ψ,s X Gp1|1S ψ,s X Gp1{pS ψ,sq0 Z pGp1q1 |S ψ,s{S 0ψ,s Z pGp1q|1.
The first expression therefore equals
|S ψ,s{S 0ψ,s Z pGp1q|1|Z pGp1q|1,
196
4. THE STANDARD MODEL
a product that also equals
|π0pS ψ,s|1|S 0ψ,s X Z pGp1q|1.
The product of the first three factors in the second expression equals the constant Cψ of (4.3.5), since
|S ψ {S 0ψ | |S ψ {S 0ψ | |Sψ |. The fourth factor σ pS 0ψ1 q in the expression has been defined (4.1.8) if S 0ψ1
is replaced by any complex, connected reductive group S1 . We recall the property (4.1.9) from §4.1, which asserts that σ pS1 q σ pS1 {Z1 q |Z1 |1 ,
for any central subgroup Z1 of S1 . We can therefore write p1 q σ pS 0ψ1 q σ S 0ψ,s {S 0ψ,s X Z pG
σpS 0ψ,sq |S 0ψ,s X Z pGp1q|.
Taking the product of the resulting two expressions, we conclude that the difference (4.4.1) equals (4.4.8)
¸
Cψ
1 sPEψ,ell
|π0pS ψ,sq|1 σpS 0ψ,sq ε1pψ1q f 1pψ1q,
where pG1 , ψ 1 q maps to the pair pψ, sq. As at this point in the discussion of the last section, we will need a lemma on signs. Its purpose is to express the sign 1 ε1 p ψ 1 q εG ψ 1 ps ψ 1 q
explicitly in terms of pψ, sq. To state it, we write xs for the image of s in Sψ . Lemma 4.4.1. The sign characters for G and G1 satisfy the relation 1 G εG ψ 1 psψ 1 q εψ psψ xs q, for any elements G1 , ψ 1 and s such that pG1 , ψ 1 q maps to pψ, sq.
We shall give the proof of this lemma in §4.6, along with that of its predecessor Lemma 4.3.1 from the last section. Assuming it for now, we make the appropriate substitution in the expression (4.4.8) we have obtained for (4.4.1). We also write f 1 pψ 1 q as fG1 pψ, sq, according to the definition (4.2.5). We thus find that (4.4.1) equals Cψ
¸
1 s Eψ,ell
P
|π0pS ψ,sq|1 σpS 0ψ,sqεGψpsψ xsq fG1 pψ, sq.
The last expression more closely resembles that of Lemma 4.3.2 if we fibre the sum over points x P Sψ . Accordingly, we write Eψ px q,
x P Sψ ,
4.4. ENDOSCOPIC TERMS
197
for the fibre of x under the canonical mapping from Eψ to Sψ . We then set Eψ,ell pxq Eψ pxq X Eψ,ell
and
1 pxq Eψ pxq X E 1 . Eψ,ell ψ,ell With this notation, we obtain the following result . Lemma 4.4.2. The difference Idisc,ψ pf q sG disc,ψ pf q equals Cψ
¸
P
εG ψ psψ xq
x Sψ
for any ψ
¸
|π0pS ψ,sq|1 σpS 0ψ,sq fG1 pψ, sq,
1 x s Eψ,ell
P
pq
P Ψr pN q and f P HrpGq.
The existence of the linear form f 1 pψ 1 q, for any G1 in the complement of Esim pGq, is part of our basic induction hypothesis. Suppose that we happen to know also that f 1 pψ 1 q is defined for any G1 P Esim pGq and ψ 1 P ΨpG1 , ψ q. This is primarily a local assumption, namely that for any valuation v, the localization ψv1 belongs to Ψ pG1v , ψv q, and Theorem 2.2.1(a) holds for pG1v , ψv1 q. It is weaker than asking that the stable multiplicity formula of Theorem 4.1.2 hold for G1 and ψ 1 . However, it does allow us to define the difference (4.4.9)
1
0p Sdisc,ψ
1 pf 1 q pf 1q Spdisc,ψ
¸
ψ 1 Ψ G1 ,ψ
P p
q
|Sψ1 |1σpS 0ψ1 qε1pψ1q f 1pψ1q
1 pf 1q and its expected value, as well as the sum between Spdisc,ψ (4.4.10)
0 G sdisc,ψ
pf q
¸
G1 Esim G
P
p q
1 pf 1 q, ιpG, G1 q 0 Spdisc,ψ
which represents the difference between sG disc,ψ pf q and its expected value. The difference 0 G sG disc,ψ pf q sdisc,ψ pf q
equals the sum over G1 P Esim pGq and ψ 1 P ΨpG1 , ψ q of the product of the right hand sides of (4.4.3) and (4.4.4). Using (4.2.5), (4.3.5) and other definitions above, we can write it as a sum similar to that of Lemma 4.4.2. Indeed, we have only to apply the discussion above to summands indexed by pairs pG1 , ψ 1 q with G1 P Esim pGq. These indices are attached to pairs pψ, sq in which s belongs to the complement of S 1ψ,ell in S ψ,ell. In particular, the existence of the linear forms f 1 pψ 1 q for G1 P Esim pGq amounts to the existence of fG1 pψ, sq, as a function defined for s in the entire domain S ψ,ell . The arguments above carry over to these summands without change. It follows that the formula of Lemma 4.4.2 remains valid with 0 sG ψ,disc pf q in G 1 place of sψ,disc pf q, and Eψ,ell pxq in place of Eψ,ell pxq.
198
4. THE STANDARD MODEL
It is convenient to change variables in the resulting sum over s. The set of S 0ψ -orbits in S ψ pxqell is invariant under translation by the point sψ . Since sψ lies in the center of S ψ , the centralizer S ψ,s equals S ψ,sψ s . A change of variables from s to sψ s therefore yields the expression Cψ
¸
¸
P
P
|π0pS ψ,sq|1 σpS 0ψ,sq εGψpxq fG1 pψ, sψ sq.
pq
x Sψ s Eψ,ell x
This is equal to the difference between Idisc,ψ pf q and 0 sG ψ,disc pf q. Suppose that in addition to being well defined, the function fG1 pψ, sq depends only on the image x of s in Sψ . This is implied by the global intertwining identity, but is again weaker. In fact, it is easily derived from basic definitions, as we shall see in the next section. The function fG1 pψ, sψ sq above can then be taken outside the last sum over s. Lemma 4.4.2, or rather its minor extension above, can then be formulated as follows. Corollary 4.4.3. Suppose that the function fG1 pψ, xq fG1 pψ, sq,
x P Sψ , s P Eψ pxq,
is defined, and depends only on x. Then the difference Idisc,ψ pf q 0 sG disc,ψ pf q equals (4.4.11)
Cψ
¸
P
1 e ψ p x q εG ψ pxq fG pψ, sψ xq,
x Sψ
where
e ψ px q
¸
P
pq
|π0pS ψ,sq|1 σpS 0ψ,sq.
s Eψ,ell x
In Corollaries 4.3.3 and 4.4.3, we now have two parallel expansions for the discrete part of the trace formula, or rather its ψ-component Idisc,ψ pf q. The comparison of these expansions will be a central preoccupation from this point on. As an elementary example of what is to come, let us consider the simplest of cases. r pN q and G P Ersim pN q are such that ψ does not Suppose that ψ P Ψ r pGq. Then belong to Ψ 0 G rdisc,ψ
G G pf q rdisc,ψ pf q tr Rdisc,ψ pf q
and
G G pf q sGdisc,ψ pf q Sdisc,ψ pf q 0Sdisc,ψ pf q , G since the expected values of rdisc,ψ pf q and sGdisc,ψ pf q are by definition equal 0 G sdisc,ψ
to 0. Strictly speaking, the premises of the two corollaries are not valid, r pGq. However, since the relevant functions are not defined in the case ψ R Ψ
4.5. THE COMPARISON
199
the expansions of Lemmas 4.3.2 and 4.4.2 hold. The coefficient Cψ in these expansions vanishes, since the factor mψ in (4.3.5) equals 0. It follows that (4.4.12) G G G tr Rdisc,ψ pf q Sdisc,ψ pf q 0Sdisc,ψ pf q, ψ R Ψr pGq, f P HrpGq. G In particular, the trace of Rdisc,ψ pf q is a stable linear form on HrpGq in this case. We of course expect that each term in the identity vanishes, but we are not yet ready to prove this. Observe that we could also have derived (4.4.12) directly from our induction hypotheses and the original expansions (4.1.1) and (4.1.2). In particular, the identity does not depend on the sign formulas of Lemmas 4.3.1 and 4.4.1.
4.5. The comparison We shall now compare the two expansions we have established. In particular, we shall try to show that for many ψ, the conditions of the parallel expansions of Corollaries 4.3.3 and 4.4.3 are valid. We recall that ψ belongs r r to the set ΨpN q Ψ GpN q of parameters for the twisted general linear r pN q GLpN q θrpN q over the global field F . The most difficult group G r ell pN q, which is to say that ψ belongs to Ψ r 2 pGq, for case is when ψ lies in Ψ some datum G (a priori unique) in Erell pN q. Its analysis will be taken up in later chapters. In this section, we assume that ψ lies in the complement of r ell pN q. Ψ The two corollaries were based on the further assumption that two functions are defined, and depend only on a point x P Sψ . Suppose that ψ r pGq, where as in §4.3 and §4.4, G is either a given datum in belongs to Ψ r pN q. Our condition that ψ lies in the complement Ersim pN q or equal to G r of Ψell pN q implies that the functions are at least defined. For the function fG pψ, uq of Corollary 4.3.3, this follows from our induction hypothesis that r, π rM y in the definition p4.2.4q1 exists for the Levi subgroup the pairing xu M G0 attached to ψ. The function fG1 pψ, sq of Corollary 4.4.3 is also amenable to induction. Its existence reduces to the analogue of Theorem 2.2.1(a) for the group M G0 , as the reader can readily verify. Alternatively, the existence of fG1 pψ, sq can be regarded as part of an analysis of the linear form f 1 pψ 1 q in the definition p4.2.5q1 that we will undertake presently (following the statements of formulas (4.5.7) and (4.5.8) below). The main part of the assumption underlying the two corollaries is that each of the two functions depends only on the image x of u or s in Sψ . To see how the comparison works, we temporarily take on a broader assumption, which includes the premise of Corollaries 4.3.3 and 4.4.3. We suppose for the moment that the global intertwining relation holds for any G in either of the corollaries. In other words, we suppose that (4.5.1)
fG1 pψ, sψ xq fG pψ, xq,
f
P HrpGq, x P Sψ ,
200
4. THE STANDARD MODEL
r pGq, or is equal to G r pN q where G is either an element in Ersim pN q with ψ P Ψ itself. According to Corollary 4.4.3, the difference
(4.5.2)
Idisc,ψ pf q 0 sG disc,ψ pf q,
equals
¸
Cψ
P
f
P HrpGq,
1 e ψ px q ε G ψ pxq fG pψ, sψ xq.
x Sψ
Corollary 4.3.3 tells us that the difference (4.5.3)
G Idisc,ψ pf q 0 rdisc,ψ pf q,
equals
¸
Cψ
P
f
P HrpGq,
iψ pxq εG ψ pxq fG pψ, xq.
x Sψ
We have now reached the point where we appeal to Proposition 4.1.1. This result, which we recall is a kind of miniature replica of the basic identity between (4.1.2) and (4.1.1), tells us that the coefficients eψ pxq and iψ pxq in the two expressions are equal. It follows from (4.5.1) that the expressions themselves are equal. The two differences (4.5.2) and (4.5.3) are therefore equal. We conclude from the definitions (4.4.10) and (4.3.8) that (4.5.4)
¸
G pf q |κG|1 tr Rdisc,ψ
G1 Esim G
P
p q
1 pf q , ιpG, G1 q 0 Spdisc,ψ
r p G q. for any G and any f P H Suppose first that G is one of the groups in Ersim pN q. Then G1 G is the only element in the set Esim pGq. The equation (4.5.4) reduces to G tr Rdisc,ψ pf q
(4.5.5) since |κG |
G 0Sdisc,ψ p f q,
G 1 in this case. In particular, the trace of Rdisc,ψ pf q is a stable r pGq. Notice that (4.5.5) has the same form as the identity linear form on H
(4.4.12) established at the end of the last section, in case ψ does not lie r pGq. Consequently, (4.5.5) is valid if G is replaced by any datum in in Ψ r Esim pN q. r pN q. The character Next, suppose that G equals G
G tr Rdisc,ψ pf q ,
P H pG q, r ell pN q means that ψ cannot then vanishes, since our condition that ψ R Ψ 0 r contribute to the discrete spectrum of GpN q GLpN q. The general iden-
tity (4.5.4) reduces to (4.5.6)
¸
G1 Esim G
P
p q
f
1 1 pf 1 q 0 ιpG, G1 q 0 Spdisc,ψ
4.5. THE COMPARISON
201
in this case. The groups G1 here represent the data in Ersim pN q we have just treated, and for which we have established (under the condition (4.5.1)) that 0 p1 G1 Sdisc,ψ pf 1 q tr Rdisc,ψ ph1q , r pG1 q whose image in SrpG1 q equals f 1 . for any function h1 P H Continuing to suppose that the general condition (4.5.1) is valid, we combine (4.5.5) and (4.5.6), and then let G range over all of the elements in Ersim pN q. Appealing to Proposition 2.1.1, or rather its global extension described in §3.4, we see that ¸
P
p q
p
q
G Ersim N
for any compatible family of functions f
P HrpGq :
p q 0,
G r ι N, G tr Rdisc,ψ f
(
G P Erell pN q .
(The groups G that represent elements in the complement of Ersim pN q in Erell pN q are part of the original definition (3.4.11), but they play no role here. For since they are composite, we can assume inductively that the G are zero.) The coefficients corresponding representations Rdisc,ψ
p
r ι N, G
q ι GrpN q, G
G are positive. The coefficients of the decompositon of Rdisc,ψ pf q into irreducible representations are of course also positive. We can therefore apply Proposition 3.5.1, the basic vanishing property we have established for compatible families. It tells us that
(4.5.7)
G pf q 0, Rdisc,ψ
f
P HrpGq,
for any G P Ersim pN q. In other words, ψ does not contribute to the discrete spectrum of G. This in turn implies that (4.5.8)
0 G Sdisc,ψ
pf q 0,
f
P HrpGq.
G In other words, the formula of Theorem 4.1.2 for Sdisc,ψ pf q is valid for any
G P Ersim pN q. We shall see that the comparison just completed has broad application. r pN q, the condiIn particular, we shall show that for many parameters ψ P Ψ r tion (4.5.1) is indeed valid for every G with ψ P ΨpGq. This will allow us to r ell pN q. conclude that the global theorems hold for the given parameter ψ R Ψ The putative identity (4.5.1) includes the condition that each side is a well defined function of x P Sψ . In other words, each side depends only on the image in Sψ of the datum in terms of which it is defined. This is of course the premise on which the two Corollaries 4.3.3 and 4.4.3 were based. It is not difficult to verify much of it directly. We shall do so, as a necessary prelude to being able to establish cases of (4.5.1) by induction.
202
4. THE STANDARD MODEL
The left hand side fG1 pψ, sψ xq of (4.5.1) can be treated uniformly for r pN q. It is defined p4.2.5q1 in terms of a semisimple G P Ersim pN q or G G element s P S ψ whose projection onto Sψ equals x. It equals f 1 p ψ 1 q f G1 p ψ 1 q , r p G q, f PH where
p G1, ψ1q is the preimage of the pair pψ, sq, s sψ s,
under the general correspondence
pG1, ψ1q ÝÑ pψ, sq. We shall establish for any G that f 1 pψ 1 q depends only on the image x of s. Note first that if s is replaced by an S 0ψ -conjugate s1 , pG1 , ψ 1 q is replaced by its image pG11 , ψ11 q under an isomorphism of endoscopic data. It follows from the definitions that
f11 pψ11 q f 1 pψ 1 q,
f11
f G1 . 1
Since any automorphism of the complex reductive group S 0ψ stablilizes a conjugate of pT ψ , B ψ q, we can choose s1 so that Intps1 q stablilizes pT ψ , B ψ q itself. With this property, the representative sx
s1
of x in S ψ is determined up to a T ψ -translate. In particular, the complex torus T ψ,x Centpsx , T ψ q0 in T ψ is uniquely determined by x. Now any point in T ψ sx can be written in the form tsx t1 tx tsx tx t1 , t P T ψ , tx P T ψ,x . It will therefore be enough to show that the linear form fx1 pψx1 q f11 pψ11 q
remains unchanged under translation of sx by any element tx P T ψ,x . x0 of T ψ,x in G p 0 is a ΓF -stable Levi subgroup of G p0 The centralizer M x x of the underlying Levi subgroup M . Since the that contains the dual M x0 , but also automorphism Intpsx q centralizes T ψ,x , it stabilizes not only M x x0 q. The M x0 -torsor some parabolic subgroup Ppx0 P P pM x x xx M
x0 Intpsx q M x
p which is in turn dual to a can therefore be treated as a Levi subset of G, r pN q Levi subset Mx of G. (Remember that we are including the case G G in our analysis. As in §2.2, we are using terminology here that originated in [A5, §1].) Now pψ, sq is the image of a pair
pψM
x
, sMx q,
ψMx
P Ψr pMxq,
sMx
P Sψ
Mx
,
4.5. THE COMPARISON
203
attached to Mx under the L-embeddings L Mx0 L G0 and L Mx L G. 1 q for Mx . In This pair is in turn the image of an endoscopic pair pMx1 , ψM x particular, we obtain an elliptic endoscopic datum Mx1 for Mx , which can be identified with a Levi subgroup of G1 . It follows that
1 pψ 1 q, f 1 pψ 1 q fM Mx x
where
1 f Mx1 pf 1 qM 1 fM x x 1 1 1 is the transfer of f to Mx . Since fMx pψMx q does not change under translation of sx by T ψ,x , the same is true of fx1 pψx1 q. We conclude that the function fG1 pψ, xq f 1 pψ 1 q fx1 pψx1 q
does indeed depend only on x. So therefore does the left hand side fG1 pψ, sψ xq of (4.5.1). The right hand side fG pψ, xq of (4.5.1) is less clear-cut. Recall that it is defined p4.2.4q1 in terms of a point u in Nψ whose projection xu onto Sψ equals x. The fibre of x under this projection is a torsor under the action of the subgroup Wψ0 of Nψ . (We have to include the subscript , since the r pN q remains a part of the discussion.) case G G The dependence of fG pψ, xq on u is twofold. It depends on its projection w wu onto Wψ , through the normalized intertwining operator in the trace
rM , ψM q IP pπM , f q , tr RP pw, π
and on the value at u r of ξr x , π rM y, an extension of the character x , πM y 1 1 rM of from Sψ to Sψ u that in turn determines the associated extension π the representation πM . Since Sψ1 Wψ0 is a normal subgroup of Nψ , we can assume that each ξr is constant on the Wψ0 -orbit of u. That is, we can arrange that 0 uq ξrpw r0 u rq ξrpu rq, ξrpw w0 P Wψ0 .
In particular, in the case that u lies in Sψ1 Wψ0 , we take ξr to be the product of the character x , πM y on Sψ1 with the trivial character on Wψ0 . We then conclude, from the earlier stages of the proof of Proposition 2.4.3 for example, that rM pw0 wq π π rM pwq, and hence that the corresponding intertwining operators for G satisfy rM , ψM q RP pw0 , π rM , ψM q RP pw, π rM , ψM q, RP pw0 w, π
w0
P Wψ0 .
The condition that fG pψ, xq depend only on x can at this point be seen to be equivalent to the requirement that the operators RP pw0 , π rM , ψM q equal 1 for 0 0 all w P Wψ . If the operators are all trivial, the condition on fG pψ, xq follows immediately from p4.2.4q1 and the conventions above. The other direction r pGq of HpGq, is perhaps less clear, since f ranges only over the subspace H 0 rM , ψM q is unitary. Since but it can be deduced from the fact that RP pw , π
204
4. THE STANDARD MODEL
we will ultimately establish the condition on fG pψ, xq by showing directly rM , ψM q are all trivial, there will be no harm in that the operators RP pw0 , π assuming henceforth that the two conditions are indeed equivalent. We will not be in a position, though, to establish either condition in general for some time. r pN q. The global diagram (4.2.3) for G r pN q Suppose however that G G maps into the local diagram (2.4.3) attached to any completion ψv . In particular, we have an injection w0 ÝÑ wv0 from Wψ0 to Wψ0v , and a decomposition rM , ψ q RP pw0 , π
â v
rM,v , ψv q. RP pwv0 , π
The factors on the right are local normalized intertwining operators for G0 GLpN q. It follows from Theorem 2.5.3 that they are all equal to the rM , ψ q is therefore also equal to 1. identity. The global operator RP pw0 , π The right hand side fG pψ, xq of (4.5.1) therefore does depend only on x, in r pN q. the case G G Consider the other case that G P Ersim pN q. As the Weyl group of 0 pS ψ , T ψ q, Wψ0 is generated by simple reflections twα0 u. Suppose that dimpT ψ q ¥ 2.
Then any wα0 centralizes a torus of positive dimension in T ψ . The centralizer p is a proper Levi subgroup M xα of G p that contains M x, and of this torus in G is dual to a Levi subgroup Mα of G that contains M . If ψα is the image in r pMα q of the chosen parameter ψM P Ψ2 pM, ψ q, w0 belongs to the Weyl Ψ α group of pS 0ψα , T ψ q. We can assume inductively that 0 RP pwα0 , π rM , ψM q RP XMα pwα ,π rM , ψα q 1.
It follows that RP pw0 , π rM , ψM q equals 1 for any w0
P Wψ0 .
Thus, the right
hand side fG pψ, xq of (4.5.1) again depends only on x, in case G P Ersim pN q and dimpT ψ q ¥ 2. It is the complementary case that dimpT ψ q 1 that causes difficulty. We shall treat some (though not all) of it by a more sophisticated comparison argument later in the section. How do the groups Sψ Sψ pGq vary, as G ranges over elements r pGq? Suppose for a moment that G is fixed. The G P Ersim pN q with ψ P Ψ centralizer Sψ Sψ pGq is then equal to a product
(4.5.9)
¹
P p q
i Iψ G
Op`i , Cq
ψ
¹
G i Iψ
P p q
Spp`i , Cq
¹
P
GLp`j , Cq ,
j Jψ
r pG q in the notation of §1.4. As we observed in §3.4, the condition that ψ P Ψ implies that the character ηG equals ηψ . If ηψ is nontrivial or N is odd, G is the only datum in Ersim pN q with ηG ηψ . If ηψ is trivial and N is even, however, there is a second datum G_ P Ersim pN q with ηG_ ηψ . The par pG_ q if and only if `i is even for each i P I pGq. rameter ψ then belongs to Ψ ψ
4.5. THE COMPARISON
With this condition, we have Iψ pG_ q the centralizer Sψ_ Sψ_ pG_ q equals
(4.5.10)
¹
P p q
Spp`i , Cq
i Iψ G
205
IψpGq and IψpG_q Iψ pGq, and
¹
G i Iψ
P p q
Op`i , Cq
ψ
¹
P
GLp`j , Cq .
j Jψ
r ell pN q in Proposition 4.5.1. Suppose that ψ lies in the complement of Ψ r pN q, and in addition, that it lies in the complement of Ψ r ell pGq for every Ψ r G P Esim pN q. Then G tr Rdisc,ψ pf q
G 0 0Sdisc,ψ pf q,
f
P HrpGq,
for any G P Ersim pN q. In other words, the global Theorems 1.5.2 and 4.1.2 hold for G and ψ. Remarks. 1. The first condition on ψ has been a de facto hypothesis for the last three sections. The second condition is new, and has been imposed in order to avoid subtleties that will have to be treated later. It does not imply r 2 pGq, for elements G the first, since it does not preclude parameters ψ P Ψ r r in the complement of Esim pN q in Eell pN q. 2. There are other global theorems besides the two mentioned in the statement of the proposition that must be established in general. However, Theorem 1.5.3 applies to parameters that are simple (and generic), r ell pN q. It is irrelevant to the case at hand. The same and therefore lie in Ψ is true of the global Theorem 4.2.2. The remaining global assertion is the intertwining relation of Corollary 4.2.1, for both the case G P Ersim pN q and r pN q. This is just the putative identity (4.5.1). If we are able to estabGG lish it directly, we will thus obtain not just the statement of the proposition, but in fact all of the global results that pertain to ψ. Proof. We shall try to establish (4.5.1) directly by induction, and thereby obtain the required assertion from the comparison above. We shall find that this is possible for many, but not all of the given parameters ψ. There will be some cases that require a more delicate comparison. To see where the problems arise, let us begin with any parameter ψ in the compler ell pN q, and see how far we can go. ment of Ψ r pN q. We have to verify (4.5.1) in both the case G P Ersim pN q and G G The identity makes sense only if the group Sψ Sψ pGq is defined, which is r pGq. This condition always holds if G G r pN q. to say that ψ belongs to Ψ r r However, it is quite possible that ψ not lie in ΨpGq for any G P Esim pN q. If this is the case, (4.5.5) is still valid for each G, as we saw in (4.4.12), and the comparison that yields the required identities (4.5.7) and (4.5.8) can be r pGq, for any of carried out. We shall therefore assume implicitly that ψ P Ψ the data G P Esim pN q we consider below. Assume for the present that
piq
dimpT ψ q ¥ 2,
G P Ersim pN q,
206
4. THE STANDARD MODEL
as in the remarks prior to the statement of the proposition. Then as we have seen in all cases, the two sides of (4.5.1) each depend only on x. Moreover, the function on the left hand side satisfies the relation
1 pψx , xq, fG1 pψ, xq f 1 pψ 1 q fx1 pψx1 q fM x
r pMx q as in the earlier remarks. It follows easily from for Mx G and ψx P Ψ the definitions that Msψ x Mx and ψsψ x ψx . The left hand side itself therefore satisfies 1 pψx , sψ xq. fG1 pψ, sψ xq fM x
On the other hand, the earlier representative sx of x in S ψ lies in the subset N ψ of S ψ . We take its T ψ -coset ux to represent x in Nψ , and on the right hand side of (4.5.1). The coset ux centralizes the complex torus T ψ,x
Centpsx, T ψ q0
in T ψ . Since T ψ,x can be identitied with the Γ-split part of the centralizer xx , the right hand side satisfies the relation of M fG pψ, xq fMx pψx , xq.
piiq
Assume also that
dimpT ψ,x q ¥ 1,
G P Ersim pN q, x P Sψ .
Since the center of any G P Ersim pN q is finite, Mx is a proper Levi subgroup r pN q. The given of G in this case. Consider the other case that G G r condition that ψ not lie in Ψell pN q means that either the multiplicity `k of some component ψk of ψ is larger than 1, or the indexing set J Jψ p 0 has an Intpsx qis not empty. In the first instance, the subgroup Sψ of G stable factor GLp`k , Cq of positive semisimple rank. In the second, Sψ has a subgroup isomorphic to C C , whose factors are interchanged by Intpsx q. r pN q, and It follows easily that the condition (ii) holds also in case G G that Mx is again proper in G. We can therefore argue by induction. More precisely, we can assume inductively that the analogue of (4.5.1) for Mx holds in all cases. It follows that
1 pψx , sψ xq fM pψx , xq fG pψ, xq. fG1 pψ, sψ xq fM x x
The identity (4.5.1) therefore holds for G. We note for future reference that r pN q, (4.5.1) in fact holds for any ψ in the complement of in case G G r ell pN q. Ψ We have established the required identity (4.5.1) for ψ, provided that the conditions (i) and (ii) on ψ hold for any G in Ersim pN q and any x in Sψ Sψ pGq. For which ψ do the two conditions not hold? Consider the product formula (4.5.9) for the group Sψ attached to a r pGq. The second and third components given G P Ersim pN q with ψ P Ψ of the product are connected complex groups. We can therefore assume that the representative sx of x described above lies in the first component,
4.5. THE COMPARISON
207
the nonconnected group defined by the product over I pGq. To check the conditions (i) and (ii), we need only describe the rank of the complex group (4.5.9), and the rank of the centralizer of sx in (4.5.9). The two conditions clearly hold unless the third component, the product over Jψ , is either trivial or consists of just one factor GLp1, Cq C . In the second case we have an infinite central subgroup of Sψ , whose centralizer is p The identity (4.5.1) then follows by induction. a proper Levi subgroup of G. r disc pGq in this case, and that (We note that ψ lies in the complement of Ψ the global theorems also follow directly from the expansions for (4.5.2) and (4.5.3).) We may therefore assume that the indexing set Jψ for ψ is empty. The two conditions remain valid unless the second component, the product over Iψ pGq, is either trivial or consists of one factor Spp2, Cq. Putting aside the second possibility for the moment, we suppose that the set Iψ pGq is also empty. The conditions (i) and (ii) are then still valid unless `i ¤ 2 for every i P Iψ pGq, or `i 3 for one i P Iψ pGq and `i 1 for the remaining indices i. Putting aside the second possibility again, we suppose that Sψ
¹
P p q
Op`i , Cq
ψ
,
`i
P t1, 2u.
i Iψ G
Recall that p qψ stands for the kernel of a character ξψ 1 or 2. If there were an element s
¹
si ,
si
ξψ pGq of order
P O p`i , C q ,
i
in the kernel, with detpsi q 1 whenever `i 2, ψ would belong to the r ell pGq of Ψ r pGq, since the centralizer of s in Sψ would then be finite. subset Ψ This outcome is expressly ruled out by the hypotheses of the proposition. On the other hand, if there is no such s, Sψ has a central torus C . In this case, (4.5.1) again follows inductively from its reduction to a proper Levi subgroup of G. We have thus seen that (4.5.1) holds for any given ψ that does not fall into one of the two cases set aside above. In the first of these cases, the index set Jψ is empty, while the product over Iψ pGq in (4.5.9) reduces simply to Spp2, Cq. Observe that the conditions (i) and (ii) remain valid in this case unless the factors in the product over Iψ pGq are all equal to Op1, Cq Z{2Z. In the second case, the index sets Jψ and Iψ pGq are both empty, while the product over Iψ pGq reduces to a product of Op3, Cq with several copies of the group Op1, Cq Z{2Z. The two outstanding cases thus take the form #
(4.5.11)
2ψ1 ψ2 ψr 1 Sψ SLp2, Cq pZ{2Zqr 1 , ψ
`
`
`
r
¥ 1,
208
and
4. THE STANDARD MODEL
#
3ψ1 ψ2 ψr 1 Sψ SOp3, Cq pZ{2Zqr , r ¥ 1, where Sψ Sψ pGq for some G P Ersim pN q, and # r, if ξψ 1, 1 r r 1, otherwise.
(4.5.12)
ψ
`
`
`
In these remaining cases, it is understood that G P Ersim pN q is some r pGq. According to the discussion leading up to datum with ψ P Ψ (4.5.10), G is uniquely determined by this condition, except for the case of (4.5.11) with r 1. In this case, however, there is a second element r pG_ q. It then follows from (4.5.11) that the group G_ P Ersim pN q with ψ P Ψ _ _ Sψ Sψ pG q equals Op2, Cq ψ . There are two possibilities. If the char-
r ell pG_ q, a acter ξψ ,_ ξψ pG_ q is 1, Sψ_ equals Op2, Cq, and ψ lies in Ψ condition on ψ that is ruled out by the proposition. We can therefore assume that ξψ ,_ 1 if r 1 in (4.5.11). In this case, Sψ_ C , and the analogue of (4.5.1) for G_ follows inductively from its reduction to a proper Levi subgroup of G_ . Let us summarize our progress to this point. We have established (4.5.1) for any ψ that satisfies the conditions of the proposition, and that does not belong to one of the two exceptional cases (4.5.11) and (4.5.12). The required global theorems for such ψ then follow from the general comparison at the beginning of the section. It remains to treat the exceptional cases by a more subtle comparison. Suppose that ψ and G are as in (4.5.11). We cannot use the earlier comparison, because we do not know that the global identity (4.5.1) holds. However, we can still apply Corollary 4.4.3 to the difference (4.5.2), since the required condition was established in general prior to the statement of this proposition. We can also apply Corollary 4.3.3 to the difference (4.5.3). Here, we do not know that the function fG pψ, xq in the resulting expression depends only on the image x of the element u P Nψ pxq used to define it. However, in the case at hand, there is only one element w wx in the subset Wψ,reg pxq of Wψ pxq that indexes the internal sum in the expression (4.3.9) of Corollary 4.3.3, and hence only one element in u ux in the corresponding subset Nψ,reg pxq of Nψ pxq. We take the liberty of setting
fG pψ, xq fG px, ux q
in order to use the notation of Corollary 4.3.3. Substituting the identity eψ pxq iψ pxq, we see that the difference (4.5.13)
0 G sdisc,ψ
G G G pf q 0rdisc,ψ pf q 0Sdisc,ψ pf q tr Rdisc,ψ pf q
4.5. THE COMPARISON
equals Cψ
¸
P
209
1 iψ pxq εG ψ pxq fG pψ, xq fG pψ, sψ xq .
x Sψ
The constant Cψ is defined in (4.3.5) as a product
mψ |κG|1|Sψ |1 mψ |Sψ |1 of positive factors. The coefficient iψ pxq satisfies Cψ
iψ pxq |Wψ0 |1
¸
P
pq
s0ψ pwq | detpw 1q|1
w Wψ,reg x
|Wψ0 |1 s0ψ pwxq | detpwx 1q|1 14 , since |Wψ0 | 2, | detpwx 1q| 2 and s0ψ pwx q 1, in this case. It is the minus sign here that will be the critical factor. From the form (4.5.11) of Sψ , we observe that the R-group Rψ is trivial. It follows that there is an isomorphism x Ñ xM from Sψ onto SψM , where ψM represents a fixed element in Ψ2 pM, ψ q. The coefficient εG ψ pxq then satisfies M εG ψ pxq εψM pxM q εψM pxM q,
a reduction one can infer either directly from the definition (1.5.6) and the x. We can therefore write fact that we can identify x with a point xM in M
1 εψ p x M q , C ψ i ψ px q ε G ψ pxq nψ mψM |SψM | M for the positive constant nψ
41 mψ mψ 1 . M
The remaining terms in our expression for (4.5.13) are the linear forms
fG1 pψ, sψ xq and fG pψ, xq. It follows from the definitions p4.2.5q1 and (2.2.6)
that
1 q fG1 pψ, xq f 1 pψ 1 q f 1 pψM
¸
rψ π M PΠ
xsψ xM , πM y fM pπM q,
M
1 q is the endoscopic pair attached to pψM , xM q. A similar where pM 1 , ψM formula holds with sψ x in place of x and xM in place of sψ xM , since sψ psψ xqM
sψ psψ xM q psψ q2xM xM .
210
4. THE STANDARD MODEL
We can therefore write ¸
Cψ
P
1 iψ pxq εG ψ pxq fG pψ, sψ xq
x Sψ
nψ mψ |Sψ |1 M
nψ
M
¸
P
rψ πM Π M
¸
¸
P
P
xM SψM πM Π rψ M
εψM pxM q xxM , πM y fM pπM q
mψM pπM q tr IP pπM , f q ,
by applying the multiplicity formula
mψM pπM q mψM |SψM |1
¸
P
εψM pxM q xxM , πM y
xM SψM
of Theorem 1.5.2 inductively to the proper Levi subgroup M of G, and substituting the usual adjoint relation fM pπM q tr IP pπM , f q
of fM . The other linear form
fG pψ, xq fG pψ, ux q
is given by p4.2.4q1 . It can be written as ¸
P
rψ πM Π M
since wx
xxM , πM y tr RP pwx, πrM , ψM q IP pπM , f q ,
wu, and xxM , πM y xur, πrM y if u ux. We can therefore write ¸ i ψ p x q εG Cψ ψ pxq fG pψ, xq P
x Sψ
nψ
¸
P
rψ πM Π M
rM , ψM q IP pπM , f q mψM pπM q tr RP pwx , π
by again applying the multiplicity formula. We have expanded (4.5.13) into a (signed) sum of two rather similar expressions. The difference (4.5.13) thus equals (4.5.14)
nψ
¸
P
rψ πM Π M
rM , ψM q IP pπM , f q . mψM pπM q tr 1 RP pwx , π
G It follows that the stable linear form 0 Sdisc,ψ pf q on the right hand side of G (4.5.13) equals the sum of tr Rdisc,ψ pf q with (4.5.14).
We have taken G to be the fixed group in Ersim pN q that is implicit in (4.5.11). We therefore write G for a general group in Ersim pN q. If ψ does r pG q, the identity (4.4.12) (with G in place of G) is valid. This not lie in Ψ condition applies to any G G, if r ¥ 2. If r 1, we have seen that there r pG q, namely G G_ . But we have also is one group G G with ψ P Ψ
4.5. THE COMPARISON
211
r ell pG_ q), seen that ξψ ,_ 1 in this case (given our assumption that ψ R Ψ and that the analogue of (4.5.1) for G follows by induction. It thus follows from (4.5.5) that 0 G G Sdisc,ψ pf q tr Rdisc,ψ pf q , f P HrpGq,
for any G P Ersim pN q distinct from G. r pN q holds for We saw earlier in the proof that the analogue of (4.5.1) for G r any ψ R Ψell pN q, and in particular, for the case (4.5.11) we are considering. It then follows from (4.5.6) that ¸ G r ιpN, G q 0 Spdisc,ψ pfrG q 0, G PErsim pN q r pN q. Substituting for each of the stable linear forms on the for any fr P H left, we see that the sum of the expression ¸ G r ιpN, G q tr Rdisc,ψ (4.5.15) pf q G PErsim pN q
with (4.5.14) vanishes, for any compatible family of functions f
P H pG q :
G
P ErsimpN q
(
.
rM , ψM q in (4.5.14) has square The global intertwining operator RP pwx , π equal to 1. Its eigenvalues are therefore equal to 1 or 1. The expression (4.5.14) is in consequence a nonnegative sum of irreducible characters. The same is true of the other expression (4.5.15). We can therefore apply Proposition 3.5.1 to the sum of the two expressions. We conclude that
RP pwx , π rM , ψM q 1,
and that
πM
P Πr ψ pM q, M
G G pf q 0Sdisc,ψ pf q 0, tr Rdisc,ψ
f
P HrpGq,
for every G P Ersim pN q. This completes the proof of the proposition for the first exceptional case (4.5.11). The argument for the remaining exceptional case (4.5.12) is similar. It is actually slightly simpler, since in this case G is the only element in Ersim pN q r pGq. In particular, there is no distinction to be made between with ψ P Ψ r 1 and r ¡ 1. The comparison is otherwise identical, and the required global theorems in this last case follow as above. Proposition 4.5.1 thus holds in all cases. Corollary 4.5.2. Suppose that ψ is as in the proposition. Then the global intertwining relation (4.5.1) of Corollary 4.2.1 is valid unless ψ falls into one of the two exceptional classes (4.5.11) or (4.5.12), in which case we have only the weaker identity (4.5.16)
¸
P
x Sψ
1 εG ψ pxq fG pψ, sψ xq fG pψ, xq
0,
f
P HrpGq,
212
4. THE STANDARD MODEL
r pGq. If G G r pN q, the identity (4.5.1) holds for any G P Ersim pN q with ψ P Ψ r ell pN q. without restriction, and indeed, for any ψ R Ψ
Proof. The assertions of the corollary were established in the course of proving the proposition. For example, we verified (4.5.1) in the generality above in order to apply the earlier comparison. In the case (4.5.11), the identity (4.5.16) follows from the fact, established at the end of the proof, that the difference (4.5.13) vanishes. The case of (4.5.12) is similar. The most interesting aspect of the last proof is the treatment of the exceptional cases (4.5.11) and (4.5.12). For example, if r ¡ 1 in (4.5.11), G is r pGq. The same goes for (4.5.12), with the only datum in Ersim pN q with ψ P Ψ r r ¥ 1. If G P Esim pN q is distinct from G, and satisfies the further condition that ξG ξψ , we know a priori that RψG 0 and 0 SψG 0. However, this does not account for the case that G G_ , in which ηG ηψ 1 and N is even. We really do need the argument above to deal with this case, as well as to compensate for not knowing (4.5.1) in case G G. Looking back at the argument, the reader can see that we have been lucky with the sign s0ψ pwx q, in which wx was the unique element in Wψ,reg pxq. If s0ψ pwx q had somehow turned out to equal 1 instead of 1, the argument would have failed, perhaps dooming the entire enterprise. The role of the sign s0ψ pwx q has broader implications. It is representative of a whole collection of arguments, of which our treatment of (4.5.11) and (4.5.12) is among the simplest and most transparent. The more elaborate comparisons will be applied in later chapters to the cases excluded by Propositon 4.5.1. In general, s0ψ pwx q is an essential ingredient of the interlocking sign Lemmas 4.3.1 and 4.4.1, and hence also Proposition 4.5.1, but which were instrumental in the general reductions of §4.3 and §4.4, and which have so far been taken for granted. This underlines the significance of the next section, in which we will establish the two lemmas in general. 4.6. The two sign lemmas Lemmas 4.3.1 and 4.4.1 are parallel. They played critical roles in the corresponding expansions of §4.3 and §4.4, which in turn led to the general results of §4.5. Both lemmas concern the basic sign character εψ εG ψ attached to ψ in §1.5. The first can be regarded as a reduction of the character to its analogue for a Levi subgroup, while the second amounts to a reduction to endoscopic groups. We shall prove them in this section, thereby completing the proof of the general reduction obtained in the last section. We revert to the setting of §4.3–§4.4. That is, we take G to be either r pN q itself. In the second case, L G and a group in Ersim pN q or equal to G S ψ are only bitorsors, under the respective groups L G0 and S ψ . Following our earlier convention, we shall understand a representation of L G or S ψ to
4.6. THE TWO SIGN LEMMAS
213
be a morphism that extends to a representation of the group L G or S ψ generated by L G or S ψ . For example, the canonical extension of εG ψ to the
r pN q is a character in this sense. That quotient Sψ of S ψ in the case G G is, it extends to a character (of order 2) on the quotient Sψ of S ψ . We will need a certain amount of preliminary discussion before we can prove the two sign lemmas. This will be useful in its own right. For we shall see that it gives some insight into the meaning of the sign characters εψ . r pGq is fixed. We assume that Theorems 1.4.1 and Suppose that ψ P Ψ 1.4.2 hold for the self-dual generic constituents of ψ. The group Lψ is then defined, and has local embeddings (1.4.14). The sign character is defined (1.5.6) in terms of the irreducible constituents
τα
λα b µα b να
of the representation τψ ps, g, hq AdG s ψrG pg, hq of the product
S ψ Lψ SLp2, Cq,
(4.6.1)
p 0 . What we need here is a decomposition ˆ of G acting on the Lie algebra g of τψ into (possibly reducible) representations whose Lψ -constituents have good L-functions. Recall that Lψ is a fibre product over k P tKψ u of L-groups L Gk . The factor L Gk comes in turn with a complex homomorphism into GLpmk , Cq. For any k, we write gk for the image in GLpmk , Cq of a point g P Lψ under the composition Lψ ÝÑ L Gk ÝÑ GLpmk , Cq. Recall also that tKψ u is the set of orbits of the involution k Ñ k _ on the set ² ² _ Kψ Iψ Jψ Jψ .
We may as well identify tKψ u with the union of Iψ and Jψ , even though choice of the subset Jψ of Kψ is not canonical. We will then write gk_
for any index k _ P J _ .
gk_ tgk1,
g
P Lψ ,
ψ
The decomposition of the representation τψ gives rise to two general families of representations of Lψ . The first consists of the representations Rkk1 pg qX
gk X tgk1 ,
g
P Lψ ,
k, k 1
P Kψ ,
on complex pmk mk1 q-matrix space. They satisfy the obvious relations
_1 Rkk
Rk_pk1q_
and Rkk1
and they include also the representations Rkpk1 q_ pg qX
gk X gk11,
g
Rk 1 k ,
P Lψ ,
k, k 1
P Kψ .
214
4. THE STANDARD MODEL
Their L-functions are the standard Rankin-Selberg L-functions Lps, Rkk1 q Lps, µk µk1 q,
where µk µk1 is the cuspidal automorphic representation of GLpmk q GLpmk1 q attached to the pk, k 1 q-component of ψ. The other family consists of the symmetric square representations Sk pg q S 2 pgk q,
P Lψ , k P Kψ ,
g
and the skew-symmetric square representations Λk pg q Λ2 pgk q,
g
They satisfy the relations
Sk_
Sk_
P Lψ , k P Kψ . Λ_ k
and
Λk _ ,
and also have familiar L-functions. We shall say that a finite dimensional representation σ of Lψ is standard if it belongs to one of these two general families. The standard representations that are self-dual consist of the following three sets Rkk1 : k, k 1 Rkk_ : k
P Iψ
(
,
P Kψ
(
P Iψ
(
,
and Sk , Λk : k
.
p k equals We recall here that Iψ is the subset of indices k P Kψ such that G a classical group SOpmk , Cq or Sppmk , Cq, rather than GLpmk , Cq. Among these self-dual representations, the ones in the second and third sets are all p k and G p k1 are of the orthogonal. Those in the first set are orthogonal if G p p same type, and symplectic if Gk and Gk1 are of opposite type. Observe that a standard representation could be reducible. If it is symplectic, however, it must be irreducible. The global L-function Lps, σ q attached to a standard representation σ of Lψ has analytic continuation, with functional equation
Lps, σ q εps, σ q Lp1 s, σ _ q.
If σ belongs to the third general family above, it is understood that εps, σ q εps, µ, σ q is the automorphic ε-factor for the cuspidal automorphic representation µ of GLpmk q attached to σ, since it is not known that the arithmetic ε-factors are the same as the automorphic ε-factors in this case. However, if σ is self-dual and symplectic, it is an irreducible Rankin-Selberg representation, for which the two kinds of ε-factors are known to be equal. In particular,
4.6. THE TWO SIGN LEMMAS
215
the ε-factor εps, σ q is a finite product of local arithmetic ε-factors, as in the definition of εG ψ. Lemma 4.6.1. There is a decomposition (4.6.2)
τψ
à κ
τκ
à κ
pλκ b σκ b νκq,
κ P Kψ ,
of τψ relative to the product (4.6.1), for standard representations σκ , irreducible representations λκ and νκ , and an indexing set Kψ with an involution such that
κ_
κ
Ø
τκ_
τκ_ .
Proof. It is clear that we can introduce the notion of a standard representation ρ for the product Aψ
Lψ SLp2, Cq
by copying the definition above for Lψ . One can then establish a decomposition à τψ pλι b ριq, ι P Iψ , ι
that satisfies the required properties of (4.6.2), but with standard representations ρι of Aψ in place of the tensor products σκ b νκ . This is a straightforward consequence of the definition (1.4.1) of ψ and the structure ˆ, which we leave to the reader. The exercise becomes of the Lie algebra g a little more transparent if we take gk_ to be the “second transpose” t g 1 , which we can do because the assertion depends only on the inner class of the automorphism. The second step is to observe that any standard representation of Aψ has a decomposition à ρι pσκ b νκq,
P
κ Kψ,ι
for standard representation σκ of Lψ and irreducible representations νκ of SLp2, Cq. If ρι is a Rankin-Selberg representation Rkk1 for Aψ , σκ is the same Rankin-Selberg representation for Lψ , while νκ ranges over the irreducible constituents of the tensor product (4.6.3)
νk b νk1
representation of SLp2, Cq. If ρι is a standard representation Sk or Λk for Aψ , σκ alternates between the corresponding two kinds of standard representation of Lψ , and νκ ranges over appropriate constituents of (4.6.3) (with k 1 k). The lemma follows.
Corollary 4.6.2. If a constituent σκ of (4.6.2) is diagonal, in the sense that it is of the form Rkk , Rkk_ , Sk or Λk , the corresponding representation νκ of SLp2, Cq is odd dimensional.
216
4. THE STANDARD MODEL
Proof. The assertion follows from the proof of the lemma, and the fact that the irreducible constituents of the tensor product (4.6.3) are odd dimensional if k 1 k. We can use the lemma to express the sign character εψ in terms of standard representations. We claim that εψ p x q εG ψ px q
± κ
det λκ psq ,
s P Sψ,
where x xs is the image of s in Sψ , and the product is over the set of 1 indices κ P Kψ such that σκ is symplectic, and satisfies ε 2 , σκ 1. The standard representations σκ parametrized by this set are irreducible, so the corresponding representations λκ of S ψ are among the factors λα in the product (1.5.6) that defines εG ψ . There could be other factors in (1.5.6), since the diagonal standard representations σκ are reducible, and might have symplectic constituents even though they are not symplectic themselves. However, these supplementary factors occur in pairs λα and λ_ α , whose contributions to (1.5.6) cancel. Indeed, either the corresponding subrepresentation τκ of τψ is not self-dual, in which case τκ_ also contributes to the product, or σκ is self-dual but not symplectic, since it is reducible, in which case its symplectic factors occur in pairs. The claim follows. We can in fact pare the product further by excluding those factors with νκ psψ q 1. These are the factors for which the irreducible representation νκ of SLp2, Cq is odd dimensional, and hence orthogonal. For if the corresponding subrepresentation τκ of τψ is not self-dual, or if it occurs in τψ with even multiplicity, we again obtain pairs of factors that cancel. If τκ is self-dual, and has odd multiplicity, it is orthogonal (since τψ is orthogonal). This implies that the representation λκ of S ψ is symplectic (since σκ is symplectic and νk is orthogonal). It thus has determinant 1, and does not contribute to the product. We can therefore write (4.6.4)
εψ pxs q
±
κ Kψ
P
det λκ psq ,
s P Sψ,
where Kψ is the subset of indices κ P Kψ such that (i) σκ is symplectic, (ii) ε 12 , σκ 1, and (iii) νκ psψ q 1. We shall work for a time with the restriction τψ,1 of τψ to the subgroup
AdG ψrG
Lψ SLp2, Cq of the product (4.6.1). Let ψM P Ψ2 pM, ψ q be a fixed embedding from Lψ to L M attached to ψ, for some fixed Levi subgroup M G0 with dual Levi Aψ
4.6. THE TWO SIGN LEMMAS
subgroup
LM
LG0. Then
AdG ψrG
AdG,M ψM ,
where AdG,M is the restriction of AdG to AdG,M
217
LM .
à
ψM
P
We fix a decomposition
pσh b νhq
h Hψ
of this representation, which we assume is compatible with both the decomposition (4.6.2) of τψ , and the root space decomposition ˆ g
à
ˆξ g
à
mp `
P
ˆα g
pM α Σ
ξ
ˆ with respect to the Γ-split component of g AM x
xqΓ Z pM
0
x. The first condition tells us that we can write of the center of M
σh b νh
σκ b νκ,
h P Hψ ,
for a surjective mapping h Ñ κ from the indexing set Hψ onto Kψ . In the p 0 , A x q (including 0), while second condition, ξ ranges over the roots of pG M p M denotes the usual set of nonzero roots. This condition means that Hψ Σ is a disjoint union of sets Hψ,ξ such that if ρξ is the restriction of AdG,M to ˆξ of g ˆ, then the subspace g (4.6.5)
ρξ ψM
à
pσh b νhq.
P
h Hψ,ξ
be the preimage of K in Hψ,ξ , and set Let Hψ,ξ ψ ˆ g ψ,ξ
à
hPH
ˆh , g
ψ,ξ
ˆh is the subspace of g ˆξ on which the irreducible representation where g µh b νh
σh b νh
ˆ of Aψ acts. The bitorsor S ψ does not generally act on the space g ψ,ξ (or for ˆ that matter on the larger root space gξ ). However, the product S ψ Aψ does act on the sum à à ˆ ˆψ,ξ ˆh , g g g ψ ξ hPH ψ
where
Hψ
º ξ
Hψ,ξ
218
4. THE STANDARD MODEL
is the preimage of Kψ in Hψ . There is consequently a S ψ -equivariant morˆ phism from S ψ into the semisimple algebra EndAψ pg ψ q. We can therefore write
ˆ εψ pxs q det s, EndAψ pg ψq ,
(4.6.6)
s P Sψ.
We shall be concerned also with the subspace p m ψ
ˆ of g ψ , and its quotient
p ˆ g ψ {mψ
gˆψ,0
à
P
pM α Σ
ˆ g ψ,α .
The group Aψ of course acts on these two spaces. So does the normalizer N ψ in S ψ of the maximal torus Tψ in S ψ . We write
xqΓ {Z pG p 0 qΓ A x { A x X Z pG p0 q Γ Z pM M M
p εM ψ puq det n, EndAψ pm ψq ,
and
{
G M
εψ
puq det
n P N ψ,
p ˆ n, EndAψ pg ψ {mψ q,
n P N ψ,
where u is the projection of n onto the quotient Nψ Then
εM ψ puq εψ
{
G M
N ψ {T ψ .
puq εψ pxuq,
u P Nψ ,
where we recall that xu is the image of u in Sψ . Consider the adjoint action of the product N ψ Aψ on the block diagonal p ψ of g ˆψ . The formula for εM subalgebra m ψ puq above matches an earlier r rq of the character εM definition of the extension εM ψM to SψM ,u , which ψM pu appears for example in (4.2.11). We can therefore write M rq ε1ψ puq, εM ψ puq εψM pu
in the notation introduced prior to the statement of Lemma 4.3.1. On the ˆ other hand, the subgroup Sψ1 of Nψ leaves invariant each of the spaces g ψ,α . 1 ˆψ,α and g ˆψ,α are dual, their contributions to Since the actions of Sψ on g
{
G M
εψ
puq cancel. It follows that the value G{M G{M εψ pwu q εψ puq
depends only on the image wu of u in Wψ . We have shown that our sign character εψ pxq εG ψ pxq satisfies a decomposition formula (4.6.7)
ε1ψ puq εψ
{
G M
p w u q εψ p x u q ,
u P Nψ .
4.6. THE TWO SIGN LEMMAS
219
In preparation for the proof of Lemma 4.3.1, we shall also investigate the global normalizing factor rψ pwq rψG pwq rP pw, ψM q, given by (4.2.10). It equals the value at λ 0 of the quotient (4.6.8) where ρP,w on
Lp0, ρP,w ψM,λ q εp0, ρP,w ψM,λ q1 Lp1, ρP,w ψM,λ q1 ,
ρ_w P |P is contragredient of the adjoint representation of LM 1
w 1 p nP w { w 1 p nP w X p nP ,
and ψM,λ represents the formal twist of the L-homomorphism ψM by a point λ in the complex vector space aM,C
X pM qF b C.
We are now using the Artin notation for L-functions introduced in §1.5, or rather its extension to finite-dimensional representations of the product Aψ , since the automorphic representation πψM in (4.2.10) will not be an explicit part of the discussion here. It is of course implicit in the parameter ψM , and we must indeed remember to give the ε-factor in (4.6.8) its automorphic interpretation. As a global normalizing factor, we know that (4.6.8) is a meromorphic function of λ, which is analytic at λ 0. This is a consequence of the general theory of Eisenstein series, and the normalization of local intertwining operators from Chapter 2. However, we will need to be more explicit. We write p P,w Σ pP as usual, where Σ
p P : wα R Σ pP αPΣ
(
Σp M denotes the set of roots in pPp, AMxq. Then ρP,w
à
P
p P,w α Σ
ρ_ α
à
P
ρα ,
p P,w α Σ
ˆ determines an isomorphism from ρ_ since the Killing form on g α to ρα . From the decomposition (4.6.5) of ρα ψM into standard representations, we see that the L-function Lps, ρP,w ψM,λ q in (4.6.5) has analytic continuation with functional equation. It follows that (4.6.8) equals
1 Lp1, ρ_ P,w ψM,λ q Lp1, ρP,w ψM,λ q .
220
4. THE STANDARD MODEL
We can thus write
1 rψ pwq lim Lp1, ρ_ P,w ψM,λ q Lp1, ρP,w ψM,λ q λ
Ñ0
λlim Ñ0 λlim Ñ0 λlim Ñ0
±
1 Lp1, ρ_ α ψM,ψ q Lp1, ρα , ψM,λ q
P
p P,w α Σ
±
Lp1 αpλq, ρ_ α ψM q Lp1
P
p P,w α Σ
±
pαpλqqa pαpλqqa
α
α
P
p P,w α Σ
αpλq, ρα ψM q1
,
where aα denotes the order of pole of Lps, ρα ψM q at s 1. It follows from (4.6.5) that ± Lps, ρα ψM q Lps, σh b νh q.
P
h Hψ,α
In particular, aα
¸
ah ,
P
h Hψ,α
where
ah ords1 Lps, σh b νh q. Moreover, since Lps, σh b νh q can be regarded as a standard automorphic L-function (of type Rkk1 , Sk or Λk , but attached to unitary, not necessarily cuspidal representations), we have the adjoint relation L s, pσh b νh q_
Lps, σh b νhq.
It follows that Since ρα (4.6.9)
ah ords1 L s, pσh b νh q_ . ρ_α , the orders aα and aα are then equal. We conclude that rψ pwq
±
±
P
P
p P,w h Hψ,α α Σ
p1qa
h
.
Observe that from the last adjoint relation, the contributions to (4.6.9) of distinct representations pσh b νh q_ and σh b νh cancel. The interior product in (4.6.9) need therefore be taken over only those h with σh_ σh . The question then is to determine the parity of ah , for any h P Hψ,α with σh_ σh . The irreducible representation νh maps the diagonal matrix 1 1 diag p|t| 2 , |t| 2 q in SLp2, Cq to the diagonal matrix
diag |t| 2 pnh 1q , |t| 2 pnh 3q , . . . , |t| 2 pnh 1q , 1
1
1
in GLpnh , Cq, where nh
dimpνhq. It follows from the definitions that n ± (4.6.10) Lps, σh b νh q L s 12 pnh 2i 1q, σh . i1 We must therefore describe the order of Lps, σh q at any real half integer. h
As a standard (unitary, cuspidal) automorphic L-function of type Rkk1 , Sk
4.6. THE TWO SIGN LEMMAS
221
or Λk , Lps, σh q is nonzero whenever Repsq ¥ 1 or Repsq ¤ 0. Moreover, its only possible poles in this region are at s 1 and s 0. The relevant half integers in (4.6.10) are thus confined to the points 0, 12 , and 1. The question separates into two cases. If dimpνh q is even, we have only to consider the order of Lps, σh q at s 12 . If dimpνh q is odd, we need to consider its order at the points s 0 and s 1. In both cases, we shall assume that the assertions of Theorem 1.5.3 can be applied inductively to the simple generic constituents of ψ. Suppose first that dimpνh q is even. Then σh cannot be diagonal, in the sense of Corollary 4.6.2. It must consequently be an irreducible, selfdual Rankin-Selberg representation of type Rkk1 . Its L-function satisfies the functional equation (4.6.11)
Lps, σh q εps, σh q Lp1 s, σh q.
It follows that Lps, σ q will have even or odd order at s 21 according k 1 to whether ε 2 , σh equals 1 or 1. The contribution of h to (4.6.9) is therefore just the sign of ε 12 , σh . Observe that if σh is orthogonal, Theorem 1.5.3(b) asserts that the sign is 1, and can be ignored. In other words, we can assume that σh is symplectic, and therefore that h lies in the subset of Hψ,α . We thus obtain the contribution to the formula (4.6.9) for Hψ,α rψ pwq of those h with dimpνh q even. It equals the product ± p1q|Hψ,α|. (4.6.12) rψ pwq
P
p P,w α Σ
Suppose next that dimpνh q is odd. Then σh is a self dual representation of Lψ , which is typically diagonal, in the sense of Corollary 4.6.2. Its L-function still satisfies the functional equation (4.6.11), but with the understanding that the ε-factor εps, σh q is given its automorphic interpretation. In particular, the orders of Lps, σh q at s 0 and s 1 are equal. It follows from (4.6.10) that the contribution of h to (4.6.9) is just 1 if dimpνh q is greater than 1, and is equal to the order of Lps, σh q at s 1 if νh is the trivial one dimensional representation of SLp2, Cq. Theorem 1.5.3(a) tells us this order, in case σh is of the form Sk or Λk . It amounts to the assertion that Lps, σh q has a pole at s 1, necessarily of order 1, if and only if the representation σh of Lψ contains the trivial representation. The same assertion has long been known in the other case that σh is of Rankin-Selberg type Rkk1 . This gives us the contribution of h to (4.6.9) in case dimpνh q is odd. It equals p1q if the representation σh b νh of Aψ contains the trivial 1-dimensional representation, and is p 1q otherwise. It is easy to characterize the total contribution to (4.6.9) of those h with ˆα,h of dimpνh q odd. The representation σh b νh of Aψ acts on a subspace g ˆα of g ˆ. It contains the trivial representation if and only if the root space g ˆα,h has nontrivial intersection with the Lie algebra of S ψ . This is to say g ˆα,h contains the corresponding root that α is a root pS ψ , T ψ q, and that g
222
4. THE STANDARD MODEL
p P,w and Hψ,α , we obtain a factor p1q in the space. As α and h range over Σ product formula (4.6.9) for rψ pwq for every positive root α of pS ψ , T ψ q that is mapped by w to a negative root. The product of these factors is equal to the ordinary sign character s0ψ pwq on Wψ introduced in §4.1. The character s0ψ pwq thus gives the contribution from those h with dimpνh q odd. This completes our analysis of the product (4.6.9) in terms of its even and odd parts. We have shown that the global normalizing factor rψ pwq rψG pwq has a decomposition
rψ pwq rψ pwq s0ψ pwq,
(4.6.13)
w
P Wψ .
Proof of Lemma 4.3.1. We must establish an identity rψG pwu q ε1ψ puq εψ pxu q s0ψ puq,
u P Nψ ,
among the four signs, where s0ψ puq s0ψ pwu q. The decompositions (4.6.7) and (4.6.13) reduce the problem to showing that the two characters rψ and
{
G M
on Wψ are equal. We are assuming that ψ and M are as in §4.3. In fact εψ we assume that M is proper in G0 , and therefore subject to our induction hypotheses, since the required assertion is trivial if M G0 . We fix w P Wψ . Let T ψ,w be the group generated in S ψ by the subset T ψ w of S ψ , and let Λ ψ be the representation of this group on the space Vψ We can then write
{
G M
εψ
EndA pgˆψ {mp ψ q. ψ
pwq det
Λ ψ pw q ,
since the determinant is independent of the preimage of w in T ψ,w . The group T ψ,w is an extension by T ψ of the cyclic group xwy generated by w. Its representation theory is well understood. For example, there is a surjective mapping Λ Ñ o, from the set of irreducible representations Λ of T ψ,w onto the set of xwy-orbits o of characters t Ñ tξ on T ψ , such that tr Λptq
¸
P
tξ ,
t P T ψ.
ξ o
It is a consequence of our definitions that tr Λ ψ pt q
¸ pM αPΣ
| tα , |Hψ,α
This leads to a decomposition Λ ψ
à
à
oα h H ψ,α
P
Λh ,
t P T ψ.
4.6. THE TWO SIGN LEMMAS
223
p M , and Λh is an irreducible where oα ranges over the xwy-orbits of roots α P Σ representation of T ψ,w that maps to oα . We can therefore write
(4.6.14)
{
G M
εψ
pw q ±
±
oα h H ψ,α
P
det Λh pwq .
An orbit oα is said to be symmetric if it contains the root pαq. If α is not symmetric, oα and oα are two distinct orbits of equal order. They give rise to pairwise contragredient sets of representations Λh , whose contributions to the formula (4.6.14) cancel. We can therefore restrict the outer product in (4.6.14) to orbits oα that are symmetric. Other factors in the product (4.6.14) can also pair off. Recall that any gives rise to an irreducible representation µh b νh of Aψ . The index h P Hψ,α self dual representations µh and νh are both symplectic, since ε 21 , µh 1 and νh psψ q 1, so their tensor product is orthogonal. On the other hand, the representation à pΛh b σh b νhq h
ˆ of T ψ,w Aψ on g ψ is invariant under the Killing form, and hence orthogonal. This does not mean that the individual representations Λh need to be orthogonal. However, the representations Λh that are not orthogonal will occur in pairs, whose contributions to (4.6.14) cancel. We can therefore take of indices h P H the interior product in (4.6.14) over the subset Hψ,α,o ψ,α such that Λh is orthogonal. Consider an orthogonal representation Λh ,
. h P Hψ,α,o
Elements of the Weyl group of a complex orthogonal group can be represented by permutation matrices. One sees from this that the image of T ψ,w under Λh is the semidirect product of the image of T ψ with the (cyclic) quotient of xwy of order |oα |. It then follows that the restriction of Λh to xwy is isomorphic to the permutation representation of xwy on oα . In particular, Λh pwq is a well defined operator, whose eigenvalues are the set of roots of 1 of order |oα |. We are assuming that the orbit oα is symmetric. It therefore has even order, from which it follows that det Λh pwq
1.
The orthogonal representations thus each contribute a sign to the product (4.6.14). The formula reduces to ± G{M (4.6.15) εψ p w q p1q|Hψ,α,o| . oα symmetric
The analysis of the product formula (4.6.12) for rψ pwq is slightly different. Since | |H |, |Hψ,wα ψ,α
224
4. THE STANDARD MODEL
the factor of α in the product depends only on the orbit oα . If oα is symmetric, we see after a moment’s thought that the integer oα
X Σp P,w
equals the number of sign changes in the sequence
pα, wα, . . . , αq p M defined Σ p P ). Since this number is odd, the (relative to the order on Σ contribtution to (4.6.12) of all the roots in a symmetric orbit is the same as the contribution of any positive root in the orbit. If oα is not symmetric, oα and oα are disjoint, and the integer oα
oα
X Σp P,w
X Σp P,w
equals the number of sign changes in the sequence
pα, wα, . . . , αq. Since the number is even in this case, the nonsymmetric orbits contribute nothing to (4.6.12). The product (4.6.12) therefore reduces to ± p1q|Hψ,α|. oα symmetric
in H occur in As we have noted, indices h in the complement of Hψ,α,o ψ,α pairs, and consequently contribute nothing to the product. The right hand side of (4.6.12) therefore reduces to the right hand side of (4.6.15). This establishes the required identity {
G M
εψ
pwq rψpwq,
and completes the proof of Lemma 4.3.1.
We observe that both assertions of Theorem 1.5.3 were needed to prove Lemma 4.3.1. They were each invoked in the preamble that led to the decomposition (4.6.13) of rψ pwq. For example, we used the assertion (b) on orthogonal ε-factors to obtain an exponent in (4.6.12) that included only indices h with σh symplectic. This reduction was essential. Had we been forced also to include orthogonal Rankin-Selberg representations σh (with a | in (4.6.12)), our product corresponding enlargement of the exponent |Hψ,α
{
G M
(4.6.12) would then not have matched the formula for εψ we obtained in the proof of the lemma. Our application of both assertions (a) and (b) is of course predicated on our induction hypothesis. In the case of (b), we can assume that the assertion holds for any pair of simple generic parameters (4.6.16) with m1
µi m2
N.
P Φr simpGiq,
Gi
P Erpmiq, i 1, 2,
4.6. THE TWO SIGN LEMMAS
225
We can actually manage with a weaker assumption on Theorem 1.5.3(b). Our proof of Lemma 4.3.1 for ψ requires only that Theorem 1.5.3(b) apply to any relevant pair ψi µi b νi , i 1, 2, of distinct simple constituents of ψ. In other words, when Theorem 1.5.3(b) implies that the contribution to (4.6.9) of a pair is 1, we require that the contribution actually be equal to 1. The pair pψ1 , ψ2 q gives rise to a RankinSelberg representation ψ1 ψ2 of Aψ , and corresponds to a root α in (4.6.9). It is relevant to Theorem 1.5.3(b) only if the Rankin-Selberg representation µ1 µ2 of Lψ is orthogonal, and the tensor product ν1 b ν2 decomposes into irreducible representations SLp2, Cq of even dimension. In fact, we see from (4.6.10) and the proof of Lemma 4.6.1 that ν1 b ν2 must decompose into an odd number of such representations for the application of Theorem 1.5.3(b) to (4.6.9) to be nontrivial. We shall use this weaker form of the induction hypothesis in the next chapter. Proof of Lemma 4.4.1. This lemma will be easier to prove than the last. We have to show that 1 εG s P S ψ,ss , ψ 1 psψ 1 q εψ psψ xs q,
where pG1 , ψ 1 q is the preimage of pψ, sq under the correspondence (1.4.11). We shall follow the general argument set out in [A9, p. 51–53]. We begin with the formula εψ p x s q εG ψ px s q
±
κ Kψ
P
det λκ psq
obtained (4.6.4) from any decomposition τψ
à
P
κ Kψ
pλκ b σκ b νκq
that satisfies the conditions of Lemma 4.6.1. We could in fact weaken these conditions. For example, a decomposition in which the representations λκ of S ψ are allowed to be reducible gives the same formula for εG ψ px s q. Notice that the point sψ belongs both the group SLp2, Cq and the centralizer Sψ . Therefore λκ psψ q νκ psψ q 1,
for any κ in the subset Kψ of Kψ . It follows that ± εG p1qdimpλκq. ψ ps ψ q κPKψ We will of course be free to apply this formula if G is replaced by the endoscopic group G1 . Suppose that s, G1 and ψ 1 are as in the putative identity. The dual p 1 is then the connected centralizer in G p 0 of the point s1 s. Its group G
226
4. THE STANDARD MODEL
ˆ1 is just the kernel of Adpsq in g ˆ. The parameter ψ 1 for G1 is Lie algebra g defined by the L-embedding ψr1 : Aψ
ÝÑ
L
G1 ,
which in turn is obtained from ψrG by restriction of the codomain to the L-subgroup L G1 of L G0 . It follows that τψ1 ,1 , the restriction of τψ1 to Aψ , is simply the mapping given by restricting the representation τψ,1 to the ˆ1 of g ˆ. For any κ P Kψ , let λsκ be the representation of S ψ1 on the subspace g p 1q-eigenspace of λκpsq. If Kψ1 denotes the set of κ P Kψ with λsκ 0, we can then write à pλsκ b σκ b νκq. τψ1 κPKψ1 This decomposition satisfies the conditions of Lemma 4.6.1 (weakened to the extent that λsκ could be reducible). It follows that ± 1 s p 1qdimpλκ q , εG ψ 1 psψ 1 q κPKψ 1
where Kψ1 is the intersection of Kψ1 with Kψ . The original formula (4.6.4) tells us that εG ψ pxs q equals the product of all the eigenvalues, counting multiplicities, of all of the operators (
λκ psq : κ P Kψ .
But the contragredient λκ Ñ λ_ κ can be treated as an involution on the set of representations λκ with κ P Kψ . It follows that if ξ is an eigenvalue not equal to p 1q or p1q, its inverse ξ 1 is another eigenvalue, but with the same total multiplicity. Therefore mp1q , εG ψ pxs q p1q
where mp1q is the total multiplicity of the eigenvalue token, the number ¸
κPK
dimpλκ q dimpλsκ q
p1q.
mp1q,
ψ
being the sum of multiplicities of eigenvalues distinct from integer. It follows that G G εG ψ psψ xs q εψ psψ q εψ pxs q
p1qmp1q
±
P
By the same
p1q, is an even
p 1qdimpλ q κ
κ Kψ
p1qmp1q
±
P
1 p 1qpdimpλ qdimpλ qq εG ψ 1 ps ψ 1 q
κ Kψ
εGψ11 psψ1 q,
κ
s κ
4.7. ON THE GLOBAL THEOREMS
227
as required. This completes the proof of Lemma 4.4.1. 4.7. On the global theorems The remaining two sections of the chapter represent a digression. The purpose of this section is to relate the global theorems we have stated with the trace formula. The discussion amounts to a series of informal remarks, which will help us to understand the formal proofs later on. In the next section, we shall sketch how our earlier comparison of spectral and endoscopic terms extends to general groups. A reader wishing to go on with the main argument, which we left off at the end of §4.5, can skip these sections and proceed directly to Chapter 5. This section is intended as more technical motivation for the global theorems stated in Chapter 1. Among other things, we shall see how the theorems are consequences of the stable multiplicity formula of Theorem 4.1.2. This will be for guidance only, since in the end we will have to establish all of the theorems together. However, it might explain how the multiplicity formula of Theorem 1.5.2, and the properties of poles and signs in Theorem 1.5.3, are in some sense implicit in the trace formula. The remarks of this section can also be regarded as an illustration of the general comparison r ell pN q that was in the simplest situation – the case of a parameter ψ P Ψ excluded earlier, and which is ultimately the hardest to prove. For this exercise, we shall assume whatever we need of the local theorems we have stated. We shall also suppose that the global seed Theorem 1.4.1 and its complement Theorem 1.4.2 are both valid, and that the resulting global parameters for groups G P Ersim pN q have the rough spectral properties that motivated the initial remarks in §1.2. Specifically, we assume that there is no contribution to the discrete spectrum of G from any parameter ψ that r 2 pGq. We shall treat the fine spectral assertions of Theorem does not lie in Ψ 1.5.2 and parts (a) and (b) of Theorem 1.5.3 in reverse order. We begin with a general global parameter (4.7.1)
ψ
ψ1 `
`
ψr ,
ψi
P Ψr simpNiq,
r ell pN q. This was implicitly excluded from earlier sections, specifically in Ψ in the comparisons of §4.3 and §4.4 that led to Proposition 4.5.1. However, with our heuristic assumptions above, the comparisons are valid for ψ, and in fact are quite transparent. They might give us a better overall view if we r pN q and f equal to a retrace some of the steps. We shall begin with G G r pN q. function fr P H Suppose for a moment that r 1. Then
ψ
ψ1 µ
b
ν
0 is simple, and has a contribution Rdisc,ψ to the relative discrete spectrum 0 of G GLpN q. We note that the sign character εG ψ is trivial in this
228
4. THE STANDARD MODEL
case, since as in Corollary 4.6.2, the tensor product of the unipotent part ν with itself is a sum of irreducible odd dimensional representations. We G G0 also recall that Rdisc,ψ , the canonical extension of Rdisc,ψ , is a product of the corresponding local extensions, according to the theory of local and global Whittaker models for GLpN q. Therefore G G Idisc,ψ pf q rdisc,ψ pf q
G tr Rdisc,ψ pf q
21 fGpψq. r pN q of either This can be regarded as an elementary analogue for G G 1 2
Theorem 1.5.2 or Theorem 4.2.2. We have used it, at least implicitly, in our earlier discussion in §4.3 and §4.4. G If r ¡ 1, Idisc,ψ pf q equals the difference (4.3.1) with which we began the comparison in §4.3, since ψ does not contribute to the discrete spectrum of r pN q. Therefore I G GG disc,ψ pf q equals the expression (4.3.2) we obtained for the difference. To analyze it, we take M to be the standard Levi subgroup of G0 GLpN q corresponding to the partition pN1 , . . . , Nr q. Then ψ is the image of a parameter ψM P Ψ2 pM, ψ q. There is one element w in the set Wψ . This element stabilizes M , and induces the standard outer automorphism on each of the factors GLpNi q of M . It follows that det w
p 1 qa
G M
1
1 r 2 .
The other coefficients in (4.3.2) satisfy mψ
|Wψ | |Sψ | 1, M
according to their definitions, and can be ignored. The fibre Nψ pwq of w in Nψ consists of one element u. We obtain rM , ψM q IP pπM , f q fG pψ, uq tr RP pw, π
fGpψq,
again by the theory of Whittaker models for GLpN q, applied this time to both M and G. The expression (4.3.2) therefore reduces to 1 r 2 rP
pw, ψM q fGpψq,
We have shown that
N Idisc,ψ pfrq Idisc,ψ pfrq
1 r rPr 2
r pN q , f GG
P HrpGq.
pw, ψMq frN pψq,
r pN q, fr P H
r 0 pN q with Levi compowhere ψ is as in (4.7.1), Pr is a parabolic subgroup of G M , w is the unique element in the Weyl set W N Wψ G r p N q, M nent M ψ
r and rPr pw, ψM q is the global normalizing factor for GpN q. We also have the stable decomposition N Idisc,ψ pfrq
¸
GPErell pN q
p
q
p q
G r ι N, G Spdisc,ψ frG .
G Our heuristic assumptions, together with (4.4.12), tell us that Sdisc,ψ van-
r 2 pGq of Ψ r ell pN q. We can therefore ishes if ψ does not belong to the subset Ψ r r 2 pG q. fix G to be the unique element in Eell pN q such that G does belong to Ψ
4.7. ON THE GLOBAL THEOREMS
229
The local Theorem 2.2.1, which we are also assuming, tells us that frN pψ q equals frG pψ q. It follows that (4.7.2)
G Spdisc,ψ pfrGq rιpN, Gq1
1 r rPr 2
pw, ψMq frGpψq,
f
P HrpN q.
Consider now the case that r 2 in (4.7.1), and that ψ1 and ψ2 are of opposite type. In other words, one is orthogonal and the other is symplectic. r 2 pGq is then of composite form The datum G with ψ P Ψ G G1 G 2 ,
Gi
P ErsimpNiq.
The stable multiplicity formula stated in Theorem 4.1.2 is global, and is not included in our heuristic assumptions. However, for our simple parameters r sim pGi q, the formula follows inductively from our discussion of the ψi P Ψ case r 1 above, with another appeal to Corollary 4.6.2 to confirm that the signs εGi pψi q equal 1. The stable multiplicity formula therefore holds for the simple parameter ψ of the product G. It takes the reduced form G pf q mψ f Gpψq, Sdisc,ψ
f
P HrpGq,
since the coefficients |Sψ |1 , σ pS 0ψ q and εG pψ q are all trivial in this case. Combining this with (4.7.2), we obtain mψ frG pψ q r ιpN, Gq1 It is easy to see that
1 2 rPr 2
pw, ψMq frGpψq,
r p N q. fr P H
r N pGq|1 1 1 2 21 |Z pGpqΓ|1 |Out 2 r N pGq| mψ , |Out p qΓ | 2 for ιpN, Gq at the end of §3.2, the fact that |Z pG given the formula for r
p
r ι N, G
q1
1 2 2
the composite G, and the definition in §1.5 of mψ . The resulting cancellation then gives rG r pN q. frG pψ q rPr pw, ψM fr P H q f pψ q, We thus obtain an identity
rPr pw, ψM q 1,
(4.7.3)
in our case of r 2, and ψ1 and ψ2 of opposite type. Actually, we should be a little careful in the last step of the justification of (4.7.3), since the mapping
fr1 fr2, fri P HrpGiq, r pN q onto H r pGq. However, an inductive application of Thedoes not take H fr
ÝÑ
frG
orem 1.5.2 to G1 and G2 tells us that the linear form f G pψ q,
f
P HrpGq,
is a sum of characters. The conclusion (4.7.3) then follows as a simple special case of Proposition 3.5.1.
230
4. THE STANDARD MODEL
The identity (4.7.3) leads to a proof of Theorem 1.5.3(b). To see this, suppose that r sim pmi q, µi P Φ i 1, 2, are simple generic parameters of the same type, either both orthogonal or both symplectic. Set ψ1 µ1 and ψ2 µ2 b ν2 , where ν2 is the irreducible two-dimensional representation of SLp2, Cq. The sum ψ of ψ1 and ψ2 then satisfies the conditions above, with N
m1
2m2 .
We calculated the global normalizing factor rPr pw, ψM q in §4.6. In the case here, Pr is the standard maximal parabolic subgroup of GLpN q of type pN1, N2q pm1, 2m2q. The representation ρPr,w ψM of Aψ in (4.6.8) acts on the Lie algebra nPr , and is the exterior tensor product of the orthogonal Rankin-Selberg representation σ of Lψ attached to µ1 µ2 with the representation ν2 of SLp2, Cq. It follows from a special case of (4.6.9), (4.6.10), and (4.6.11), specifically the description below (4.6.11) of the contribution of the representation σ σh to (4.6.9), that rPr pw, ψM q ε
It then follows from (4.7.3) that (4.7.4)
ε
1 2,
1 2, σ
qε
1 2,
µ 1 µ2 .
µ1 µ2 q 1.
This is the assertion (b) of Theorem 1.5.3, with µ1 µ2 and pm1 , m2 q in place of φ1 φ2 and pN1 , N2 q. The assertion (a) of Theorem 1.5.3 is harder, even with the heuristic assumptions we have allowed ourselves. Its formal proof will be one of our long term goals. We shall make only a few general remarks here, just to give a sense of the direction of the later proof. Theorem 1.5.3(a) concerns the case r 1. Suppose that ψ φ ber sim pGq of simple generic parameters, for a simple datum longs to the set Φ r G P Esim pN q, as in the assertion. The statement was of course motivated by the expected properties of global L-functions. If φ is orthogonal, according to the criterion of Theorem 1.4.1, the group Lφ Lψ is an extension p φ . In this case, the representaof WF by the complex orthogonal group G tion S 2 φrG of Lφ contains the trivial representation, and the L-function Lps, φ, S 2 q would be expected to have a pole at s 1. Similarly, if φ is symplectic, the L-function Lps, φ, Λ2 q would be expected to have a pole at s 1. We will eventually have to establish these criteria by harmonic analysis and the trace formula. The technique relies on the introduction of a supplementary parameter ψ ψ`ψ r pN q, with N 2N . in Ψ r ell pN q, since The global parameter ψ does not lie in the elliptic set Ψ r it has a repeated factor. Let G P Esim pN q be the twisted endoscopic
4.7. ON THE GLOBAL THEOREMS
231
r pG q, and such that G p and G p are of the datum such that ψ lies in Ψ same type, either both orthogonal or both symplectic. The quadratic characters ηψ and ηG attached to ψ and G are both equal to 1. We therefore have the companion datum G_ in Ersim pN q, and ψ is also conr pG_ q. The groups G and G_ share a maxtained in its parameter set Ψ imal Levi subgroup M . It is isomorphic to GLpN q, and comes with a canonical element ψ ,M P ΨpM , ψ q (which we may as well identify with ψ ). Let ρ ρpG q and ρ_ ρpG_ q be the adjoint representations of x GLpN, Cq on the Lie algebras n and n_ of the unipotent radicals M p and G p _ . (We caution of corresponding maximal parabolic subgroups of G _ ourselves that ρ does not denote the contragredient of ρ here.) Then ρ_ p is orthogonal and Λ2 if G p is symis of the same type as G, namely S 2 if G plectic, while ρ is of type opposite to G. On the other hand, the centralizer groups satisfy Sψ_ Sψ pG_ q Spp2, Cq
and Sψ
Sψ pG q
#
Op2, Cq,
if N is even,
SOp2, Cq, if N is odd.
It follows that the Lie algebra of Sψ_ intersects n_ in a 1-dimensional space, while the Lie algebra of Sψ intersects n only at t0u. These properties are direct consequences of the structure of the Lie alp and G p _ . They reduce the problem to the behaviour of global gebras of G normalizing factors. To be more specific, we have first to note that the heuristic assumptions of this section preclude a contribution from ψ to the discrete spectrum of either G or G_ . The analogue for either of these groups of the expression (4.3.2) is therefore simply the discrete part of the corresponding trace formula, represented by the first term in the difference (4.3.1). Consider then the expression for G_ _ Idisc,ψ pf _q Idisc,ψ pf _q, f _ P HrpG_q, given by the analogue of (4.3.2). The double sum in (4.3.2) collapses to one simple summand in this case, of which the various terms are easy to describe explicitly. In particular, by (4.6.9) and (4.6.10), the analogue of the global normalizing factor rP pw, ψM q reduces to the sign p1qa_ , a_ ords1 Lps, ρ_ ψ q. This sign is supposed to equal p1q, since it comes from the L-function that is expected to have a simple pole. One can also describe all of the terms in the expression for Idisc,ψ pfr q, N
fr
P HrpN q,
232
4. THE STANDARD MODEL
given by its analogue of (4.3.2). In this case, the relevant global normalizing factor does in fact equal p1q. We choose f _ and fr so that they transfer to the same function in SrpG_ q. It then follows easily from the global intertwining relation for G_ (which reduces to the local intertwining relation we r pN q (which we already know) that the are assuming) and its analogue for G _ r analogues for f and f of the linear form fG pψ, uq in (4.3.2) are equal. We then see immediately that the two expressions are equal up to a multiple that is positive if and only if Lps, ρ_ ψ q does have a pole (necessarily simple) at s 1. A little more thought reveals that the absolute value of this multiple is actually equal to the coefficient r ιpN , G_ q. The required assertion (a) is therefore equivalent to the undetermined sign in an identity Idisc,ψ pfr N
q rιpN
_ , G_ q Idisc,ψ pf _ q
being positive. How might one use the existence of this identity to determine the sign? We can write _ _ pf _ q, pf _q Sdisc,ψ Idisc,ψ
since the only elliptic endoscopic datum for G_ through which ψ in G_ itself. For similar reasons, we can write
factors
_
_ pfrG q. q rιpN , G q Spdisc,ψ pfrG q rιpN , G_q Spdisc,ψ Now the factor σ pS 0ψ q in the putative stable multiplicity formula for Sdisc,ψ
(4.7.5) Idisc,ψ pfr N
vanishes, since the group
S 0ψ
S ψ pG q0 GLp1, Cq
has infinite center. We therefore expect that Sdisc,ψ vanishes (as a linear r pG q). If it does, we see that form on H Idisc,ψ pfr
q rιpN
_ , G_ q Idisc,ψ p f _ q,
given that fr and f _ transfer to the same function in SrpG_ q. In other words, the identity above will indeed have the required positive sign. Our heuristic assumptions do not include the stable multiplicity formula of Theorem 4.1.2. However, it is possible to establish that Sdisc,ψ vanishes from the assumptions we do have, and the expression for Idisc,ψ pf
G q Idisc,ψ pf q,
f
P HrpG q,
given by the analogue of (4.3.2). If N is odd, the argument is easy. In this case, the group S ψ
S 0ψ
has
r disc pG q of Ψ r pG q infinite center, and ψ does not belong to the subset Ψ defined in §4.1, from which we see that Idisc,ψ pf q vanishes. Since G G is not an endoscopic group for G in this case (it is actually a twisted
4.7. ON THE GLOBAL THEOREMS
233
endoscopic group), ψ does not factor through any proper endoscopic datum for G . It follows that Sdisc,ψ pf
q Idisc,ψ pf q 0,
as required. If N is even, the heuristic argument is more complicated. In this case, the expression for Idisc,ψ pf q contains a sign
p1qa , a ords1 Lps, ρ ψ q, which equals p1q if Theorem 1.5.3(a) if false. One combines this informa_ _ tion, and the fact that the corresponding sign p1qa for Idisc,ψ pf _q then equals
1, with the endoscopic decomposition
1 q Sdisc,ψ pf q ιpG , G1 q Spdisc,ψ pf 1 q G G, and its analogue (4.7.5) for GrpN q, G
Idisc,ψ pf
for G and G1 Since
and G_ .
1 Sdisc,ψ
G G Sdisc,ψ b Sdisc,ψ , r sim pGq is simple, we find an expression we can evaluate easily given that ψ P Ψ that the first decomposition gives a formula for Sdisc,ψ pf q. An analysis of
the three terms in the second decomposition then leads to a contradiction, namely that (4.7.5) cannot hold if the two signs are as above. In other words, Theorem 1.5.3(a) must be valid. These last remarks have been quite sketchy. They are intended to serve as a heuristic argument in support of assertion (a) of Theorem 1.5.3. There is no point in being more precise now, since we will later have to revisit the argument in more detail and greater generality. As we will see, the formal proof of the assertion will be much more difficult without our heuristic assumption that ψ contributes nothing to the discrete spectra of G and G_ . Having discussed the two assertions of Theorem 1.5.3, we take ψ as in (4.7.1), and assume that both assertions are valid (where applicable) for the simple constituents ψi of ψ. This allows us to apply the two sign lemmas r pN q and proved in §4.6 to ψ. We shall use the lemmas, both for G G G P Ersim pN q, to derive the stable multiplicity formula of Theorem 4.1.2. With ψ fixed, we take G to be the unique datum in Erell pN q such that ψ r 2 pGq of Ψ r ell pN q. We of course want to assume that G lies in the subset Ψ is simple, the case we are studying. The first step is to apply Lemma 4.3.1, r pN q in place of G, to the global normalizing factor in (4.7.2). We with G obtain N rψ rPr pw, ψM pwq rrψN pwq ε˜1ψ puq q r
ε˜Nψ pxuq sr0ψ puq ε˜Nψ pxuq, r ψ pwq. We are using the fact where u is the unique element in the fibre N that
ε˜1ψ puq ε˜ψM puq 1,
234
4. THE STANDARD MODEL
following an earlier remark in the special case r sr0ψ
puq pwuq 1,
1, and that
sr0ψ
since the group Srψ pN q0 is abelian. The point x xu is the unique element in the set Sψ pN q. It represents a point s P Srψ pN q such that the pair pG, ψ q is the preimage of pψ, sq under the correspondence (1.4.11). The next step r pN q, G in place of pG, G1 q, to the sign is to apply Lemma 4.4.1, with G
p q pxq. We obtain
r N G
ε˜N ψ p x q εψ
ε˜N ˜N ˜N ψ pxu q ε ψ px q ε ψ ps ψ x q
εGψpsψ q εGpψq, since the point sψ lies in the connected group Srψ pN q Srψ0 pN q. The identity (4.7.2) then takes the form
G pfrGq rιpN, Gq1 Spdisc,ψ
1 r G ε 2
pψq frGpψq,
r pN q. fr P H
The remaining terms in the identity are easily treated. Given the formula p qΓ t1u in the case for r ιpN, Gq at the end of §3.2, and the fact that Z pG here that G is simple, we obtain
p
r ι N, G
r N pGq|1 q1 12 |Z pGpqΓ|1 |Out
The resulting power
1 r 1
2
1
r N pGq|. 2 |Out
can then be related to the finite group
Sψ pGq Sψ
S ψ Sψ {Z pGpqΓ
attached to G and ψ. One finds that
r N pG, ψ q| |Sψ |1 |Out
1 r 1 , 2
r N pG, ψ q being stabilizer of ψ in Out r N pGq, after first noting that with Out the product r N pGq| |Z pGpqΓ| |Out
p equals SOpN, Cq (N odd), equals 1, 2 or 4 in the respective cases that G SppN, Cq or SOpN, Cq (N even). We then note that the integer mψ in the putative multiplicity formula satisfies
mψ
r N pGq| |Out r N pG, ψ q|1 , |Out
according to the definition in §1.5. Finally, by the local Theorem 2.2.1, which we are taking for granted, we can write frN pψ q frG pψ q. It follows that
G Spdisc,ψ pfrGq mψ |Sψ |1 εGpψq frGpψq,
This in turn can be written as (4.7.6)
G Sdisc,ψ pf q mψ |Sψ |1 εGpψq f Gpψq,
f
P HrpN q. f
P HrpGq,
4.7. ON THE GLOBAL THEOREMS
235
r pN q onto SrpGq. since the mapping fr Ñ frG takes H The identity (4.7.6) is the stable multiplicity formula of Theorem 4.1.2 r 2 pGq, since the constant σ pS 0 q equals 1 for the for the parameter ψ P Ψ ψ trivial group S 0ψ . We can therefore apply its analogues from Corollary 4.1.3 to the data G1 in the endoscopic expansion
Idisc,ψ pf q
¸
G1 PEell pGq
1 ιpG, G1 q Spdisc,ψ p f 1 q,
f
P HrpGq,
for G. The right hand side becomes a double sum ¸¸
(4.7.7)
G1
ψ1
ιpG, G1 q |Sψ1 |1 ε1 pψ 1 q f 1 pψ 1 q,
over G1 P Eell pGq and ψ 1 P ΨpG1 , ψ q. The analysis of this expression then follows the general discussion in §4.4 that led to Lemma 4.4.2. It will be useful to revisit briefly the argument in the simpler case here, especially r 2 pGq was formally ruled out of the earlier discussion. since the case ψ P Ψ (See [K5, §12] and [K6, p. 191].) p The outer sum in (4.7.7) can be taken over G-orbits of elliptic endoscopic 1 1 p p qΓ data G , in which the corresponding element s P G is treated as a Z pG p 1 -orbits of homomorphisms ψ 1 coset. The inner sum is over the set of G L 1 from Aψ to G that map to ψ. We can replace it by the coarser set of p so long as we multiply the orbits under the stabilizer AutG pG1 q of G1 in G, summand by the quotient
|OutGpG1q| |OutGpG1, ψ1q|1.
p 1 -orbit) in the finite group Again, OutG pG1 , ψ 1 q is the stabilizer of ψ 1 (as a G p1 . OutG pG1 q AutG pG1 q{G
The first coefficient in (4.7.7) equals p 1 qΓ |1 |OutG pG1 q|1 . ιpG, G1 q |Z pG
We can therefore write (4.7.7) as a double sum (4.7.8)
¸
pG1 ,ψ1 q
|Sψ1 |1 |Z pGp1qΓ|1 |OutGpG1, ψ1q|1 ε1pψ1q f 1pψ1q
p over G-orbits of pairs. The main step is to apply the correspondence (1.4.11) to the indices pG1, ψ1q. We thereby transform the sum in (4.7.8) to a double sum over p G-orbits of pairs pψG , sG q, where ψG is an L-homomorphism from Aψ to L G that maps to ψ, and s is an element in the centralizer S G ψG . The set p of G-orbits of ψG is the finite set ΨpG, ψ q of order mψ . We can therefore remove the sum over ψG , identifying each ψG with ψ, if we multiply the summand by mψ . The sum over sG then becomes a sum over elements s
236
4. THE STANDARD MODEL
in the underlying finite abelian group Sψ S ψ . It remains to express the summand in (4.7.8) in terms of ψ and s. It follows easily from the definitions that p1 OutG pG1 , ψ 1 q Sψ,s {Sψ,s X G
S ψ,s{S ψ,s X Gp1, where (
g P Sψ : gsg1 s , s P S ψ , p qΓ Centps, S ψ q, S ψ,s Sψ,s {Z pG Sψ,s
and p1 G
Gp1{Z pGpqΓ.
Since S ψ is abelian, we also have S ψ,s and
p1 S ψ,s X G
S ψ Sψ , Sψ pG1q{Z pGpqΓ.
The quotient of this second group by the subgroup p 1 qΓ Z pG
equals
p 1 qΓ Sψ pG1 q{Z pG
Z pGp1qΓ{Z pGpqΓ S ψ pG1q Sψ pG1q Sψ1 .
We thus obtain a reduction
|Sψ1 |1 |Z pGp1qΓ|1 |OutGpG1, ψ1q|1 |Sψ |1 for the product of the first three coefficients in (4.7.8). Applying Lemma 4.4.1 to the fourth coefficient, we obtain 1 G ε1 p ψ 1 q εG ψ 1 psψ 1 q εψ psψ sq εψ psψ sq. Finally, we can write the fifth term as f 1 pψ 1 q
¸
P
rψ π Π
xsψ s, πy fGpπq,
by applying Theorem 2.2.1 to each completion ψv of ψ. We have now expressed all five factors in the summand of (4.7.8) in terms of ψ and s. We can therefore replace the double sum over G1 and s1
4.7. ON THE GLOBAL THEOREMS
237
by the simple sum over s P Sψ described above. We obtain Idisc,ψ pf q mψ |Sψ |1
mψ |Sψ |1
¸ rψ π PΠ
¸
P
P
s Sψ π Π rψ
¸ ¸
P
εψ psψ sq xsψ s, π y fG pπ q
εψ pxq xx, π y fG pπ q
π x Sψ
mψ pπ q fG pπ q,
where (4.7.9)
¸
mψ pπ q
#
x , πy εψ 1 εψ ,
mψ ,
if
0,
otherwise.
On the other hand, we can write
G Idisc,ψ pf q tr Rdisc,ψ pf q ,
f
P HrpGq.
This is because ψ does not contribute to any of the terms with M G in the expansion (4.1.1) of Idisc,ψ pf q. We have thus established the multiplicity formula (4.7.10)
G pf q tr Rdisc,ψ
¸
P
rψ π Π
mψ pπ q fG pπ q,
f
P HrpGq,
postulated by Theorem 1.5.2, under the heuristic assumptions we have made. In so doing, we have shown that the sign character εψ pxq is forced on us by the known dependence of global intertwining operators on L-functions for GLpN q. It has perhaps been difficult to follow the stream of informal remarks of this section. Let us recapitulate what we have done. After taking on some plausible assumptions, we have sketched a proof of four main global assertions – the triviality of orthogonal ε-factors (Theorem 1.5.3(b)), the existence of poles of L-functions (Theorem 1.5.3(a)), the stable multiplicr 2 pGq (Theorem 4.1.2), and the spectral ity formula for parameters ψ P Ψ r 2 pGq (Theorem 1.5.2). The stable multiplicity formula for parameters ψ P Ψ multiplicity formula is at the heart of things. Had we taken it as an assumption (albeit a less plausible one), we could easily have established the other three assertions. In particular, under this condition, our justification of the spectral multiplicity formula above can be regarded as a formal proof. We shall state this as a lemma for future use. Lemma 4.7.1. Assume that the stable multiplicity formula (4.7.6) holds r 2 pGq. Then the spectral multiplicity for any N ¥ 1, G P Ersim pN q and ψ P Ψ formula (4.7.10) also holds for any N , G and ψ. Our proof of the lemma is represented by the elementary argument above, beginning in the paragraph following (4.7.6). The condition on the
238
4. THE STANDARD MODEL
stable multiplicity formula is to be interpreted broadly. It implicitly includes r 2 pGq and the Theorems 1.4.1, 1.4.2 and 2.2.1 (in the definition of the set Ψ G linear form f pψ q), and also includes the condition that there be no conr 2 pGq (by tribution to the discrete spectrum of G from parameters not in Ψ deduction from (4.7.6)). This last global condition is among the most difficult we will have to prove. It is one of the reasons that the simple arguments of this section have to be replaced by the rigorous proofs that will occupy the rest of the volume. We will appeal to Lemma 4.7.1 several times later, in each case at the end of an extended argument that establishes the other conditions. 4.8. Remarks on general groups We shall conclude with a brief description of how some of the earlier discussion of the chapter might be extended. A general heuristic comparison of spectral and endoscopic terms was the topic of [A9]. This followed the special case of the “generic” discrete spectrum treated in [K3, §10–12]. The article [A9] is not so easy to read, for among other reasons, the fact that it was written without the benefit of the volume [KS] of Kottwitz and Shelstad. We are in a stronger position here. In addition to [KS], we now have the results of the earlier sections of this chapter on which to base our general remarks. The special cases treated in these sections, namely the groups r pN q that are the main focus of G P Ersim pN q and the twisted group G G the volume, are clearly simpler. For a start, they come with a well defined substitute Lψ for the global Langlands group. In addition, the locally trivial 1-cocycles that complicate the general theory are not present. There is also the further simplification that the groups Sψ are all abelian. However, we can still sketch how some of our arguments extend to the general case, referring as necessary to the relevant sections of [A9]. For this discussion, we take G to be a general triplet pG0 , θ, ω q over the global field F . Following [KS, 2.1] (but with the notation of §3.2), we form the dual set p 0 L θ, pG p 0 θp G G p 0 . We also form the L-set which is a bitorsor under the dual group G L
p WF GG
LG0 Lθ,
which is an L-bitorsor under the L-group L G0 . Recall that L θ L θω is an p 0 to L G0 , which depends on ω. It extension of the automorphism θp from G p when we form its semidirect product with WF . Recall must be used in G also that an endoscopic datum G1 for G represents a 4-tuple pG1 , G 1 , s1 , ξ 1 q that satisfies (analogues of) the conditions (2.1.1)–(2.1.4) of [KS]. We are p We observe that in the taking s1 here to be a semisimple element in G. notation here, the condition (2.1.4a) of [KS] asserts that s1 ξ 1 pg 1 q a1 pw1 q ξ 1 pg 1 q s1 ,
g1
P G1,
4.8. REMARKS ON GENERAL GROUPS
239
where w1 is the image of g 1 in WF , and a1 pwq is a locally trivial 1-cocycle p 0 q. from WF to Z pG We must of course assume that we have a suitable substitute for the Langlands group LF . We take it to be an object in the category of locally compact topological groups. We suppose that we have at our disposal an extension LF ÝÑ WF ,
with compact connected kernel KF , that takes the place of LF . We could of course assume simply that LF is the actual Langlands group LF . In practice, LF will be a larger group, or rather a group that contains some quotient of LF over WF that is relevant to the endoscopic study of G. For example, we will suggest later in §8.5 how to replace all of our complex groups Lψ with one locally compact group LF . We suppose that along with LF , we are also given a distinguished family Φbdd pGq of L-homomorphisms φ : LF
ÝÑ
L
G0 .
p 0 , and be determined up These objects should have bounded image in G to the general equivalence relation introduced in [K3]. Namely, two Lhomomorphisms φ and φ1 are equivalent if
(4.8.1)
φ1 puq z puq g φpuq g 1 ,
u P LF ,
p 0 , and z is the pullback of a locally trivial 1-cocycle where g belongs to G p 0 q. For any φ P Φ pGq, we set Sφ pG0 q equal to the group from WF to Z pG bdd p 0 such that of elements s P G
(4.8.2)
sφpuq z puq φpuq s,
u P LF ,
where z is again the pullback of a locally trivial 1-cocycle. Our guiding intuition is to be based on the following expectation: there should be a canonical L-homomorphism
ÝÑ
LF
LF
from the actual Langlands group to LF , determined up to conjugacy by the kernel KF , such that the restriction mapping from Φbdd pGq to Φbdd pGq is a bijection that preserves the centralizers Sφ pG0 q. Given the set Φbdd pG0 q, we define ΨpGq to be the family of equivalence classes of L-homomorphisms ψ : AF
ÝÑ
L
G0 ,
AF
LF SU p2q,
whose restriction to LF lies in Φbdd pG0 q, and for which the centralizer set Sψ
S ψ pG q
is not empty. Equivalence is defined here by the evident analogue of (4.8.1). The set Sψ pGq is defined as in (4.8.2), but with ψ in place of φ, and s ranging
240
4. THE STANDARD MODEL
p rather than G p 0 . The elements ψ in ΨpGq are then to be over elements in G treated as the global parameters for G. We would actually need to assume more broadly that LF comes also with a distinguished family Φbdd pH q of special parameters for any group H that arises in the endoscopic study of G. For example, H could be a Levi subgroup M of G0 , or a twisted endoscopic group G1 for G, or perhaps an r 1 of G1 . In fact, we would typically permit G itself to auxiliary extension G vary over some suitable family that allows for induction arguments. This of course is exactly what we have done in the special cases we are treating in this volume. In general, we assume that the sets Φbdd pH q are functorial with respect to endoscopic embeddings among the associated L-groups. We then define the general families of parameters ΨpH q and the centralizers Sψ pH q for H, as we did above for G. Finally, we suppose that LF comes with local homomorphisms
ÝÑ
LFv
LF
over the corresponding Weil groups, which are defined as usual up to conjugation. We then obtain localization mappings
P ΨpGq, from ΨpGq to the local parameter sets Ψ pGv q. We can of course form the subset ΨpG, χq of global parameters in ΨpGq that are attached to a given central character datum pXG , χq for G. The localizations then provide a ψ
ÝÑ
ψv ,
mapping
ψ
ψ ÝÑ tpψ q, cpψ q , ψ P ΨpG, χq, as in the special case of §3.3. We recall that tpψ q is the nonnegative real number given by the norm of the imaginary part of the archimedean infinitesimal character of ψ, and that c P CA pG, χq is the equivalence class of families of conjugacy classes in L G attached to unramified places v of F . For any such pair pt, cq, we define ΨpG, t, cq to be the set of ψ P ΨpG, χq such that tpψ q, cpψ q pt, cq.
If c belongs instead to some quotient CrA pG, χq of CA pG, χq, as in §3.3, we can of course define ΨpG, t, cq to be the union of sets associated to the points in the fibre of c in CA pG, χq. We can also define the set r 1 , ξr1 , t, cq, ΨpG1 , t, cq ΨpG
r 1 , ξr1 q, as the union of for any element G1 P E pGq with auxiliary datum pG r1 , χ r1 , χ subsets of ΨpG r1 q attached to points in the fibre of c in CA pG r1 q. The general goal is of course to understand the discrete part of the trace formula, and the spectral information it contains. In §3.3, we observed that it has a canonical decomposition
Idisc,t pf q
¸
P p
c CrA G,χ
q
Idisc,t,c pf q,
f
P H p G q,
4.8. REMARKS ON GENERAL GROUPS
241
into c-components. It is therefore enough to understand any such component, and specifically, to compare the c-analogues of the spectral and endoscopic expansions (4.1.1) and (4.1.2). In our special case of groups G P Ersim pN q, where we can appeal to strong multiplicity 1 for GLpN q and its generalization in Theorem 1.3.2, we indexed c-components in (3.3.12) r pN q. This is not possible and (3.3.13) directly in terms of parameters ψ P Ψ in general. Instead, one must somehow decompose the linear forms (4.8.3)
tr MP,t,c pw, χq IP,t,c pχ, f q ,
f
P H p G q,
and (4.8.4)
1 Spdisc,t,c pf 1 q,
f
P H pG q ,
in the general c-analogues of (4.1.1) and (4.1.2) into contributions from parameters ψ P ΨpG, t, cq. The question is embedded in the larger problem of establishing general global intertwining relations for (4.8.3) and stable multiplicity formulas for (4.8.4). There is no reason to expect that we could establish global intertwining relations in advance. They will no doubt have to be proved by local-global methods, which ultimately rely on the identity of (4.1.1) and (4.1.2). At least this is how we will be dealing with the groups G P Ersim pN q in this volume. On the other hand, we might often expect to have stable multiplicity formulas already in hand before beginning the comparison. One reason is that they apply only to connected quasisplit groups, while the general object G is much broader. We will not see this distinction for much of the present volume, since the groups G P Ersim pN q of interest in the first eight chapters are in fact quasisplit. Our proof of the stable multiplicity formula for these groups will have to come out of the comparisons of the expansions (4.1.1) and (4.1.2), and will be one of our most difficult tasks. Once we do have a proof, however, it will be much easier to deal with inner forms of G. We shall therefore assume that the linear form (4.8.4) satisfies the stable multiplicity formula of §4.1, or rather the adaptation of (4.1.12) to this general object. We thus assume that for our general triplet G, we can write
1 Spdisc,t,c pf 1 q
¸
ψ 1 Ψ G1 ,t,c
for any G1
P p
q
1 1 pf 1 q, Spdisc,ψ
P EellpGq, where 1 1 pf 1 q |Sψ1 |1 σpS 0 1 q ε1 pψ1 q f 1 pψ1 q. (4.8.5) Spdisc,ψ ψ The coefficients |Sψ1 |1 and σ pS 0ψ1 q in (4.8.5) have already been defined. The 1 sign character εG ψ 1 in the coefficient
1 ε1 p ψ 1 q ε G ψ 1 ps ψ 1 q
242
4. THE STANDARD MODEL
is defined exactly as in (1.5.6). The linear form f 1 pψ 1 q is supposed to be a product ± 1 1 ± 1 f 1 pψ 1 q fv pψv q, f1 fv , v
v
r 1 Fv , attached to the localizations ψ 1 of stable characters on the groups G v
p q
of ψ 1 . Part of our assumption here is that we have already defined these local objects. Given the stable multiplicity formula (4.8.5), the general c-analogue of the endoscopic expansion (4.1.2) will be more accessible than that of the spectral expansion (4.1.1). We shall sketch what extensions of the analysis of §4.4 are needed to deal with it. We do not have to be concerned with the induction hypotheses that complicated the earlier discussion. We do have to address the main complication of the general case here, the fact that the set Sψ can be larger than the centralizer of the image of ψ. For example, p 0 q of G p 0 , rather than just its subgroup of Sψ contains the full center Z pG Γ-invariants. We define the associated two quotients of Sψ by Sψ
(4.8.6)
Sψ {Z pGp0q
and
π0pS ψ q Sψ {Sψ0 Z pGp0q. The sets Sψ Sψ pGq, S ψ S ψ pGq and Sψ Sψ pGq are then bitorsors under the respective complex groups Sψ Sψ pG0 q, S ψ S ψ pG0 q and r pN q and G G r Sψ Sψ pG0 q. In our special cases G P Esim pN q, G G p SOp2, Cq), Sψ does equal the centralizer, and this (and so long as G Sψ
notation matches what we have been using. In the general case, we write Cψ Cψ pG0 q for the actual centralizer p 0 , following Kottwitz [K3, §10]. We then define the of the image of ψ in G associated two quotients C ψ C ψ pG0 q and Cψ Cψ pG0 q of Cψ by Cψ
Cψ {Z pGp0qΓ
and
p 0 qΓ . Cψ π0 pC ψ q Cψ {Cψ0 Z pG These are normal subgroups of finite index in the corresponding quotients S ψ and Sψ , since p 0 q X Cψ Z pG
and
Z pGp0qΓ
p 0 q S 0 Z pG p 0 q. Cψ0 Z pG ψ
In particular, the connected components C 0ψ and S 0ψ are equal. We thus obtain a canonical injection S ψ {C ψ
Sψ {Cψ Ñ ã
p0 q . ker1 F, Z pG
We note again that in the special cases above, we have C ψ S ψ and Cψ Sψ , and therefore no need for this supplementary notation.
4.8. REMARKS ON GENERAL GROUPS
243
In the general case, the set S ψ retains its central role. For example, the chain (4.1.13) of subsets of ΨpGq is defined in terms of S ψ , as before. We can of course define a similar chain of subsets of ΨpG, t, cq, as well as an analogous chain of subsets of ΨpG1 , t, cq for any endoscopic datum G1 r 1 , ξr1 q. It is the connected group S 0 attached to S ψ , with auxiliary datum pG ψ or rather its analogue for a parameter ψ 1 , that determines the coefficient σ pS 0ψ1 q in (4.8.5). (We were actually anticipating the subsequent definition (4.8.6) of the quotient S ψ1 when we stated (4.8.5).) Since this coefficient vanishes if the center of S 0ψ1 is infinite, we can take the sum in the formula
1 for Spdisc,t,c pf 1q above over the subset Ψdisc pG1 , t, cq ψ 1
(
P ΨpG1, t, cq : |Z pS ψ1 q| 8
of ΨpG1 , t, cq , or better, the smaller subset (4.8.7)
Ψs-disc pG1 , t, cq ψ 1
(
P ΨpG1, t, cq : |Z pS 0ψ1 q| 8
of “stable-discrete” parameters. For general perspective, we note that if we were to form the set Ψs-ell pG1 , t, cq Ψell pG1 , t, cq X Ψs-disc pG1 , t, cq
of “stable-elliptic” parameters in Ψell pG1 , t, cq, it would reduce simply to the subset Ψ2 pG1 , t, cq. We are assuming that c is any class in some given quotient CrA pG, χq. The c-analogue of the endoscopic expansion is the formula ¸
G1 Eell G
P p q
for the c-component
1 pf 1 q ιpG, G1 q Spdisc,t,c
Idisc,t,c pf q,
f
P H pG q ,
of Idisc,t pf q given by (3.3.10). We therefore have an expansion (4.8.8)
¸¸ G1
ψ1
1 1 pf 1 q ιpG, G1 q Spdisc,ψ
for Idisc,t,c pf q, where G1 is summed over Eell pGq, and ψ 1 is summed over Ψdisc pG1 , t, cq. The coefficient ιpG, G1 q is given in (3.2.4) as (4.8.9) p 0 q 1 ker1 F, Z pG p 1 q Z pG p 1 qΓ 1 OutG pG1 q1 , π0 pκG q1 ker1 F, Z pG
1 1 pf 1 q is defined by (4.8.5). The heart of §4.4 was a transformawhile Spdisc,ψ tion of the double sum over pG1 , ψ 1 q to a double sum over pψ, sq, according to the correspondence (1.4.11). The general process here is a little more complicated. It will be instructive for us to describe the steps in symbolic form, although this should not disguise the fact that we are basically repeating discussion from §4.4.
244
4. THE STANDARD MODEL
The general correspondence (1.4.11) can be regarded as an equivariant bijection
Y pG q ÝÑ p 0 -sets. The domain X pGq is the set of pairs x pG1 , ψ 1 q, between left two G 1 where G belongs to the set E pGq of endoscopic data for G, taken up to the image ξ 1 pG 1 q of G 1 in L G and up to translation of the associated semisimple p by Z pG p 0 q, while ψ 1 is an actual L-homomorphism from A to element s1 P G ψ X pG q
r 1 , which factors through the L-embedding ξr1 . The codomain the group L G is the set of pairs y pψ, sq, where ψ belongs to the set F pGq of actual L-homomorphisms from Aψ to L G0 , and s is a semisimple element in S ψ . The projections of the two kinds of pairs onto their first components yield a larger diagram
X pG q
Ó E pG q
ÝÑ
Y pG q
Ó F p Gq
p 0 -equivariant mappings. of G We note that the correspondence restricts to a bijection between the 0 p G -invariant subsets
Xdisc,t,c pGq x pG1 , ψ 1 q : G1 and
Ydisc,t,c pGq y
pψ, sq :
P EellpGq, ψ1 P Fs-discpG1, t, cq
(
(
P FdiscpG, t, cq, s P S ψ,ellq of X pGq and Y pGq. We are writing Eell pGq, Fdisc pG, t, cq and Fs-disc pG1 , t, cq here for the respective preimages of Eell pGq, Ψdisc pG, t, cq, and Ψs-disc pG1 , t, cq r 1 , ξr1 q. This bijection also fits into a diagram, as in sets E pGq, F pGq and F pG above, but with E pGq and F pGq replaced by Eell pGq and Fdisc pG, t, cq. p 0 -sets X pGq, Xdisc,t,c pGq, etc. all descend to mapThe mappings of G p 0 zzX pGq, G p 0 zzXdisc,t,c pGq, etc., of pings among the corresponding sets G 0 0 0 p -orbits. The set G p zzEell pGq of G p -orbits in Eell pGq equals Eell pGq, by G 0 p p 0 -orbits in Fdisc pG, t, cq projects definition. The set G zzFdisc pG, t, cq of G onto Ψdisc pG, t, cq, but this mapping is not generally bijective. However, it 1 0 p follows from the definitions that the group ker F, Z pG q acts transitively ψ
on its fibres, and that the the injective image of stabilizer of any ψ equals 1 0 0 p p Sψ {Cψ in ker F, Z pG q . The fibre of ψ in G zzFdisc pG, t, cq therefore has order (4.8.10)
p 0 Sψ 1 Cψ , ker1 F, Z G
p q
since |Sψ | |Sψ |. That said, we see that the mappings all fit together into a general diagram
4.8. REMARKS ON GENERAL GROUPS
p 0 zzXdisc,t,c pGq G
Ó
> Gp0zzY
p 0 zzEell pGq G
(4.8.11)
disc,t,c
Ó
245
p Gq
p 0 zzFdisc pG, t, cq G
} Eell pGq
Ó
Ψdisc pG, t, cq.
The diagram (4.8.11) is the focal point for our transformation of the endoscopic expression (4.8.8). Speaking in symbolic terms, we need to follow the path that leads from the lower left hand corner of the diagram to the lower right hand corner. The first step will be to change the double sum in p 0 -orbits in Xdisc,t,c pGq. This requires two (4.8.8) to a sum over the set of G 1 changes in the inner sum over ψ . The variable ψ 1 in (4.8.8) is supposed to be summed over the set r 1 , ξr1 , t, cq. Ψs-disc pG1 , t, cq Ψs-disc pG
We can instead sum it over the larger set r 1 zzFs-disc pG r 1 , ξr1 , t, cq G p
p 1 zzFs-disc pG1 , t, cq, G
if we multiply the summand by the product (4.8.12)
p 1 1 Sψ1 Cψ1 1 . ker1 F, Z G
p q
r 1 of (4.8.10), Indeed, this product equals the inverse of the analogue for G r 1 Ñ G1 imply that the mapping since the conditions on the extension G p1 q ker1 pF, G
ÝÑ
r1 q ker1 pF, G p
is an isomorphism. The variable G1 in (4.8.8) is supposed to be summed over the set p 0 zzEell pGq Eell pGq G in the lower left hand corner of the diagram. This is what we want. However, p 0 is the extension AutG pG1 q of G p 1 , rather than G p1 . the stabilizer of G1 in G 1 This means that we should really be summing ψ over the set of AutG pG1 qp 1 -orbits. We can do this, so long as we orbits in Fdisc pG1 , t, cq, rather than G multiply the summand by the number
|OutGpG1q| |OutGpG1, ψ1q|1 p 0 -orbits in the given AutG pG1 q-orbit of ψ 1 . As earlier, OutG pG1 , ψ 1 q is of G p 0 -orbit, in OutG pG1 q. the stabilizer of ψ 1 , regarded as a G (4.8.13)
We have shown how to express (4.8.8) as an iterated sum over the base p 0 zzEell pGq and its pointwise fibres in (4.8.11). That is, we can write set G
246
4. THE STANDARD MODEL
(4.8.8) as a sum over orbits x pG1 , ψ 1 q in the upper left hand set in (4.8.11), so long as the summand is rescaled by the product of (4.8.12) and (4.8.13). We write this in turn as a sum over elements y pψ, sq in the upper right hand set in (4.8.11), with the expectation of being able to express the summand in terms of the bijective image pψ, sq of pG1 , ψ 1 q. In other words, p 0 zzFdisc pG, t, cq and we express (4.8.8) as an iterated sum over the base set G its pointwise fibres in (4.8.11). The second step is to write this as a double sum over ψ and s in more familiar sets. As it presently stands, the expression is an iterated sum over orbits of p 0 and orbits of elliptic elements L-homomorphisms ψ P Fdisc pG, t, cq under G 0 p . We can take the first sum over s P S ψ,ell under the stabilizer of ψ in G ψ in the quotient Ψdisc pG, t, cq, provided that we multiply the summand by p 0 is the centralizer Cψ . the order (4.8.10) of its fibre. The stabilizer of ψ in G The second sum is over the set of orbits in S ψ,ell under Cψ , or equivalently, the set of orbits under the quotient C ψ of Cψ . However, we would prefer to take a sum over the set S 0ψ zzS ψ,ell
E pS ψ,ellq Eψ,ell of orbits in S ψ,ell under the subgroup C 0ψ S 0ψ of C ψ . The C ψ -orbit of s is bijective with the quotient of C ψ by the subgroup C ψ,s
Centps, C ψ q.
The C 0ψ -orbit of s is bijective with the quotient of C 0ψ by the subgroup C ψ,s
Centps, C 0ψ q.
We can therefore take the second sum over orbits s in Eψ,ell , if we multiply the summand by the quotient
|C ψ,s{C ψ,s| |C ψ {C 0ψ |1,
which is to say the quotient (4.8.14)
|C ψ,s{C ψ,s| |Cψ |1.
We have now established that the double sum over G1 and ψ 1 in the endoscopic expression (4.8.8) can be replaced by a double sum over ψ P Ψdisc pG, t, cq and s P Eψ,ell , provided that the summand is multiplied by the product of the four factors (4.8.10), (4.8.12), (4.8.13) and (4.8.14). The summand itself becomes the product of these four factors with (4.8.9) and the right hand side of (4.8.5). Observe that four of these six factors are given in terms of the pair x pG1 , ψ 1 q, rather than its image y pψ, sq. As in §4.4, some of the components of these factors will cancel from the product. Others can be expressed directly in terms of y. For example, the denominator in (4.8.13) equals
|OutGpG1, ψ1q| |Cψ,s{Cψ,s X Gp1 Z pGp0qΓ|,
4.8. REMARKS ON GENERAL GROUPS
where Cψ,s
tg P Cψ :
gsg 1
247
s u Cy
is the preimage of C ψ,s in Cψ . In particular, (4.8.13) specializes to the factor (4.4.5) in §4.4. We conclude that (4.8.8) can be written as the sum over ψ P Ψdisc pG, t, cq and s P Eψ,ell of the product of two expressions
|Cψ,s{Cψ,s X Gp1Z pGp0qΓ|1 |Cψ1 |1 |Z pGp1qΓ|1 |C ψ,s{C ψ,s|1
(4.8.15) and
|π0pκGq|1 |Sψ |1 σpS 0ψ1 q ε1pψ1q f 1pψ1q, pψ, sq is the image of x pG1, ψ1q under
(4.8.16)
in which y the bijection in (4.8.11). The expressions (4.8.15) and (4.8.16) are the two factors (7.9) and (7.10) in [A9]. They also reduce to the corresponding two expressions of §4.4, where the group Cψ is equal to Sψ . (The multiplicity mψ in the second expression in §4.4 does not appear here, since ψ now represents an element r pGq.) As in the special in a set ΨpGq rather than an equivalence class in Ψ cases of §4.4, the product of the two expressions simplifies. We shall outline the steps. The product of the first two factors in (4.8.15) has a reduction
|Cψ,s{Cψ,s X Gp1 Z pGp0qΓ|1 |Cψ1 |1 |C ψ,s{C 0ψ,s Z pGp1qΓ|1,
as in the discussion in §4.4. (See also [A9, p. 48–49].) The entire expression (4.8.15) can then be written as
|C ψ,s{C 0ψ,s Z pGp1qΓ|1 |Z pGp1qΓ|1 |π0pC ψ,sq|1 |C 0ψ,s X Z pGp1qΓ|1 |π0pS ψ,sq|1 |S 0ψ,s X Z pGp1qΓ|1,
since C 0ψ,s
S 0ψ,s. In the expression (4.8.16), the coefficient p 1 qΓ p q Γ 0 σ S 0 { S 0 X Z pG σ pS 0ψ1 q σ Sψ1 {Z pG ψ,s ψ,s
satisfies an identity
p 1 qΓ | , σ pS 0ψ1 q σ pS 0ψ,s q |S 0ψ,s X Z pG
by (4.1.9). Lemma 4.4.1, whose proof in §4.6 carries over to the general case, allows us to write the coefficient ε1 pψ 1 q in (4.8.16) as r1 G ε1 p ψ 1 q εG ψ 1 p s ψ 1 q εψ p s ψ x s q . Finally, we can again write (4.8.17)
f 1 pψ 1 q fG1 pψ, sq,
to denote the dependence of the linear form in (4.8.16) on ψ and s. All of the components in (4.8.15) and (4.8.16) have now been expressed in terms of ψ and s. With no further need to refer to the original pair
248
4. THE STANDARD MODEL
x pG1 , ψ 1 q, we can allow x to denote an element in the group Sψ of connected components of S ψ . We can then write the endoscopic expression (4.8.8) as a triple sum (4.8.18) |π0 pκG q|1 |Sψ |1
¸¸¸ ψ
x
|π0pS ψ,sq|1 σpS 0ψ,sq εGψpsψ xq fG1 pψ, sq
s
over ψ P Ψdisc pG, t, cq, x P Sψ and s P Eψ,ell pxq. The transformation (4.8.18) of the endoscopic expression (4.8.8) is the general analogue of the formula of Lemma 4.4.2 (with allowance for our slightly different use of the symbol ψ). We would again expect the linear form fG1 pψ, xq fG1 pψ, sq, x P Sψ , s P S ψ,ss pxq, to depend only on the component x of s. If this is so, we can compress the sum over s in (4.8.18) to the coefficient epxq eψ pxq of Proposition 4.1.1. After a change of variables in the sum over x, we will then be able to write (4.8.18) as a double sum (4.8.19)
|π0pκGq|1 |Sψ |1
¸¸ ψ
1 e ψ p x q εG ψ pxq fG pψ, sψ xq
x
over ψ P Ψdisc pG, t, cq and x P Sψ . This is the general analogue of the expression (4.4.11) of Corollary 4.4.3. The c-discrete part Idisc,t,c pf q of the trace formula is thus equal to the elementary endoscopic expression (4.8.19). The other half of the problem would be to extend the analysis of §4.3 to the spectral expansion of Idisc,t,c pf q. The formal aspect of this process can be carried out. For example, the commutative diagram (4.2.3) makes sense for any parameter ψ in the general set ΨpGq. However, one is immediately forced to deal with the linear form (4.8.3). One would need to decompose (4.8.3) explicitly into suitable linear forms fG pψ, uq, ψ P Ψdisc pG, t, cq, u P Nψ , defined explicitly in terms of local normalized intertwining operators. This would be the general analogue of Corollary 4.2.4. We would again expect the linear form fG pψ, xq fG pψ, uq,
x P Sψ , u P Nψ pxq,
to depend only on the image x of u in Sψ . If this is so, the methods in §4.3 would lead to an elementary spectral expression for Idisc,t,c pf q as a double sum (4.8.20)
|π0pκGq|1 |Sψ |1
¸¸ ψ
i ψ px q ε G ψ pxq fG pψ, xq
x
over ψ P Ψdisc pG, t, cq and x P Sψ . We will say no more about the putative expression (4.8.20), except to point out that it is the general analogue of the expression (4.3.9) of Corollary 4.3.3. In particular, the coefficient ipxq iψ pxq in the expression equals
4.8. REMARKS ON GENERAL GROUPS
249
the coefficient epxq eψ pxq in (4.8.19), by Proposition 4.1.1. Moreover, the linear form fG pψ, xq in (4.8.20) ought to be equal to the linear form fG1 pψ, sψ xq in (4.8.19), by some general version of the global intertwining relation. This would establish a term by term identification of the two expressions (4.8.19) and (4.8.20). We have noted that the chain of subsets (4.1.13) of ΨpGq also makes sense in this general setting. Of most significance is the subset Ψ2 pGq, which ought to govern the discrete spectrum. To be more specific, we could say that the comparison of (4.8.19) and (4.8.20) has two purposes. One is to prove that parameters in the complement of Ψ2 pGq do not contribute to the discrete spectrum. The other is to compute the contribution of parameters ψ that do lie in Ψ2 pGq. The expected contribution of ψ will be a multiplicity formula for representations π in a global packet Πψ , in terms of associated characters xx, πy on the finite group Sψ . This would be the natural generalization of the multiplicity formula (4.7.9) of Theorem 1.5.2, with allowance made for the larger group Sψ , the fact that Sψ could be nonabelian, and the possibility of G being a twisted group. It could also be regarded as the degenerate case of the proposed global intertwining relation. Given the stable multiplicity formula (4.8.5) for ψ P ΨpGq, one would try to deduce the actual spectral multiplicity formula for representations π by a comparison of (4.8.19) with (4.8.20), as in our special case discussed at the end of the last section. We add one final comment – on the title of this chapter. It seems apt, despite being a possible case of cultural appropriation! The contents of the chapter do represent a model of sorts. The model is standard, in the sense that it governs the general endoscopic comparison of trace formulas. However, we are still a very long way from being able to apply it to arbitrary groups. We can hope that the results we obtain for the groups G P Ersim pN q, combined with similar results for other classical groups, will lead to an endoscopic classification of a much broader class of groups, which share only the property of being endoscopically related to general linear groups. What about more general groups? Langlands has outlined a remarkable, though still speculative, strategy for applying the trace formula to cases of functoriality that are not tied to endoscopy [L13], [L14], [FLN]. His proposal is probably best suited to the stable trace formula. If it works, it might ultimately yield a general stable multiplicity formula as a byproduct. One could then imagine applying this formula in the standard model, as above, to a general endoscopic classification of representations.
CHAPTER 5
A Study of Critical Cases 5.1. The case of square integrable ψ In the last chapter, we studied the contribution to the trace formula of many global parameters ψ. We did so by applying induction arguments to what we called the standard model. The culmination of these efforts was Proposition 4.5.1, which established the stable multiplicity formula in a number of cases. It also yielded the multiplicity formula of Theorem 1.5.2 for these cases, with its interpretation as a vanishing condition. We recall that the stable multiplicity formula is the assertion of Theorem 4.1.2. It is at the heart of all of our global theorems. We have now established r pN q that do not lie in Ψ r ell pN q, or in any of the it for global parameters ψ P Ψ r r r subsets Ψell pGq of ΨpN q attached to elements G P Esim pN q. In this chapter we shall begin the study of the cases that remain. These cases are of course the most important. They are also the most difficult. The questions become increasingly subtle as we consider successive subsets of parameters in the chain (5.1.1)
r sim pGq Ψ r 2 p Gq Ψ r ell pGq, Ψ
G P Ersim pN q,
attached to a given G. The answers will draw upon new techniques, which we will introduce in this and future chapters. These will be founded on localglobal methods, in which local and global techniques are used to reinforce each other. We begin by setting up a general family of global parameters, on which we will later impose local constraints. Throughout the chapter, we will let Fr
(5.1.2)
8 º
N 1
FrpN q
denote a fixed family of global parameters ψ
`1ψ1 `
in the general set r Ψ
8 º
`
`r ψr
r pN q. Ψ
N 1
We assume that Fr is closed under the projections ψ
ÝÑ
1 ¤ i ¤ r,
ψi , 251
252
5. A STUDY OF THE CRITICAL CASES
and under direct sums
pψ1, ψ2q ÝÑ
ψ1 ` ψ2,
ψ1, ψ2
r P F.
In other words, Fr is the graded semigroup generated by its simple components ψ µ b ν, ψ P Frsim , with the grading
deg ψ
N,
ψ
P Ψr pN q,
r We have obviously written inherited from Ψ.
Fr X Ψr sim We shall also write Frell , Fr2 pN q, Frsim pGq, etc., for the intersection of Frsim
here. r Fr with the corresponding subset of Ψ. Throughout this chapter, the field F will remain global, and N will again be fixed. We shall assume inductively that all of the local and global theorems hold for any ψ P Fr with degpψ q N . This is of course the blanket induction hypothesis we have carried up until now. We are introducing r since it is in the limited context of such it formally within the family F, families that we will eventually establish the general local theorems. Its interpretation here will be for the most part obvious. There is one point in the induction hypothesis for Fr that does call for an explanation. It concerns the assertion of Theorem 1.5.3(b). Consider a pair ψi µi b νi , i 1, 2,
r Motivated by the remark of distinct self-dual, simple parameters in F. following the proof of Lemma 4.3.1 in §4.6, we shall call pψ1 , ψ2 q an ε-pair, for want of a better term, if the following condition holds: the generic pair pµ1, µ2q is orthogonal, in that it satisfies the condition imposed on the pair pφ1, φ2q in Theorem 1.5.3(b), and the tensor product ν1 b ν2 is the sum of an odd number of irreducible representations of SLp2, Cq of even dimension. The corresponding sum
(5.1.3)
ψ
ψ1
`
ψ2 ,
which we will call an ε-parameter, is a self-dual, elliptic parameter for a general linear group. If w is the associated twisted Weyl element for the relevant maximal Levi subset, it follows from the analysis of (4.6.9) and (4.6.10) in the proof of Lemma 4.3.1 that the corresponding global normalizing factor satisfies (5.1.4)
rψ pwq εp 21 , µ1 µ2 q.
Theorem 1.5.3(b) asserts that this number equals 1. Our induction hypothesis for Fr will be that the assertion is valid for any ε-pair pψ1 , ψ2 q with degpψ1 q
degpψ2 q N.
5.1. THE CASE OF SQUARE INTEGRABLE ψ
253
It then follows from the remark in §4.6 mentioned above that Lemma 4.3.1 is valid for any ψ P FrpN q that is not an ε-parameter. To make further progress, we shall also take on the following temporary hypothesis on the elements in Frell pN q. Assumption 5.1.1. Suppose that G P Erell pN q, and that ψ belongs Fr2 pGq. Then there is a unique stable linear form f
ÝÑ
f G pψ q,
r pGq with the general property on H
frG pψ q frN pψ q,
together with a secondary property that
f
P HrpGq,
r pN q, fr P H
f G pψ q f S pψS qf O pψO q, in case
f
P HrpGq,
G GS G O , ψ ψS ψO ,
and fG
fS fO
are composite. The reader will recognize in this hypothesis a global analogue of the local assertion of Theorem 2.2.1(a). It would obviously follow from the local assertion, but this of course has yet to be established. We note that if ψ lies in the complement of Fr2 pGq in FrpGq, the global assertion can be reduced to the corresponding assertion for a Levi subgroup of GLpN q, which then follows from our induction hypothesis. In other words, the assertion of Assumption 5.1.1 is valid more generally for any ψ P FrpGq. (If G GS GO , we have naturally to define FrpGq as the product of FrpGS q and FrpGO q, rather than as a subset of FrpN q.) This is the way Theorem 2.2.1(a) was stated. We need to pause here for some further discussion of the condition ψ P Fr2 pGq in Assumption 5.1.1. For it raises a second point concerning our r If ψ is not simple generic, which is to say induction hypothesis for F. r sim pN q of self dual cuspidal automorphic that it does not lie in the set Φ representations of GLpN q, the datum G P Erell pN q such that ψ belongs to Fr2 pGq is given by the induction hypothesis and the original constructions of §1.4. However, we will obviously have to deal also with parameters that are simple generic. According to the global theorems we have stated, there will be three r sim pGq of Φ r sim pN q attached to a equivalent ways to characterize the subset Φ given G P Ersim pN q. Theorem 1.4.1, which we have used for the formal definir sim pGq in terms of the discrete spectrum of G. Theorem tion, characterizes Φ
254
5. A STUDY OF THE CRITICAL CASES
r pGq, describes 1.5.3(a), together with the fact that ηψ ηG for any ψ P Φ r sim pGq in terms of the poles of L-functions. The third characterization Φ is provided by the stable multiplicity formula of Theorem 4.1.2. It tells us r sim pGq is the subset of elements ψ P Φ r sim pN q such that the linear that Φ G r form Sdisc,ψ on HpGq is nonzero. We have of course not proved any of these
theorems, regarded as conditions for the generic elements in Frsim pN q. But we could ultimately still take the assertion of any one of them as a definition of the subset Frsim pGq of Frsim pN q. The third characterization is the least elementary. However, with its roots in harmonic analysis, it would be the most natural one to work with as we try to extend the global part of our induction hypothesis here to N . We shall therefore take it as the basis for a temporary definition, which we will later show is equivalent to the one in §1.4. Suppose that Fr is generic, in the sense that it is contained in the subset r If G belongs to Ersim pN q, we could simply of generic global parameters in Φ. define Frsim pGq by the third condition above, namely as the subset of eleG 0. For we know from (3.3.14) that if ψ ments ψ P Frsim pN q with Sdisc,ψ
belongs to Frsim pN q, there is a G P Ersim pN q with this property. Indeed, the left hand side of (3.3.14) is nonzero, while it follows from Proposition 3.4.1 and Theorem 1.3.2 that the contribution to the right hand side of (3.3.14) of any datum in the complement of Ersim pN q in Erell pN q vanishes. We expect G P Ersim pN q to be uniquely determined by ψ. We know this to be so if ηψ 1 or N is odd. But if ηψ 1 and N is even, we would not be able to rule out the possibility at present that ψ could lie in two different sets Frsim pGq. A more serious problem with this definition is the possibility of a conflict with Assumption 5.1.1 in some of our later constructions. We shall therefore adopt a slightly more flexible (though ultimately equivalent) convention. r we simply assume that we have attached global For the generic family F, subsets
0u of Frsim pN q to the simple endoscopic data G P Ersim pN q, so that ¤ (5.1.6) Frsim pN q Frsim pGq. (5.1.5)
Frsim pGq tψ
P FrsimpN q : P
G Sdisc,ψ
p q
G Ersim N
Assumption 5.1.1, as it applies to simple pairs
pG, ψq,
G P Ersim pN q, ψ
P FrsimpGq,
represents an axiom that is imposed afterwards. Once it is granted, we could then make our temporary definition precise by taking Frsim pGq to be the set G of all parameters ψ P Frsim pN q such that Sdisc,ψ 0, and such that the linear form frN pψ q transfers to G. Assumption 5.1.1 for simple pairs pG, ψ q then
5.1. THE CASE OF SQUARE INTEGRABLE ψ
255
becomes the assertion (5.1.6). In working with this temporary definition, we must keep in mind the hypothetical possibility that the inclusion (5.1.5) might be proper, or the possibility that of the union (5.1.6) might not be disjoint. But in any case, we have now agreed on an ad hoc convention for the simple parameters in FrpN q, thereby introducing a new definition of the sets Fr2 pGq to which Assumption 5.1.1 is supposed to apply. We shall resolve r at the end it in terms of the original definition (for a particular family F) of the chapter. If Fr is nongeneric, in the sense that it contains elements in the compler in Ψ, r we will assume that the relevant global theorems hold for ment of Φ the generic part µ of any simple parameter ψ P Frsim . That is, we assume r sim pGq are equivalent for such parameters. that the three ways of defining Φ r r The subsets F2 pGq and F pGq of Fr are then all well defined. We have thus refined the global induction hypothesis, as it applies to r We observe that the two adjustments conparameters in a given family F. cern the two parts of Theorem 1.5.3. Part (a) of the theorem represents an induction hypothesis for generic families that is assumed to have been resolved in the nongeneric case. Part (b) represents an induction hypothesis in the nongeneric case that is irrelevant for generic families. We will ultimately study the two kinds of families Fr separately, the generic case being the topic of Chapter 6, and the nongeneric case the topic of Chapter 7. In this chapter, however, we will generally be able to treat the two cases together. Having assumed inductively that for a given family Fr the relevant theorems hold for elements ψ P FrpN q with N N , we must try to show that they hold also for parameters ψ P FrpN q. This is a long term proposition. It will not be settled in general until the later chapters, after we have constructed families Fr with specified local conditions. In the meantime, we shall see what can be deduced from the induction hypotheses, when used in combination with Assumption 5.1.1. In this section, we shall look at the basic case that ψ lies in Frell pN q. With our convention above for the simple generic elements, Frell pN q is the union over G P Erell pN q of the sets Fr2 pGq. However, this temporary convention still leaves open the question of how to assign a meaning to the centralizers SψG attached to simple pairs pG, ψ q, with ψ a generic element in Frsim pGq. In situations where pG, ψ q is understood to have been fixed, we shall implicitly agree that (5.1.7)
Sψ
G
Sψ
#
1,
H,
if G
G,
otherwise,
for any G P Ersim pN q. This is what we expect, since G should be the unique datum with ψ P Frsim pGq. In any case, the difference 0 G Sdisc,ψ pf q 0 Sdisc,ψ pf q, f P HrpGq,
256
5. A STUDY OF THE CRITICAL CASES
pf q and its expected value will then be defined as in (4.4.9) between Sdisc,ψ (with G in place of G1 ). We will of course also be interested in the case that ψ is not simple generic. In this case, SψG is given by induction, and pf q is then defined by Assumption 5.1.1. We fix the linear form 0 Sdisc,ψ
ψ P Frell pN q, and take G to be a fixed datum in Erell pN q with ψ P Fr2 pGq. We remind ourselves that G is uniquely determined by ψ, except possibly in the case that ψ is simple generic. The centralizer set r pN q Srψ pN q Sψ G
r pN q is a connected abelian bi-torsor. It follows that the global infor G r pN q and ψ reduces simply to the assertion of Astertwining relation for G sumption 5.1.1. In particular, the stated conditions of Corollaries 4.3.3 and r pN q in place of G) both hold. In the case of Corollary 4.3.3, 4.4.3 (with G there is also an implicit condition, namely that the proof of Lemma 4.3.1 is valid under the induction hypotheses now in force. If we assume that ψ is not an ε-parameter (5.1.3), this condition holds, and both corollaries are r pN q and ψ. We then see from the identity of Proposition 4.1.1, valid for G which is elementary in this case, that the corresponding expressions (4.3.9) r pN q, the difference between and (4.4.11) are equal. It follows that for fr P H r Idisc,ψ pf q and the linear form r pN q G pfrq 0rdisc,ψ pfrq equals the difference between Idisc,ψ pfrq and the linear form 0 N rrdisc,ψ
r pN q pfrq 0sGdisc,ψ pfrq. r r pN q0 GLpN q, we know that 0 rrN Since G disc,ψ pf q 0. It follows from the r pN q in place of G) that definition (4.4.10) (again with G ¸ pfr q 0. r (5.1.8) ιpN, G q 0 Spdisc,ψ 0 N srdisc,ψ
G Ersim N
P
p q
If G belongs to the complement of Ersim pN q in Erell pN q, we know from G G our induction hypothesis that 0 Sdisc,ψ pf q, the difference between Sdisc,ψ pf q and its expected value, vanishes. One of our long term goals is to show that the same is true if G belongs to Ersim pN q, or indeed if G is replaced by any element G P Ersim pN q. In other words, we would like to establish the stable multiplicity formula of Theorem 4.1.2 for the given ψ, and any G P Ersim pN q. The next lemma gives some preliminary information. Lemma 5.1.2. Assume that for the given pair
pG, ψq,
G P Erell pN q, ψ
P Fr2pGq,
5.1. THE CASE OF SQUARE INTEGRABLE ψ
257
ψ is not an ε-parameter (5.1.3). Then the stable multiplicity formula (4.1.11) holds for G and ψ if and only if for every G
pf q 0 S pf q 0, Sdisc,ψ disc,ψ
P ErsimpN q with G G.
f
0S disc,ψ G r f vanishes
Proof. One direction is clear. If
P HrpGq,
0 for every G
G in r pN q. The for any fr P H
G Ersim pN q, (5.1.8) tells us that 0 Spdisc,ψ p q required assertion in this direction already being known if G is composite, we r pN q to can assume that G P Ersim pN q. The correspondence fr Ñ frG from H 0 G r S pGq is then surjective, by Corollary 2.1.2. It follows that Sdisc,ψ vanishes,
r N pGq-symmetric stable linear form. In other words, the stable as an Out multiplicity formula for G p f q, Sdisc,ψ
is valid. G Conversely, suppose that 0 Sdisc,ψ ¸
But if G
G G
f
P HrpGq,
0. Then (5.1.8) reduces to pfr q 0. r ιpN, G q 0 Spdisc,ψ
P ErsimpN q is distinct from G, (4.4.12) tells us that pf q tr R pf q, 0 r p G q , f P H Sdisc,ψ pf q Sdisc,ψ disc,ψ r pG q unless ψ is generic and simple, in which case since ψ does not lie in Ψ
the convention (5.1.7) leads to the same outcome. Applying Proposition 3.5.1 to the resulting formula ¸
G G
r ι N, G tr Rdisc,ψ f
p
q
p q 0,
in which tfr u is replaced by any compatible family of functions tf u, we deduce that
pf q tr Rdisc,ψ
pf q 0, Sdisc,ψ
G
G,
as required.
Corollary 5.1.3. Suppose that G and ψ are as in the lemma, and that one of the further conditions (i) N is odd, (ii) ηψ 1, or (iii) G R Ersim pN q, holds. Then (5.1.9) G while Sdisc,ψ
G Sdisc,ψ pf q mψ |Sψ |1 εGpψq f Gpψq,
f
P HrpGq,
0 for every G P ErsimpN q distinct from G. In other words, the stable multiplicity formula holds for ψ and any G P Ersim pN q.
258
5. A STUDY OF THE CRITICAL CASES
G Proof. If (iii) holds, the stable multiplicity formula for Sdisc,ψ pf q is a consequence of our induction hypotheses, as we noted above. It then follows G from the lemma that Sdisc,ψ 0 for all G G, which is to say, every G. G We can therefore assume that G P Ersim pN q. Recall that Sdisc,ψ vanishes unless the characters ηψ and ηG on ΓF attached to G and ψ are equal. But r 2 pGq, and we know that ηG determines G uniquely as a ηψ ηG , since ψ P Ψ simple twisted endoscopic datum, if it is nontrivial or if N is odd. In these G cases then, Sdisc,ψ 0 for every G P ErsimpN q distinct from G. The required stable multiplicity formula (5.1.9) for G then follows from the lemma.
The remaining case is for N even, ηψ 1, and G P Ersim pN q. It is not so easily resolved. There is now a second group G_ P Ersim pN q with ηG_ ηψ ηG , and even though ψ does not belong to Fr2 pG_ q (except G_ possibly when ψ is generic and simple), we cannot say a priori that 0 Sdisc,ψ vanishes. In this case, G and G_ are split groups, whose dual groups are SppN, Cq and SOpN, Cq. We write L for the group, isomorphic to GL 12 N , that represents a Siegel maximal Levi subgroup of both G and G_ . For convenience, we shall also write r N pG q|, orpG q |Out
G
P ErsimpN q,
an integer equal to either 1 or 2. There are two transfer mappings f and
ÝÑ
fL
fL,
f
P HrpGq
f_
ÝÑ f _,L fL_, f _ P HrpG_q, r pGq and H r pG_ q respectively to the space S pLq I pLq. from H
It is not hard to see from the definitions that they have a common image in S pLq, which we denote by Sr0 pLq. Lemma 5.1.4. Suppose that G and ψ are as in Lemma 5.1.2, and also that N is even, ξψ 1, and G P Ersim pN q. Then there is a linear form hL on Sr0 pLq such that
ÝÑ
hL pΛq,
hL
P Sr0pLq,
G pf q mψ |Sψ |1 εGpψq f Gpψq orpGq f LpΛq, Sdisc,ψ
and while for every G
G_ Sdisc,ψ pf _q orpG_q f _,LpΛq,
pf q 0 S pf q 0, Sdisc,ψ disc,ψ
f
f_
P HrpG_q,
f
P HrpGq,
P ErsimpN q distinct from G and G_.
P HrpGq,
5.1. THE CASE OF SQUARE INTEGRABLE ψ
259
Proof. If G P Ersim pN q is distinct from G and G_ , the character ηG is distinct from ηψ , and
Sdisc,ψ
0Sdisc,ψ 0.
The corresponding summand in the identity (5.1.8) thus vanishes, and the left hand side of (5.1.8) reduces to a sum of two terms. It follows from (3.2.6) that r ιpN, Gq orpGq r ιpN, G_ q orpG_ q since |Z pGpqΓ| |Z pGp_qΓ| 1. The identity then takes the form G G_ (5.1.10) orpGq1 0 Sdisc,ψ pf q orpG_q1 0Sdisc,ψ pf _q 0, for any compatible pair of functions f, f _ : f
P HrpGq,
f_
P HrpG_q
(
.
As members of a compatible family of functions, f and f _ are not independent. Their stable orbital integrals have the same values on Levi subgroups that are common to G and G_ . But any such Levi subgroup is conjugate to a subgroup of L. Since this is the only obstruction to the independence of f and f _ , the two summands on the left hand side of (5.1.10) can differ only by the pullback of a linear form on Sr0 pLq. In other words, we can write 0 G_ G_ Sdisc,ψ pf _ q Sdisc,ψ pf _q orpG_q f _,LpΛq, f _ P HrpG_q,
where Λ represents a linear form on Sr0 pLq. G The required formula for Sdisc,ψ pf q then follows from (5.1.10), and the specialization 0 G Sdisc,ψ
G pf q mψ |Sψ |1εGpψq f Gpψq pf q Sdisc,ψ
of the definition (4.4.9).
Corollary 5.1.5. With the conditions of the lemma, the linear form f_
ÝÑ orpG_q f _,LpΛq, f _ P HrpG_q, r pG_ q of a unitary character on G_ pAq. is the restriction to H r pG_ q (except possibly when ψ is Proof. Since ψ does not belong to Ψ generic and simple), (4.4.12) tells us that 0 G_ G_ G_ Sdisc,ψ pf _ q Sdisc,ψ pf _q tr Rdisc,ψ pf _ q . G_ pf _ q, the corollary follows. Since the given linear form equals Sdisc,ψ
The last two lemmas and their corollaries came with the condition that ψ P Erell pN q not be an ε-parameter (5.1.3). We shall now see what happens in this supplementary case.
260
5. A STUDY OF THE CRITICAL CASES
Lemma 5.1.6. Suppose that for the given pair
pG, ψq,
G P Erell pN q, ψ
P Fr2pGq,
ψ is an ε-parameter (5.1.3). Then
pf q tr Rdisc,ψ
pf q, 0 0Sdisc,ψ
f
P HrpGq,
for every G P Ersim pN q. Moreover, the generic components µ1 and µ2 of ψ1 and ψ2 satisfy ε 21 , µ1 µ2 1, in accordance with Theorem 1.5.3(b). Proof. We are assuming that ψ is a sum of an ε-pair pψ1 , ψ2 q of simple parameters. By assumption, the generic components µ1 and µ2 are either both orthogonal or both symplectic, while the unipotent components ν1 and ν2 of ψ1 and ψ2 are of opposite type. Therefore ψ1 and ψ2 are themselves of opposite type. This implies that the datum G G1 G2 is composite. It follows that (5.1.11)
frG pψ q tr πψ pfrG q
tr πψ pfr1q
tr πψ2 pfr2 q ,
1
r pN q, fr P H
rψ where πψ πψ1 b πψ2 is the automorphic representation in the packet Π for G obtained from Assumption 5.1.1 and our induction hypothesis. We would like to argue as in the proof of Lemma 5.1.2. However, we do not know that the identity (5.1.8) is valid. The problem is that we r pN q in place of G) for the εdo not have a proof of Lemma 4.3.1 (with G parameter ψ. This alters the contribution of Corollary 4.3.3 to (5.1.8). We must describe the resulting modification of the identity. With the limited information presently at our disposal, we can make use r pN q in place of G) only if the summand in (4.3.9) of Corollary 4.3.3 (with G is multiplied by the quotient N 1 sN,0 puq1 rψN pwu q εN,1 ψ puq εψ pxu q ψ
r pN q of the unproven identity of Lemma of the two sides of the analogue for G 4.3.1. There is of course only one summand, since the centralizer SrψN is
connected. Given that SrψN is also an abelian torsor, we deduce that N,0 N εN,1 ψ puq εψ pxu q sψ puq 1.
The remaining factor in the quotient satisfies rψ pwu q rψN pwq ε
1 2,
µ 1 µ2 ,
according to (5.1.4). By Assumption 5.1.1, the linear forms in (4.3.9) and (4.4.11) again satisfy the global intertwining relation fG1 pψ, sψ xq fG1 pψ, xq fG pψ, xq,
5.1. THE CASE OF SQUARE INTEGRABLE ψ
261
r pN q and fr again in place of G and f . This equals the trace but with G (5.1.11). We can now compare the modified form of (4.3.9) with the corresponding expression (4.4.11). We will then be led to an explicit, possibly nonzero contribution of the composite group G P Erell pN q to the sum (5.1.8). In fact, by substituting the formulas for the coefficients above, we find that the left hand side of (5.1.8) equals
Cψ1 εp 21 , µ1 µ2 q 1 tr πψ pfrG q ,
(5.1.12)
where Cψ1 is the product of the positive constants Cψ and eψ pxq. This is the modification of (5.1.8), an identity that could also have been obtained directly from first principles rather than the general formulas of Corollaries 4.3.3 and 4.4.3. We have shown that the left hand side of (5.1.8) equals (5.1.12). In the summand of G P Ersim pN q in (5.1.8), we can use (4.4.12) to write 0
(5.1.13)
pf q tr R pf q , Sdisc,ψ disc,ψ
f
P HrpGq,
r pG q. In the identity itself, we can replace the since ψ does not belong to Ψ family tfr u by an arbitrary compatible family of functions (
f
P HrpGq : G P ErellpN q , writing f f in case G G. We conclude that ¸ pf q C 2 tr πψ pf q 0, r ιpN, G q tr Rdisc,ψ ψ G Ersim N
P
p q
for the nonnegative coefficient Cψ2
Cψ1 1 εp 21 , µ1 µ2q .
The left hand side of this expression can be written as a nonnegative linear combination of irreducible characaters. Proposition 3.5.1 then tells us that each of the coefficients vanishes. In particular, the right hand side of (5.1.13) equals 0. This gives the first assertion of the lemma. From the vanishing of the remaining coefficient Cψ2 , and the fact that Cψ1 is strictly positive, we see also that 1 ε 21 , µ1 µ2 0. This is the second assertion of the lemma. Lemma 5.1.6 is an important complement to the previous lemmas and their corollaries. It completes our discussion of square integrable parameters ψ P Fr2 pGq attached to composite endoscopic data G P Erell pN q. We now know that these parameters do not contribute to the discrete spectrum of any simple endoscopic datum G P Ersim pN q. Lemma 5.1.6 also gives a reduction of our induction hypothesis, as it applies to Theorem 1.5.3(b). To be precise, we have shown that the number (5.1.4) attached to an ε-pair pψ1 , ψ2 q is equal to 1 whenever degpψ1 q degpψ2 q N.
262
5. A STUDY OF THE CRITICAL CASES
With this step, we can retire the terms “ε-pair” and “ε-parameter” from our lexicon! 5.2. The case of elliptic ψ In the last section, we looked at square-integrable parameters (5.2.1)
ψ
P Ψr 2pGq :
(
G P Erell pN q .
We resolved what remained to be done when G lies in the complement of Ersim pN q in Erell pN q. We shall now add some observations for the other critical case, that of elliptic parameters (5.2.2)
ψ
P Ψr ellpGq :
(
G P Ersim pN q ,
or more precisely, parameters in (5.2.2) that lie in the complement of (5.2.1). Recall that the sets (5.2.1) and (5.2.2) can both be identified with subsets of r pN q. Their union is precisely the set of parameters that were not treated Ψ by the general induction arguments of Chapter 4. Their intersection, the set of square integrable parameters ψ for simple data G, will be the most difficult family to treat. Among other things, we will eventually have to show that the linear form Λ of Lemma 5.1.4 vanishes. r ell pGq that are not square-integrable have The elliptic parameters ψ P Ψ their own set of difficulties. These stem from the global intertwining relation for G, since we are not yet in a position to prove this identity. We will G pf q, which therefore not be able to establish the predicted formula for Sdisc,ψ reduces to 0 in this case, or to establish the expected property that the G pf q vanishes. Another consequence is that we will not representation Rdisc,ψ yet be able to resolve the case r 1 of (4.5.11) left open from §4.5. In this section, we shall put together what can be obtained from the standard model. We will introduce formulas to which the unresolved intertwining relation contributes an obstruction term. These formulas will eventually be strengthened to yield what we want. They will also be applied slightly differently in the next section to give more information about the square integrable parameters of §5.1. For the rest of the chapter, we will be following the global notation of the last section (and the previous two chapters). In particular, F is a global r pGq are of course to be understood as field, while G P Erell pN q and ψ P Ψ global objects over F . In this section, we will be assuming that G is simple. It will then be convenient to denote the complement of an embedded set in r sim pGq is the (5.1.1) in its successor by the appropriate superscript. Thus Ψ 2 2 r sim pGq in Ψ r 2 pGq, while Ψ r pGq is the complement of Ψ r 2 p Gq complement of Ψ ell r in Ψell pGq. We therefore have a decomposition (5.2.3)
r ell pGq Ψ r sim pGq Ψ
²
r sim pGq Ψ 2
²
r 2 p G q. Ψ ell
r 2 pGq that we will be looking at in this section. It is the set Ψ ell
5.2. THE CASE OF ELLIPTIC ψ
263
We will ultimately be focused on a family Fr of parameters (5.1.2), on which we will later impose local constraints. Our present concern will therefore be with the subset 2 Frell pGq Fr X Ψr 2ellpGq,
G P Ersim pN q,
r There is actually no compelling reason to work with F r in much of this of F. section, since Assumption 5.1.1, the one axiom we have imposed beyond 2 pG q. our blanket induction hypothesis, is irrelevant to parameters in Frell 2 pGq, since it is in this However, we may as well state the results for Frell context that they will be applied. Suppose then that N and G P Ersim pN q are fixed, and that ψ is a given 2 pGq. Then ψ is characterized by the conditions parameter in the set Frell #
(5.2.4)
2ψ1 2ψq ψq 1 ψr , Sψ Op2, Cqq Op1, Cqrq ψ , q ¥ 1,
ψ
`
`
`
`
`
along with the requirement that the Weyl group Wψ contain an element w in
the regular set Wψ,reg . Recall that p qψ denotes the kernel of a sign character, which was defined in terms of the degrees Ni of the simple components ψi of ψ. The condition on the existence of w is that there be an element in Sψ whose projection onto any factor Op2, Cq belong to the nonidentity connected component. This holds either if q r, or if there is an even number of indices i with Ni odd. We may as well write Sψ,ell
(
x P Sψ : Eψ,ell pxq H
for the subset of components in the quotient Sψ of Sψ that contain elliptic elements. Then Sψ,ell is also the set of components x which contain a regular Weyl element wx P Wψ,reg pxq. This element is unique. r M pG q. We write M for a Levi subgroup of G such that ψ belongs to Ψ We can then identify the multiplicity free parameter ψM
ψ1 ψr
r 2 pM, ψ q. We can also write with an element in Ψ
M
GLpN1q GLpNq q G
as in the local notation (2.3.4). Then if ψ
ψq
N
Nq
and
1
1
`
`
ψr
Nr ,
the group G Gψ represents the unique element in Ersim pN q such that ψ lies in Fr2 pG q. Our condition q ¥ 1 is of course because ψ does not lie in Fr2 pGq. It implies that M is proper in G, and that we can apply our induction hypotheses to M and ψM .
264
5. A STUDY OF THE CRITICAL CASES
The first step is to apply the results of Chapter 4 to the component r pN q, as we did in the last section. The centralizer set S rψ pN q for G r pN q G r is connected, for any parameter ψ at all, so the quotient Sψ pN q contains r pN q and ψ is still one element x. The global intertwining relation for G r pN q in place of G, which valid in this case. It is the formula (4.5.1), with G r ell pN q, we established inductively for any parameter in the complement of Ψ near the beginning of the proof of Proposition 4.5.1. (It was of course the r ell pN q that we treated in the last section, where we relied exceptional set Ψ on Assumption 5.1.1.) In particular, the conditions of Corollaries 4.3.3 and r 4.4.3 apply to GpN q, ψ , and the corresponding terms in the two expansions (4.3.9) and (4.4.11) are equal. The expansions are themselves then equal, and the two corollaries yield the familiar identity 0 N rrdisc,ψ
pfrq 0srNdisc,ψ pfrq,
Since 0 N rrdisc,ψ
r pN q. fr P H
N pfrq 0 pfrq tr Rdisc,ψ
in this case, it follows from the definition (4.4.10) that ¸
(5.2.5)
G Ersim N
P
p q
r ι N, G
p
pfr q 0. q 0Spdisc,ψ
2 p Gq This is the same as the identity (5.1.8), but for the parameter ψ P Frell rather than the parameter in Fr2 pGq of §5.1. We shall use it in much the same way. The next lemma is stated in terms of a compatible family
f
(5.2.6)
P HrpGq :
G
P ErellpN q
(
,
or more precisely, the subset of functions in a compatible family parametrized by the subset Ersim pN q of Erell pN q. If G G, we write f f . In case N is even and ηG 1, so that there is a second group G_ P Ersim pN q with ηG_ 1, we will also write f _ f , if G G_ . Lemma 5.2.1. Suppose that for the given pair
pG, ψq,
G P Ersim pN q, ψ
P Frell2 pGq,
the index r in (5.2.4) is greater than 1. Then the sum ¸
(5.2.7)
G Ersim N
P
equals (5.2.8)
c
p q
¸
P
p
q
p q
1 εG ψ pxq fG pψ, sψ xq fG pψ, xq ,
x Sψ,ell
for a positive constant c (5.2.6).
r ι N, G tr Rdisc,ψ f
cpG, ψq, and any compatible family of functions
5.2. THE CASE OF ELLIPTIC ψ
265
Proof. The general strategy of proof is pretty clear by now. We need to establish a formula for the sum (5.2.7). We have a formula for the parallel sum on the left hand side of (5.2.5). For each index of summation G P Ersim pN q, we have then to obtain an expression for the difference (5.2.9)
pf q tr Rdisc,ψ
pf q, 0Sdisc,ψ
f
P HrpGq,
of the corresponding two summands. We shall apply the standard model, specifically Corollaries 4.3.3 and 4.4.3, to the pair pG , ψ q. We observe that the two corollaries would have been easy to establish directly in the case at hand, since the centralizer S ψ S G ψ for (5.2.9) is considerably simpler than the general object S ψ treated in §4.3 and §4.4. The most important case is that of G G. It is responsible for the linear forms fG1 pψ, sψ xq and fG pψ, xq in (5.2.8). We recall that these objects are defined by (4.2.4) and (4.2.5). They satisfy the conditions of Corollaries 4.4.3 and 4.3.3, with pG, ψ q being our given pair. In other words, they are well defined functions of the component x in Sψ . For the first function fG1 pψ, sψ xq, this fact was established under general conditions prior to the proof of Proposition 4.5.1. For the function fG pψ, xq, it follows from the special nature of ψ, specifically, the fact that there is only one element wx in the Weyl set Wψ pxq. We can therefore apply Corollaries 4.3.3 and 4.4.3 to our pair pG, ψ q. r pGq, the difference Together, they tell us that for any f P H G pf q 0sGdisc,ψ pf q Idisc,ψ
G pf q 0sGdisc,ψ pf q 0rdisc,ψ
equals Cψ
¸
P
G G pf q pf q 0rdisc,ψ Idisc,ψ
1 iψ pxq εG ψ pxq fG pψ, sψ xq fG pψ, xq .
x Sψ
On the one side of the identity, we can write 0 G rdisc,ψ
G G pf q 0sGdisc,ψ pf q tr Rdisc,ψ pf q 0Sdisc,ψ pf q.
This follows from the definitions, since the expected value of the trace of G Rdisc,ψ pf q is 0. It also follows from the definitions that the coefficient iψ pxq on the other side vanishes unless x belongs to the subset Sψ,ell of Sψ . In this case, we have iψ pxq |Wψ0 |1
¸
P
pq
s0ψ pwq | detpw 1q|1
w Wψ,reg x
1 s0ψ pwxq | detpwx 1q|1 1 1 12 q . The other side of the identity therefore equals Cψ
1 q 2
¸
P
x Sψ,ell
1 εG ψ pxq fG pψ, sψ xq fG pψ, xq .
266
5. A STUDY OF THE CRITICAL CASES
This gives a formula for (5.2.9) in case G G. Suppose that G is an element in Ersim pN q that is distinct from G. We r pG q, claim that the difference (5.2.9) vanishes. If ψ does not belong to Ψ this is just the identity (4.4.12). In fact, we already know that both sides of r pG q. (5.2.9) vanish unless ηG ηψ . Suppose then that ψ does belong to Ψ Then ηG ηψ ηG .
Since G and G are distinct elements in Ersim pN q, this implies that ηG G G_ , r q, and Sψ_
Sψ pG_q
1,
q
Spp2, Cq .
By assumption, r q ¥ 2. We are therefore dealing with a case that was treated by induction in §4.5. Indeed, the pair pG_ , ψ q satisfies the condition
_
_
dimpT ψ q dimpT ψ,x q q
¥ 2,
_
_
x P Sψ_ ,
_
since the subtorus T ψ,x of the maximal torus T ψ of S ψ defined in §4.5 _ actually equals T ψ in this case. As in the discussion of the conditions (i) and (ii) in the proof of Proposition 4.5.1, we see that the global intertwining relation (4.5.1) holds for pG_ , ψ q. The difference (5.2.9) therefore vanishes in the remaining case G G_ , by Corollaries 4.3.3 and 4.4.3. The last step is to apply the identity (5.2.5). We can replace the functions tfr u in (5.2.5) by any compatible family (5.2.6). Substituting the formulas we have obtained in the cases G G and G G, we conclude that (5.2.7) equals (5.2.8), with ιpN, Gq Cψ cr
1 q 2 .
Remark. The only property of the constant c in (5.2.8) we will use is its positivity. We could of course write it more explicitly. It follows from (3.2.6) and (4.3.5) that cr ιpN, Gq Cψ
1 q 2 1
r N pGq| mψ |Sψ |1 1 q 21 |Out 2 1 1 1 q 1 r 2 |OutN pG, ψq| |Sψ | 2 ,
and one can check that this equals 1 if n is even.
q 1 2
r ε
, where ε
0 if N is odd and
We have to treat the case of r q 1 in (5.2.4) separately. The under2 pGq of lying object of course remains a parameter ψ in the complement Frell Fr2 pGq, for some G P Ersim pN q. However, we now suppose that it falls into the special case #
(5.2.10)
ψ 2ψ1 Sψ Op2, Cq
5.2. THE CASE OF ELLIPTIC ψ
267
1.
of (5.2.4). This implies that N is even and that ηG pair of split groups G and G_ in Ersim pN q, with Sψ_
We thus have a
Sψ pG_q Spp2, Cq.
In this case, the group M
GLpN1q
can be identified with a Levi subgroup of both G and G_ . If G equals either G or G_ , the subset
Sψ,ell
(
x P Sψ : Eψ,ell pxq H
of the group Sψ Sψ pG q consists of one element, which we denote simply by x1 . In each case, the Weyl set Wψ,reg px1 q consists of one element w1 . Lemma 5.2.2. Suppose that for the given pair
pG, ψq,
G P Ersim pN q, ψ
P Frell2 pGq,
the index r in (5.2.4) equals 1. Then the sum of ¸
(5.2.11)
G Ersim N
P
and (5.2.12)
p
q
p q
1 8
pf _qM pψ1q fG__ pψ, x1q
1 8
fG1 pψ, sψ x1 q fG pψ, x1 q ,
equals (5.2.13)
p q
r ι N, G tr Rdisc,ψ f
for any compatible family of functions (5.2.6). Proof. The proof is essentially the same as that of the last lemma. For each G P Ersim pN q, we have to obtain an expression for the difference (5.2.9) that we can substitute into the identity (5.2.5). In particular, we have to apply Corollaries 4.3.3 and 4.4.3 to the data G G and G G_ . We require separate discussion here simply because we know less about the contribution from G G_ . That is, the inductive arguments of §4.5 are not strong enough to establish the expected global intertwining relation for G_ . This will account for the extra term (5.2.12). We can say that the conditions of the two corollaries are applicable to both G and G_ . The justification follows from the remarks at the beginning of the proof of the last lemma, except in the application of Corollary 4.3.3 to G_ . In this case, we simply adopt the convention used to treat (4.5.11) in the proof of Proposition 4.5.1, which relies on the fact that the element wx P Wψ,reg pxq introduced earlier is unique. Once again, we shall apply both corollaries together. Assume that G is one of the groups G or G_ . For any r pG q, the difference f P H 0
pf q 0 s pf q tr R pf q rdisc,ψ disc,ψ disc,ψ
pf q 0Sdisc,ψ
268
5. A STUDY OF THE CRITICAL CASES
then equals
Cψ
¸
x Sψ
P
iψ pxq εG ψ px q
pf q1G pψ, sψ xq fG pψ, xq ,
where Cψ is the constant (4.3.5) attached to G . It follows easily from Corollary 4.6.2 and the original definition (1.5.6) that the sign character εG ψ equals 1. Since iψ pxq vanishes unless x belongs to Sψ,ell , a subset of Sψ that consists of the one element x1 , the last expression therefore reduces to
pf q1G pψ, sψ x1q fG pψ, x1q . If G G, the identity component of the group S ψ S ψ is abelian. We
(5.2.14)
Cψ iψ px1 q
have
iψ px1 q 21 as in the proof of the last lemma. Since
p
r ι N, G
q
21 ,
q Cψ 21 18 ,
the product of r ιpN, Gq with (5.2.14) in this case is equal to the expression (5.2.13), with f f . If G G_ , the group S ψ S _ ψ is connected, and simple of rank 1. We obtain iψ px1 q |pWψ_ q0 |1
¸
_ x1 w Wψ,reg
P
p q
s0ψ pwq | detpw 1q|1
12 s0ψ pwxq | detpwx 1q|1 14 , as in the proof of the case (4.5.11) in Proposition 4.5.1. We also have the reduction pf q1G pψ, sψ x1q pf q1G pψ, 1q pf qM pψ1q in (5.2.14). Since r ιpN, G_ q Cψ_ 41 18 , ιpN, G_ q with (5.2.14) in this case is equal to p1q times the product of r the expression (5.2.12), with f _ f . The minus sign here is critical. It is what forces us to place (5.2.12) and (5.2.13) on opposite sides of the proposed identity. We continue the argument at this stage as in the proof of the last lemma. If G belongs to Ersim pN q, but is not equal to G and G_ , ψ does not belong r pG q. The difference (5.2.9) then vanishes by (4.4.12) (or simply from to Ψ the fact that ηG ηψ ). It remains only to apply (5.2.5), with a general compatible family (5.2.6) in place of the functions tfr u. Substituting the formulas we have obtained in the cases G G, G G_ , and G distinct from G and G_ , we conclude that the sum of (5.2.11) and (5.2.12) equals (5.2.13), as required. We have so far worked exclusively with parameters of degree equal to the fixed integer N of our running induction hypotheses. This will not be sufficient. We will soon have to consider some elliptic parameters of degree
5.2. THE CASE OF ELLIPTIC ψ
269
greater than N . For such objects, we will of course not be able to rely on the results of this section. We will need to see what can be salvaged. Suppose that ψ `1 ψ1 ` ` `r ψr is a general parameter in our family FrpN N
`1N1
q, with degree `r N r
greater than N . We may as well assume that ψ belongs to the subset Frdisc pN q of FrpN q. In other words, the simple components ψi of ψ are self-dual. We define ð ψ , ψi , an auxiliary parameter in Frell pN N
,
`i odd
,
q, where
¸
Ni .
`i odd
r pG q, for a given simple datum Suppose that ψ belongs to Ψ G P Ersim pN q. Following earlier notation, we write M for a Levi subr M pG q. Then group of G such that ψ belongs to Ψ 1 1 M GLpN1 q`1 GLpNr q`r G , ,
where `1i r`i {2s is the greatest integer in `i {2, and G , is the unique r 2 pG , q. We can identify the element in Ersim pN , q such that ψ , lies in Ψ corresponding parameter `1 `1 ψ ,M ψ11 ψr1 ψ ,
r 2 pM , ψ q. We will naturally have an interest in the with an element in Ψ r pG q attached to ψ . It can be written as the stable linear form on H pullback r p G q, f G pψ q f M pψM q, f PH of a stable linear form on M pAq. This linear form will thus be well defined by our Assumption 5.1.1 on FrpN q, so long as N , ¤ N . There is one part of the standard model that can be established for many ψ . It is a variant of (5.2.5), in which the sum is taken over Erell pN q instead of its subset Ersim pN q. For any G P Erell pN q, the stable linear form Sdisc,ψ
r pG q is defined by (3.3.9), (3.3.13) and Corollary 3.4.2. Its expected on H value is given by the analogue of Corollary 4.1.3, as a linear combination of linear forms f p ψ q, f P SrpG q, ψ P ΨpG , ψ q. These objects are again well defined by Assumption 5.1.1 whenever N , ¤ N . We can therefore define the stable linear form 0
Sdisc,ψ p f q,
f
P HrpGq,
270
5. A STUDY OF THE CRITICAL CASES
as the difference between Sdisc,ψ pf q and its expected value, so long as N , ¤ N .
P FrdiscpN q is as above, and that Nr ¤ N.
Lemma 5.2.3. Suppose that ψ (5.2.15)
N1
Then
¸
(5.2.16)
G PErell pN q
r ι N , G
p
q 0Spdisc,ψ pfrq 0,
r pN fr H
q.
Proof. We are taking the sum in (5.2.16) over the full set Erell pN q for the obvious reason that there is only limited scope for the inductive application of Theorem 4.1.2. That is, we cannot say that 0 Spdisc,ψ pfr q vanishes for general elements G in the complement of Ersim pN q. We do know that this linear form is well defined, since the condition (5.2.15) obviously implies that N , ¤ N . We also know that Lemma 4.3.1 is valid for the pair r pN q, ψ , which is to say that the application of Theorem 1.5.3 to the G proof of Lemma 4.3.1 in §4.6 remains valid. The assertion (a) of Theorem r pN q, since in this case it is only Rankin1.5.3 is not actually relevant to G Selberg L-functions that contribute to the factor s0ψ pwq in (4.6.13). The condition on the other assertion (b) still holds, by virtue of its extension in Lemma 5.1.6 and the condition (5.2.15). Lemma 4.4.1 is of course also valid r pN q, ψ , since there were no implicit conditions in its proof from for G §4.6. With these remarks, we can see that the discussion preceding the special r pN q applies here as well. In particular, the conditions case (5.2.5) for G of Corollaries 4.3.3 and 4.4.3 remain in force, and the analogues of these r corollaries hold for the general case GpN q, ψ at hand. Moreover, the r analogues for GpN q, ψ for the expressions (4.3.9) and (4.4.11) are equal. It follows that 0 N rrdisc,ψ
pfrq 0srNdisc,ψ pfrq,
r pN fr P H
q,
where 0 srdisc,ψ pfrq equals the left hand side of (5.2.16). Since N
0 N rrdisc,ψ
pfrq 0,
we obtain the required identity (5.2.16).
5.3. A supplementary parameter ψ In this section, we shall return to the “square integrable” parameters
P FrellpN q of §5.1. Recall that ψ takes the general form (5.3.1) ψ ψ1 ψr , ψi FrsimpNiq, and lies in the subset Fr2 pGq of Frell pN q attached to some G P Erell pN q. (Recall
ψ
`
`
also that G is unique, except possibly if ψ is generic and simple. In the
5.3. A SUPPLEMENTARY PARAMETER ψ
271
latter case, we simply fix a G with ψ P Frsim pGq and treat it as if it were unique, knowing that this is what we will eventually establish.) Our goal is G to establish the stable multiplicity formula for Sdisc,ψ pf q. In §5.1, we found that the proof was rather direct under any of the conditions N odd, ηG 1 or G Ersim pN q. However, the remaining case of N even and G P Ersim pN q split is considerably harder. We will need to establish it for the relevant parameters in Frell pN q in order to complete the induction argument of this chapter. In the special case that ψ is simple generic, we will also need to deal with the first part (a) of Theorem 1.5.3. This is the condition on the poles of Lfunctions, which we recall is intimately tied up with the definition of Frsim pGq and the corresponding special case of the stable multiplicity formula. It represents a second critical global assertion that will have to be established in the context of the family Frell pN q. The other global assertions are less pressing. They will be resolved later, in terms of the stable multiplicity formula and the local results we establish. We recall that the second part (b) of Theorem 1.5.3 has already been resolved for FrpN q. It was established as one of the assertions of Lemma 5.1.6. To deal with these questions, we can assume that G P Ersim pN q is simple. Given the parameter ψ P Fr2 pGq, we set N
N1
N,
where the constituents ψi of ψ have been ordered so that N1 i. We then introduce the supplementary parmeter (5.3.2)
2ψ1
ψ
`
ψ2
`
`
¤ Ni for each
ψr
in FrpN q. There is a unique element G in Ersim pN p contains the product and such that G
q such that ψ P Ψr pG q,
p1 G pG pψ G 1
p G, where G1 Gψ is the element in Ersim pN1 q such that ψ1 Frsim pG1 q. Then ψ lies in the subset Frell pG q of FrpG q. 1
is contained in
We will need to work with two maximal Levi subgroups of G . The first applies to the case of Neven and ηψ 1, in which we have a maximal Levi subgroup L GL 21 N of G. We then have the corresponding Levi subgroup L G1 L GL 12 N G1 of G . It is adapted to the decomposition ψ
ψ1
`
ψ
ψ
`
ψ1
of ψ, and comes with the linear form fL
ÝÑ
f L pψ1 Λq,
f
P HrpG q,
272
5. A STUDY OF THE CRITICAL CASES
where Λ is the linear form attached to L in Lemma 5.1.4. The second is a special case of the general definition prior to Lemma 5.2.3. This is the Levi subgroup M GLpN1 q G , , G , P Ersim pN , q, r M pG of G such that ψ belongs to Ψ
ψ
2ψ1
`
ψ
,
,
q. It is adapted to the decomposition ψ , ψ2 ψr `
`
of ψ, and comes with the linear form
P HrpG q, in which ψ is identified with the product ψ1 ψ , . f
ÝÑ
f M pψ q,
f
One of our ultimate goals is to show that Λ vanishes. An essential step will be to establish analogues of Lemmas 5.2.1 and 5.2.2 for the parameter ψ . These will be needed for the induction argument that is to drive the proof of the local theorems over the next two chapters. They will also be used directly for the general proof of the global theorems in Chapter 8. As with their predecessors, we shall state the lemmas in terms of a compatible family of functions (
f
P HrpGq : G P ErellpN q , with the same notation f f and f _ f in the cases G G and G G_ . We shall again deal separately with the cases r ¡ 1 and r 1. Suppose first that r ¡ 1. This is the case that ψ lies in the complement sim r F2 pGq of Frsim pGq in Fr2 pGq. The problem is to understand the linear form (5.3.3)
Λ of Lemma 5.1.4, and to show ultimately that it vanishes. We can therefore assume that N is even and ηψ 1, as in the earlier lemma, and in particular that G is split. We then have the maximal Levi subgroups L and L of G and G respectively. To deal with this case, we shall also have to strengthen our induction hypothesis. We shall assume that in dealing with the square integrable parameter here, we have already been able to treat the elliptic parameters of §5.2. Lemma 5.3.1. Suppose that for the given pair
pG, ψq,
G P Ersim pN q, ψ
P Fr2pGq,
the index r in (5.3.1) is greater than 1, while N is even and ηψ 1. Assume also that the global theorems are valid for parameters in the complement of Frell pN q in FrpN q. Then the sum (5.3.4)
¸
G Ersim N
P
p q
equals (5.3.5)
c
r ι N , G tr Rdisc,ψ
p
¸
P
x Sψ
q
G
εψ ,ell
px q
pf q
b f L pψ1 Λq
fG1 pψ , sψ xq fG pψ , xq ,
5.3. A SUPPLEMENTARY PARAMETER ψ
for positive constants b bpG , ψ patible family of functions (5.3.3).
q and c cpG
273
,ψ
q, and any com-
Proof. This is the first of two lemmas in which we have to work with parameters of rank greater than N . The proofs require extra discussion, since they are beyond the scope of our running induction hypotheses. The supplementary arguments are usually not difficult, but they do contain a number of points to be checked. We shall include more detail here than in the next lemma. We may as well start with a general compatible family (5.3.3). We then r pN q that for any G P Erell pN q has the same image fix a function fr P H in SrpG q as the corresponding function f in (5.3.3). We can then work interchangeably with the function fr or the family (5.3.3). This will simplify the notation at times, particularly when G equals G. In principle, the lemma should simply be the formula for (5.3.6)
¸
G Ersim N
P
p q
r ι N , G tr Rdisc,ψ
p
q
pf q
that would be the direct analogue for ψ of Lemma 5.2.1. In particular, the coefficient in (5.3.5) is to be the analogue c
12 rιpN
,G
q Cψ
of the coefficient c in the corresponding expression (5.2.8) of Lemma 5.2.1. However, the induction assumptions hold only proper subparameters of ψ. We will see that most proper subparameters of ψ can either be ruled out by general considerations, or expressed in terms of proper subparameters of ψ. For the ones that cannot, we will be able to use Λ to describe the defect in the associated terms. In other words, we will keep track of what changes the existence of Λ makes in the arguments used to derive the earlier formula. The discussion will again center around a sum (5.3.7)
¸
G Erell N
P p q
r ι N , G
p
q 0Spdisc,ψ pfrq,
r pN fr P H
q,
over endoscopic data. Recall that this is the left hand side of the formula (5.2.16), established for more general ψ in Lemma 5.2.3. In the case at hand, the sum N1 N2 Nr in (5.2.15) equals N . The formula therefore holds, and (5.3.7) vanishes. The problem is to analyze the summands in (5.3.7). We shall apply what we can of the standard model for G to the stable linear form 0 G Sdisc,ψ pf q 0 Sdisc,ψ pf q, f P HrpGq,
where f maps to the function fr P SrpG q in (5.3.7). The problem now is complicated by the fact that G ranges over Erell pN q, rather than the subset Ersim pN q of simple data whose analogue for N indexed the sum in
274
5. A STUDY OF THE CRITICAL CASES
the original formula (5.2.5). We recall that 0 Sdisc,ψ pf q is defined as the difference between the original stable form Sdisc,ψ pf q, and its expected formula (5.3.8)
¸
ψ PF pG ,ψ q
|Sψ |1 σpS 0ψ q εpψq f pψq
given by the twisted analogue of Corollary 4.1.3. Consider a general element G
G1 G 2 , in Erell pN q. Thus, N N 1
P ErsimpN k q, k 1, 2,
Gk
N 2 , and G1 and G2 represent the two data denoted GS and GO in Assumption 5.1.1. We can write
p5.3.9paqq and
p5.3.9pbqq
0
pf q Sdisc,ψ
¸
Sdisc,ψ pf q
¸
ψ
ψ
pf q Sdisc,ψ
0
pf q, Sdisc,ψ
where the sum in each case is over parameters ψ
ψ1 ψ2,
ψk
with
P FrpN k q, k 1, 2,
ψ ψ1 ` ψ2. The decomposition (5.3.9(a)) follows from (3.3.9), (3.3.13) and Corollary 3.4.2. Since the expected value (5.3.8) of the left hand side of (5.3.9(a)) has a similar decomposition, with the summand of any ψ in the complement of FrpG , ψ q taken to be 0, the difference does satisfy (5.3.9(b)). We observe also that the summand of ψ in (5.3.9(a)) has an obvious product decomposition (5.3.10)
pf q S 1 1 pf 1 q S 2 2 pf 2 q, Sdisc,ψ disc,ψ disc,ψ
if f f 1 f 2 . We can assume that the two nonnegative integers N 1 and N 2 in the partition of N attached to G satisfy N 1 ¤ N 2 . The most important cases will then be the trivial partition (i) N 1
0, N 2 N
,
in which G lies in the subset Ersim pN (ii) N 1
q of simple data, and the partition
N1, N 2 N
attached to ψ1 and ψ. Before we discuss these, however, we shall first take care of the others.
5.3. A SUPPLEMENTARY PARAMETER ψ
275
If 0 N 1 N1 , there is no index of summation ψ in (5.3.9), since N1 ¤ Ni for each i. It follows that (5.3.11)
0
Sdisc,ψ pf q 0,
in this case. In the remaining case that N 1 ¡ N1 , both N 1 and N 2 are less than N . We can then apply our induction hypothesis to the factors on the right hand side of (5.3.10). This tells us that each factor equals its expected value, and so therefore does the product. It follows that the summands in (5.3.9(b)) all vanish, and that (5.3.11) holds in this case as well. Thus, the summand of G in (5.3.7) vanishes unless the corresponding partition N N 1 N 2 satisfies (i) or (ii). Consider the case (ii). This of course does not characterize G uniquely. Moreover, there are several indices ψ ψ 1 ψ 2 in (5.3.9(b)) attached to the given G . Among these, it is the pair (5.3.12)
pG, ψq pG1 G_, ψ1 ψq,
G1
Gψ , 1
that is the most significant. We shall deal first with the other pairs. Assume that pG , ψ q is any pair associated with the partition (ii) that is not equal to (5.3.12). If the parameter ψ ψ 1 ψ 2 is not equal to ψ1 ψ, the second component ψ 2 contains the factor ψ1 with multiplicity two. This means that ψ 2 lies in the complement of Frell pN q in FrpN q. The given condition of the lemma then tell us that the global theorems are valid 2 2 for pG2 , ψ 2 q, and in particular, that the factor Sdisc,ψ 2 pf q in (5.3.10) equals its expected value. Our induction hypothesis tells us that the other factor 1 1 1 N is less Sdisc,ψ 1 1 pf q in (5.3.10) also equals its expected value, since N than N . It follows that the summand of ψ in (5.3.9(b)) vanishes. Assume next that ψ ψ1 ψ, but that the datum G G1 G2 is not equal to G1 G_ . If G1 G1 the parameter ψ 1 ψ1 does not lie FrpG1 q. It then follows by induction that 0 1 Sdisc,ψ1
1 pf 1q Sdisc,ψ pf 1q 0, 1
and therefore that the summand of ψ again vanishes. On the other hand, if G1 equals G1 , the second component G2 cannot equal G. This is because p 1 and G p are both orthogonal or both symplectic (by definition, since the G p 1 and G p 2 of same is true of the parameters ψ1 and ψ), whereas the factors G p must be of opposite type. Since we the dual twisted endoscopic group G have ruled out G2 G_ , the only other possibility is a datum that satisfies ηG2 In this case, 0 G2 Sdisc,ψ2
ηψ 1. 2
G pf 2q Sdisc,ψ pf 2q 0, 2
2
by (3.4.7), and so the summand of ψ in (5.3.9(b)) vanishes once again.
276
5. A STUDY OF THE CRITICAL CASES
Suppose then that pG , ψ q equals (5.3.12). The decomposition (5.3.10) takes the form pf q S G1 pf1 q S G_ pf _ q, G1 Gψ1 , Sdisc,ψ disc,ψ disc,ψ1 where f
f1 f _. The first factor becomes G pf1G q f1G pψ1q, Sdisc,ψ 1
1
1
1
with an inductive application of Theorem 4.1.2 to G1 . According to Lemma 5.1.4, the second factor satisfies G_ Sdisc,ψ pf _q orpG_q f _,LpΛq. It follows that
pf q S pf q orpG_ q f L pψ1 Λq, Sdisc,ψ disc,ψ G since the expected value of Sdisc,ψ pf q is 0. This provides the contribution to the sum (5.3.7) of G G1 G_ . In fact, from what we have shown, it represents the only contribution from composite endoscopic data G . To be precise, the sum of those terms in (5.3.7) corresponding to data G in the complement of Ersim pN q in Erell pN q equals 0
p
r ι N , G1
(5.3.13)
G_q orpG_q f L pψ1 Λq.
It remains to consider the partition (i) attached to simple data
G P Ersim pN q. This represents the point at which we began the proof of Lemma 5.2.1, the model we are following from §5.2. For each such G , we have to obtain an expression for the difference
pf q tr Rdisc,ψ
(5.3.14)
p f q, 0Sdisc,ψ
f
P HrpGq,
of the corresponding two summands in (5.3.6) and (5.3.7). The most important case is the datum G G . It has the property that ηG equals the sign character
ηη ηψ ηψ of η . It is in fact the only element in Ersim pN q with this property unless N1 is even and ηψ 1, in which case G is split, and there is a second split datum G_ P Ersim pN q. The remaining elements G P Ersim pN q can be ηG
1
1
1
ignored, since 0
Sdisc,ψ pf q Sdisc,ψ pf q tr Rdisc,ψ pf q
0,
by (3.4.7). In particular, for the elements G P Ersim pN q distinct from G and G_ , the difference (5.3.14) vanishes. Suppose first that G G_ (so in particular, N1 is even and ηψ1 1). We claim that (5.3.14) vanishes in this case as well. Since ψ does not lie r pG_ q, we would expect both terms in (5.3.14) to vanish, but we are of in Ψ course not in a position to prove this. We also cannot appeal to (4.4.12), r pN q. since it applies only to parameters in Ψ
5.3. A SUPPLEMENTARY PARAMETER ψ
277
To study (5.3.14), we need to apply the expansions (4.1.1) and (4.1.2) (with G_ in place of G). The proper spectral terms in (4.1.1) correspond to proper Levi subgroups M _ of G_ of which ψ contributes to the discrete M_ spectrum, in the sense that the representation Rdisc,ψ is nonzero. Upon reflection, we see that there is no M _ with this property. Indeed, any M _ is either attached to a partition of N that is not compatible with ψ , in which case we apply Corollary 3.4.3, or M _ is a product of groups for which we can combine our induction hypothesis with the fact that the set r 2 pM _ , ψ q is empty. It follows from (4.1.1) that Ψ G_ G_ Idisc,ψ pf _ q tr Rdisc,ψ pf _ q 0. The proper endoscopic terms in (4.1.2) are indexed by composite endoscopic data pG_ q1 in Erell pG_ q. After further reflection, we see that there is no com1 is nonzero. Indeed, posite pG_ q1 for which the associated linear form Sdisc,ψ 1 _ either the two factors of pG q give a partition of N that is incompatible with ψ , in which case we apply Proposition 3.4.1, or they are data for which we can combine our induction hypothesis with the fact that the set r G_ q1 , ψ is empty. It follows from (4.1.2) that Ψ G_ G_ Idisc,ψ pf _ q Sdisc,ψ pf _ q 0. The vanishing of (5.3.14) then follows from the fact that the expected value G_ of Sdisc,ψ is 0. Suppose finally that G G . In this case we have again to apply (4.1.1) and (4.1.2) (with G in place of G). The expansions will now contain proper terms that are nonzero, so we will want to work with the refinements represented by Corollaries 4.3.3 and 4.4.3. This will lead to a formula for the value (5.3.15)
tr Rdisc,ψ pf q G
G p f q, 0Sdisc,ψ
f
P HrpG q,
of (5.3.14) at G G . The connected group Sψ0 is just SOp2, Cq, so we are still dealing with a rather elementary form of the standard model. However, we will have to be careful in applying our induction hypotheses to derive analogues of the corollaries for ψ , since the condition N
deg ψ ¡ N
is more serious in this case. The proper spectral terms for G in (4.1.1) are attached to proper Levi subgroups. We claim that these terms vanish for Levi subgroups that are not conjugate to the group M described prior to the statement of the lemma. We recall that the orthogonal or symplectic factor G , of M is the unique r 2 pG , q contains the subparameter datum in Ersim pN , q, such that Ψ ψ
,
ψ2
`
`
ψr .
278
5. A STUDY OF THE CRITICAL CASES
The claim follows easily from the original classification Theorem 1.3.2 for GLpN q, given the structure of Levi subgroups of G , and the nature of ψ . This leaves the terms attached to conjugates of M . Corollary 4.3.3 is designed express the sum of such terms. With a little thought, and a brief review of our later justification of Lemma 5.2.1, we see that its proof in §4.3 extends to G , since it requires only that the degree N , of ψ , be less than N . The expansion (4.1.1) therefore behaves as expected. We see that the difference
Idisc,ψ pf q tr Rdisc,ψ pf q , G
(5.3.16)
G
f
P HrpG q,
equals the corresponding expansion (4.3.9) of Corollary 4.3.3. Arguing then as in the proof of Lemma 5.2.1, we conclude that if we take the product of (5.3.16) with p1q times the corresponding coefficient r ιpN , G q in (5.3.6), we obtain the contribution of the linear forms fG pψ , xq to the given expression (5.3.5). The proper endoscopic terms for G in (4.1.2) are attached to composite endoscopic data G1
G11 G12,
G1k
P ErsimpNk1 q, k 1, 2,
in Erell pG q. Corollary 4.4.3 is designed to express the sum of such terms. However, its extension to G would require the stable multiplicity formula for the linear forms (5.3.17)
1 1 p f 1 q S 1 1 pf 1 q S 1 1 p f 1 q, Sdisc,ψ 1 2 disc,ψ disc,ψ 1
attached to parameters ψ1
ψ11 ψ21 ,
ψk1
such that
2
P FrpNk1 q, k 1, 2,
ψ11 ψ21 , r pG1 q. It is possible to treat most composand functions f 1 f11 f21 in H 1 1 ite pairs pG , ψ q with our induction hypotheses, augmented by the given assumption on parameters in the complement of Frell pN q in FrpN q. In fact, ψ
`
after a moment’s consideration, we see that the stable multiplicity formula holds in all cases, with the possible exception of the pair
pG1 , ψ1 q pG1 G, ψ1 ψq.
Moreover, Lemma 5.1.4 can be applied to this exceptional pair. It tells us in this case that the linear form (5.3.17) equals the difference between its expected value and the linear form orpGq f L pψ1 Λq
(5.3.18)
Recalling our justification of Lemma 5.2.1 as needed, we see that apart from this defect, the proof of Corollary 4.4.3 in §4.4 extends to G . It follows that the sum of the difference (5.3.19)
Idisc,ψ pf q 0 Sdisc,ψ pf q, G
G
f
P HrpG q,
5.3. A SUPPLEMENTARY PARAMETER ψ
279
with (5.3.18) equals the corresponding expansion (4.4.11). Arguing then as in the proof of Lemma 5.2.1, we conclude that if we add the expression
p
q orpGq f L pψ1 Λq to the product of (5.3.19) with r ιpN , G q, we obtain the rest of the given expression (5.3.5), namely the contribution from the linear forms fG1 pψ , sψ xq. r ι N ,G
(5.3.20)
We have now finished our discussion of the difference (5.3.15). It is of course equal to the difference between (5.3.19) and (5.3.16). Our conclusion ιpN , G q and (5.3.15) equals is that the sum of (5.3.20) with the product of r the entire expression (5.3.5). This in turn completes our discussion of the full set simple endoscopic data G P Ersim pN q. For we have already seen that the difference (5.3.14) vanishes if G G . Therefore the sum of (5.3.20) with the sum over G of the product of r ιpN , G q and (5.3.14) equals (5.3.5). It thus follows that the sum of (5.3.20) with the spectral sum (5.3.6) equals the sum of (5.3.5) with the contribution of the simple data G to the endoscopic sum (5.3.7). We observed earlier that the contribution of the remaining composite data G to (5.3.7) equals (5.3.13). The sum of the two terms (5.3.13) and (5.3.20) equals b f L pψ1 Λq, for the positive coefficient b
rιpN
, G1 G_ q orpG_ q
p
r ι N ,G
q orpGq.
This represents the supplementary summand in the original given expression (5.3.4). We have now shown that (5.3.4) equals the sum of (5.3.5) with the endoscopic sum (5.3.7). We have also observed that (5.3.7) vanishes, by Lemma 5.2.3. We have therefore completed the proof of Lemma 5.3.1, at last. Suppose now that r 1. Then ψ ψ1 belongs to the subset Frsim pGq of the Fr2 pGq. In this case, ψ 2ψ, so that N 2N is even and ηψ 1. In particular, we have the two split groups G and G_ in Ersim pN q. The maximal Levi subgroup M GLpN q of G becomes also a maximal Levi subgroup of G_ . This case carries the added burden of the assertion of Theorem 1.5.3(a) on the poles of L-functions. We shall set δψ 1 if ψ is not generic, or in case ψ is generic, the Langlands-Shahidi L-function Lps, ψ, ρ_ q attached to a maximal parabolic subgroup P _ M N _ of G_ has a pole at s 1. This is what Theorem 1.5.3(a) predicts. We set δψ 1 if ψ is generic, and it is the L-function Lps, ψ, ρ q attached to a maximal parabolic subgroup P M N of G that has a pole at s 1. In other words, δψ 1 if Theorem 1.5.3(a) does not hold for ψ. We recall that G is defined in terms of the group G Gψ assigned to ψ. We recall further that G is defined by induction unless ψ is (simple) generic,
280
5. A STUDY OF THE CRITICAL CASES
in which case it was taken to be some preassigned element in Ersim pN q such that ψ lies in the subset Frsim pGq of Frsim pN q specified by the temporary definition of §5.1. In the latter instance, there remains some possible ambiguity in the choice of G. It will be resolved by the stable multiplicity formula, or equally well, the formula δψ 1. We will of course have to establish both formulas. If N is even and ηψ 1, we have the two split groups G and G_ in Ersim pN q, with a common maximal Levi subgroup L GLp 12 N q. In this case, we must also be concerned about the possible existence of the linear form Λ. Recall that
_ pf _ q orpG_ qf _,L pΛq, Sdisc,ψ
f_
P HrpG_q,
and that the right hand side represents a unitary character on G_ pAq. It will be convenient also to write G pf q orpGq f GpΓq, Sdisc,ψ
P HrpGq,
f
r pGq. The formula of Lemma 5.1.4 then for a stable linear form Γ on H reduces to G pf q orpGq f Gpψq orpGq f LpΛq, Sdisc,ψ
since the coefficients in the formula satisfy
mψ |Sψ |1 εG pψ q orpGq 1 1 orpGq,
in the case at hand. We then obtain f G pψ q f G pΓq
As in the case of r L
f L pΛ q ,
f
P HrpGq.
¡ 1 above, we also have the maximal Levi subgroup G L G GL 12 N GL 12 N q G
of G . It comes with the stable linear form f
ÝÑ
f L p Γ Λ q,
f
P HrpG q.
Lemma 5.3.2. Suppose that for the given pair
pG, ψq,
G P Ersim pN q, ψ
the index r in (5.3.1) equals 1, while N of (5.3.21)
¸
G Ersim N
P
p q
and 1 8
(5.3.22) equals (5.3.23)
1 8
P Fr2pGq, is even and ηψ 1.
r ι N , G tr Rdisc,ψ
p
q
pf q
pf _qM pψq δψ fG__ pψ
, x1 q
1 2
fG1 pψ , sψ x1 q δψ fG pψ , x1 q ,
Then the sum
f L p Γ Λq
5.3. A SUPPLEMENTARY PARAMETER ψ
281
for any compatible family of functions (5.3.3), and elements x1 in Lemma 5.2.2.
P Sψ
,ell
as
Proof. The lemma will again be a modification of its counterpart from §5.2. We have thus to see what can be salvaged of the proof of Lemma 5.2.2. This time there will be modifications introduced by the sign δψ , as well as from the possible existence of Λ. As in the proof of the last lemma, we shall work interchangeably with a general compatible family (5.3.3) and a fixed r pN q, which for any G P Erell pN q have the same image in function fr P H SrpG q. The sign δψ is not hard to keep track of. The formula of Corollary 4.3.3 is based on the inductive assumption that the sign obeys Theorem 1.5.3(a). Its direct analogues for G and G_ would hold only if δψ equals 1. If we look to see what changes δψ makes if the proof of Lemma 5.2.2 is applied to ψ , we observe that we need only insert a coefficient δψ in the negative summands in the expressions (5.2.12) and (5.2.13) (or rather their analogues for G_ and G ). The expressions so obtained are just (5.3.22) and (5.3.23). Our task then is to describe what change the possible presence of Λ makes in the analogue ¸
(5.3.24)
G Ersim N
P
p q
r ι N , G tr Rdisc,ψ
p
q
pf q
of (5.2.11). It amounts to a re-examination of the relevant parts of the proof of Lemma 5.3.1, specifically the finer analysis of the summands in ¸
G Erell N
P p q
r ι N , G
p
q 0Spdisc,ψ pfrq.
The sum is the left hand side of (5.2.16). It vanishes, since the condition of Lemma 5.2.3 remains valid, so we can write (5.3.24) as the sum of ¸
(5.3.25)
G Ersim N
P
with (5.3.26)
¸ G
p q
r ι N , G
r ι N , G
p
p
q trpRdisc,ψ pf qq 0Spdisc,ψ pf q
q 0Spdisc,ψ p f q,
G
P ErellpN q ErsimpN q.
This was of course also the foundation of the proof of Lemma 5.3.1. Having outlined the argument for this last lemma in considerable detail, we can afford to be briefer here. The effect of Λ will be seen in the endoscopic contributions to (5.3.26) or (5.3.25) of proper products G1 G2 in any of the sets Erell pN q, Erell pG q or Erell pG_ q. (Products from the sets Erell pN q and Erell pG q were denoted G1 G2 and G11 G12 q in the proof of Lemma 5.3.1, while products from the set Erell pG_ q were seen to be innocuous.) The presence of Λ will be a
282
5. A STUDY OF THE CRITICAL CASES
consideration only if both G1 and G2 are taken from the set tG, G_ u. We have then to describe the change Λ makes in the expected formula for G1 G2 1 2 Sdisc,ψ pf1 f2q Sdisc,ψ pf1q Sdisc,ψ pf2q,
P HrpGk q, where G1 G2 is the group G G G_ in Erell pN q, the group G1 G G in Eell pG q, and the group pG_ q1 G_ G_ in Eell pG_ q. (5.3.27)
fk
Now the formula for (5.3.24) that would be given by the analogue of Lemma 5.2.2 for ψ (modified by the signs δψ as above) is just the sum of the expressions (5.3.22) and (5.3.23). However, this formula is predicated on the compound linear form S pf1 f2 q in each of the three cases of (5.3.27) being equal to its expected value. Suppose that the discrepancy 0 S pf1 f2 q is in fact nonzero. In the first case, it contributes a nonzero summand to (5.3.26), which must then be added to (5.3.24) as a correction term for the formula to continue to hold. In the second and third cases, it alters the proper pf q in (5.3.25). endoscopic component of the corresponding term 0 Spdisc,ψ The change in the associated summand of (5.3.25) must then be subtracted from (5.3.24) as a correction term, for the formula to remain valid. These remarks are of course with the understanding that the function f1 f2 in each case has the same image in SrpG1 G2 q as the chosen compatible family. In the first case, (5.3.27) equals the product of orpGqorpG_ q with the linear form f1G pΓq f2L pΛq f GL pΓ Λq f L pΓ Λq. The expected value of this linear form is 0. We have then to add a term equal to the product of the coefficient r ι N , G or G or G_
p
with
q p q p q 14
f L pΓ Λ q
to (5.3.24). In the second case, (5.3.27) equals the product of orpGq2 with the linear form f1G pΓq f2G pΓq f GG pΓ Γq. The expected value of this linear form equals f G G p ψ ψ q
f1Gpψq f2Gpψq f1GpΓq f1LpΛq f2GpΓq f2LpΛq f GGpΓ Γq f LGpΛ Γq f GLpΓ Λq f LLpΛ Λq.
We note that
f LG pΛ Γq f GL pΓ Λq f L pΓ Λq,
since f GG acquires a symmetry from the automorphism in AutpG , G1 q that interchanges the two factors of G1 GG. We must therefore subtract
5.3. A SUPPLEMENTARY PARAMETER ψ
the product of the coefficient
p
r ι N ,G
q rιpG
, G Gq orpGq2
283
18
with the difference
f GG pΓ Γq f GG pψ ψ q 2f L pΛ Γq
f LL pΛ Λq
from (5.3.24). In the third case, (5.3.27) equals the product of orpG_ q2 with the linear form f1L pΛq f2L pΛq f LL pΛ Λq. The expected value of this linear form is 0. We must therefore subtract the product of the coefficient r ι N , G_ ι G_ , G_
p
qp
G_q orpG_q2 18
with
f LL pΛ Λq from (5.3.24). We note that it is the coefficient 18 rather than 14 that appears in the latter two cases, since the associated two endoscopic data G G and G_ G_ each come with an extra automorphism that interchanges the two factors. Combining the three cases, we see that the sum total of what must be added to (5.3.24) is the linear form
1 4 1 2
f L p Γ Λq
1 8
f L p Γ Λ q.
2f L pΓ Λq
f LL pΛ Λq
18 f LLpΛ Λq
Its sum with (5.3.24) is the given expression (5.3.21). The formula of Lemma 5.2.2, adjusted for the lack of information we have about ψ , thus tells us that (5.3.21) equals the sum of (5.3.22) and (5.3.23). The proof is complete. From the last proof, we can extract formulas for the stable distributions attached to pG , ψ q and pG_ , ψ q. We record them here for use later, when we come to the general global classification in Chapter 8. Corollary 5.3.3. Under the conditions of the lemma, we have Sdisc,ψ pf q tr Rdisc,ψ pf q G
G
1 4
orpG
q δψ fG pψ
, x1 q f GG pΓ Γq
and G_ G_ Sdisc,ψ pf _ q tr Rdisc,ψ pf _ q 14 orpG_ q δψ fG__ pψ , x1 q f _,LL pΛ Λq . Proof. These formulas are implicit in the proof of the lemma. If G is equal to either G or G_ , it would not be hard to compute the difference
Sdisc,ψ pf q tr Rdisc,ψ pf q
directly from the general expansions (4.1.1) and (4.1.2). There will be one spectral term, corresponding to the Levi subgroup M of G or G_ , and one
284
5. A STUDY OF THE CRITICAL CASES
endoscopic term, corresponding to the proper endoscopic datum G1 of G or pG_ q1 of G_ . The point is that we have already done the calculations, and can in principle read these terms off from the lemma. The spectral terms are in the second summands of (5.3.23) and (5.3.22) respectively, the contributions of the terms with coefficient pδψ q. These summands must be taken over to the left hand side of the relevant part of the formula given by the lemma, and then multiplied by the inverse of the coefficient ιpN , G q that occurs in (5.3.21). Since 1 8
r ι N , G
p
q1 14 orpGq,
we see that the spectral term in the resulting formula for
pf q, Sdisc,ψ
(5.3.28) will be if G
G
1 4
, and
orpG
G
P tG
q δψ fG pψ
41 orpG_q δψ fG__ pψ
, G_ u,
, x1 q , x1 q
if G G_ . The endoscopic terms do not come from the first summands in (5.3.23) and (5.3.22). For these summands represent only expected values. Moreover in the case of (5.3.22), the summand corresponds to the simple endoscopic datum G_ for G_ rather than pG1 q_ . However, the two endoscopic terms are easily extracted from the proof of the lemma. They are essentially the formulas for (5.3.27), in the cases that G1 G2 equals G1 and pG_ q1 . We of course have again to place each of these ingredients on the left hand side of the relevant part of the formula of the lemma. Accounting for the coefficients, and inserting the extra sign p1q in the case of (5.3.22), we then see that the endoscopic term in the formula for (5.3.28) will be
41 orpG q f GGpΓ Γq
if G
G
if G_
G_. The stated formulas follow.
, and
14 orpG_q f LLpΛ Λq
Remark. If we change the conditions N even and ηψ 1 in the statement of the lemma, the formula simplifies. For without these conditions, Λ does not exist, and the last summand in (5.3.21) can be taken to be 0. In other words, (5.3.21) reduces to the sum (5.3.24) over G . In the particular case that N is odd, the term (5.3.23) can also be taken to be 0, since the set Sψ ,ell is then empty. It follows from a simplified form of the proof of the lemma that the sum of (5.3.24) with (5.3.22) vanishes if N is odd, and equals (5.3.23) if N is even and ηψ 1.
5.4. GENERIC PARAMETERS WITH LOCAL CONSTRAINTS
285
5.4. Generic parameters with local constraints Our aim is to apply the discussion of the last three sections to the proof of the local theorems. We shall do so in due course, by constructing particular families Fr that among other things, satisfy the conditions of §5.1. We begin here by imposing some ad hoc local conditions on the members of a general r with a view to strengthening the results obtained so far. family F, Suppose that Fr
º N
FrpN q,
r pN q, FrpN q Ψ
r We assume as in §5.1 that is a family of parameters in the general set Ψ. r F is the graded semigroup generated by its subset Frsim of simple elements. However, we shall also take on a temporary hypothesis of a different sort. r which we state as It consists of three local constraints on elements in F, r follows in terms of notation for F introduced in §5.1 and a positive integer N.
Assumption 5.4.1. There is a finite, nonempty set V V pFrq of archimedean valuations of F for which the following three conditions hold. (5.4.1)(a) Suppose that ψ r 2 p G v q. v P V . Then ψv P Ψ
P
Frsim pGq, for some G
P
Ersim pN q, and that
(5.4.1)(b) Suppose that ψ P Fr2sim pGq, for some G P Erell pN q. Then there is r pG q for any G P Ersim pN q a valuation v P V such that ψv does not lie in Ψ v v p p G. over v with G v 2 pGq for some G P E rell pN q. Then there is (5.4.1)(c) Suppose that ψ P Frdisc a valuation v P V such that the kernel of the composition of mappings
Sψ
ÝÑ
Sψv
ÝÑ
Rψv ,
Sψ
Sψ pGq,
contains no element whose image in the global R-group Rψ regular.
Rψ pGq is
The intent of the first condition (5.4.1)(a) is that Theorem 1.4.2 should hold for simple generic parameters ψ P Frsim pN q and valuations v P V . The condition in fact will allow us to bypass the possible ambiguity of the provisional definition of Frsim pGq, adopted for families Fr in §5.1. The remaining two conditions contain technical assertions that are less clear. They will make better sense once we see how to apply them later in this section, and after we have constructed families for which they hold in §6.3. We fix a family Fr that satisfies Assumption 5.4.1. We may as well assume at this point that the elements in Fr are all generic, since it is to the
286
5. A STUDY OF THE CRITICAL CASES
local classification of tempered representations in Chapter 6 that they will be applied. However, we will continue to denote them by ψ (rather than φ). This preserves the notation of the last three sections, and will also ease the transition to the family that will be applied to nontempered representations in Chapter 7. We do not assume a priori that the elements in Fr satisfy Assumption 5.1.1. We instead fix the positive integer N , and as in §5.1, assume inductively that all the local and global theorems hold for any ψ P Fr with degpψ q N . We shall combine this inductive property with Assumption 5.4.1 to establish the original conditions in Assumption 5.1.1. Lemma 5.4.2. Suppose that G P Eell pN q, and that ψ belongs to Fr2 pGq. Then the conditions of Assumption 5.1.1 hold for the pair pG, ψ q. Proof. The assertion of course includes the case that ψ is simple. Expanding on the remark concerning (5.4.1)(a) above, we recall that Frsim pGq was defined provisionally in (5.1.5) and (5.1.6) as a set of paramG is eters ψ P Frsim pN q for which the corresponding stable linear form Sdisc,ψ nonzero. The problem is that for a given ψ, we do not know yet that this characterizes the datum G P Ersim pN q uniquely. We do know from our remarks in §3.4 that if G P Ersim pN q is another such datum, and if ηG is vanishes. Moredistinct from the quadratic character ηψ ηG , then Sdisc,ψ
r ell,v pN q, and that Gv is over, (5.4.1)(a) asserts that for any v P V , ψv lies in Φ r pGv q contains ψv . This condition will the unique datum in Erv pN q such that Φ fail if G is replaced by G , unless Gv equals Gv , which in turn implies that p G. p Since the properties G p G p and ηG ηG imply that G G, G the condition (5.4.1(a)) does indeed characterize the set Frsim pGq. If ψ is not simple, we recall that the condition ψ P Fr2 pGq (or the more general condition ψ P FrpGq) is defined inductively. In this case, our inr pGv q, for any valuation v of duction hypotheses imply that ψv belongs to Φ F. Given ψ and G, we write the set of valuations of F as a disjoint union
V
²
U
²
Vun ,
where V is given by Assumption 5.4.1 and U is a finite set, and where pG, ψ q is assumed to be unramified at every v in the remaining set Vun . We will apply the twisted endoscopic decomposition (3.3.14) of Corollary 3.3.2 to a function fr frV frU frun ,
r pN q , fr P H
that is compatible with this decomposition. In so doing, we will first fix the functions frV
¹
P
v V
frv ,
frv
P Hrv pN q,
5.4. GENERIC PARAMETERS WITH LOCAL CONSTRAINTS
and frun
¹
287
frv ,
P
v Vun
r U pN q to vary. We are writing H r v pN q here for the and then allow frU P H r r Hecke module on GpN, Fv q, and HU pN q for the C-tensor product over v P U r v p N q. of H We are assuming that ψ belongs to Fr2 pGq. We shall use (5.4.1)(b) to define frV . This condition, together with the stronger condition (5.4.1)(a) in case ψ lies in the subset Frsim pGq of Fr2 pGq, tells us that there is a valuation r pG q, for any datum G in Ersim pN q w P V such that ψw does not lie in Φ w w p G. p We then choose a function frw P H r w pN q such that over w with G w G frw,N pψw q is nonzero, but so that the image fw frw w of frw in SrpGw q vanishes for any such Gw . The existence of frw follows easily from Proposition 2.1.1, and the trace Paley-Wiener theorems [CD] and [DM] at w. At the other places v P V , we choose frv subject only to the requirement that frv,N pψv q not vanish. The function frV then has the property that its value
frV,N pψV q
¹
frv,N pψv q
v
is nonzero, but its transfer frV
¹
frvGv
P p G. p vanishes for any G P Ersim pN q with G v V
The other function to be fixed is frun . We take it simply to be any decomposable function frun
¹
P
frv
v Vun
r N, K r un pN q such that the value in the unramified Hecke algebra H
frun,N pψun q
¹
P
frv,N pψv q,
v Vun
r un pN q stands here for the product over v is nonzero. Of course, K r v pN q of GLpN, Fv q. the standard maximal compact subgroups K We have thus to apply the identity
(5.4.2)
N Idisc,ψ pfrq
¸
G PErell pN q
r ι N, G Spdisc,ψ fr
p
q
p q
to the variable function (5.4.3)
fr frV frU frun ,
frU
P HrU pN q.
P Vun of
288
5. A STUDY OF THE CRITICAL CASES
We have not asked that the given datum G be simple. However, if G P Erell pN q is a datum that is not simple, and distinct from G, we can say that G Sdisc,ψ Sdisc,ψ 0. This follows from the application of our induction hypothesis to the comr 2 pGq, ψ cannot also belong to posite factors of G , since as an element in Ψ r pG q. Suppose next that G P Ersim pN q, but is again distinct from G. If Ψ p G, p we obtain G
pfr q Sp pfr fr fr q 0, Spdisc,ψ U un disc,ψ V
p by our choice of frV . If G
p we have G, ηG ηG ηψ ,
r pGq. This implies that S given that ψ lies in Ψ disc,ψ 0, as we agreed in §3.4. The summands in (5.4.2) with G G therefore all vanish, and we obtain
(5.4.4)
N G Idisc,ψ pfrq rιpN, Gq Spdisc,ψ pfrGq.
We shall compare the last formula with the spectral expansion of pfrq. This expansion is the analogue of (4.1.1) for GrpN q, described first at the end of §3.3. It is a sum of terms parametrized by Levi subgroups
N Idisc,ψ
0 M
GLpN1q GLpNr q
of GLpN q. From the classification of automorphic representations of GLpN q (taken up to conju(Theorem 1.3.3), we know that there is at most one M gacy) for which the corresponding term is nonzero. It is then a consequence of the definitions (or if one prefers, the twisted global intertwining relation discussed at the end of §4.2) that this term is a nonzero multiple of the linear form frN pψ q frV,N pψV q frU,N pψU q frun,N pψun q
cpψV , ψV q frU,N pψU q, un
where cpψV , ψVun q is a nonzero constant. Combined with (5.4.4), this tells us that frU,N pψU q is a nonzero multiple of the linear form G G Spdisc,ψ pfrGq Spdisc,ψ pfrVG fUG frunG q,
and hence depends only on frUG . Therefore, frU,N pψU q is the pullback of a linear form on SrpGU q. We are now free to take frV to be a variable function. It follows from the conditions (5.4.1)(a) and (5.4.1)(b), and the discussion of the specific function frV above, that the corresponding linear form frV,N pψV q is also the pullback of a linear form on SrpGV q. Enlarging the finite set U , if necessary, r un pN q. Having we can assume that frun is the characteristic function of K
5.4. GENERIC PARAMETERS WITH LOCAL CONSTRAINTS
289
treated the function frU , we conclude that frN pψ q is indeed the pullback of a linear form on SrpGq. In other words, we can write (5.4.5)
frN pψ q frG pψ q,
r pN q, fr P H
r pN q where the right hand side represents the restriction to the image of H r of some linear form on S pGq. If G is simple, the linear form on SrpGq is uniquely determined by (5.4.5), since Corollary 2.1.2 tells us that the twisted transfer mapping is surjective. r pG q. In particular, it can be identified with a unique stable linear form on H If G GS GO is composite, and ψ ψS ψO is the corresponding decomposition of ψ, we define the linear form on SrpGq as the tensor product of the linear forms on SrpGS q and SrpGO q given by ψS and ψO . We then have to check that this definition is compatible with (5.4.5). In other words, we need to show that
(5.4.6)
frG pψ q frG pψS
ψO q,
r pN q, fr P H
where the left hand side is defined by (5.4.5), and the right hand side is defined by the linear forms attached to ψS and ψO . To establish (5.4.6), we return briefly to the discussion surrounding the N function (5.4.3) above. For this particular function fr, we know that Idisc,ψ pfrq is on the one hand equal to
p
q
p q
G r ι N, G Spdisc,ψ frG ,
and on the other, is a constant multiple of the value at fr of the left hand side of (5.4.5). This constant is easy to compute, as an elementary special case of the general remarks in §4.7 for example. The reader can check that ιpN, Gq1 equals the product of mpψ q with |S |1 . In other its product with r words, the stable multiplicity formula G pfrGq mpψq |Sψ |1 frGpψq Spdisc,ψ
holds for the given function fr. We are using the fact that ψ is generic here, so there is no ε-factor in the computation, and no sign εG pψ q. Suppose again that G GS GO and ψ ψS ψO . Our general induction hypothesis then tells us that G S 0 Spdisc,ψ pf Gq Spdisc,ψ pf S q Spdisc,ψ pf O q S O
mpψS q |Sψ |1 f S pψS q mpψO q |Sψ |1 f O pψO q, for any decomposable function f G f S f O in SppGq. We thus have two S
O
G formulas for Spdisc,ψ . Replacing f G by the image frG of the function (5.4.3), and noting that
mpψ q |Sψ |1
mpψS q |Sψ |1 mpψO q |Sψ |1, S
O
we see that (5.4.6) holds if fr is the function (5.4.3). Now the local analogue r pGV q (this of (5.4.6) holds if fr is replaced by an arbitrary function frV P H
290
5. A STUDY OF THE CRITICAL CASES
is Theorem 2.2.1(a) for the archimedean valuations v P V , which is part of what we are taking for granted from [Me] and [S8]), or by the unit frun in the G pψ q Hecke algebra (by definition). Since the components frVG pψV q and frun un are nonzero for the functions frV and frun in (5.4.3), the local analogue of r pGU q. With (5.4.6) holds also if fr is replaced by an arbitrary function frU P H this fact in hand, we replace the fixed component frV in (5.4.3) by a general r pGV q. The required identity (5.4.6) then follows for a general function in H r pN q, since it holds for its local factors frV , frU decomposable function fr P H r and fun . This completes the last step in the proof of the lemma. Remark. The assertion of the lemma holds more generally if ψ is any element in FrpGq. For as we remarked after stating Assumption 5.1.1, the assertion for parameters ψ in the complement of Fr2 pGq follows from the corresponding assertion for a Levi subgroup of GLpN q.
We shall now establish some global identities for FrpN q. Keep in mind that FrpN q is composed of generic global parameters with rather serious local constraints. For this reason, the identities we obtain will not be general enough to have much interest in their own right. Their role is intended rather to be local. They will drive the local classification we are going to establish in the next chapter. We will return to global questions later in Chapter 8, where having armed ourselves with the required local results, we will be able to establish the global theorems in general. We shall prove four lemmas, which represent partial resolutions of the four main lemmas from §5.2 and §5.3, and apply to parameters in the family FrpN q. We continue to denote these generic parameters by ψ rather than φ, in order to match the notation from the two earlier sections. Lemma 5.4.3. Suppose that
pG, ψq,
G P Ersim pN q, ψ
P Frell2 pGq,
is as in Lemma 5.2.1, but with Fr being our family of generic parameters that satisfy Assumption 5.4.1. Then for every G r p G q. f PH
pf q 0 0 S pf q, Rdisc,ψ disc,ψ
P
f
P HrpGq,
Ersim pN q, while the expression (5.2.8) vanishes for any
Proof. Given the pair pG, ψ q, we choose a place v P V pFrq that satisfies (5.4.1)(c). We then consider the expression (5.2.8) in Lemma 5.2.1, for a decomposable function f
fv f v ,
fv
P HrpGv q,
fv
r pGq, and a corresponding decomposition in H
ψ
ψv ψ v
P Hr GpAv q ,
5.4. GENERIC PARAMETERS WITH LOCAL CONSTRAINTS
291
of ψ. The main terms in (5.2.8) become fG pψ, xq fv,G pψv , xv q fGv pψ v , xv q and
1 pψv , xv q pf v q1 pψv , xv q, fG1 pψ, xq fv,G G
by (4.2.4) and (4.2.5). Since ψ is generic, we also have reductions sψ and εψ pxq 1. The expression (5.2.8) therefore equals (5.4.7)
c
¸
1
1 pψv , xv q f v pψv , xv q fv,G pψv , xv q . pf v q1G pψv , xv q fv,G G
P
x Sψ,ell
We will derive the required properties from Corollary 3.5.3. This entails identifying (5.4.7) with the supplementary expression (3.5.12) of the corollary. We have therefore to expand the linear forms in fv from (5.4.7) in terms of the basis T pGv q described at the beginning of the proof of Proposition 3.5.1. The element xv in (5.4.7) is the image of x in Sψv . We shall write rv pxq for its image in the local R-group Rψv . We shall also identify ψv with a fixed r 2 pMv , ψv q, where Mv is a fixed Levi subgroup over local preimage in the set Ψ Gv such that this set is not empty. (We are thus following the local notation (2.3.4) rather than letting Mv denote the localization of the global Levi subgroup M attached to ψ, a group that could properly contain Mv .) The linear form fv,G pψv , xv q in (5.4.7) then equals (5.4.8)
¸
P
p q
r ψv Mv πv Π
xxrv , πrv y tr RP
v
rv , ψv IPv pπv , fv q , rv pxq, π
in the notation (2.4.5). The other term in (5.4.7) that depends on fv is the linear form 1 pψv , xv q. We are not assuming the local intertwining relation of Theofv,G rem 2.4.1, which asserts that it equals fv,G pψv , xv q. However, our assumption that ψv is archimedean and generic, and hence that Theorem 2.2.1 holds, leads to a weaker relation between the two linear forms. For if the assertion (b) of this theorem is combined with the theory of the R-groups Rpπv q, atr ψ pMv q and reviewed in §3.5, one obtains tached to representations πv P Π v an endoscopic characterization of the irreducible constituents of the induced representation IPv pπv q. They correspond to characters on the group Sψv pGq whose restriction to the subgroup Sψv pMv q equals the character attached to 1 pψv , xv q has an expansion πv . It is then not hard to see that fv,G (5.4.9)
¸
P
p q
r ψv Mv πv Π
xxrv , πrv y tr RP1
v
rv , ψv IPv pπv , fv q , rv pxq, π
where rv , ψv q ε1πv prv q RPv prv , π rv , ψv q, RP1 v prv , π
πv
P Π ψ p Mv q , v
rv
P Rψ , v
292
5. A STUDY OF THE CRITICAL CASES
for a (sign) character rv
επv prv q επv prv q1
ÝÑ
on the 2-group Rψv . This is included in Shelstad’s endoscopic classification of representations for real groups, as we will recall (6.1.5) in the next section. The local intertwining relation would tell us that the character επv is trivial for each πv . As it applies to the parameter ψv , Theorem 2.2.1(b) can thus be summarized as the existence of a natural isomorphism from the endoscopic R-group Rψv onto any of the associated representation theoretic R-groups Rpπv q. A triplet τv Mv , πv , rv pxq , πv P Πψv pMv q, or rather its orbit under the local Weyl group W0Gv , can therefore be identified with an element in the basis T pGv q. It follows from the definition (3.5.3) that we can write (5.4.8) and (5.4.9) respectively as fv,G pψv , xv q and
1 pψv , xv q fv,G
¸
P
p q
r ψv Mv πv Π
¸
P
apπv , xv q fv,G Mv , πv , rv pxq
a1 pπv , xv q fv,G Mv , πv , rv pxq ,
p q and a1 pπv , xv q
r ψ v Mv πv Π
for coefficients apπv , xv q επv rv pxq apπv , xv q. The expression (5.4.7) can therefore be written as ¸
(5.4.10)
P p q
dpτv , f v q fv,G pτv q,
τv T G v
where dpτv , f v q equals the sum c
¸¸ x
a1 pπv , xv q pf v q1G pψ v , xv q apπv , xv q fGv pψ v , xv q
πv
r ψ pMv q such that the triplet over elements x P Sψ,ell and πv P Π v Gv Mv , πv , rv pxq belongs to the W0 -orbit represented by τv . Our expression (5.4.10) for (5.2.8) matches the supplementary expression (3.5.12) of Corollary 3.5.3 (with the group G1 in (3.5.12) being the group G here). Observe also that any x P Sψ,ell maps to a regular element in the global R-group Rψ . It follows from the condition (5.4.1(c)) on v that the image rv pxq of x in the local R-group Rψv in nontrivial. In other words, the coefficient dpτv , f v q vanishes if τv is represented by a triplet of the form pMv , πv , 1q. We can therefore apply Corollary 3.5.3 to the formula given by Lemma 5.2.1. We find that the coefficents dpτv , f v q all vanish, as do the multiplicities of any irreducible constituents of representations Rdisc,ψ in (5.2.7). It follows that the expression (5.2.8) vanishes , as required, and
5.4. GENERIC PARAMETERS WITH LOCAL CONSTRAINTS
293
that the representations Rdisc,ψ are all zero. Finally, as we observed in the proof of Lemma 5.2.1, the difference pf q 0 Sdisc,ψ pf q, Rdisc,ψ
G
P ErsimpN q,
f
P HrpGq,
vanishes unless G G , in which case it equals (5.2.8). Having just shown that (5.2.8) itself vanishes, we can say that the difference vanishes also in the case G G. We conclude that
p f q 0 0 S pf q , Rdisc,ψ disc,ψ
G
f
P ErsimpN q,
as required. This completes the proof of the lemma.
P HrpGq,
Lemma 5.4.4. Suppose that
pG, ψq,
G P Ersim pN q, ψ
P Frell2 pGq,
is as in Lemma 5.2.2, with Fr again being our family of generic parameters that satisfy Assumption 5.4.1. Then
pf q 0 0 S pf q, Rdisc,ψ disc
f
P HrpGq,
for every G P Ersim pN q, while the expressions (5.2.13) and (5.2.12) vanish r pGq and f _ P H r pG_ q respectively. for any functions f P H
Proof. This is the case of a parameter (5.2.4) with r 1. Apart from the analysis of the new term (5.2.12), which is attached to G_ and M , the proof is similar to that of the last lemma. For the given pair pG, ψ q, we choose v P V pFrq so that the condition (5.4.1)(c) holds. We then write the expression (5.2.13) in a form (5.4.10) that matches the supplementary expression (3.5.12) of Corollary 3.5.3. To deal with (5.2.12), we can argue as in the treatment of the parameter (4.5.11) from the proof of Proposition 4.5.1. In the simple case here, the r 2 pM, ψ q for the Levi subgroup M GLpN q is restricted parameter ψM P Ψ just the simple component ψ1 of ψ. We can therefore identify ψM with the automorphic representation πψ1 of GLpN q. It follows from the definitions of its two terms that (5.2.12) equals (5.4.11)
1 8
rψ1 , ψ1 qq IP _ pπψ1 , f _ q . tr p1 RP _ pw1 , π
This is essentially the analogue for G_ of the general expression (4.5.14) obtained in the proof of Proposition 4.5.1. As in the earlier discussion of (4.5.14), we observe that (5.4.11) is a nonnegative integral combination of irreducible characters. Its sum with the first term (5.2.11) in the formula given by Lemma 5.2.2 is therefore of the general form (3.5.1). Since the other expression (5.2.13) is of the form (3.5.12), we can apply Corollary 3.5.3 to the formula. We conclude that the expressions (5.2.13) and (5.2.12) both vanish, as required, and that the representations Rdisc,ψ are all zero. We then recall that as in the proof of Lemma 5.2.2, the difference
pf q 0 Sdisc,ψ pf q, Rdisc,ψ
G
P ErsimpN q,
f
P HrpGq,
294
5. A STUDY OF THE CRITICAL CASES
equals a positive multiple of either (5.2.13) or (5.2.12) if G is equal to the associated group G or G_ , and vanishes otherwise. The difference therefore vanishes in all cases, and
p f q 0 0 S pf q, Rdisc,ψ disc,ψ
G
P ErsimpN q,
f
P HrpGq,
as required.
Lemma 5.4.5. Suppose that
pG, ψq,
G P Ersim pN q, ψ
P Fr2pGq,
is as in Lemma 5.3.1, for our family Fr of generic parameters with local constraints. Then the linear form Λ in (5.3.4) vanishes. Proof. Recall that Λ was defined in Lemma 5.1.4 as a linear form on Sr0 pLq. It represents the defect in the stable multiplicity formula for each of the pairs pG, ψ q and pG_ , ψ q. The lemma thus asserts that the stable multiplicity formula is valid for either of these pairs. We observe also that the premise of Lemma 5.3.1 holds here. It amounts to the first vanishing assertion of Lemma 5.4.3 for parameters in the complement of Frell pN q in FrpN q. (It was this global theorem that was required in the proof of Lemma 5.3.1.) Therefore Lemma 5.3.1 is valid in the case at hand. The supplementary term (5.4.12)
b f L pψ1 Λq,
L
G1 L, f P HrpG q,
in (5.3.4) is clearly the main point. If this linear form were a nonnegative linear combination of characters in f , the entire expression (5.3.4) would be of the general form (3.5.1), and we could argue as in the proof of Lemma 5.4.3. According to Corollary 5.1.5, the pullback of Λ from LpAq to G_ pAq is a unitary character. However, it is not clear that the pullback of Λ to GpAq has the same property. To sidestep this question, we introduce the composite _ r endoscopic datum G_ 1 G1 G in Eell pN q. The group L , regarded itself as an endoscopic datum in ErpN q, has a global Levi embedding into both r _ G and G_ 1 . If f1 P HpG1 q is the associated function in the compatible L family (5.3.3), it follows from the definitions that f1 equals f L . The supplementary term in (5.3.4) therefore equals (5.4.13)
L
b f1
pψ1 Λq,
f1
P HrpG_1 q,
G_ 1
G1 G_ .
Since ψ1 is assumed to be a proper subparameter of ψ, its degree N1 is less than N . It follows from our induction hypothesis that the stable linear form r pG1 q attached to ψ1 is a nonzero unitary character. Corollary 5.1.5 on H L then tells us that f1 pψ1 Λq represents a positive multiple of a unitary character (possibly 0) in f1 . It follows that if we replace the supplementary term (5.4.12) by (5.4.13), the expression (5.3.4) will indeed be of the general form (3.5.1).
5.4. GENERIC PARAMETERS WITH LOCAL CONSTRAINTS
295
With this modification, we choose v P V pFrq so that the condition (5.4.1)(c) holds for the pair pG , ψ q. Following the proof of Lemma 5.4.3, we then write the expression (5.3.5) in Lemma 5.3.1 as a sum (3.5.12) for which the required condition of Corollary 3.5.3 holds. We can therefore apply Corollary 3.5.3 to the formula provided by Lemma 5.3.1. It then follows that the coefficients in the irreducible decomposition of the modified form of (5.3.4) vanish, as does the entire expression (5.3.5). In particular, the sumr pG1 q attached to mand (5.4.13) vanishes. Since the stable linear form on H ψ1 is nonzero, this can only happen if Λ vanishes, as claimed. Lemma 5.4.6. Suppose that
pG, ψq,
G P Ersim pN q, ψ
P FrsimpGq,
is as in Lemma 5.3.2, for our family Fr of generic parameters with local constraints. Then the linear form Λ in (5.3.21) vanishes, while the sign δψ in (5.3.22) and (5.3.23) equals 1. Proof. We will have to draw on the techniques of all of the last three lemmas, since all of the corresponding unknown quantities occur here. We will also have to include something more to take care of the sign δψ . However, the basic idea is the same. We shall apply Corollary 3.5.3 to the formula provided by Lemma 5.3.2. We choose v P V pFrq so that (5.4.1)(c) holds for the pair pG , ψ q. As in the proof of Lemma 5.4.3, we then write the expression (5.3.23) on the right hand side of the formula of Lemma 5.3.2 in the general form (3.5.12). The fact that there is now an unknown sign δψ in the expression is of no consequence. For the condition on the coefficients in (3.5.12) required for Corollary 3.5.3 remains a consequence of our given condition (4.5.1)(c). The left hand side of the formula is the sum of two expressions (5.3.21) and (5.3.22). The first of these contains the supplementary summand (5.4.14)
1 2
f L p Γ Λ q,
L
G L, f P HrpG q.
Following the proof of the last lemma, we introduce the composite endo_ in Erell pN q, with a corresponding function scopic datum G_ 1 G G r pG_ q in the compatible family (5.3.3). Since L again has a global f1 P H 1 Levi embedding into both G and G_ 1 , the supplementary summand (5.4.14) equals (5.4.15)
1 2
L
f1
pΓ Λq,
f1
P HrpG_1 q,
G_ 1
G G_ .
It then follows from Corollary 5.1.5 and the definition of Γ that f1 pΓ Λq represents a positive multiple of a unitary character (possibly 0) in f1 . The expression (5.3.21), written with (5.4.15) in place of (5.4.14), is consequently of the general form (3.5.1). As for the second expression (5.3.22), we note that the parameter ψ ,M for M GLpN q can be identified with the L
296
5. A STUDY OF THE CRITICAL CASES
automorphic representation πψ of GLpN q. Appealing to the definitions, as in the proof of Lemma 5.4.4, we can write the expression in the form (5.4.16)
1 8
rψ , ψ qq IP _ pπψ , f _ q , tr p1 δψ RP _ pw_ , π
f_
P HrpG_q.
Since δψ equals either p 1q or p1q, (5.4.14) is a nonnegative linear combination of characters on the group G_ pAq. It follows that the sum of (5.3.21) and (5.3.22) can be written as an expression of the general form (3.5.1). We can thus apply Corollary 3.5.3 to the formula given by Lemma 5.3.2. We then see that, among other things, the expressions (5.4.14) and (5.4.16) each vanish. From the vanishing of (5.4.14), we deduce that either Γ or Λ equals zero. Suppose that Γ is the factor that vanishes. The linear form f G pψ q f G pΓq
f L pΛ q f L pΛ q ,
f
P HrpGq,
r pGq is then induced from the Levi subgroup LpAq. This contradicts the on H local condition (5.4.1(a)) for the simple parameter ψ. It is therefore Λ that vanishes, as claimed. The operator δψ RP _ pw_ , πψ , ψ q in (5.4.16) is unitary, and commutes with IP _ pπψ , f _ q. The vanishing of (5.4.16) then implies that this operator equals 1. In particular, our normalized intertwining operator here is a scalar, as we expect from the fact that the centralizer group Sψ_ is isomorphic to SLp2, Cq. To show that it actually equals 1, we appeal to the relation between normalized intertwining operators and Whittaker models. Since the parameter ψ is generic, case (iii) of Corollary 2.5.2 tells us that
RP _ pw_ , πψ,v , ψv q 1
for any v. Therefore
RP _ pw_ , πψ , ψ q
¹
RP _ pw_ , πψ,v , ψv q 1.
v
We conclude that the sign δψ equals 1, and therefore that the second assertion of the lemma also holds. In the last lemma, N is even and ηψ equals 1. If one of these conditions is not satisfied, the formula of Lemma 5.3.2 becomes simpler, as we noted at the end of the last section. In particular, Λ is not defined, and does not contribute a summand to (5.3.21). If N is odd, the summands in (5.3.23) are also not defined, and the term (5.3.23) does not exist. However, the formula of Lemma 5.3.2 is otherwise valid, and the sign δψ is invariably present in the term (5.3.22). The simplified analogue Lemma 5.4.6 asserts that δψ 1. It is proved as above. We can now clarify the temporary definition of the sets Frsim pGq adopted in §5.1. We shall show that the three possible ways to characterize Frsim pGq are all equivalent.
5.4. GENERIC PARAMETERS WITH LOCAL CONSTRAINTS
297
Corollary 5.4.7. Suppose that ψ belongs to the subset Frsim pN q of our family Fr of generic parameters with local constraints, and that G belongs to Ersim pN q. Then the following conditions on pG, ψ q are equivalent. G r pGq does not vanish. (i) The linear form Sdisc,ψ on H (ii) Theorem 1.4.1 holds for ψ, with Gψ G. (iii) The global quadratic characters ηG and ηψ are equal, and the global L-function condition δψ 1 holds. Proof. The first assertion Λ 0 of Lemma 5.4.6 tells us that condition (i) characterizes G uniquely in terms of ψ. If G is any element in Ersim pN q, we know that
pf q tr R pf q , Sdisc,ψ disc,ψ
f
P HrpGq,
since the proper terms in the expansions (4.1.1) and (4.1.2) all vanish by our induction hypotheses. It follows that the condition of Theorem 1.4.1 characterizes the postulated datum Gψ uniquely, and that Gψ G. The conditions (i) and (ii) are therefore equivalent. The third assertion δψ 1 of Corollary 5.4.7 (together with (3.4.7), and the remark above in case N is odd or ηψ 1) tells us that (i) implies (iii). But we know that for the given ψ, there is at most one G that satisfies (iii). The conditions (i) and (iii) are therefore also equivalent. The condition (i) was the basis for our temporary definition in §5.1. It has been the pivotal condition for our arguments in this chapter. The condition (ii) was used for the original definition from Chapter 1. Knowing now that the three conditions are equivalent for the family Fr of this section, we can define Frsim pGq to be the set of ψ P Frsim pN q such that any one of the conditions holds for pG, ψ q. It is clear from the discussion that Frsim pN q is a disjoint union over G P Ersim pN q of the sets Frsim pGq. We shall add a comment to what we have established in this section. The preceding Lemmas 5.4.3, 5.4.4, 5.4.5 and 5.4.6 apply to the respective formulas of Lemmas 5.2.1, 5.2.2, 5.3.1 and 5.3.2. We recall that the four formulas became increasingly complex as each one took on another unknown quantity. The four unknowns were the global intertwining relation for G (or G ), the global intertwining relation for G_ (or G_ ), the linear form Λ, and the sign δψ . By the time we reached the last Lemma 5.3.2, all four of these unknowns were present as terms in the associated formula. Their resolution, under the local constraints of this section, rests exclusively on Corollary 3.5.3 of Proposition 3.5.1. The basic principle behind Proposition 3.5.1 can be described as follows: if a linear combination of reducible characters vanishes, and if the coefficients are all positive, then the characters are themselves equal to zero. This is an obvious simplification, but it serves to illustrate the main idea for the proofs of the last four lemmas. The critical point is that the unknown quantities, or rather the ones that do not go into the supplementary expression (3.5.12)
298
5. A STUDY OF THE CRITICAL CASES
of Corollary 3.5.3, must all occur with the appropriate signs. They do – most fortunately for us! This phenomenon seems quite remarkable. We have seen it before, in the treatment of the parameters (4.5.11) and (4.5.12) of Proposition 4.5.1 and in Lemma 5.1.6, and we will see it again, most vividly perhaps in our completion of the global classification in §8.2. I do not have an explanation for it.
CHAPTER 6
The Local Classification 6.1. Local parameters The next two chapters will be devoted to the proof of the local theorems. Our methods will be global. Given the appropriate local data, we shall construct a family of global parameters to which we can apply the results of Chapter 5. Since we are aiming for local results, we take F to be a local field in Chapters 6 and 7. We will then use the symbol F9 to denote the auxiliary global field over which the global arguments take place. This follows a general convention of representing global objects by placing a dot over symbols that stand for the given local objects. To obtain a general idea of the strategy, consider a local parameter
`1ψ1 ` ` `r ψr , ψi P Ψr simpGiq, Gi P ErsimpNiq, r sim pGi q and Ersim pNi q are understood to be over the local field where the sets Ψ F . We are proposing to construct suitable endoscopic data Gi P Ersim pNi q r sim pGi q over a global field F such that and parameters ψi P Ψ pF, Gi, ψiq pFu, Gi,u, ψi,uq, 1 ¤ i ¤ r, ψ
9
9
9
9
9
9
9
at some place u of F9 . This will lead to a family r F 9
Frpψi, . . . , ψr q 9
9
`91 ψ9 1 ` ` `9r ψ9 r : `9i
(
¥0
of global parameters to which we can try to apply the methods of Chapter 5. In this chapter we will treat the case that ψ φ is generic. We will r9 has local constraints (5.4.1), where show that the resulting global family F
r q is a set of archimedean places of F9 . In the next chapter, we will V V pF deal with general parameters. There we will establish local constraints that 9
r q being a set of p-adic are roughly parallel to (5.4.1), but with V V pF places. We shall prepare for this analysis with some remarks on local Langlands parameters, which we will apply presently to the localizations φ9 v of global 9
r These fall into three categories. We will have separate parameters φ9 P F. observations for the completions F9v at F9 at v u, at nonarchimedean places v u, or at archimedean places v u. 9
299
300
6. THE LOCAL CLASSIFICATION
The first remarks are aimed at general Langlands parameters over F F9u . They pertain thus to the objects for which we are trying to establish the local theorems. Since the main questions are nonarchimedean, we assume for the time being that F is p-adic. Our purpose is to introduce a temporary modification of one of our definitions. It will be a local p-adic analogue of the temporary global definition we adopted after Assumption 5.1.1, and later resolved in terms of the original definition in Corollary 5.4.7. At issue is the definition of the local r p G q. parameter sets Φ We begin by stating one more theorem, which can be regarded as a local analogue of Theorem 1.4.1. This last of our collection of stated theorems amounts to a special case of Theorem 2.2.1, just as the initial Theorem 1.4.1 was a special case of Theorem 1.5.2. We will resolve it, along with the generic case of all the other local theorems, in §6.8.
r sim pN q is a simple generic local paramTheorem 6.1.1. Suppose that φ P Φ r eter. Then there is a unique Gφ P Eell pN q such that
frN pφq frGφ pφq,
r pN q, fr P H
for a linear form frGφ pφq on SrpGq. Moreover, Gφ is simple.
r pGq attached to a given We consider now the definition of the set Φ simple datum G P Ersim pN q. As in the global case, there will be three equivr sim pGq of simple local parameters alent ways to characterize the subset Φ r sim pN q. Our original definition gives Φ r sim pGq as the subset of irrein Φ ducible N -dimensional representations of LF that factor through the image of L G in GLpN, Cq. The second characterization is in terms of poles of local L-functions, and will be postponed until §6.8. The third is provided by r sim pGq as the subset of local parameters Theorem 6.1.1. It characterizes Φ r sim pN q for which the datum Gφ of the theorem equals G. The first φPΦ characterization is the simplest. However, the local Langlands parameters for G are quite removed from the local harmonic analysis that governs the classification of Theorem 2.2.1. The third is more technical, but is formulated purely in terms of local harmonic analysis. For this reason, we shall temporarily replace the original definition by one that is based on Theorem 6.1.1. We shall make it parallel to the temporary global definition from the beginning of §5.1. r cusp pN q be the subspace of functions fr P H r pN q such that for each Let H G r r r G P Eell pN q, f lies in the subspace Scusp pGq of SrpGq. It follows from Proposition 2.1.1 that the mapping
fr
ÝÑ
à
P p q
G Erell N
frG ,
r cusp pN q, fr P H
r cusp pN q in I r pN q descends to an isomorphism from the image Ircusp pN q of H r r onto the direct sum over G P Eell pN q of the spaces Scusp pGq. If φ is any
6.1. LOCAL PARAMETERS
r pN q, we can then write element in Φ
(6.1.1)
frN pφq
¸
P p q
G Erell N
frG pφq,
301
r cusp pN q, fr P H
for uniquely determined linear forms frG pφq on each of the spaces Srcusp pGq. The formula (6.1.1) is a rough local analogue of the global decomposition (3.3.14). It tells us that we can attach local subsets (6.1.2)
r sim pGq tφ P F rsim pN q : f G pφq 0 for some f Φ
P HrcusppGqu
r sim pN q to the simple endoscopic data G P Ersim pN q so that of Φ
(6.1.3)
r sim pN q Φ
¤
P
p q
G Ersim N
r sim pGq. Φ
To be definite, and in minor contrast to the global convention of §5.1, we r sim pGq by taking the inclusion in (6.1.2) to be equality. That is, we define Φ r sim pGq to be the set of all parameters φ P Φ r sim pN q such that the define Φ G r cusp pGq does not vanish, as on the right hand side linear form f pφq on H of (6.1.2). This temporary definition will be in force throughout Chapter 6. During this period, the local theorems will be interpreted accordingly. For example, Theorem 6.1.1 will now include the assertion that the union (6.1.3) is actually disjoint. If we assume inductively that the local theorems r pN1 q with N1 (including Theorem 6.1.1) are all valid for parameters in Ψ r pGq in terms of the fundamental sets Φ r sim pG1 q N , we can define the full set Φ r with G1 P Esim pN1 q, exactly as we did for the global case in §1.4. This in turn allows us to describe the centralizers Sφ by the global prescription of §1.4 and the convention (5.1.7), and then to introduce the subsets r sim pGq Φ r 2 pG q Φ r ell pGq Φ r pG q , Φ
according to the later global definitions of §4.1. We will thus temporarily forget about general Langlands parameters for r pN q the p-adic group G (though we will still be able to treat elements in Φ as representations of LF ). We will come back to them in §6.8, after proving the local theorems in their interpretation above. We will then be able to r pGq in terms of the original one. This resolve the temporary definition of Φ will yield the general local theorems for generic φ in their original form. The next remarks are very elementary. They concern special elements r pGq, which will still be identified with local Langlands parameters, and in Φ which will apply to localizations φ9 v at p-adic places v of F9 distinct from u. The parameters we have in mind are spherical, in the sense that they are attached to the p-adic spherical functions of [Ma]. In particular, they are unramified if the given group G P Ersim pN q over F is unramified. The group G is quasisplit over F . It is unramified, by definition, if it splits over an unramified extension of F . This is the case unless N is even, p SOpN, Cq, and the quadratic character ηG is ramified. The group G
302
6. THE LOCAL CLASSIFICATION
p SOpN, Cq and ηG 1. Let itself is split over F unless N is even, G K be our fixed maximal compact subgroup of GpF q. Then K is special in general, and hyperspecial if G is unramified. An irreducible representation of GpF q is said to be K-spherical (or simply spherical) if its restriction to K contains the trivial representation. A local Langlands parameter φ will r φ contains a K-spherical representation. be called spherical if its L-packet Π We recall the description of these familiar objects. Let M0 be the standard minimal Levi subgroup of G. Then M0 is a x0 is the group of maximally split, maximal torus over F , whose dual M p diagonal matrices in G. Spherical parameters factor through the image of L M in L G, and in particular descend to the quotient W of L . Let 0 F F
φ0 : W F
ÝÑ
L
M0
x0 WF M
be the canonical splitting of L M0 , which is to say the identity mapping of WF to the second factor of L M0 . If λ is a point in the complex vector space aM0 ,C
X pM0qF b C,
we write |w|λ for the point in the complex torus
xΓ q0 X pM0 qF b C pM 0 absolute value |w| of an element
AM x0
attached to λ and the w P WF . The mapping φλ pwq φ0 pwq |w|λ , w P WF , is then a Langlands parameter for M0 , whose image in ΦpGq we continue to denote by φλ . The elements in the family
tφλ : λ P aM ,Cu 0
are the spherical parameters for G. Under the Langlands correspondence for the torus M0 , the spherical parameter φλ P ΦpM0 q maps to the spherical character πλ ptq eλpHM0 ptqq ,
t P M0 p F q ,
on M0 pF q. If λ is purely imaginary, πλ is unitary and φλ has bounded image x0 . Treating φλ as an element in ΦpGq, we write Πφ in this case for the in M λ set of irreducible constituents of the induced representation IP0 pπλ q, where P0 P P pM0 q is the standard Borel subgroup of G. This is the L-packet of φλ . For any λ, it contains a unique K-spherical representation. Our final remarks concern archimedean parameters. These will be applied to localizations φ9 v at the archimedean places v of F9 distinct from u. The complex case presents little difficulty, and we will in any case be choosing the global field F9 in §6.2 to be totally real. We therefore assume for the rest of this section that F R, with G P Ersim pN q continuing to be a simple endoscopic datum over F . We shall review a few points in Shelstad’s general classification of the representations of GpF q in terms of local Langlands r p G q. parameters φ P Φ
6.1. LOCAL PARAMETERS
303
The archimedean valuations v u of F9 will in fact play a central role in the global arguments of this chapter. To prepare ourselves, it will be helpful r 2 pGq attached to discrete series to review the archimedean parameters φ P Φ representations of GpF q. This amounts to an explicit specialization of some of the observations in §1.2. The local Langlands group for F is just the real Weil group WF WR . It is generated by the group C and an element σF , with relations σF2 1 and z P C . σF z σF1 z, A parameter φ : WF Ñ L G, when composed with our mapping of L G into GLpN, Cq, becomes a selfdual, N -dimensional representation of WF . We are interested in the case r 2 pGq, which is to say that the group that φ lies in Φ Sφ
SφpGq Cent φpWF q, Gp
is finite. As we noted in the general discussion of §1.2, this is equivalent to the condition that the irreducible constituents of the representation of WF attached to φ all be distinct and self-dual. Recall [T2] that the irreducible representations of WF are all of one or two dimensions. There are two self-dual one-dimensional representations, the trivial representation and the sign character of R pWF qab . The irreducible self-dual representations of WF of dimension two are parametrized by positive half integers µ P 21 N, and are given by
z
1 µ ÝÑ pzz 0 q pzz01qµ
and
,
z
P C ,
0 1 σF ÝÑ p1q2µ 0 . The image of any such representation lies in Op2, Cq or Spp2, Cq, according to whether µ is an integer or not. To construct all the parameters in Φ2 pGq, we simply select families of distinct irreducible representations of WF that p are of the same type (either symplectic or orthogonal) as G. To be more precise, let p rN {2s be the greatest integer in N {2. If p r 2 pGq is a direct sum of p distinct two-dimensional G SppN, Cq, any φ P Φ representations of symplectic type. It can therefore be identified with a set µφ
pµ1, . . . , µpq,
P 12 N N, p SOpN, Cq, with N odd, If G
µi
of p distinct, positive, proper half integers. r 2 pGq is a direct sum of p distinct two-dimensional repa parameter φ P Φ resentations of orthogonal type and a uniquely determined one-dimensional representation. It can therefore be identified with a set µφ
pµ1, . . . , µpq,
µi
P N,
304
6. THE LOCAL CLASSIFICATION
of p distinct positive integers. The one-dimensional representation is trivial p SOpN, Cq, with if p is even, and is the sign character if p is odd. If G r N even, we of course want the set Φ2 pGq to be nonempty. We therefore assume that G is split if p is even, and nonsplit if p is odd. There remain two possibilities in this case, which we characterize by setting p1 equal either r 2 pGq can then be identified with a set to p or p 1. A parameter φ P Φ µφ
pµ1, . . . , µp1 q,
µi
P N,
of p1 distinct positive integers. It corresponds to a direct sum of p1 distinct two-dimensional representations of orthogonal type, augmented by the sum of the trivial and sign characters in case p1 p 1. The notation µφ was used before. In the proof of Lemma 2.2.2, it was N that represented the infinitesimal character of φ used for a vector in 12 Z p1 r as an element in Φell pN q. In this section, µφ is a vector in 12 N (where 1 r p p in the first two cases above) attached to φ P Φ2 pGq. Having long ago r 2 pGq with a subset of Φ r ell pN q, we must now be prepared agreed to identify Φ to identify µφ with the vector 2p1 µφ ` pµφ q P 12 Z 21 Z N N
in 12 Z . In any case, µφ in this section represents the infinitesimal charr 2 pGq. We shall say that φ is in general position acter of φ as an element in Φ if its infinitesimal character is highly regular, in the sense that the positive half-integers (6.1.4)
µi , |µi µj | : 1 ¤ i j
¤ p1
(
are all large. p SOpN, Cq, for N even or odd, GpF q We should also note that if G has a central subgroup of order 2. In this case, we can speak of the central character of φ, with the understanding that it refers to the representations r φ . One shows that it is trivial if and only if in the corresponding L-packet Π the integer cφ
¸
µi
i
is even. It would not have been difficult to describe more general Langlands parameters in explicit terms. (See [L11, §3] for the case of a general group over R.) We have treated the square integrable (archimedean) parameters r 2 pGq because they are the objects we will use in the next section to φPΦ construct automorphic representations. We recall in any case that Theorem r pGq (assuming the 2.2.1 is valid for general archimedean parameters φ P Φ r pN q and G r of the work in progress two special cases for the twisted groups G by Mezo and Shelstad, which would also give Theorem 2.2.4).
6.1. LOCAL PARAMETERS
305
The results for generic archimedean parameters (including their pror pN q and G) r do not include the local intertwining jected analogues for G relation of Theorem 2.4.1 (or of Theorem 2.4.4). They do, however, implicitly contain a weaker form of the identity. It is a consequence of Shelstad’s proof of the generic case fG1 pφ, xq f 1 pφ1 q
¸
P
rφ π Π
xx, πy fGpπq,
r bdd pGq, x P Sφ , φPΦ
of the formula (2.2.6) of Theorem 2.2.1(b), specifically the fact that the pairing xx, π y in this formula is compatible with the short exact sequence (2.4.9). In particular, if φM P Φ2 pM, φq is a square integrable parameter that maps to φ, Πφ is a disjoint union over πM P ΠφM of sets ΠπM pGq on which the pφ of Rφ acts simply transitively. As the irreducible constituents dual group R of induced representations IP pπM q, these sets are in turn compatible with the self-intertwining operators RP pw, π rM , φM q, in the sense that follows from the work of Harish-Chandra [Ha4, Theorem 38.1] and Knapp and Stein [KnS, Theorem 13.4]. (See [A10, (2.3)], for example, and the general review in Chapter 3 prior to Lemma 3.5.2.) It then follows from [A10, (2.3)], the formula above for fG1 pφ, xq, and the definition (2.4.5) of fG pφ, xq (which includes the definition of the factor xu r, π ry on the right hand side of (2.4.5)) that (6.1.5)
fG pφ, xq
¸
rφ π PΠ
επM pxq xx, π y fG pπ q,
x P Sφ ,
r φ , and επ is the pullback to Sφ where πM is the projection of π onto Π M M of a character on Rφ . The as yet unknown characters εψM pxq of course are what weaken (6.1.5). They free the identity from its dependence on the finer normalization of intertwining operators in §2.3–§2.4, and the particular pairing xx, π y. Shelstad’s proof of the formula for fG1 pφ, xq is contained in [S3] and [S6]. It follows from her proof of [S3, Corollary 5.3.16 and Theorem 5.4.27], the discussion of R-groups in [S3, §5.3], and her conversion of the spectral transfer factors of [S5] to a pairing xx, π y represented by the specialization of [S6, Theorem 7.5] to connected quasisplit groups. It is clear that the resulting identity (6.1.5) is indeed a weaker version of the local intertwining identity. It reduces Theorem 2.4.1, in the case of archimedean F and generic pψ on φ ψ, to a question of base points for the simply transitive action of R r π pGq. We recall, however, that Shelstad has also shown each of the sets Π M r φ , then x , π y 1 that if π is the unique pB, χq-generic representation in Π [S6, Theorem 11.5]. It then follows from Theorem 2.5.1(b) that επM p q 1 in this case. The formula (6.1.5) will be our starting point for the general proof of Theorem 2.4.1. Recall that the mapping rφ Π
Ñ
ã
Spφ
306
6. THE LOCAL CLASSIFICATION
for G is generally a proper injection. In using the archimedean parameters with our global arguments, we will need the following lemma. Lemma 6.1.2. Suppose that G P Ersim pN q is a simple endoscopic datum r pGq is a corresponding generic over our archimedean field F , and that φ P Φ r φ in Spφ generates Spφ . parameter. Then the image of Π Proof. We will prove the lemma with G being any connected quasisplit group over F . We will also take the opportunity to recall the initial foundations of real endoscopy, on which the results of this chapter ultimately depend. We first choose a parameter φM P Φ2 pM, φq, for some Levi subgroup M of G. As we noted in the case G P Ersim pN q above, Πφ is a disjoint union over pφ acts simply transitively. πM P ΠφM of sets ΠπM pGq on which the group R The set of characters x , π y on Sφ attached to representations π P ΠπM pGq pφ , where ξ is the character x , πM y on Sφ . From the is thus a coset ξ R M short exact sequence (2.4.9) for G, it is clear that we need only show that the injective image of ΠφM generates SpφM . We may therefore assume that M G, or in other words, that φ lies in Φ2 pGq. Since Φ2 pGq is assumed not to be empty, G has a discrete series, and therefore a maximal torus T over F that is anisotropic modulo Z pGq. As in the examples for G P Ersim pN q described explicitly above, φ factors through the image of L T under an admissible L-embedding into L G. Moreover, as a general consequence of the fact that φ belongs to Φ2 pGq, the centralizer p is contained in Tp. This implies that Sφ TpΓ and Sφ of its image in G p qΓ . It is then not hard to see from the definitions that there is a TpΓ {Z pG canonical isomorphism from the dual Spφ of Sφ onto the finite 2-group (6.1.6)
Σ_ pG, T q{Σ_ pG, T q X tµ σµ : µ P X pT q, σ
P Γ F u,
where Σ_ pG, T q is the set of co-roots for pG, T q. (This assertion relies on
the fact that T {Z pGq is anisotropic, and hence that the norm of any element in Σ_ pG, T q vanishes.) Following Langlands and Shelstad, we write
DpT q W pG, T q{W GpRq, T pRq ,
where W GpRq, T pRq is the subgroup of the full Weyl group induced by elements in GpRq. This set is bijective with Πφ . More precisely, one sees from Harish-Chandra’s classification of discrete series, and the resulting natural grouping of these objects into packets according to their infinitesimal characters, that if a base point π0 P Πφ is fixed, there is a canonical bijection π
ÝÑ
invpπ0 , π q,
from Πφ to DpT q. We also write
E pT q im H 1 pF, Tsc q
π
ÝÑ
P Πφ ,
H 1 pF, T q ,
6.2. CONSTRUCTION OF GLOBAL REPRESENTATIONS π9
307
where Tsc is the preimage of T in the simply connected cover of the derived group of G. This group is canonically isomorphic with Spφ . For by the remarks in [K5, §1.1] based on Tate-Nakayama duality, one can write Γ ^ E pT q im π0 pTpsc q
ÝÑ π0pTpΓq^ TpΓ{Z pGpqΓ ^ Spφ.
But the mapping g
ÝÑ
g σ pg q1 ,
σ
P ΓF ,
P G normalizes T , descends to a canonical bijection ker H 1 pF, T q ÝÑ H 1 pF, Gq D pT q Ý Ñ from DpT q onto a set that is in fact contained in E pT q. (See [L8, p. 702].) If π0 P Πφ is fixed, the mapping of DpT q into E pT q thus gives an embedding
in which g
of Πφ into Spφ . We take π0 to be the unique pB, χq-generic representation in Πφ . It then follows from [S6, Theorem 11.5] that the embedding of Πφ into Spφ gives the pairing xx, πy, x P Sφ, π P Πφ,
of Theorem 2.2.1(b). (See also [S5, §13].) We are trying to show that the image of the set Πφ in Spφ generates the group Spφ . It follows from Shelstad’s initial observation [S2, Lemma 2.1] on the set DpT q that the image in (6.1.6) of the coroot α_ of any noncompact root of pG, T q is contained in the image of Πφ . We have only to show that such coroots generate the group (6.1.6). Having recalled the background, it will now be very easy for us to prove the lemma. We choose a base for the roots ΣpG, T q of pG, T q by taking the chamber in the Lie algebra of T that contains a Harish-Chandra parameter of π0 . It then follows from [V1, Theorem 6.2(f)] that the simple roots are all noncompact. The simple coroots α_ P Σ_ pG, T q therefore map into the image of Πφ in (6.1.6). Any coroot β _ P Σ_ pG, T q is of course an integral combination of simple coroots. The group (6.1.6) is therefore generated by the images of simple coroots, and hence also by the image of Πφ . We conclude that Spφ is generated by the image of Πφ , as claimed. 6.2. Construction of global representations π9 We now begin the construction of the global objects needed to establish the local theorems. The methods of this section will be representation theoretic. They will also be more elementary than in earlier chapters, to the extent that they do not depend on the running induction hypotheses we have been carrying since §4.3. We shall construct global representations from local ones as an exercise in the invariant trace formula. We fix a simple endoscopic datum G P Ersim pN q over our local field F . The main goal of this section will be to inflate any square-integrable
308
6. THE LOCAL CLASSIFICATION
representation of GpF q to an automorphic representation. The squareintegrability condition rules out the case that F C, so we assume that F is either real or p-adic. We have first to identify F with a localization of some global field F9 . The construction of suitable global fields from local fields is well known, but we may as well formulate what we need as a lemma, for the convenience of the reader. Lemma 6.2.1. Assume that the local field F is either real or p-adic, and that r0 is a positive integer. Then there is a totally real number field F9 with the following properties. (i) F9u F for some valuation u of F9 . (ii) There are at least r0 Archimedean valuations on F9 . Proof. This is a simple exercise in Galois theory. The main point is to construct a totally real field F9 so that (i) holds. There is no problem if F R, since we can then set F9 Q. We can therefore take F to be a finite extension of the p-adic field Qp . Then F Qp pαq, where α is a root of an irreducible monic polynomial q over Qp of degree n. We identify the space of monic polynomials of degree n over a field E with E n . Since F is open in the union of the finite set of extensions of Qp of degree n, we can replace q by any polynomial in some open neighborhood Up of q in Qnp . Consider also the set of monic polynomials of degree n over R. The subspace of such polynomials with distinct real roots is an open subset U8 of Rn . Since the diagonal image of Qn in Rn Qnp is dense, the intersection Qn X pU8 Up q
is nonempty. We can therefore assume that the coefficients of q lie in Q, and that the roots of q in C are distinct and real. We then see that F9 Qpαq is a totally real field, and that F F9u for some place u of F9 over p. To construct a totally real field with many Archimedean places, we re? ? place F9 by a field F9 p q 1 , . . . , q k q, where q1 , . . . , qk are distinct (positive) prime numbers. These primes can be arbitrary if F R, but must be ? chosen so that Qp p q i q Qp if F is p-adic as above. There are infinitely many such primes, as one sees readily from Dirichlet’s theorem on primes in arithmetic progression and the law of quadratic reciprocity (or its second ? ? supplement in case p 2). The field F9 p q1 , . . . , qk q then satisfies the required conditions. We can now construct the automorphic representation. We are assuming that F is real or p-adic, and that G is the fixed datum in Ersim pN q. We take F9 to be a totally real global field with F9u F , as in the lemma, for which the set S8 of archimedean places is large. We then set S8 puq S8 Y tuu
6.2. CONSTRUCTION OF GLOBAL REPRESENTATIONS π9
and u S8
309
S8 tuu.
To compactify the notation, it will be convenient to denote other objects associated with F9 by a dot. For example, we have the adeles A9 over F9 , the 9 endoscopic sets Er pN q
E GrpN q 9
9 and Er v pN q
E
r v pN q over F9 and F9 v , G 9
r9 pN q for any valuation v on F9 , as well as the Hecke modules H
H GrpN q r v pN q H G r v pN q on G r pN, Aq and G r pN, Fv q. We can write ErpN q and H r pN q H r u pN q as usual for corresponding objects attached to Er u pN q and H F Fu . Lemma 6.2.2. Suppose that π P Π2 pGq is a square integrable representation of GpF q. We can then find a simple endoscopic datum G P Er sim pN q, together with an automorphic representation π of GpAq that occurs discretely 2 in L GpF qzGpAq , with the following properties: (i) pGu , πu q pG, π q. (ii) For any valuation v R S8 puq, πv is spherical. u , π is a square integrable representation of GpF q (iii) For any v P S8 v v whose Langlands parameter φv P Φ2 pGv q is in general position. 9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
Proof. We have first to define the endoscopic datum G9 over F9 . The p
p This determines G uniquely complex dual group G9 must be equal to G. p if G SppN, Cq, and up to a quadratic id`ele class character η9 ηG9 if p SOpN, Cq. There are two kinds of local constraints on η. G 9 The component η9 u of η9 at u must equal the quadratic character ηG on F determined by the given local endoscopic datum. The other constraints are at u , and are imposed by the requirement that the archimedean places v P S8 9 9 GpFv q have a discrete series. The question is relevant here only if N is even p SOpN, Cq. As we noted in §6.1, the requirement is that and G
η9 v
εpv ,
where p 12 N , and εv is the sign character on Fv R . We take η9 ηG9 to be any quadratic id`ele class character with the given finite set of local constraints. (We take η9 to be equal to 1 if all of the constrained local characters η9 v equal 1.) This determines an endoscopic datum G9 over F9 such that G9 u G, and such that at each v P S8 , the real group G9 v pF9v q has a discrete series. 9 To construct the automorphic representation π, we turn to a natural generalization of an argument applied by Langlands to the group GLp2q [L9, p. 229]. We shall apply the trace formula to a suitable function f9 on G9 pAq. (See also [CC] for a similar application of the twisted trace formula for GLpN q.)
310
6. THE LOCAL CLASSIFICATION
Let
K9 8,u
¹
K9 v
R pq
v S8 u
be a standard maximal compact subgroup of G9 pA9 8,u q, the group of points G9 with values in the finite ad`eles of F9 that are 0 at u. We take f98,u to be the characteristic function of K9 8,u on G9 pA9 8,u q. At u, we take f9u P HpGq to be a pseudocoefficient fπ of π. In other words, f9u has the property that for any irreducible, tempered representation π of GpF q, the relation #
(6.2.1)
if π π, 0, otherwise,
1, f9u,G pπ q tr π pfπ q
u , we will take a function holds. At the remaining set of places S8 u f98 pxu8q
on the group
u G9 pF98 q
¹
v
P8
¹ v
f9v pxv q
Su
P8 Su
G9 pF9v q
9
obtained by letting each fv be a stable sum of pseudocoefficients corresponding to a Langlands parameter φv P Φ2 pG9 v q in general position. In other words, ¸ f9v fπv ,
P
πv Πφv
where fπv is a pseudocoefficient of the square integrable representation πv of G9 pFv q. The existence of pseudocoefficients follows from [BDK] in the p-adic case, and [CD] in the archimedean case. Let Z9 8,u be the intersection of K9 8,u with (the diagonal image in G9 pA9 8,u q of) the center of G9 pF9 q. This actually equals the center of G9 pF9 q, a group of p SOp2, Cq. In case G p does equal order 1 or 2, except in the abelian case G 9 9 to SOp2, Cq, the existence of discrete series implies that GpFv q is compact u or equals u, and therefore that Z9 8,u is a finite if v either belongs to S8 u f9 on G 9 pF 9 u q GpF q be constant group. We require that the function f98 u 8 on (the diagonal image of) Z9 8,u . We are also requiring that φv be in genu . Finally, it will be convenient to fix a place eral position for each v P S8 u v P S8 , and then require that regularity of the infinitesimal character of φv dominate that of the other parameters φv1 . We can meet these conditions by first choosing the parameters φv1 and the corresponding function u,v f98
¹
u,v v 1 S8
P
f9v1
so that the product f8 f9u is constant on Z9 8,u . At the place v, we then define φv in terms of its infinitesimal character by setting 9u,v
µ φv
nµv
6.2. CONSTRUCTION OF GLOBAL REPRESENTATIONS π9
311
where µv is fixed, and n is a large integer such that µφv represents an element in Φ2 pG9 v q whose central character on Z9 8,u is trivial. The resulting function
f8u,v fv fu
u 9 f98 fu
9
9
9
then has the required properties. We shall apply the trace formula to the function u f9pxq f98 pxu8q f9upxuq f98,upx8,uq,
9 q, x P G9 pA
in HpG9 q. Since f9 is cuspidal at more than one place, we can use the simple 9 version [A5, Theorem 7.1] of the invariant trace formula for G. Since f9 is both cuspidal and stable at some archimedean place, the invariant orbital integrals on the geometric side vanish on conjugacy classes that are not semisimple. (See [A6, Theorem 5.1].) The geometric side of the trace formula therefore reduces to a finite sum (6.2.2)
¸
9 q f9 pγ q vol G9 γ pF9 qzG9 γ pA G9
γ
of invariant orbital integrals f9G9 pγ q
» 9
p qz p q 9
9
9
Gγ A G A
f9px1 γxq dx,
taken over the semisimple conjugacy classes γ in G9 pF9 q that are R-elliptic at each place in S8 . u chosen above. The infinitesimal character Consider the place v P S8 of φv , or more precisely, of any of the representations in the packet of φv that define f9v as a sum of pseudo-coefficients, depends linearly on n. It follows from fundamental results of Harish-Chandra [Ha1, Lemma 23], [Ha2, Lemmas 17.4 and 17.5] that (6.2.3)
f9G pγ q cpγ q nqpGγ q , 9
where cpγ q is independent of n, and q pG9 γ q
1 2
dimpG9 γ q rankpG9 γ q .
The constant cpγ q vanishes for all but a finite set of γ, but is nonzero at γ 1. The exponent q pG9 γ q is maximal when the centralizer G9 γ of γ in G9 9 which is to say that γ is central. In this case, equals G, f9G9 pγ q f9G9 p1q, u . The dominant term in (6.2.2) therefore corresponds by our condition on f98 to γ 1. It follows that (6.2.2) is nonzero whenever n is sufficiently large. The geometric expansion (6.2.2) equals the spectral side of the trace formula. The parabolic terms in [A5, Theorem 7.11(a)], namely those corresponding to proper Levi subgroups M in the general expansion (3.1.1),
312
6. THE LOCAL CLASSIFICATION
vanish on functions that are pseudocoefficients of discrete series. (See [A6, p. 268].) The spectral side therefore reduces simply to the trace G tr Rdisc pf9q 9
¸
mdisc pπ9 q f9G pπ9 q
π9
of f9 (operating by right convolution) on the automorphic discrete spectrum 9 of G. It follows that for n sufficiently large, there is an irreducible representation π9 of G9 pAq with positive multiplicity mpπ9 q, and with f9G9 pπ9 q tr π9 pf9q
0.
It remains to establish that the automorphic representation π9
â v
π9 v
π8u b πu b π8,u 9
9
9
has the properties (i) – (iii). At first glance, these properties appear to be consequences of the definition of f9, and the fact that (6.2.4)
u tr f98 pπ9 8u q tr f9upπ9 uq tr f98,upπ9 8,uq
tr πpf q 0. 9
9
For example, (ii) follows immediately from the definition of f98,u as the characteristic function of K9 8,u . However, something more is needed for (i) and (iii). The problem is that the definition of a pseudocoefficient (6.2.1) (in the case of (i)) does not rule out the possibility that the localization π9 u could be a nontempered representation whose character on the elliptic set matches that of π. We can treat (iii) by an argument based on the infinitesimal character. 9 As a constituent of the automorphic discrete spectrum of G, π9 is unitary, u and so therefore are its localizations π9 v . If v belongs to S8 , π9 v has the same infinitesimal character as φv , since the trace of π9 v pf9v q is nonzero when f9v is the stable sum of pseudocoefficients for φv above. It follows from basic estimates for unitary representations, and the fact that φv is in general position, that π9 v must be tempered. Appealing again to the definition of f9v , we conclude that π9 v itself belongs to the L-packet Πφv , for the Langlands parameter φv P Φ2 pG9 v q in general position we have chosen. This is the condition (iii). Our proof of (iii) does not apply to (i). For the infinitesimal character of π9 u need not be in general position if F R, and is also too weak to help us if F is p-adic. We shall turn instead to a separate global argument, which would also give a different proof of (iii). While the argument is somewhat more sophisticated, it makes use only of the results of §3.4, and therefore does not rely on any induction hypothesis. Corollary 3.4.3 tells us that π9 occurs in the subspace L2disc,ψ9 G9 pF9 qzG9 pA9 q
L2disc GpF qzGpAq 9
9
9
9
6.2. CONSTRUCTION OF GLOBAL REPRESENTATIONS π9
313
of the discrete spectrum attached to a (uniquely determined) global parameter ψ9
P Ψr pN q. Given ψ, we then use the formula (3.3.14) to write 9
9
N Idisc, pfrq ψ9
¸
9
9 r ι N, G Spdisc, fr , ψ9
p
r ell N G E
P p q
q
p q
r pN q. fr P H 9
9
9
The quadratic character ηψ9 attached to ψ9 satisfies
ηπ ηG ,
ηψ9
9
9
according to the definitions. It follows that the stable linear form Sdisc, ψ9
attached to any G P Er ell pN q vanishes unless ηG equals ηG9 . This yields the usual simplification in the right hand side of the last formula. To reduce 9
9 it further, we have to specialize the function fr in the global Hecke module
r pN q. H We will take decomposable functions 9
fr 9
(6.2.5)
¹
frv 9
fru8 fru fr8,u
f9v
f8u fu f 8,u
v
9
9
9
and f9
(6.2.6)
¹ v
9
9
9
r pN q and H r pG9 q respectively so that for any v, fr and f9v have the same in H v 9
9
images in SrpG9 v q. We impose no further conditions on the components fru and f9u at u. However, we will constrain the other components of f9 as above. We take the component f98,u off9 away from S8 puq to be the characteristic 9 8, q . If v belongs to S u , we take f9 to be the r G9 pA function of K9 8,u in H v 8 r pG9 v q of the stable sum of pseudoimage in the symmetric Hecke algebra H 9
coefficients for φv . In addition, we choose frv so that its image frv in SrpGv q 9
9
vanishes for any Gv P Er ell,v pN q distinct from G9 v . The existence of frv in this case is a consequence of Proposition 2.1.1, as we noted at a similar juncture in the proof of Lemma 5.4.2. 9
9
P S8u implies that the global transfer p p G. fr of fr vanishes for any G P Er ell pN q with G Combined with the The condition on a function frv at v 9
9
9
9
9
property of ηG above, it tells us that
pfr q 0, Spdisc, ψ9 9
314
6. THE LOCAL CLASSIFICATION
for any G reduces to
P ErellpN q distinct from G. The formula given by (3.3.14) therefore 9
9
N G Idisc, pfrq rιpN, G9 q Spdisc, pfrGq ψ9 ψ9 9
9
9
9
G rιpN, Gq Sdisc, pf q. ψ 9
9
9
9
9 We shall combine this with the expansions (4.1.1) and (4.1.2) for G. 1 Suppose G P Eell pG9 q is a global elliptic endoscopic datum for G9 that is 9 p 1 is then composite, the localization distinct from G. Since the dual group G 1 1 u , the transfer f91 of Gv of G at any v is distinct from G9 v . If v belongs to S8 v r pG9 v q of the stable f9v to SrpG1v q therefore vanishes, since f9v is the image in H sum of pseudocoefficients for the square integrable parameter φv P Φ2 pG9 v q. It follows that 1 f91 f9G 0.
The proper terms in the endoscopic expansion (4.1.2) of Idisc,ψ9 pf9q therefore vanish. Similarly, the proper terms in the spectral expansion (4.1.1) of Idisc,ψ9 pf9q also vanish, as we noted earlier in the proof. It follows that tr Rdisc,ψ9 pf9q
G Idisc,ψ pf q Sdisc, pf q . ψ 9
9
9
9
9
Combining this with the reduction above, we conclude that (6.2.7)
tr Rdisc,ψ9 pf9q
N pfrq. rιpN, Gq1 Idisc, ψ 9
9
9
We can regard the two sides of (6.2.7) as linear forms in the corresponding pair of variable functions f9u and fru . From (6.2.4) and the fact that π9 is a constituent of Rdisc,ψ9 , we know that if f9u is a pseudocoefficient of π, the left hand side of (6.2.7) is nonzero. Indeed, it is a nontrivial sum of irre9 q, taken at the positive definite function ducible characters on the group G9 pA f9. The same is therefore true of the right hand side. It then follows from the u of f9 in terms of φ , together with the relation of definition at any v P S8 v v 9
the corresponding function frv with ψ9 v , that the infinitesimal characters of ψ9 v and φv correspond. At this point we are taking for granted a property of archimedean infinitesimal characters, namely that their twisted transfer is compatible with the Kottwitz-Shelstad transfer of functions. The property can be regarded as a part of the work [Me], [S8] in progress. It could also be treated as a more elementary but nonetheless interesting exercise in twisted transfer factors, following its untwisted analogue [A13, (2.6)], for example. The infinitesimal character of ψ9 v is by definition that of the Langlands parameter φψ9 v . It cannot be in general position if ψ9 v is not generic. On the other hand, we have agreed that it equals the infinitesimal character of 9
r v pN q. Since the original parameter φv , regarded as a generic element in Ψ 9 φv was chosen to be in general position, ψv must therefore be generic. The 9
6.2. CONSTRUCTION OF GLOBAL REPRESENTATIONS π9
315
nongeneric component of any global parameter is of course unaffected by localization. It follows that the global parameter φ9 ψ9 is generic. Its localization φ9 u ψ9 u is consequently also generic. We have shown that as a self-dual, local parameter for GLpN q, φ9 u ψ9 u is generic. We will need to know further that it is tempered. We have therefore to show that as an N -dimensional representation of the local Langlands group LF LF9v , φ9 u is unitary. Consider the expansion (4.1.1) for the term N N Idisc, pfrq Idisc, pfrq φ9 ψ9 9
9
on the right hand side of (6.2.7). Appealing to the general discussion of §4.3, or simply to the local definition (2.2.1) and the properties in §2.5 of local intertwining operators for GLpN q, we see that it is a scalar multiple of
9 9 frN pφ9 q. As a function of the variable component fru hand side of (6.2.7) therefore takes the form
cu fru,N pφ9 u q, 9
(6.2.8)
cu
P HrpN q of fr, the right 9
P C.
Since the left hand side of (6.2.7) does not vanish if the corresponding funcr pGq is a pseudocoefficient of π, the coefficient cu is nonzero. In tion f9u P H addition, there is a (twisted) cuspidal function fru 9
P
r cusp pN q such that H
r cusp pN q is the subspace of functions fru,N pφ9 u q is nonzero. (Recall that H r pN q such that fr belongs to Srcusp pG q for every G P Erell pN q, or fr P H r equivalently, such that frM 0 for any proper Levi subset M of GpN q. 9
r cusp pN q follows from Proposition That we can take fru above to be in H 9 r ell pN q of Φ r pN q. Indeed, the 2.1.1.) We conclude that φu lies in the subset Φ r pN q attached to a local parameter in the complement of linear form on H r r cusp pN q. We have been making free use here of the Φell pN q vanishes on H local Langlands correspondence for GLpN q. In particular, the original interr ell pN q in §1.2 tells us that as an N -dimensional representation pretation of Φ of LF , φ9 u decomposes into irreducible constituents that are self-dual and multiplicity free. It follows that φ9 u is unitary. In other words, the linear 9
r pN q is tempered. form fru,N pφ9 u q in fru P H We have shown that the right hand side (6.2.8) of (6.2.7) is tempered in 9
9
the variable component fr of fru . We want to conclude that the same is then true of the left hand side of (6.2.7). At this point, it would be convenient to be able to say that the local twisted transfer mapping 9
fr fru 9
ÝÑ
frG
r pN q onto SrpGq is represented by a correspondence that extends to from H the underlying Schwartz spaces. A proof of this property would no doubt be accessible. It is perhaps already known, but I do not have a reference. We
316
6. THE LOCAL CLASSIFICATION
need only the special case represented by the twisted tempered character in (6.2.8), and this is not hard to obtain directly. Let me give a very brief sketch of the argument. The twisted tempered character in (6.2.8) has exponents. They are attached to the chamber a r of any given cuspidal parabolic subset Pr of P
r pN q, and are linear forms λ on the complex vector space a r . If we G P ,C
evaluate the character at functions in the subspace Crreg pN q of Schwartz r pN, F q with strongly regular support, we find that the Pr functions on G exponents satisfy the inequality Re λpH q
¤ 0,
H
P aPr .
On the other hand, it is easy to describe the image of Crreg pN q under the transfer mapping fr Ñ frG . Given Corollary 2.1.2 and the explicit nature of r pN q, G , we see that the image of Crreg pN q is the twisted transfer factors for G the space of functions of δ whose restriction to any maximal torus T pF q GpF q lies in the subspace of CrN -reg pT q of Schwartz functions of strongly r pN q-regular support. The variable δ here represents an Out r N pGq-orbit of G r pN q-regular, stable conjugacy classes in GpF q. It is then not hard strongly G to show that the P -exponents implicit in the left hand side of (6.2.7) satisfy the analogue of the inequality above. This in turn implies that the left hand side of (6.2.7) does indeed represent a tempered linear form in the variable component f f9u of f9. Since the global representation π9 is a constituent of the representation Rdisc,φ9 on the left hand side of (6.2.7), its local u-component π9 u must then be tempered. It remains only to apply (6.2.4) one more time. If f9u is specialized to a pseudocoefficient of the original representation π of GpF q, we see that f9u,G pπ9 u q tr π9 u pf9u q
0,
for the irreducible representation π9 u of GpF q we now know is tempered. It then follows from the definition of a pseudocoefficient that π9 u π, as required. We have therefore established the remaining condition (i) of Lemma 6.2.2. Our proof of the lemma is at last complete. 9 π 9 q are as in the stateCorollary 6.2.3. Suppose that pF, G, π q and pF9 , G,
ment of the lemma, and that φ9 P obtained from π9 in the proof of the r N pG9 v q-orbit of the) of F9 , the (Out 9 Πφ9 v . In particular, the archimedean
r pN q is the generic global parameter Φ lemma. Then for any valuation v u localization π9 v lies in the local packet parameters fixed at the beginning of the 9
9 construction represent localizations of φ.
Proof. We choose a pair pfr, f9q of functions (6.2.5) and (6.2.6) as in the proof of lemma, but with two minor differences. We take f9u to be the fixed r pG9 u q, and we take pseudocoefficient fπ rather than a variable function in H 9
6.2. CONSTRUCTION OF GLOBAL REPRESENTATIONS π9
317
f98,u to be a variable function in the K9 8,u -spherical Hecke algebra, rather than the unit. With the two functions being otherwise exactly as before, the earlier conclusions remain valid. In particular, the identity (6.2.7) holds. 9 Moreover, the right hand side of (6.2.7) is a nonzero multiple of frN pφ9 q. It follows that
frN pφ9 q cpφ9 q tr Rdisc,φ9 pf9q , 9
(6.2.9)
for a nonzero constant cpφ9 q. The right hand side of (6.2.9) is nonzero as a linear form in f98,u , since f9G9 pπ9 q is nonzero if f98,u is a unit. Since f98,u is allowed to vary, we see r 9 . If v from the left hand side that for any v R S8 puq, π9 v does belong to Π φv u belongs to S8 , we combine (6.2.9) with the twisted archimedean character
relations for GLpN q and the fact that frv transfers to f9vG . It follows that π9 v r 9 in this case as well. belongs to Π φv 9
9
Recall that we have been carrying an unproven lemma from §2.3. It is Lemma 2.3.2, which we used to bring global methods to bear on the normalization of local intertwining operators. We can now see that Lemma 2.3.2 is a special case of the lemma we have just established (with the factor G of §2.3 in place of the group G here). Its two assertions (i) and (ii) are just the corresponding assertions of Lemma 6.2.2. Lemma 2.3.2 has therefore now been proved. Returning to Lemma 6.2.2 itself, we note from the early part of the 9 F 9 q depends only on the given local construction that the global pair pG, pair pG, F q. In particular, it is independent of the local representation π. Bearing this in mind, we formulate an important corollary that is based on a separate hypothesis. Corollary 6.2.4. Suppose that F , G, F9 and G9 are as in Lemma 6.2.2, and that the local theorems (interpreted as in §6.1) are valid for generic r pN1 q over F with N1 ¤ N . Then for any simple local parameters φ1 P Φ r sim pG9 q with r sim pGq, there is a simple global parameter φ9 P Φ parameter φ P Φ the following properties. (i) pF9u , G9 u , φ9 u q pF, G, φq. (ii) For any valuation v R S8 puq, the localization φ9 v is spherical. u , φ9 is a local parameter in general position in (iii) For any v P S8 v r 2 pG9 v q. Φ
P Πr 2pGq with r tr π pfu q frG u pφq f u,N pφq, r pGq and fr P H r pN q with the same image in SrpGq. for any functions fu P H u Proof. Given φ, we obtain a representation π 9
9
9
9
9
This follows from our assumption that the local theorems hold for G. We can then construct the automorphic representation π9 of G9 pA9 q of the lemma.
318
6. THE LOCAL CLASSIFICATION
r pN q In proving Lemma 6.2.2, we obtained a global parameter φ9 ψ9 in Ψ from Corollary 3.4.3, which we then showed was generic. We will see in a moment that φ9 is simple. Since π9 is a constituent of the representation G9 9 this will tell us that G 9 satisfies the condition of the Rdisc, attached to φ, φ9 global datum Gφ9 of Theorem 1.4.1. The condition (i) applies to the valuation v u. In the proof of the theorem, we showed that 9
fru,N pφ9 u q 0, 9
if fru and f9u have the same image in SrpGq, and f9u is a pseudocoefficient of π. The local theorems we are assuming tell us that the generic parameter φ9 u is uniquely determined by this condition. Since the condition also holds for φ, we conclude that φ9 u φ. This completes the proof of (i). It also tells us that the local parameter φ9 u is simple. Therefore the global parameter φ9 is also simple, as an element in 9
r pN q. Φ We have shown that G9 satisfies the condition of the global datum Gφ9 9
imposed on the simple datum φ9 by Theorem 1.4.1. If G P Er ell pN q is another global datum that satisfies this condition, one can show that Gu equals G9 u . 9
r pN q with fr pφ9 u q 0, taking One applies (3.3.14) to any function fr P H u,N account of Propositions 2.1.1 and 3.4.1. If we grant this property, we can then combine the resulting identity 9
p G
9
9
Gpu Gp Gp
of dual groups with the identity ηG
9
ηφ ηG 9
9
9 of quadratic characters to deduce that G G. This is the uniqueness r sim pG9 q, as claimed. assertion of Theorem 1.4.1. It implies that φ9 P Φ The conditions (ii) and (iii) apply to valuations v of F9 that are distinct from u. They follow from Corollary 6.2.3.
Remarks. 1. We have not yet made any induction assumption on the global r sim pG9 q is based on the original theorems. Our claim that φ9 belongs to Φ definition provided by conditions of Theorem 1.4.1. We did not fully justify the uniqueness assertion of this theorem in the proof above. There is really no call to do so. For we will impose a global induction hypothesis in §6.4, in r of the kind considered the context of a generic family of global parameters F in §5.4, which will include whatever is needed. We will construct the family 9
r in the next section. In the process, we will apply Corollary 6.2.4 as F 9
p
stated, knowing that φ9 does determine G9 by the given conditions G9
Gp
6.3. CONSTRUCTION OF GLOBAL PARAMETERS φ9
319
and ηG9 ηφ9 , and that this will give the uniqueness assertion of Theorem 1.4.1 once the global induction hypothesis is imposed. 2. In the proof of Lemma 6.2.2, we chose a variable archimedean parameter φv in order to work with the simple formula (6.2.3). A reader familar with the identities of Harish-Chandra on which (6.2.3) is based will observe that our condition of general position is sufficient. Suppose that φv
P Φr 2pGv q, 9
v
P S8u ,
are archimedean parameters in general position, and that the product of p SOp2, Cq, or simply their corresponding central characters (on Z 8,u if G Z pF9 q otherwise) with that of φ is trivial. Then we can choose the global u. parameter φ9 in Corollary 6.2.4 so that φ9 v φv for each v in S8 The point here is that the formula (6.2.3) is just a special case of a more general formula, in which the variable n is replaced by a tensor product over v of vectors composed of the half integers (6.1.4) attached to the infinitesimal characters of the parameters φv . Our condition on the general position of each φv means that these half integers are all larger than some preassigned constant. This forces the terms with γ central to dominate the others, and the sum (6.2.2) to be nonzero. 3. There are other variants of Lemma 6.2.2, which could be proved in the same way. For example, we could construct the automorphic representation π9 so that π9 v equals a given square-integrable representation for every v in some finite set V of p-adic places, and so that π9 v is as in (ii) for each v not in S8 puq Y V . Similarly, we could arrange for the global parameter φ9 of r 2 pG9 v q at each v P V , Corollary 6.2.4 to be equal to a prescribed element in Φ and to be as in (ii) at each v outside S8 puq Y V . 6.3. Construction of global parameters φ9 Our goal for Chapter 6 will be to establish the local theorems for Langlands parameters in the general set r Φ
º
r pN q , Φ
N
taken over the fixed local field F . To this end, we now take on the natural induction hypothesis that will carry us through the chapter. We fix N , and assume inductively that the local theorems (interpreted as in §6.1) all hold r with degpφq N . for generic parameters φ P Φ In this section, F is assumed to be real or p-adic, as in §6.2. We fix an elliptic endoscopic datum G P Erell pN q over F , which in general need not be in the subset Ersim pN q of simple data. We then fix a local parameter φ `1 φ1
(6.3.1)
r pGq, with simple components in Φ
φi
P Φr simpGiq,
` ` `r φr Gi
P ErsimpNiq.
320
6. THE LOCAL CLASSIFICATION
(Our requirement here that the simple components be self-dual means that φ lies in the subset r disc pGq tφ P Φ r pGq : |Z pS φ q| 8u Φ
r pGq. We observe that this is the local, generic analogue of correspondof Φ ing global set in the chain (4.1.13).) We assume that φ is not simple, or equivalently, that each Ni is less than N . The induction hypothesis above then tells us that the data Gi P Ersim pNi q are determined by the uniqueness assertion of Theorem 6.1.1. For clarity we also assume that G is in fact r ell pN q, or in other words, if `i ¡ 1 for some simple if φ does not belong to Φ r r pN q, which is implicit in i. This insures that ΦpGq injects as a subset of Φ r pG q . the assertion that the compound parameter φ in (6.3.1) lies in Φ According to the induction hypothesis, we can apply Corollary 6.2.4 to the pairs pGi , φi q. If F9 is the totally real field of Lemma 6.2.1, we will obtain global pairs
pG i , φ i q , 9
9
G9 i
P ErsimpNiq, 9
φ9 i
P Φr simpGiq, 9
that satisfy the conditions (i)–(iii) of the lemma. These will determine a
9 global endoscopic datum G9 P Er ell pN q, which will be simple if some `i such that the resulting global parameter
φ9 `1 φ9 1
(6.3.2)
`
¡ 1,
`r φ9 r
`
9 for r pG9 q. Our aim is to establish some finer properties of φ, belongs to Φ 9 suitable choices of its simple components φi . The case that each `i 1 in (6.3.1) is at the heart of things. However, we will also have to be concerned with higher multiplicities in dealing with the local intertwining relation. We fix a Levi subgroup M of G such that the r 2 pM, φq is nonempty. For a given choice of the global datum G, 9 we will set Φ 9 9 r 9 also choose a global Levi subgroup M such that Φ2 pM , φq is nonempty. The general notation, first introduced for the global parameter ψ of Lemma 5.2.3, is relevant here. For example, if
N
¸
Ni
`i odd
and φ
à
φi ,
`i odd
we have
GLpN1q`1 GLpNr q`1 G, where `1i r`i {2s, and G is the unique element in Erell pN q such that φ r 2 pG q. We can then identify the local parameter lies in in Φ 1 `1 (6.3.3) φM φ1 φ`r φ M
r
1
1
r
6.3. CONSTRUCTION OF GLOBAL PARAMETERS φ9
321
r 2 pM, φq. Similarly, we can identify the corresponding with an element in Φ global parameter 1 `1 (6.3.4) φ9 M φ9 11 φ9 `rr φ9
9 q. r 2 pM 9 ,φ with an element in Φ The properties we will use are summarized in the multiple conditions of the statement of the following proposition.
Proposition 6.3.1. Given the local objects G, φ, M and φM over F as in 9 M 9 φ, 9 and (6.3.1) and (6.3.3), we can choose corresponding global objects G, 9 9 φM over F as in (6.3.2) and (6.3.4) such that the following conditions are satisfied. (i) There is a valuation u of F9 such that
pFu, Gu, φu, Mu, φM,uq pF, G, φ, M, φM q, 9
9
9
9
9
and such that the canonical maps Sφ9
9 M
and Sφ9
ÝÑ
SφM
ÝÑ
Sφ
are isomorphisms. (ii) For any valuation v outside the set S8 puq, the local Langlands parameter φ9 v `1 φ9 1,v ` ` `r φ9 r,v is a direct sum of quasicharacters of F9v , while the corresponding decomposition of the subparameter φ9 1,v
` ` φr,v 9
contains at most one ramified quasicharacter. u . Then for any v P V , the parameters φ9 (iii)(a) Set V S8 i,v lie in r 9 Φ2 pGi,v q, and are in relative general position. Moreover, the canonical mapping Πφ9 M,V ÝÑ Spφ9 M SpφM , obtained from the combined places v P V , is surjective. (iii)(b) Suppose that each `i equals 1. Then there is a v r G for some G property that if φv lies in Φ v v p equals G. p G 9
p q
P
p q
PV
with the
9 Er sim,v N , the dual group
v
(iii)(c) Suppose that some `i is greater than 1. Then there is a v such that the kernel of the composition of mappings Sφ9
ÝÑ
Sφ9 v
ÝÑ
Rφ9 v
contains no element whose image in the global R-group Rφ9 to Rφ,reg . 9
PV
RφpGq belongs 9
9
322
6. THE LOCAL CLASSIFICATION
Remarks. 1. The statement of the proposition does contain a rather large number of conditions. I have tried to organize them in a way that makes sense. For example, they represent refinements of the conditions (i)–(iii) of Corollary 6.2.4. In addition, the three parts (a)–(c) of (iii) are parallel to the three conditions (5.4.1)(a)–(5.4.1)(c) of Assumption 5.4.1. 2. The first assertion of (iii)(a) means that the integers (6.3.5)
r º
pµφ q tµi,v,k : 1 ¤ k ¤ pi rNi{2s, 1 ¤ i ¤ ru 9
9
i,v
i 1
are all large, and that their differences are large in absolute value. The second assertion of (iii)(a) is that for any character ξ on the abelian group Sφ9 M , there is a representation π9 V
â
P
π9 v ,
π9 v
P Πr φ
v V
9
M,v
,
r9 in the product Π φM,V of local archimedean packets such that
ξ ps q
¹
P
xs, πv y,
s P Sφ9 M .
9
v V
3. The conditions (b) and (c) of (iii) are perhaps more palatable here than in the original forms (5.4.1)(b) and (5.4.1)(c), since we have now seen their application to the lemmas of §5.4. Proof. We choose the totally real field F9 according to Lemma 6.2.1. The only requirements are that F9u F for a fixed place u, and that F9 have sufficiently many real places. We will then apply Corollary 6.2.4 inductively to each of the local pairs pGi , φi q, as agreed above. This will yield global
r sim pG9 i q. endoscopic data G9 i P Er sim pNi q and generic global parameters φ9 i P Φ 9 Once chosen, the global pairs pG9 i , φi q determine a global endoscopic datum 9
p together with a generic global parameter φ9 P Φ r pG9 q G9 P Er ell pN q with G9 G, 9 9 as in (6.3.2). They also determine a global Levi subgroup M of G with 9
p
9 q as in x, together with a generic global parameter φ9 M P Φ r pM 9 ,φ M9 M (6.3.4). The conditions G9 u G, φ9 u φ, M9 u M and φ9 M,u φM in (i) follow directly from the corresponding conditions of Corollary 6.2.4 for the pairs pG9 i, φ9 iq. The isomorphisms of centralizers in (i) are also consequences of the construction. They follow easily from the mappings
x
LG
i
9 > LG i
∨
∨
Γ
Γ9 and the definitions of the local and global centralizers. >
6.3. CONSTRUCTION OF GLOBAL PARAMETERS φ9
323
Before we consider the remaining conditions (ii) and (iii)(a)–(iii)(c), we 9 entails a number of recall that the construction of each φ9 i , and hence of φ, choices. These include a choice of quadratic id`ele class character η9 i ηG9 i for
p i SOpNi , Cq. If Ni is even, η9 i determines the outer twisting each i with G of G9 i . In this case, the local values η9 i,v of η9 i are predetermined at each v in the set S8 puq. If Ni is odd, η9 i determines the L-embedding of L Gi into GLpNi , Cq. Its local value is fixed at u, but is otherwise arbitrary. We impose two further local conditions on the family tη9 i u. The first is a constraint on the characters η9 i with Ni odd, at the supplementary archimedean places. We fix a set of distinct places
Vo in the set V
tvi :
Ni oddu
S8u . We then require that for any v P V , #
p 1 ppεεv qqp , , v i
(6.3.6)
η9 i,v
i
if v if v
vi , vi ,
where εv is the nontrivial quadratic character on F9v R , and pi 12 pNi 1q. This of course requires that there be at least as many places in V as there are indices i with Ni odd, one of our reasons for insisting that F9 have sufficiently many real places. The second constraint applies to all characters η9 i , at the places v R S8 puq. We require that for any such v, there be at most one i such that η9 i,v ramifies, and for good measure, that the ramification be tame. This is certainly possible. It is well known that one can choose a quadratic extension of any number field with arbitrarily prescribed localizations at finitely many places. If we choose the characters η9 i successively with increasing i, we can insist that η9 i,v be unramified at any place v R S8 puq which divides 2, or at which η9 j,v ramifies for some j i. We can now establish (ii). Suppose that v lies in the complement of S8 puq. Corollary 6.2.4 tells us that for any i, the parameter φ9 i,v in the
r pG9 i,v q of Φ r v pNi q is spherical. According to the remarks in §6.1, this subset Φ means that it is a direct sum of pNi 1q unramified quasicharacters of F9v , together with another quasicharacter that is unramified if and only if η9 i,v is unramified. In particular, the sum φ9 v is a direct sum of quasicharacters, as claimed. Our condition above on η9 i,v implies further that there is at most one i such that φ9 i,v contains a ramified quasicharacter. This gives the second claim of (ii). It remains to establish the property (iii), with its three parts (a)–(c), for u . We first recall that for any v P V , we have a the valuations v in V S8 decomposition 9
φ9 i,v
pi à
k 0
φ9 i,v,k
324
6. THE LOCAL CLASSIFICATION
for any index i, where
φ9 i,v,k : 1 ¤ k
(
¤ pi rNi{2s
are irreducible, self-dual, two-dimensional representations of WF9v , and φ9 i,v,0 is a one-dimensional quadratic character that occurs only when Ni is odd. In case Ni is odd, the two-dimensional representations φ9 i,v,k are orthogonal, and hence have a nontrivial determinant. It follows that pi ¹
detpφi,v q
φ9 i,v,0
9
k 1
ηi,v pεv qp
detpφ9 i,v,k q1 #
i
9
εv , 1,
if v vi , otherwise,
where vi P V is the valuation fixed above. For any i, the two dimensional representations ( φ9 i,v,k : 1 ¤ k ¤ pi , 1 ¤ i ¤ r are all mutually inequivalent, as we recall from their origins in the proof of Lemma 6.2.2. Apart from a harmless condition on the central character, the only constraint was a regularity condition on the infinitesimal characters of these parameters. For a given v, we choose the parameters successively with increasing i. We can then insist that the infinitesimal character of φ9 i,v be highly regular, in a sense that dominates the condition already imposed on the parameters φ9 j,v with j i. The first assertion of (iii)(a) is then valid. The second assertion of (iii)(a) requires more discussion. We are certainly free to replace M9 by the subgroup G9 , since the general linear factors of M9 are connected. In fact, we may as well assume simply that M G G, to save notation. In other words, we shall assume that the group Sφ Sφ9 is finite. We have then to show that the global objects can be chosen so that the mapping (6.3.7)
Πφ9 V
ÝÑ
Spφ9 ,
Sφ9
Sφ{Z pGpqΓ, 9
9
is surjective. Given Lemma 6.1.2, the main step is to verify that the dual mapping Sφ9
(6.3.8)
pSφ{Z pGpqΓ ÝÑ 9
9
Sφ9 V
¹
P
v V
Sφ9 v
is injective. The center Z pG9 qΓ p
Z pGpqΓ , 9
v
v
P V,
plays no role here, so it will be enough to verify that the mapping Sφ9
ÝÑ
π0 pSφ9 V q
¹
P
π0 pSφ9 v q
v V
is injective. We have therefore to show that if s component Sφ09 for each v P V , then s 1. v
P Sφ lies in the identity 9
6.3. CONSTRUCTION OF GLOBAL PARAMETERS φ9
325
We are regarding φ9 v as an N -dimensional representation of WF9v WR . Since its two dimensional constituents are distinct, they each contribute a discrete factor Op1, Cq to the centralizer Sφ9 v . The one-dimensional constituents are parametrized by the subset I o of indices i with Ni odd. They of course need not be distinct. In fact, for any v P V , they are all trivial unless v vi for some i P I o , in which case the representation corresponding to i is the sign character εv of F9v R . It is these repeated factors that define the identity component of Sφ9 v . In particular, in the case v vi , we see that Sφ09 SOp|I o | 1, Cq. v
The intersection of Sφ9 with Sφ09 is given by the factors of Sφ9 attached to
the subset I 1 of indices i P I o with Ni 1, since the factors corresponding to other indices i P I o map to nontrivial components of Sφ9 v attached to two-dimensional representations of WF9v . If v vi1 , for some i1 P I 1 , the intersection of Sφ9 with Sφ09 takes the form v
v
¹
tiPI 1 : ii1 u
Op1, Cq
φ9 v
Op1, Cqk ,
k
|I 1| 2,
where the subscript on the left follows the notation defined prior to (1.4.8), and the exponent on the right is understood to be 0 if |I 1 | ¤ 2. The intersection of these groups, as i1 ranges over I 1 , is trivial. In particular, any element in the intersection of Sφ9 with all of the groups Sφ09 is trivial. v We have shown that the mapping (6.3.8) is injective. The dual mapping of abelian character groups Spφ9 V
ÝÑ
Spφ9
is therefore surjective. Recall that we have an injection Πφ9 V
¹
P
pΠφ q Ñ 9
v
ã
v V
Spφ9 V
¹
P
v V
pSpφ q. 9
v
It follows from Lemma 6.1.2, applied to each field F9v , that the image of the set Πφ9 V generates the finite group Spφ9 V . Its image under the mapping (6.3.7) therefore generates Spφ9 . To insure that the image of (6.3.7) actually equals Spφ9 , we simply enlarge F9 . That is, we carry out the entire argument anew,
with a totally real extension F9 1 in place of F9 , and with φ9 1v1 φ9 v for each place v 1 of F9 1 over v. (In allowing different completions φ9 1v1 to be equal, we are actually applying the variant of Corollary 6.2.4 described in Remark 2 following its proof.) If F9 1 is large enough, the global parameter φ9 1 obtained as above gives rise to a composition Πφ9 1
V
1
ÝÑ
SpφV
ÝÑ
Spφ9
Spφ1 9
326
6. THE LOCAL CLASSIFICATION
of surjective mappings. Denoting F9 1 and φ9 1 by the original symbols F9 and 9 we obtain global data for which the mapping (6.3.7) is indeed surjective. φ, This completes the proof of the second assertion of (iii)(a). Consider next the condition (iii)(b). Then each `i equals 1, and the group Sφ is finite. This is the case in which the datum G P Erell pN q need not be simple. If φ has any one-dimensional constituents, we take v v1 vi1 , for any fixed index i1 P I 1 with Ni1 1. Otherwise, we take v to be r pG q, for some simple any fixed valuation in V . Suppose that φ9 v lies in Φ v p equals SOpN, Cq, the two-dimensional datum Gv P Esim pN q over F9v . If G v constituents of φ9 v are all orthogonal, since they are mutually inequivalent. By our local induction hypothesis, the localization assertion of Theorem 1.4.2 is valid for each G9 i . (See also Remark 1 after the proof of Corollary p i G9 i equals SOpNi , Cq, for any i 6.2.4.) This implies that the group G with Ni ¡ 1. The same being trivially true if Ni 1, it follows that p
p SOpN, Cq G p , G v
p SppN, Cq. Then all of as required. The other possibility is that G v the two-dimensional constituents of φ9 v are symplectic, which implies that p i equals SppNi , Cq for any i with Ni ¡ 1. If there were any remaining G one-dimensional constituents of φ9 v , one of them, namely
φ9 i1 ,v,0
ηi ,v , 9 1
would be equal to εv , while all of the others would be trivial. Under such circumstances, the image of φ9 v cannot be contained in SppN, Cq. Therefore, p i equals SppNi , Cq for there are no one-dimensional representations, and G all i. It follows that p p , G9 SppN, Cq G
and in particular, that G9 P Er sim pN q is simple. We have established (iii)(b). Consider finally the condition (iii)(c). Then some `i is greater than 1, and the group Sφ is infinite. In this case, G is required to be simple. We choose v as above, namely to be vi1 if there is an index i i1 with Ni 1, and to be any fixed valuation in V otherwise. Let x be an element in Sφ9 whose image in the global R-group Rφ9 lies in Rφ,reg . We have to show that 9 its image in the local R-group Rφ9 v is nontrivial. The fact that Rφ,reg is 9 nonempty implies that Sφ9 is of the form 1 Op2, Cqq Op1, Cqr . 9
In particular, `i ¤ 2 for each i. If there is an i with Ni 1 and `i 2, we choose the index i1 above so that `i1 2. In this case, i1 contributes a factor Op2, Cq to the local group Sφ9 v . The image of x in Rφ9 then projects onto the nonidentity component of this factor, and is nontrivial as required. If i1 cannot be chosen in this way, there is an i with Ni ¡ 1 and `i 2. In this case, i contributes a product of several factors Op2, Cq to Sφ9 v . The image
6.3. CONSTRUCTION OF GLOBAL PARAMETERS φ9
327
of x in Rφ9 v then projects onto the product of nonidentity components, and again is nontrivial as required. This establishes the last of the conditions (iii)(c), completing the proof of the lemma. The construction of Proposition 6.3.1 will allow us to apply global methods to the proof of the local theorems. The essential case is that of p-adic F . However, we will still have work to do in the case F R. This will be based on a variant of the construction, which entails a more restrictive choice of the global field F9 . Lemma 6.3.2. Suppose that the given local objects F , G, φ, M and φM are such that F R, and such that the two dimensional constituents
tφ i :
(6.3.9)
Ni
2u
of φ are in relative general position. Then we can choose global objects F9 , 9 9 G, φ, M9 and φ9 M that satisfy the conditions of Proposition 6.3.1, and so that (6.3.10)
φ9 i,v
φi ,
1 ¤ i ¤ r, v
P S8u .
Proof. The first step is to choose the global field F9 . Since F R, we can start with the rational field Q. We will then take F9 to be the totally ? ? real Galois extension Qp q1 , . . . , qk q of Q, for distinct (positive) prime numbers q1 , . . . , qk , as in the proof of Lemma 6.2.1. We need to take k to be large, in order that the set S8 of archimedean valuations of F9 be large. Once k is fixed, we choose the primes qi to be large relative to the degree 2k of F9 over Q, and so that
1pmod 4q. Then the discriminant of F equals q1 qk , so that F {Q is unramified at any qi
9
9
2k .
prime p that is not large relative to Taking u to be any fixed archimedean 9 valuation of F , we obtain F9u F R, as required.
The next step is to choose global endoscopic data G9 i P Er ell pNi q with G9 i,u Gi . This amounts to the choice of quadratic id`ele class characters η9 i p i SOpNi , Cq. The degrees Ni here for F9 , which are 1 by definition unless G are always equal to 1 or 2, since F R. The archimedean local conditions 9
(6.3.11)
η9 i,v
ηi ,
v
P S8u , 1 ¤ i ¤ r,
are forced on us by (6.3.10). As in the proof of Proposition 6.3.1, we chose η9 i successively with increasing i so that ηi,v is unramified at any place v R S8 such that η9 j,v ramifies for some j i. In addition to this condition, we can also insist that η9 i,v be unramified for any v R S8 whose norm is not large relative to 2k . We choose each η9 i subject to these conditions, and thereby obtain the required global endoscopic data G9 i
P ErpNiq. 9
328
6. THE LOCAL CLASSIFICATION
p i SOp2, Cq. The archimedean localConsider an index i such that G izations (6.3.11) of η9 i then each equal the nontrivial quadratic character εR of R . It follows that the quadratic extension E9 i of F9 attached to η9 i by class field theory is totally imaginary. The group G9 i is of course the corresponding anisotropic form of SOp2q. Let Z9 i8 Z 8 pG9 i q be the (central) subgroup of ¹ G9 i pF9v q, G9 i,8 v PS8 defined as in the proof of Lemma 6.2.2. It is the diagonal image in G9 i,8 of the intersection of G9 i pF9 q with the maximal compact subgroup K9 8 of G9 i pA9 8 q. We shall show that Z9 i8 equals Z{2Z. We shall first observe that Z9 i8 is isomorphic to the cyclic subgroup C9 i of roots of unity in E9 i . Obviously
G9 i pF9 q tx P E9 i : σ pxq x1 u,
where σ generates the Galois group E9 i {F9 . Therefore Z9 i8 is the group of units u of E9 i such that σ puq u1 . The Dirichlet unit theorem asserts that the group of all units is the direct product of C9 i with a free abelian group of rank r [Cassels, §18]. Since the group Z9 i8 is finite, it can contain no unit of infinite order, and is therefore contained in C9 i . Since E9 i is totally complex, σ acts by complex conjugation on Z9 i8 . Therefore Z9 i8 does equal C9 i Let m ¥ 2 be the positive integer such that C9 i is generated by a primitive mth root of unity 2πi ζm e m .
Then Qrζm s is the maximal cyclotomic subfield of E9 i . If m is greater than 2, there is a prime number p ¤ m that ramifies in Qrζm s, and hence also in E9 i . On the other hand, we know from the choice of F9 and η9 i that E9 i {Q can ramify at p only if p is very large. Indeed, it follows from the construction that any such p must be large relative to the degree 2k 1 of E9 i over Q. But the degree φpmq of Qrζm s{Q both divides 2k 1 , and approaches infinity as m becomes large. This contradicts the existence of p. Therefore m 2, and E9 i contains only the square root p1q of unity. We have therefore established that Z9 i8 C9 i Z{2Z. We note in passing that our ramification condition on the primes qi is more stringent than necessary. The argument we have just given would apply to any totally real extension F9 {Q that ramifies only at primes that are large relative its degree.
r sim pG9 i q. We now use Corollary 6.2.4 to construct global parameters φ9 i P Φ k Since |S8 | 2 is even, the product of |S8 |-copies of the central character of φi on the group Z9 i8 Z{2Z equals 1. According to Remark 2 following Corollary 6.2.4, we can then choose the simple parameters φ9 i so that 9
6.3. CONSTRUCTION OF GLOBAL PARAMETERS φ9
329
(6.3.10) holds. This construction is not the same as that of the proof of Proposition 6.3.1. For example, the earlier property (6.3.6) is incompatible with the requirement (6.3.10) here. However, the conditions (i), (ii) and (iii) of the proposition remain valid for the resulting compound parameters φ9 and φ9 M here. Indeed, the assertions (iii) that relied on (6.3.6) in the earlier construction are now trivial consequences of (6.3.10). The lemma follows. Remark. In following parts of the proof of Proposition 6.3.1, we have made the argument above a little more elaborate than necessary. It would have been sufficient to take E9 i
F Qi, 9
9
for a suitable imaginary quadratic extension Q9 i of Q. However, the argument we have given could conceivably be of interest in its own right. The primary objects in both Proposition 6.3.1 and Lemma 6.3.2 are the simple global parameters 1 ¤ i ¤ r.
φ9 i ,
They are in turn given by Corollary 6.2.4, which relies on the application of the simple trace formula in Lemma 6.2.2 that is the source of the underlying global construction. Once chosen, they determine the required global pair pG,9 φ9 q from the given local pair pG, φq. We took the secondary objects G and φ as the given data in the proposition for the obvious reason that they are the objects to which the local theorems apply. Let us put them aside for a moment. Suppose instead that we have been given only the simple constituents φi of φ, and that we have constructed corresponding simple global parameters φ9 i by Corollary 6.2.4. We can then form the associated family
Frpφ1, . . . , φr q
r F 9
(6.3.12)
9
9
`91 φ9 1
`
`
`9r φ9 r : `9i
¥0
(
of compound global parameters. The supplementary constraints we imposed on the parameters during the proofs of Proposition 6.3.1 and Lemma 6.3.2 9 are independent of G and G. More precisely, the primary objects can be chosen independently of any pair
pG, φq, 9
9
φ9 P F9 pG9 q, G9 P Er ell pN9 q, N9
(with G9 being simple if φ9
9
R
r ell pN9 q). Φ
¥ 0,
The conditions (iii)(a)–(iii)(c)
r satisfies Assumption 5.4.1, with of Proposition 6.3.1 then imply that F
V
9
V pFr q being the set of archimedean places S8u . 9
r9 able to apply the global results of §5.4 to F.
We will therefore be
330
6. THE LOCAL CLASSIFICATION
6.4. The local intertwining relation for φ We turn now to our main task, the proof of the local theorems. We are treating local generic (Langlands) parameters φ in this chapter. Our general goal is to apply the inductive procedure of Chapter 5 to the global parameters constructed in the last section. In this section, we shall establish the local intertwining identity of Theorem 2.4.1. We are taking F to be a local field. If F is the complex field C, the case (ii) of Corollary 2.5.2 tells us that the relevant intertwining operator equals 1. Theorem 2.4.1 then follows from the definitions, and the fact that any centralizer Sφ is connected in this case. We can therefore assume that F is real or p-adic as in the last two sections. We also have the simplification of Lemma 2.4.2, which reduces the problem to the case that φ ψ belongs to r 2 pM q of Φ r pM q. We will therefore use the notation introduced the subset Φ r pGq, and prior to Proposition 2.4.3, in which φ represents a parameter in Φ r pM, φM q is a fixed pair with φM P Φ2pM, φq. We fix a datum G P Ersim pN q over the local field F and a local Langlands parameter φ `1 φ 1 ` ` `r φ r
r pGq, as in (6.3.1). In this section we will assume that some `i is greater in Φ r 2 pGq. Given G and φ, we than 1, or in other words, that φ does not lie in Φ 9 M 9 9 φ, 9 and φ choose the global objects F9 , G, M that satisfy the conditions of Proposition 6.3.1. Having fixed the simple global constituents φ9 1 , . . . , φ9 r of 9 we then form the family φ, r F 9
Frpφ1, . . . , φr q 9
9
of global parameters (6.3.12). As we have noted, the conditions of Assump-
r and V S u from the conditions (iii) of Proposition tion 5.4.1 follow for F 8 6.3.1. The construction of the global objects relies on the induction hypothesis from §6.3 that the local theorems are valid for all parameters of degree less than N . To this, we now add the formal global induction hypothesis promised in Remark 1 following Corollary 6.2.4. We assume that the global 9
r theorems hold for global parameters of degree less than N in the set F. The induction arguments of Chapter 5 therefore apply to the parameters in 9
r In particular, they will allow us to extend our global hypothesis to the F. 9
r pN q of this section. The first step in the global compound parameter φ9 P Φ induction argument, the case of simple parameters, will be resolved in §6.7. We studied the global intertwining relation in §4.5. We established it directly, by induction arguments that can clearly be carried out in the context 9
r in the cases outlined in Corollary 4.5.2. of global parameters in the set F, As we remarked after this corollary, the reduction also carries over without change to the local intertwining relation of Theorem 2.4.1. In other words, 9
6.4. THE LOCAL INTERTWINING RELATION FOR φ
331
the local intertwining relation holds for our generic parameter φ, unless it r ell pG q for some G P Ersim pN q, or unless it belongs to the local belongs to Φ analogue for φ of one of the two exceptional classes (4.5.11) or (4.5.12). In each of these cases, it suffices to consider elements s and u that map to a point x in the subset Sφ,ell of Sφ . (We understand Sφ,ell to be the local analogue of the global set Sψ,ell introduced near the beginning of §5.2.) We 9 observe that Proposition 4.5.1 also yields the other global theorems for φ, unless φ falls into the first of the three cases above. We thus have three exceptional cases to deal with. We shall study them together. In each case, we shall apply the results of Chapter 5 (and the 9 assertion of Proposition 4.5.1) to the global parameter φ. Keep in mind 9 9 that since φ ψ is generic, the point sψ9 and the sign character εG are both ψ9 trivial. r ell pGq is characterized by the local generic form of The case of φ P Φ (5.2.4). The conditions are #
φ 2φ1
` ` 2φq ` φq 1 ` ` φr , Sφ Op2, Cqq Op1, Cqrq φ , q ¥ 1,
(6.4.1)
with the requirement that the Weyl group Wφ contain an element w in the regular set Wφ,reg . The other two cases are represented by (4.5.11) and (4.5.12) (which is to say the local generic forms of (4.5.11) and (4.5.12) with φ and ` in place of ψ and `). In all three cases, the associated global pair pG,9 φ9 q satisfies an identity (6.4.2)
¸
P
9 x 9 x 9 q f9 9 pφ, 9q f9G19 pφ, G
0,
x Sφ,ell
r pG9 q, f9 P H
where x9 is the isomorphic image of x in Sφ9 . This follows from Lemmas 5.4.3 and 5.4.4 in the case (6.4.1), and Corollary 4.5.2 (together with the assertion of Lemma 5.4.4) in the cases (4.5.11) and (4.5.12). Recall that if q r 1 in (6.4.1), there is a second group G_ P Ersim pN q such that r pG_ q, and such that S _ Spp2, Cq. The corresponding global pair φPΦ φ pG9 _, φ9 q actually belongs to the second case (4.5.11) (with pG, ψq equal to pG,9 φ9 _q). It represents that part of the specialization to r 1 of (4.5.11) that was excluded by the hypotheses of Proposition 4.5.1. Since this was later covered by Lemma 5.4.4, in the assertion that (5.2.13) vanishes, we can include it now in our discussion of (4.5.11). In the cases (4.5.11) and (4.5.12), we set q 1. We assume that f9
¹ v
f9v
332
6. THE LOCAL CLASSIFICATION
is a decomposable function. The summand in (6.4.2) can then be decomposed into a difference of products ¹
1 pφ9 v , x9 v q f9v, G9
¹
v
f9v,G9 pφ9 v , x9 v q
v
over all valuations v. In order to remove the places v outside S8 puq from this equation, we have first to establish a very special case of the local intertwining relation. Lemma 6.4.1. Suppose that F is nonarchimedean. Then the local intertwining relation holds in case r ¤ 3 and Ni
1,
1 ¤ i ¤ r.
Proof. The constraints we have imposed here are obviously pretty stringent. Applying them to the cases (6.4.1), (4.5.11) and (4.5.12), we see that the group M is particularly simple. In fact, M is abelian except in the case (4.5.12), with r 3, where we have G
Mder Spp2q.
Notice that this last case is the setting of Corollary 2.5.2(v). One could perhaps use the earlier corollary to give a direct local proof of the lemma, but there would still be a number of supplementary points that I have not tried to verify. We shall instead give a less direct proof based on the application of Lemma 2.5.5. We must show that the objects G, M , φ and φM over F satisfy the conditions (i)–(iii) of Lemma 2.5.5. The condition (ii) follows from [KS, §5.3]. (We assume here that the stable distribution on M pF q attached to φM in [LL] is the same as the linear form given by Theorem 2.2.1(a), an induction hypothesis we will resolve in Lemma 6.6.2.) Before we consider the condition (i) on char F , we shall establish the final condition (iii) of Lemma 2.5.5. The global identity (6.4.2) in the restricted case here is based on the id`ele class characters η9 i π9 i , 1 ¤ i ¤ r, of order 1 or 2, which are obtained in Proposition 6.3.1 from the given u of characters ηi πi on F . Suppose that v lies in the set S8 S8 archimedean places. Since a completion η9 v at v must be equal to p 1q or p1q, the representations in Πr φ9 v are all induced from a minimal parabolic subgroup. It then follows from the remark on [S6, Lemma 11.5] following (6.1.5) that the characters επ9 M ,v p q in (6.1.5) are all trivial. Therefore
1 pφ9 v , x9 v q f9v,G pφ9 v , x9 v q, f9v, G9
The identity (6.4.2) becomes ¸
P
x Sφ,ell
¹
R
v S8
1 pφ9 v , x9 v q f9v, G9
¹
R
v S8
x P Sφ .
f9v,G9 pφ9 v , x9 v q f98 pφ9 8 , x9 8 q,
6.4. THE LOCAL INTERTWINING RELATION FOR φ
where
f98 pφ9 8 , x9 8 q
¹
P
v S8
333
f9v,G9 pφ9 v , x9 v q.
It is a direct consequence of the condition (iii)(a) of Proposition 6.3.1 that r pG9 8 q to functions the linear mapping from functions f98 in H x
ÝÑ
f98 pφ9 8 , x9 8 q,
on Sφ is surjective. This implies that (6.4.3)
¹
R
v S8
1 pφ9 v , x9 v q f9v, G9
¹
R
v S8
x P Sφ ,
f9v,G9 pφ9 v , x9 v q,
for any x P Sφ,ell . 9 It follows from the fundamental lemma and Theorem 2.5.1(b) that almost all of the factors on each side of (6.4.3) equals 1. From the definitions and the local correspondence for M9 v (which we are assuming by induction), it follows that for any v R S8 puq and x P Sφ,ell , we can choose f9v so that the corresponding factor on the right hand side of (6.4.3) is nonzero. This allows us to reduce (6.4.3) to an identity in f f9u and φ φ9 u . We obtain (6.4.4)
fG pφ, xq epxq fG1 pφ, xq,
for a complex number epxq eu px9 u q
¹
R pq
v S8 u
x P Sφ,ell , f
P HrpGq,
1 pφ9 v , x9 v q f9 9 pφ9 v , x9 v q1 f9v, v,G G9
that is independent of f . The identity (6.4.4) would seem to be a little weaker than the condition (iii) of Lemma 2.5.5. However, with the elliptic elements x P Sφ,ell acounted for, the condition follows easily for a general pair ps, uq by induction from the usual arguments of descent. If we impose the condition (i) that charpF q 2, φ will then satisfy the three requirements (i), (ii) and (iii) of Lemma 2.5.5. The lemma then tells us that the coefficient epxq equals 1, and therefore that the local intertwining relation is valid if charpF q 2. It remains to deal with the case that charpF q 2. From (6.4.3), (6.4.4), and what we have just established, we see that ¹
P
ev px9 v q 1,
v S2
where S2 is the set of valuations of F9 with charpF9v q 2. The totally real field F9 constructed in the proof of Lemma 6.2.1 has the property that F9v F for each v P S2 . A slightly more sophisticated construction, which we leave to the reader, would give another totally real field F9 with the property F9v F for every v P S2 , but such that
|S2| |S2|
1 2k
1.
334
6. THE LOCAL CLASSIFICATION
(The point is that any totally real field has a totally real extension of any given degree, in which a given prime splits completely.) In applying Proposition 6.3.1 to both F9 and F9 , we can choose the quadratic characters η9 i for any valuations v P S2 and v P S not equal and η9 i so that η9 i,v η9 i,v 2 to the distinguished valuations u P S2 and u P S2 . It follows that ev px9 v q ev px9 v q eu pxq,
for any such v and v , where eu pxq conclude that
eu pxq
¹ v S2
P
and therefore that
e px q e px q
x P Sφ,ell ,
0 is a fixed complex number.
ev px9 v q
¹
P
ev px9 v q
1
We
1,
v S2
¹
eu pxq
¹
v u
P
ev px9 v q 1.
v S2
The local intertwining relation for charpF q (6.4.4).
2 then follows again from
We now return to the general case. We are trying to extract the local intertwining relation from the global identity (6.4.2), which we recall applies to the global parameter φ9 attached to φ by Proposition 6.3.1. We shall exploit the different factors of the terms of (6.4.2) attached to a decomposable ± function f9 f9v . The first step is to remove the factors at places v R S8 puq. We claim that they satisfy the local intertwining relation
1 pφ9 v , x9 v q f9 9 pφ9 v , x9 v q, f9v, v,G G9
(6.4.5)
v
R S8puq.
To see this, we recall that φ9 v is a direct sum of quasicharacters of F9v by Proposition 6.3.1(ii). If one of these constituents is not self-dual, φ9 v does not r 2 pM 9 q. It is then easy to see that x does not factor through an element in Φ v v belong to Sφ9 v ,ell , or equivalently, that the Weyl element wx does not belong to Wφ9 v ,reg . The identity (6.4.5) then follows inductively by reductions similar to those in §4.5 prior to Proposition 4.5.1. If all of the constituents of φ9 v are self-dual, it follows from the second assertion of Proposition 6.3.1(ii) that there are at most three distinct constituents. The local pair pG9 v , φ9 v q then satisfies the condition of Lemma 6.4.1, and so (6.4.5) again follows. Since we can choose the functions f9v in (6.4.5) so that each side is nonzero, we can remove these factors from the global identity (6.4.2). The identity becomes (6.4.6)
¸
P
x Sφ,ell
¹
P pq
v S8 u
1 pφ9 v , x9 v q f9v, G9
¹
P pq
v S8 u
f9v,G9 pφ9 v , x9 v q
0.
For what remains to be proved of the local intertwining relation postulated in Theorem 2.4.1, the main step turns out to be the case of certain archimedean parameters φ. There ought to be a local proof in this case,
6.4. THE LOCAL INTERTWINING RELATION FOR φ
335
perhaps following from the methods of [KnZ2] and [S4]–[S6]. However, this still seems illusive, to me at least, so we shall continue to apply the global methods above. We do know that there is nothing further to prove in case F C. We shall therefore assume for the present that F R. The degrees of the irreducible constituents φi of φ are then equal to 1 and 2. r 2 pM, φq, and we We are writing φM as usual for a fixed element in Φ identify SφM with a subgroup of Sφ . The action of SφM on Sφ by translation stabilizes the subset Sφ,ell , and gives a simply transitive action x
ÝÑ
x P Sφ,ell , xM
x xM ,
P Sφ
M
,
of SφM on Sφ,ell . In the case F R under present consideration, we already have much information about the associated L-packets. According to the rφ results of Shelstad reviewed in §6.1, we have the archimedean packets Π r and ΠφM of irreducible representations, and a surjective mapping π Ñ πM r φ to Π r φ . We also have a canonical pairing x , y on Sφ Π r φ such from Π M that (6.4.7)
fG1 pφ, xq
and (6.4.8)
fG pφ, xq
¸
rφ π PΠ
¸
P
rφ π Π
xx, πy fGpπq,
x P Sφ ,
εpπM q xx, π y fG pπ q,
x P Sφ,ell ,
r pGq in the associated archimedean Hecke algebra. In for any function f P H the second equation, x is restricted to the subset Sφ,ell of Sφ , and
εpπM q επM pxq 1
is the corresponding restriction of the character επM in (6.1.5), which der φ . The local intertwining relation pends only on the image πM of π in Π M asserts that εpπM q equals 1.
Lemma 6.4.2. Assume that F R, and that φ is in general position. Then there is an element ε1 P SφM such that fG pφ, xq fG1 pφ, x ε1 q,
f
P HrpGq, x P Sφ,ell.
Proof. The lemma asserts simply that the sign takes the special form εpπM q xε1 , πM y,
πM
P Πr φ
M
.
In other words, εpπM q can be identified with the restriction to the image r φ in Spφ of a character on Spφ . Since it treats only a pretty severe of Π M M M specialization of both F and φ, the lemma would appear to be a pretty minor step. In point of fact, it represents essential preparation for the general argument. Its proof amounts to an exercise in the Fourier analysis on a finite abelian group.
336
6. THE LOCAL CLASSIFICATION
9 M 9 9 φ, 9 ,φ 9 We choose global data pF9 , G, M q as in Lemma 6.3.2. Then F is a totally real field, whose set S8 of archimedean places is large. The valuation u belongs to S8 , and we have
pFv , Gv , φv , Mv , φM,v q pF, G, φ, M, φM q, 9
for each v (6.4.9)
9
9
9
9
P S8. The identity (6.4.6) becomes ¸ ¹ 1 pφ, xq ¹ fv,G pφ, xq 0, fv,G 9
9
P
x Sφ,ell
P
P
v S8
v S8
for a test function f98
¹
P
v S8
f9v ,
f9v
P HrpGq,
r pG q. which is now a finite product of functions from the same space H r If f P HpGq is fixed, we can regard the function (6.4.7) of x P Sφ as an r φ . In particular, inverse Fourier transform of the function fG pπ q of π P Π 1 we can recover fG pπ q from fG pφ, xq as a Fourier transform on Sφ . It will be convenient to take a partial Fourier transform, relative to the variable represented by the SφM -torsor Sφ,ell in Sφ . Bearing in mind the general lack of a canonical splitting for the sequence
1 we define
ÝÑ
SφM
ÝÑ
fG1 pφ, ξ q |SφM |1
for any character ξ
Sφ
ÝÑ
¸
P
ÝÑ
Rφ
1,
fG1 pφ, xq ξ pxq1 ,
x Sφ,ell
P Spφ. The function apξ q fG1 pφ, ξ q,
ξ
P Spφ,
obviously satisfies the equivariance relation
apξ ξR q ξR prell q1 apξ q,
P Rpφ, pφ of Spφ , where rell r pxq is the image under translation by the subgroup R in Rφ of (any) x P Sφ,ell . We can therefore write fG1 pφ, xq a_ pxq, x P Sφ,ell ,
(6.4.10)
for the inverse transform (6.4.11)
a _ px q
¸
P
ξ SpφM
a pξ q ξ px q,
ξR
x P Sφ,ell .
pφ , The last summand is obviously invariant under translation of ξ P Spφ by R pφ . and therefore does represent a function of ξ in the quotient SpφM Spφ {R We define the partial Fourier transform fG pφ, ξ q of fG pφ, xq in the same way. It is then easy to see that
fG pφ, ξ q εpξ q fG1 pφ, ξ q,
ξ
P Spφ,
6.4. THE LOCAL INTERTWINING RELATION FOR φ
for the function εp ξ q
#
εpπM q, if ξ 0,
x , πM y, for some πM P Πr φ
337
M
,
otherwise,
pφ . (As above, we are making no distinction between a on SpφM Spφ {R p pφ -invariant function on Spφ .) The original function function on SφM and an R is therefore the inverse transform
fG pφ, xq pεaq_ pxq
of the product
pεaqpξq εpξq apξq εpξq fG1 pφ, ξq.
These definitions are of course examples of formal operations in finite abelian Fourier analysis. In general, any function a on Spφ that satisfies (6.4.10) has an inverse transform (6.4.11) on Sφ,ell . It satisfies the inversion formula apξ q |SφM |1
which specializes to
¸
P
x Sφ,ell
¸
P
x Sφ,ell
a_ pxq |SφM | ap1q.
Any two functions have a convolution
pa1 a2qpξq which satisfies
a_ pxq ξ pxq1 ,
¸
P
ξ1 SpφM
a1 pξ1 q a2 pξ11 ξ q,
pa1 a2q_ a_1 a_2 .
We shall use these operations to transform (6.4.9) to the assertion of the lemma. We order the valuations in S8 as v1 , . . . , vn , and set ai pξ q avi pξ q f9v1 i ,G pφ, ξ q,
ξ
P Spφ,
for 1 ¤ i ¤ n. The left hand side of (6.4.9) can then be written n ¹
¸
P
x Sφ,ell
¸
i 1
a_ i px q
n ¹
pεaiq_pxq
i 1
pa1 anq_pxq pεa1q pεanq _pxq
x
|Sφ | pa1 anqp1q pεa1q pεanq p1q ¸ a1 pξ1 q an pξn q εpξ1 qa1 pξ1 q εpξn qapξn q |Sφ | M
M
|Sφ |
ξ1 ,...,ξn
¸
M
ξ1 ,...,ξn
1 εp ξ 1 q εp ξ n q a 1 p ξ 1 q a n p ξ n q ,
338
6. THE LOCAL CLASSIFICATION
where the last two sums are taken over the set
(
pξ1, . . . , ξnq P pSpφ qn : ξ1 ξn 1 . Suppose that pξ1 , . . . , ξn q is a multiple index such that for each i, ξi belongs r φ in Spφ . We can then choose the functions fv P H r p Gq to the image of Π so that for any ξ P Spφ , ai pξ q is nonzero if ξ lies in the coset of ξi , and is M
M
9
M
i
zero otherwise. This follows from the results of Shelstad, by which Rφ is rφ . identified with the representation theoretic R-group RπM of any πM P Π M The functions f9vi serve to isolate the summand of pξ1 , . . . , ξn q as the only nonvanishing term in our expression for the left hand side of (6.4.9). Since the right hand side of (6.4.9) vanishes, we conclude that εpξ1 q εpξn q 1,
(6.4.12)
if ξ1 ξn
1.
r φ . In fact, The character ξ 1 in SpφM belongs to the image of Π M it follows from [S6, Theorem 11.5] that ξ corresponds to the pBM , χM qr φ . Applying the identity (2.5.4) of Theorem generic representation πM P Π M 2.5.1(b) for the pB, χq-generic constituent of IP pπM q to the definition of the sign εpπM q, we see that εp1q 1. We are assuming that the integer n |S8 | in (6.4.12) is fixed and large. Having now established that εp1q 1, we are free to replace n by any integer m ¤ n, since we can always set ξi 1 for i ¡ m. We can therefore assume also that the mapping ε preserves any relation ξ1 ξm 1, m ¥ 1,
r φ in Spφ . Since this image generamong elements ξ in the image of Π M M ates SpφM , by condition (iii)(a) of Proposition 6.3.1, ε extends uniquely to a character on the group SpφM . Therefore
εp ξ q ξ p ε1 q ,
ξ
P Spφ
M
,
P Sφ . It follows from (6.4.7) and (6.4.8) that ¸ fG pφ, xq εpπM q xx, π y fG pπ q
for a unique element ε1
M
P
rφ π Π
¸
rφ π PΠ
¸
P
rφ π Π
xε1, πM y xx, πy fGpπq xx ε1, πy fGpπq fG1 pφ, x ε1q,
as required. Lemma 6.4.3. The formula of the last lemma holds with ε1
1.
Proof. We are taking φ to be a parameter (6.4.1), (4.5.11) or (4.5.12) p q R. Our assumption is that φ is in general position, in the sense that its simple constituents φi with Ni 2 are in relative general
r G over F in Φ
6.4. THE LOCAL INTERTWINING RELATION FOR φ
339
position. In the proof of the last lemma, we applied the global construction of Lemma 6.3.2. In particular, we worked with identical archimedean parameters
φi ,
φ9 i,v
(6.4.13)
1 ¤ i ¤ r, v
P S8,
attached to the global field F9 . In the proof of this lemma, we shall use the p-adic form of the construction in Proposition 6.3.1. This will give us supplementary information about φ. The logic of the construction will actually be slightly convoluted, since we will choose the objects of the proposition in a different order. The irreducible degrees Ni of φ equal 1 or 2. We introduce new degrees by setting # Nj , if j ¤ q, 7 Nj Nq 1 Nr , if j q 1,
where j ranges from 1 to pq 1q. In principle, we will be applying Proposition 6.3.1 to a local parameter
`1φ71 ` ` `q φ7q ` φ7q
φ7
with simple components φ7j
P Φr simpG7j q,
G7j
1,
P ErsimpNj7q,
over a p-adic field F 7 , and multiplicities `1 , . . . , `q that match those of φ.
However, we will need to choose endoscopic data G9 7j P Er sim pNj7 q over a global field F9 7 before we introduce φ7 . We may as well let F9 7 be the global field F9 chosen in the proof of the last lemma. For each j ¤ q, we then take G9 7j to be the global endoscopic datum G9 j over F9 that was constructed also in the proof of the last lemma. The remaining datum G9 7q 1 is determined by the requirements that 9
G9 7q p
and
1
Gpq 1 Gpr Gpq 1 Gpr 9
9
η9 q7
1 ηG7 9
q 1
r ¹
η9 i .
i q 1
(By convention, η9 i 1 in case G9 i SppNi , Cq.) Once these objects are chosen, we fix a nonarchimedean place u7 of F9 such that for each j, the local endoscopic datum G7j G9 7j,u7 p
is elliptic, and hence simple, over the local field F 7 tion is needed only in case G9 7j p
Fu7 . (This last condi9
SOp2, Cq.) We then choose local parameters
r sim pN 7 q over the p-adic field F 7 whose central character on the finite φ7j P Φ j 8 9 group Z Z 8 pG9 j q is trivial. The existence of φ7 follows easily from our j
j
340
6. THE LOCAL CLASSIFICATION
r pN 7 q induction hypothesis that the local theorems hold for parameters in Φ j over F 7 . Once we have the local parameters φ7j and the global data G9 7j , we choose
r sim pG9 7 q over F9 according to Corollary 6.2.4. In global parameters φ9 7j P Φ j fact, by Remark 2 following Corollary 6.2.4, we can choose them so that
φ9 7
j,v
#
φj , φq 1 ` ` φr ,
if j if j
¤ q, q
1,
for any v P S8 . This is because the product over v of the associated central characters with that of φ7j will be trivial on Z9 j8 . The global pairs pG9 7j , φ9 7j q then determine a global endoscopic datum G9 7 of §6.3, together with a global parameter
`1φ71
φ9 7
(6.4.14)
9
r pG9 7 q such that in Φ
`
`
P ErsimpN q, as at the beginning 9
`r φ9 7q
`
φ9 7q
1
φ9 7v
φ, v P S8. 7 The global objects G and φ9 7 are not those of the proof of Proposition 6.3.1, since the archimedean components of φ9 7 are all equal. They nonetheless satisfy the conditions of the proposition (with M9 7 and φ9 7M obtained from G9 7 and φ9 7 as at the beginning of §6.3). The conditions are in fact obvi9
ous consequences of the restricted nature of the multiplicities `i in (6.4.14). For example, in the cases (6.4.1) or (4.5.11) that the multiplicities are all 2, the datum G9 7q 1 is just the factor G9 7 of M9 7 that is classical, namely an
orthogonal or symplectic group. Since the associated parameter φ9 7q 1 φ9 7 is then simple, the group Sφ9 7 Sφ7 in condition (iii)(a) actually equals 1, M M making the condition trivial. The other conditions are either equally clear or as in the proof of the proposition. It therefore follows that the relation (6.4.6) holds for φ9 7 . We shall apply it to the case that the group Sφ7 is M trivial. With the assumption that Sφ7 t1u, the sum over x P Sφ7 ,ell in (6.4.6) M reduces to a single element x1 . We identify x1 with the base point in the larger set Sφ,ell . Since M9 v7 M for any v P S8 , the identity reduces in this case to (6.4.15)
¹
f9v,G pφ, x1 ε1 q fG7 pφ7 , x1 q
¹
f9v,G pφ, x1 q pf 7 q1G pφ7 , x1 q,
v
v
for any function
¹ v
f9v f 7 ,
f7
P HrpG7q,
f9v
P HrpGv q, v P S8. 9
Suppose that the point ε1 is not equal to 1. Then the linear forms r pGq are linearly independent. This follows fG pφ, x1 q and fG pφ, x1 ε1 q in f P H
6.4. THE LOCAL INTERTWINING RELATION FOR φ
341
from the definition (2.4.5) of fG pφ, xq, the disjointness of the irreducible constituents of induced tempered representations, and the fact that the image of r φ generates Spφ . Let f fv be a variable function in H r p Gq H r pG v q Π 1 1 M M at some fixed place v1 P S8 . At the other places v P S8 , and also at the place u7 , we fix functions so that the contributions to the left hand side of (6.4.15) do not vanish. It follows that fG pφ, x1 q c1 fG pφ, x1 ε1 q,
f
P HrpGq,
for a constant c1 that is independent of f . This is a contradiction. We have established the required condition that ε1 1, for parameters φ such that the group Sφ9 7 equals t1u. This includes the basic elliptic case M
(6.4.1), and the first supplementary case (4.5.11). The remaining supplementary case (4.5.12), in which q 1 and `1 3, is slightly different. In this case, the classical factor G9 7 of M9 7 is larger than the group G9 7q 1 G9 72 . The associated global parameter equals φ9 7
φ7q ` φ7q 1 φ71 ` φ72, 9
9
9
9
and is not simple. If it satisfies the further two conditions that N1 and N27 is odd, the rank
N17 1
2 N17 N27 is even, and the character ξ 7 on the group Op1, Cq Op1, Cq is nontrivial. ψ We then see from (1.4.9) that Sφ7 is isomorphic to Z{2Z, and also that there N
3N1
N2
9
Nr
M
9
M
p The quotient S 7 of is a nontrivial central element in the dual group G. φ9 M Sφ9 7 is therefore trivial, and we still have ε1 1. If these two conditions are M not met, however, Sφ9 7 has order 2, and the argument fails. M
It therefore remains to establish the lemma in the case (4.5.12), under the further restriction that either N17 2 or N27 is even. Since there are only two self-dual, one-dimensional characters on F R_ , this restriction implies that if Ni N17 1, then Ni 2 for every i ¡ 1. There are several ways to proceed. The simplest perhaps is to introduce a new parameter φ5
2φ1 ` φ51 ` φ2 ` ` φr , φ51 P Φr simpN1q, r pGq over F R. That is, we replace one of the three copies of φ1 in the in Φ
original parameter φ with a simple parameter φ51 that is different from all the others. If N1 1, for example, the last restriction allows us to set φ51 equal to φ1 εR . If we apply Proposition 6.3.1 to φ5 , we obtain a parameter φ9 5 over the global field F9 . In fact, one finds that φ9 5 can be chosen so that φ9 5
v
#
φ, if v φ5 , if v
v0 , v0 ,
342
6. THE LOCAL CLASSIFICATION
for any v P S8 , where v0 is a fixed valuation in S8 , and so that the conditions of Proposition 6.3.1 remain valid for φ9 5 . Then for any v v0 , φ9 5v is an archimedean parameter for which we have already established the lemma. We leave the reader to check that the resulting application of (6.4.6) yields the required property ε1 1 for the remaining parameter φ φ9 5v0 . The last two lemmas represent the main step. They yield the local intertwining relation of Theorem 2.4.1 in the special case that F R and φ is in general position. With this information, it will be possible to deduce the general case. r p Gq Proposition 6.4.4. Theorem 2.4.1 is valid for generic parameters φ P Φ over the general local field F .
Proof. As we agreed at the beginning of the section, the theorem holds unless φ belongs to one of the three exceptional cases (6.4.1), (4.5.11) and (4.5.12), and the given elements s and u map to a point x in the set Sφ,ell . We must allow F to be either real or p-adic, since the parameters φ over R treated above were required to be in general position. We therefore take r pGq to be an arbitrary generic parameter over the given field F , of the φPΦ form (6.4.1), (4.5.11) or (4.5.12). We shall apply Proposition 6.3.1 directly to φ. From the objects G, φ, 9 M 9 9 φ, 9 and φ M and φM over F attached to φ, we obtain global objects G, M over F9 that satisfy the conditions of the proposition. We then obtain the identity (6.4.2) from the results of §5.4 and Corollary 4.5.2, and its reduction (6.4.6) implied by Lemma 6.4.1. It follows that the sum ¸
P
x Sφ,ell
¹
f1 9
v,G
P
u v S8
pφv , xv q 9
9
f 1 pφ, xq
¹
G
P
u v S8
fv,G pφv , x9 v q fG pφ, xq 9
9
vanishes, for any decomposable function ¹ v
P8
f9u f
Su
f8u f, 9
f9v
P HrpGv q, f P HrpGq. 9
u , the parameter φ9 P Φ r pG9 v q is in general position. It For any v P S8 v follows from Lemmas 6.4.2 and 6.4.3 that
1 pφ9 v , x9 v q f9v,G pφ9 v , x9 v q, f9v,G
x P Sφ,ell .
The last identity then reduces to a linear relation ¸
P
x Sφ,ell
u f98 pφ, xq fG1 pφ, xq fGpφ, xq ,G9
in f , with coefficients u f98 pφ, xq ,G9
¹
P
u v S8
f9v,G9 pφ9 v , x9 v q.
0
6.5. ORTHOGONALITY RELATIONS FOR φ
343
u , the coefficients are linearly independent. This follows As linear forms in f98 from the definition (2.4.5), the disjointness of constituents of induced tempered representations, and the condition (iii)(a) of Proposition 6.3.1. We therefore come to the conclusion that
fG1 pφ, xq fG pφ, xq,
x P Sφ,ell , f
P HrpGq.
This is the identity to which we had reduced Theorem 2.4.1. We have now established it simultaneously for each of the three critical cases (6.4.1), (4.5.11) and (4.5.12). Therefore Theorem 2.4.1 holds for each of the three cases, and hence for any generic parameter. Corollary 6.4.5. The canonical self-intertwining operator attached any r pGq, w P W 0 and πM P Π r φ satisfies φPΦ M φ (6.4.16)
RP pw, π rM , φq 1.
rM of πM Proof. As we noted near the end of §2.5, the extension π is canonical in this case, and so therefore is the intertwining operator in (6.4.16). To establish (6.4.16), it suffices to consider the two special cases represented by the local analogues of (4.5.11) and (4.5.12). We shall sketch a proof for these cases that combines what we have just established with another global observation. From the local objects G, φ, M and φM over F , we obtain global objects 9 M 9 9 φ, 9 and φ 9 G, M over F . It is not hard to see from the global identity (4.5.14) (which holds for both (4.5.11) and (4.5.12)) and the condition (iii)(a) of Proposition 6.3.1 that rM , φq epwq 1, RP pw, π
for a complex number epwq. The right hand side of the local intertwining relation (2.4.7) then satisfies fG pφ, uq fG pφ, wuq epwq fG pφ, uq,
since u and wu have the same image in Sφ . It follows that epwq 1.
6.5. Orthogonality relations for φ We are working towards the local classification of tempered representations. We will continue to use global methods, and in particular, the constructions of global data in §6.2 and §6.3. However, the global methods will have to be supplemented with some new local information. In this section, we shall stabilize the orthogonality relations given by the local trace formula, and as well as their twisted analogues. In particular, we shall use the local intertwining relation we have now established to refine the general results of [A11, §1–3]. We suppose henceforth that our local field F is nonarchimedean. We are assuming inductively that the local theorems are valid for generic parameters of degree less than our fixed positive integer N . The theorems now have the
344
6. THE LOCAL CLASSIFICATION
interpretation of §6.1. However, we recall that this entails essentially no change in their application. Let us be a little more explicit about the last point. Suppose that φ lies r sim pN q and Φ r pN q. We can then use our induction in the complement of Φ hypothesis to define a substitute of the local Langlands group LF , exactly as in the global case treated in §1.4. It is a complex reductive group
ÝÑ ΓF over ΓF , which comes with a GLpN, Cq-orbit of L-embeddings into L GLpN q. r pGq of Φ r pN q if and only if one If G belongs to Ersim pN q, φ lies in the subset Φ of these embeddings factors through the L-image of L G in L GLpN q. The Lφ
local centralizer sets Sφ , S φ and Sφ are then given as usual in terms of the p image of Lφ in L G. We can also form the local set ΦpG, φq of G-orbits of L L-homomorphisms from Lφ to G that map to φ. This set has order mpφq |ΦpG, φq|
equal to 1 or 2, like its global analogue. r ell pGq of elliptic paramOur concern in this section will be the subset Φ r pGq. These are the elements φ P Φ r pGq such that the subset eters in Φ S φ,reg,ell
ts P S φ,ss : |S φ,s| 8u
of S φ is nonempty. We have taken the liberty here of writing S φ,reg,ell
S φ,reg X S φ,ell,
where
S φ,reg ts P S φ,ss : S 0φ,s is a torusu conforms to the usual definition of regular semisimple. It is then clear that the two characterizations of S φ,reg,ell are equivalent. It is also easy to see that if S φ,reg,ell is nonempty, it actually equals S φ,ell . Observe that φ is elliptic if and only if there is an elliptic pair
pG1, φ1q,
G1
P EellpGq, φ1 P Φr 2pG1q, r pGq equals φ. In fact, it follows from the in Φ
such that the image of φ1 definitions that our basic correspondence
pG1, φ1q ÝÑ pφ, sq
defines a bijection between local, generic, elliptic variants Xell pGq tpG1 , φ1 q : G1
P EellpGq,
φ1
P F2pG1qu
and
Yell pGq tpφ, sq : φ P Fell pGq, s P S φ,ell u of the general sets introduced in §4.8. This descends to a bijection between their quotients Xell pGq tpG1 , φ1 q : G1
P EellpGq,
φ1
P Φr 2pG1qu
6.5. ORTHOGONALITY RELATIONS FOR φ
and
345
r ell pGq, x P Sφ,ell u. Yell pGq tpφ, xq : φ P Φ Our primary goal is to describe the linear form
(6.5.1)
ÝÑ f 1pφ1q, f P HrpGq, pG1, φ1q P XellpGq, pφ, sq. This is the content of Theorem 2.2.1(b), f
in terms of the central assertion of the local theorems. The local intertwining relation provides a r 2 pGq in Φ r ell pGq, solution of the problem when φ lies in the complement of Φ as we saw in Proposition 2.4.3. Part of the remaining problem, in which r 2 pGq, will be to complete the inductive definition of f 1 pφ1 q by φ lies in Φ establishing Theorem 2.2.1(a). This includes Theorem 6.1.1, which will be r sim pGq of Φ r 2 p Gq needed to refine our temporary definition of the subset Φ r r and complete the inductive definition of the sets Φ2 pGq and Φell pGq. We will deal with these matters in §6.6 and §6.7. For the orthogonality relations of this section, it is best to work with the set Tell pGq that was part of the proof of Proposition 3.5.1. Recall that Ttemp pGq is the set of GpF q-orbits of triplets τ pM, σ, rq, where M is a Levi subgroup of G, σ is a unitary, square-integrable representation of M pF q, and r is an element in the R-group Rpσ q of σ. The subset Tell pGq consists of (orbits of) triplets τ such that r belongs to the subset Rreg pσ q of elements in Rpσ q for which the determinant dpτ q detpr 1qaM
is nonzero. Extending the convention we are using in this volume (in contrast r N pG q to that of [A10]), we write Trtemp pGq and Trell pGq for the sets of Out orbits of elements in Ttemp pGq and Tell pGq respectively. We recall that there is a linear form fG pτ q attached to any τ
pM, σ, rq.
For general groups G, fG pτ q is determined only up to a scalar multiple, which depends on various choices related to the normalization of intertwining operators. Having made such choices for G P Ersim pN q in Chapter 2, we may now be more precise, even though this is not really necessary for orthogonality relations. The case that M G is not at issue, since fG pτ q is then just equal to fG pσ q. If M is proper in G, we apply our induction r 2 pM q. We obtain a unique hypothesis to the tempered representation σ P Π r 2 pM q, whose packet contains the reprebounded, generic parameter φM P Φ r pGq we denote by φ. We can then sentation πM σ, and whose image in Φ set (6.5.2)
rM , φq IP pπM , f q , fG pτ q tr RP pr, π
f
P HrpGq,
in the notation (2.4.2) (but with π there written as πM here). We note that the R-group RpπM q of πM defined in [A10] is a priori different from the R-group Rφ of φ defined in §2.4. We have of course agreed to identify the stabilizer Wφ of φM with a subgroup of W pM q, the
346
6. THE LOCAL CLASSIFICATION
Weyl group that contains the stabilizer W pπM q of πM . It is a consequence of the disjointness of tempered L-packets for M that Wφ contains W pπM q. On the other hand, it follows from the form (2.3.4) of M , the corresponding form of the Weyl group W pM q, and our induction assumption that Theorem 2.2.4(b) holds for the factor G of M , that any element in Wφ stabilizes πM . Therefore Wφ equals W pπM q, in the generic case at hand. Moreover, we now know that elements in the subgroup Wφ0 of Wφ give scalar intertwining operators for the induced representation IP pπM q. It follows that Wφ0 is contained in the subgroup W 0 pπM q of W pπM q. We thus have a surjective mapping (6.5.3)
Rφ
Wφ{Wφ0 ÞÑ
W pπM q{W 0 pπM q RpπM q,
πM
P Πr φ
M
,
r pGq. At the beginning of the which makes sense for any parameter φ P Φ next section, we shall observe that this mapping is an isomorphism, and hence that RpπM q equals Rφ . The essential part of the study of (6.5.1) turns out to be the special case r cusp pGq of cuspidal functions in H r pGq. It in which f lies in the subspace H follows from [Ka, Theorem A] that the mapping that sends fG P Icusp pGq to the function τ ÝÑ fG pτ q, τ P Tell pGq,
on Tell pGq is a linear isomorphism from Icusp pGq onto the space of complex valued functions of finite support on Tell pGq. (This result is also a special case of the orthogonality relations in [A10, §6].) A similar assertion applies r pGq-invariant functions in Icusp pGq, if we replace to the space Ircusp pGq of Out Tell pGq by Trell pGq. Suppose that pG1 , φ1 q and pφ, sq are corresponding pairs in the elliptic sets Xell pGq and Yell pGq. Observe that G1 equals G if and only r 2 pGq of Φ r ell pGq. When if s equals 1, in which case φ maps to the subset Φ G this is so, we suppose that φ is such that the linear form f pφq is defined, according to the prescription of Theorem 2.2.1(a), since this is the case not covered by our induction hypothesis. In all cases, we then have an expansion (6.5.4)
f 1 p φ1 q
¸
P p q
τ Trell G
cφ px, τ q fG pτ q,
f
P HrcusppGq,
for uniquely determined complex numbers cφ px, τ q that depend only on the image x of s in Sφ,ell . The expansion (6.5.4) was a central object of study in [A11], but with the different notation ∆pφ1 , τ q cφ px, τ q. The article [A11] was aimed at general groups, in which the family of functionals f G ÝÑ f G pφq, φ P Φ 2 pG q, could be defined only as an abstract basis of the space of stable linear forms on Icusp pGq, rather than by local Langlands parameters. In particular, there
6.5. ORTHOGONALITY RELATIONS FOR φ
347
were no groups S φ in [A11] associated to the linear forms φ, and hence no pairs pφ, sq. This accounts for the earlier notation, and the cruder results of that paper. The local intertwining relation gives a description of the coefficients in r 2 pGq in Φ r ell pGq. In this case, φ and (6.5.4) if φ lies in the complement of Φ Sφ are given by (6.4.1). We can then take
φ1 φq φr r 2 pM, φq that was part of the discussion of the last to be the parameter in Φ φM
section. Since F is nonarchimedean, our induction hypothesis gives a perfect r φ . We form the subset pairing x , y on SφM Π M Trφ,ell pGq τ
pM, πM , rq :
πM
P Πr φ
M
, r
P Rreg pπM q
(
of Trell pGq attached to φ, a set that could a priori be empty. The coefficients in (6.5.4) can then be expressed in terms of the pairing
xx, τ y xx, πrM y,
pM, πM , rreg q, x , πrM y is an extension of the character τ
on Sφ,ell Trφ,ell , where as in §2.4, x , πM y on SφM to the SφM -torsor Sφ,ell. To see this, we consider the local r cusp pGq. After exintertwining relation (2.4.7), as f varies over the space H panding the two sides of (2.4.7), according to the definitions (2.4.6), (6.5.4), (2.4.5) and (6.5.2), we need only compare the coefficients of fG pτ q. We see that (6.5.5)
cφ px, τ q
#
xx, τ y,
0,
if τ P Trφ,ell pGq, otherwise,
r 2 pG q. for any x P Sφ,ell and τ P Trell pGq, and for the given parameter φ P Φ ell We note that the two sides of (6.5.5) have a parallel dependence on the extension π rM of the component πM of τ , a choice we have not built into our notation for τ . We turn now to orthogonality relations. The starting point is the canonical Hermitian inner product
I pf, g q IppfG , gG q
»
p q
Γell G
fG pγ q gG pγ q dγ
on Ircusp pGq. Recall that Γell pGq denotes the set of elliptic conjugacy classes in GpF q, and dγ is a canonical measure that is supported on a set of strongly regularly classes in Γell pGq. According to the local trace formula, this inner product has a spectral expansion (6.5.6)
I pf, g q
¸
P p q
τ Trell G
mpτ q |dpτ q|1 |Rpτ q|1 fG pτ q gG pτ q,
where mpτ q denotes the number of elements in Tell pGq that map to τ , as an equivalence class tpM, σ, rqu in Trell pGq, and Rpτ q is the R-group Rpσ q. The
348
6. THE LOCAL CLASSIFICATION
formula (6.5.6) is a specialization of [A10, Corollary 3.2], in which we have written gG pτ q pg qG pτ _ q. The R-groups are all abelian here, and the 2-cocycles introduced in [A10, p. 86] all split. The supplementary coefficient mpτ q occurs in (6.5.6) because we have summed over Trell pGq instead of Tell pGq. r pN q. There is a A similar identity holds for the twisted component G canonical twisted Hermitian inner product
p q p
I fr, gr
on the space Ircusp pN q
Ip frN , grN
q
»
p q
r ell N Γ
frN pγ q grN pγ q dγ
r pN q , where frN Icusp G
frGr pN q is the usual
r cusp pN q in the space I rcusp pN q, while image of a given function fr P H 0 r ell pN q Γell G r pN q is the set of elliptic G r pN, F q-orbits in G r pN, F q, and Γ r ell pN q that is supported on the strongly regdγ is a canonical measure on Γ r ular classes in Γell pN q. This inner product also has a spectral expansion. It r pN q for G r pN q of the is a sum of terms over the analogue Trell pN q Tell G 0 r pN q GLpN q, Trell pN q is our set set Trell pGq. By the special properties of G r r Φell pN q Φell GpN q of multiplicity free elements
φ φ1
` ` φr
r pN q. The twisted spectral expansion is then in Φ
(6.5.7)
I pfr, grq
¸
P
p q
r ell N φ Φ
|dpφq|1 frN pφq grN pφq,
where |dpφq| Twisted orthogonality relations for GLpN q, which include the cases treated in [W4], are a part of work in progress by Waldr pN q. Like its global anaspurger on the twisted local trace formula for G r pN q in place of G), the discrete part of the twisted logue (3.1.1) (with G local trace formula contains a sum over elements w P W pM qreg . In the case r pN q0 is the Levi subgroup such that φ belongs to the set at hand, M G r M pN q Φ M G r pN q , and the sum over W pM qreg can be restricted to the Φ elements wreg that act on the space aM Rr , and for which 2r .
| detpwreg 1qa | |dpφq| 2r . For any G P Ersim pN q, there is also a stable Hermitian inner product » G G p S pf, g q S pf , g q f G pδ q g G pδ q dδ M
p q
∆reg G
on the space Srcusp pGq. We recall that ∆ell pGq denotes the set of elliptic, stable conjugacy classes in GpF q, and dδ is a canonical measure supported on the strongly regular classes in ∆ell pGq. The definition obviously extends to products of groups, such as the compound elements in Erell pN q and Eell pGq.
6.5. ORTHOGONALITY RELATIONS FOR φ
349
It can be used to stabilize the inner products I pf, g q an I pfr, grq. One obtains two expansions (6.5.8) and (6.5.9)
I pf, g q I pfr, grq
for local coefficients and
p
r ι N, G
¸
G1 PEell pGq
¸
P p q
G Erell N
ιpG, G1 q Sppf 1 , g 1 q,
f, g
p
r cusp pN q, fr, gr P H
q p
q
r ι N, G Sp frG , grG ,
P HrcusppGq,
ιpG, G1 q |Z pG1 qΓ |1 |OutG pG1 q|1
r N pGq|1 q ι GrpN q, G 12 |Z pGpqΓ|1 |Out
analogous to their global counterparts described at the end of §3.2. The linear mappings (6.5.10) and (6.5.11)
Ircusp pGq
ÝÑ
Ircusp pN q
ÝÑ
À
à
G1 PEell pGq
à
P p q
G Erell N
À
Srcusp pG1 q
Srcusp pGq,
frG are thus isometric isomorphisms, f 1 and frN Ñ 1 G G relative to the natural inner products on the right hand spaces. For the ordinary expansion (6.5.8), we refer the reader to [A11, §2], which is the restriction to cuspidal functions of the general stabilization of [A16, §10]. The treatment of the twisted expansion (6.5.9) will be similar. We need to derive an explicit spectral expansion for S pf, g q. We shall use the spectral expansion (6.5.7) of the left hand side of (6.5.9) to deduce information about the right hand side. However, in order to accommodate the limited understanding we have at this point, we need to adopt some temporary terminology. r ell pN q is a cuspidal lift if there is a We shall say an element φ P Φ r G P Eell pN q such that given by fG
(i)
Ñ
frN pφq frG pφq,
r cusp pN q, fr P H
where frG pφq is the linear form on Srcusp pGq defined by (6.1.1), and (ii)
r 2 pG q. φPΦ
It is a consequence of (6.1.1) that G is uniquely determined by (i). If φ r sim pN q of Φ r ell pN q, then satisfies both (i) and (ii), and lies in the subset Φ r r G lies in the subset Esim pN q of Eell pN q. This is a formal consequence of
350
6. THE LOCAL CLASSIFICATION
r sim pGq of Φ r 2 pGq. Conversely, if our temporary definition of the subset Φ r sim pN q satisfies (i), and the corresponding datum G lies in Ersim pN q, φPΦ (ii) itself follows from the definition. r ell pN q is a cuspidal lift, but Theorem 2.2.1(a) asserts that every φ P Φ we have yet to establish this. If φ is not simple, there is a unique G with r 2 pGq by our induction assumption, but it remains to establish the first φPΦ condition (i). This amounts to the secondary assertion (2.2.4) of Theorem 2.2.1(a). In any case, we can certainly assume inductively that for each r ell pN 1 q is a cuspidal lift. N 1 N , every element φ1 P Φ Suppose that we are given a subset r c pN q Φell pN q Φ ell
that consists entirely of cuspidal lifts. Then Φcell pN q is a disjoint union over G P Erell pN q of the subsets r 2 p G q. r c pN q X Φ r c p Gq Φ Φ 2 ell
As in the untwisted case, one can show that the mapping frN
ÝÑ
frN pφq,
r ell pN q, frN φPΦ
P IrcusppN q,
is a linear isomorphism from Ircusp pN q onto the space of complex valued funcr ell pN q Trell pN q. Let I rc pN q denote the tions of finite support on the set Φ cusp r ell pN q are supported subspace of elements in Ircusp pN q that as functions on Φ r c pN q. Then for any G P Erell pN q, the mapping (6.5.11) sends on the subset Φ ell c c pG q. pN q onto a subspace of SrcusppGq, which we shall denote by Srcusp Ircusp c r We note that Scusp pGq can be identified with the space of functions of finite r c pG q. support on Φ 2
Proposition 6.5.1. Suppose that G P Erell pN q, and that f and g are funcr cusp pGq such that f G and g G lie in Src pGq. Then tions in H cusp S pf, g q
(6.5.12)
¸
P p q
rc G φ Φ 2
mpφq |Sφ |1 f G pφq g G pφq.
Moreover, the formula (6.5.8) can be written as I pf, g q
(6.5.13)
¸
mpφq |Sφ |1
¸
f 1 p φ 1 q g 1 p φ 1 q,
xPSφ,ell p q where in the last summand, pG1 , φ1 q corresponds to pφ, xq.
P
r ell G φ Φ
r cusp pN q such that frN and g rN Proof. Let fr and gr be functions in H G G are the preimages of f and g under the mapping (6.5.11). Then the only nonvanishing term on the right hand side of (6.5.9) is that of the given group G. Using (6.5.7) to rewrite the left hand side of (6.5.9), we obtain ¸
P
p q
rc N φ Φ ell
|dpφq|1 frN pφq grN pφq rιpN, Gq S pf, gq.
6.5. ORTHOGONALITY RELATIONS FOR φ
351
By assumption, each φ in the sum on the left is the pullback of a linear form on Srcusp pGq, which we are also denoting by φ. That is frN pφq grN pφq f G pφq g G pφq.
From the explicit formula for r ιpN, Gq above, it is easy to check that
|dpφq|1 rιpN, Gq1 mpφq |Sφ|1,
r c pGq. The identity (6.5.12) follows. Observe that we can for any φ P Φ 2 r c pGq to Φ r 2 pGq, since enlarge the index of summation in this formula from Φ 2 c elements in the subspace Srcusp pGq of Srcusp pGq can be regarded as functions r 2 pGq that vanish on the complement of Φ r c pGq. In fact, we are free to on Φ 2 r 2 pGq by the subset Φ2 pGq of G-orbits p replace Φ of local parameters (rather r pGq-orbits), so long as we remove the coefficient mpφq. than Out To establish the second assertion, we apply the formula we have just obtained to any G1 P Eell pGq. If G1 G, we can assume inductively that r 2 pG1 q is a cuspidal lift. In other words, f 1 and g 1 are images every element in Φ r of functions in Hcusp pG1 q that satisfy the condition of the proposition, with r 2 pG1 q. We can therefore write (6.5.8) as r c p G1 q Φ Φ 2
I pf, g q
¸
G1 Eell G
P p q
ιpG, G1 q
¸
r 2 G1 φ1 Φ
P p q
mpφ1 q |Sφ1 |1 f 1 pφ1 q g 1 pφ1 q.
The process of converting this to the formula (6.5.13) is an elementary local analogue of the discussion of the global multiplicity formula in §4.7, itself a very simple case of the standard model of §4.3–4.4. Indeed, the last double sum is over pairs pG1 , φ1 q in the local set Xell pGq defined at the beginning of the section. We have only to rewrite it in terms of the bijective image pφ, xq of pG1, φ1q in the second local set YellpGq. From the explicit formula for ιpG, G1 q, it is easy to see that mpφq |Sφ |1
ιpG, G1q mpφ1q |Sφ1 |1.
The identity (4.5.13) follows.
We shall actually use (6.5.13) in its interpretation as a set of orthogonality relations for the matrix coefficients tcφ,x pτ qu in (6.5.4). We of course have to require that the linear form (6.5.4) attached to a given pair pφ, xq in Yell pGq be defined, which is to say that for the corresponding pair pG1 , φ1 q r 2 pG1 q is a cuspidal lift. By induction, this is in Xell pGq, the element φ1 P Φ r 2 pG q only at issue if x 1, which in turn implies that φ lies in the subset Φ r ell pGq. To describe the orthogonality relations, we write of Φ apφ, xq mpφq1 |Sφ |
and
bpτ q mpτ q1 |dpτ q| |Rpτ q|,
τ
P TrellpGq.
352
6. THE LOCAL CLASSIFICATION
Corollary 6.5.2. Suppose that yi
pφi, xiq,
i 1, 2,
are two pairs in Yell pGq such that the associated, linear forms (6.5.4) are defined. Then (6.5.14)
¸
P p q
τ Trell G
bpτ q cφ1 px1 , τ q cφ2 px2 , τ q
#
apφ1 , x1 q, 0,
if y1 y2 , otherwise.
Proof. We identify Ircusp pGq with the space of finitely supported functions of tτ u, and we identify the right hand side of (6.5.10) with the space of finitely supported functions of an index set ty u that includes all pairs pφ, xq P YellpGq for which (6.5.4) is defined. Then the isomorphism (6.5.10) is given by a matrix C tcpy, τ qu such that cpy, τ q cφ px, τ q if y pφ, xq. We can assume that the basis of SrpGq defined by the appropriate subset of ty u is orthogonal. We can then write the two expressions (6.5.6) and (6.5.13) for I pf, g q as an orthogonality relation ¸
apy q1 cpy, τ1 q cpy, τ2 q bpτ1 q1 δτ1 ,τ2
y
for the matrix C, where A tapy qu and B tbpτ qu are the invertible diagonal matrices that represent the two inner products. The matrix form of this identity is C A1 C B 1 , which becomes CBC A upon inversion. The required formula follows. As a preliminary application of the orthogonality relations of the corollary, we shall show that the expansion (6.5.4) simplifies in the case not r 2 pG q. covered by (6.5.5), namely that φ P Φ
Lemma 6.5.3. (a) Suppose that pφ, xq is a pair in Yell pGq such that φ lies in r 2 pGq of Φ r ell pGq, and such that the associated linear form (6.5.4) the subset Φ is defined. Then (6.5.15)
f 1 p φ1 q
(b) Suppose that
¸
P p q
r2 G π Π
cφ px, π q fG pπ q,
f
P HrcusppGq.
yi pφi , xi q, i 1, 2, are two pairs in Yell pGq that both satisfy the conditions of (a). Then (6.5.16) # ¸ mpφ1 q1 |Sφ1 |, if y1 y2 , 1 mpπ q cφ1 px1 , π q cφ2 px2 , π q 0, otherwise. r
P p q
π Π2 G
6.6. LOCAL PACKETS FOR COMPOSITE φ
353
Proof. The first part (a) asserts that the definition (6.5.4) simplifies r 2 pGq. We have to show cφ px, τ1 q 0 for any in the case that φ belongs to Φ r 2 pGq in Trell pGq. fixed element τ1 in the complement of Π r We know that τ1 lies in a packet Tφ1 ,ell , for a unique parameter φ1 in r 2 pGq in Φ r ell pGq. We shall apply the formula (6.5.5) for the complement of Φ the coefficients cφ1 px1 , τ q,
x1
or rather the inversion
|Sφ1,ell|1
¸
P
P Sφ ,ell, τ P TrellpGq, 1
xx1, τ1y cφ px1, τ q
#
1
x1 Sφ1 ,ell
1, if τ τ1 , 0, otherwise,
of this formula that is a consequence of the definition of the pairing It allows us to write bpτ1 q cφ px, τ1 q
¸
bpτ q cφ px, τ q |Sφ1 ,ell |1
P p q |Sφ ,ell|1 τ Trell G
¸
1
x1
xx1, τ1y
¸
¸
P
x , y.
xx1, τ1y cφ px1, τ q 1
x1 Sφ1 ,ell
bpτ q cφ px, τ q cφ1 px1 , τ q.
τ
Since φ1 φ, Corollary 6.5.2 tells us that the last sum over τ vanishes. Since bpτ1 q 0, we then see that cφ,x pτ1 q vanishes, as required. The definition (6.5.4) thus reduces to the required formula (6.5.15) of (a). Having established (a), we substitute (6.5.15) into the formula (6.5.14) of Corollary 6.5.2. The required formula (6.5.16) of (b) follows. 6.6. Local packets for composite φ We are now ready to start proving the theorems that give the local classification. They consist of the five local theorems we stated in Chapters 1 and 2, specialized to generic local parameters ψ φ. We have already established the local intertwining relation Theorem 2.4.1 for pairs pG, φq. r φ described by What remains is the classification of general L-packets Π Theorems 1.5.1 and 2.2.1. We will also have to deal with the supplementary r φq. These will be treated afterwards Theorems 2.2.4 and 2.4.4 for pairs pG, in §6.8, as will the questions in §6.1 related to Theorem 6.1.1. We note again that in the archimedean case, the local classification has been established. With the exception of our formulation of Theorem 2.4.1, the theorems for archimedean parameters φ are included in the general results of Shelstad for real groups (and work in progress by Mezo for twisted real groups). We recall that our proof of Theorem 2.4.1 for generic parameters, completed by global means in §6.4, applies to all local fields F . We thus continue to assume that the local field F is nonarchimedean, as in the last section. We also maintain our induction hypothesis that the
354
6. THE LOCAL CLASSIFICATION
r of degree less than the fixed local theorems hold for generic parameters in Φ integer N . Suppose that G is a fixed datum in Ersim pN q. We are trying to classify the irreducible tempered representations of GpF q. In particular, we r N pGq-orbits of) irreducible representar φ of (Out want to attach a packet Π r tions to any φ P Φbdd pGq, which satisfies the endoscopic character identity (2.2.6) of Theorem 2.2.1. r 2 pGq in Suppose for the moment that φ lies in the complement of Φ r bdd pGq. Then φ is the image of a parameter φM P Φ r 2 pM, φq, for a proper Φ Levi subgroup M of G. The assertion of Theorem 2.2.1(a) actually applies to data G in the larger set ErpN q. However, as we have noted before, the usual argument of descent on any such G reduces the statement to a corresponding assertion for M , which then follows from our induction hypothesis. For Theorem 2.2.1(b), we can appeal to Proposition 2.4.3, since we have established the local intertwining relation of Theorem 2.4.1 for generic parameters. We r φ of (orbits of) of irreducible tempered representations obtain a packet Π r φ that satisfies the endoscopic character of GpF q, and a pairing on Sφ Π identity (2.2.6). Therefore, Theorem 2.2.1 holds for φ. So does Theorem 1.5.1(a), since apart from the assertion for unramified pairs pG, π q that we verified in §6.1, it represents only a less precise form of Theorem 2.2.1. As for this remaining assertion, Corollary 2.5.2(iv) tells us that the relevant intertwining operators are trivial. The local intertwining relation then tells us that the associated unramified character x , π y is trivial. However, since we are dealing with tempered representations, we have r φ also satisfies the supplemore to establish. We must show that the packet Π mentary conditions of Theorem 1.5.1(b). These are tied to the question on the relation between the endoscopic and representation theoretic R-groups Rφ and RpπM q, raised in our discussion of (6.5.3) from the last section. The question is now easy to resolve. We can identify (6.5.10) as a mapping from the space of finitely supported functions of τ P Trell pGq to the space of finitely supported functions of pφ, xq P Yell pGq. It follows from (6.5.4) that the mapping factors through the projection (6.5.3), or rather the restriction of (6.5.3) to the subset Rφ,reg of Rφ . But we also know that the mapping (6.5.10) is an isomorphism, and so in particular, is surjective. It follows that the projection (6.5.3) takes Rφ,reg bijectively onto Rreg pπM q, for any r φ . This statement amounts to the assertion that the set Tφ,ell is πM P Π M r ell pGq. It might appear to be insignificant, since not empty if φ belongs to Φ r ell pGq, and is in fact empty Rφ,reg consists of only one element if φ lies in Φ otherwise. Nonetheless, its application to Levi subgroups L M gives us what we want. Indeed, any element in Rφ belongs to the subset RφL ,reg r ell pLq that maps to φ. It attached to some L and some parameter φL P Φ follows that the projection (6.5.3) is an isomorphism, and hence that RpπM q rφ . equals Rφ for any πM P Π M
6.6. LOCAL PACKETS FOR COMPOSITE φ
355
In the proof of Proposition 3.5.1, we described the general classification of Πtemp pGq by harmonic analysis. This characterizes Πtemp pGq as the image of a bijection
t π u, pL, σ, ξq ÝÑ ξ (
L P L, σ
P Π2,temppLq, ξ P Π Rpσq ,
where the left hand side represents a W0G -orbit of triplets, and the right hand side is given by the irreducible constituent pξ b πξ q of the character
pr, f q ÝÑ
tr RP pr, π rM , φq IP pπM , f q
of Rpσ q HpGq. Our goal here is to establish the finer endoscopic classification of Theorem 1.5.1. In the proof of Proposition 2.4.3, we constructed r φ of φ over Π r temp pGq. It is a priori a multiset, equipped with a the packet Π p mapping to Sφ given by the irreducible constituents pξ b π q of the character
px, f q ÝÑ fGpφ, xq
¸
P
rφ πM Π M
xx, πrM y tr RP pwx, πrM , φq IP pπM , f q
r pGq. We have just seen that we can identify RpπM q with the Rof Sφ H group Rφ Sφ {Sφ1 of φ. It follows that the two kinds of classification are compatible. In particular, the packet structure of the second imposes an endoscopic, character theoretic interpretation on the first. The original classification, for its part, yields finer properties of the packets. This is the implication we need to exploit here. It tells us that the r φ are multiplicity free. Combined with the properties of Π rφ elements in Π M that are part of our induction hypothesis, it also confirms that the mapping r φ to Spφ is a bijection. Finally, it tells us that the complement of Π r 2 p Gq from Π r temp pGq is a disjoint union over local parameters φ in the complement of in Π r r bdd pGq of the packets Π r φ . These are the conditions of Theorem Φ2 pGq in Φ 1.5.1(b), for the p-adic case at hand. We have established
Proposition 6.6.1. Theorems 1.5.1 and 2.2.1 hold for generic parameters r 2 pGq in Φ r bdd pGq. φ in the complement of Φ Having dealt with the complementary set, we assume from this point on r 2 pGq. Then that φ lies in Φ (6.6.1)
φ φ1
` ` φr ,
P Φr simpGiq,
φi
Gi
P ErsimpNiq,
for distinct simple parameters φi . In this section, we shall treat the case that r sim pGq of Φ r sim pGq. r ¡ 1, or equivalently, that φ lies in the complement Φ 2 9 9 From Proposition 6.3.1 we obtain corresponding global objects pF , G, φ9 q that satisfy the given list of conditions. In particular, φ9 φ9 1
φr r 2 pGq. Moreover, the set is a global parameter which lies in Φ ( r F rpφ1 , . . . , φr q `1 φ1 (6.6.2) F `r φr : `i ¥ 0 `
`
9
9
9
9
9
9
9
`
`
9
9
9
356
6. THE LOCAL CLASSIFICATION
is a family of global parameters that satisfies the conditions of Assumption 5.4.1. We can therefore apply the relevant lemmas of Chapter 5 to the pair pG,9 φ9 q. In order to extract local information for pG, φq from the global properties 9 q, we need to be able to remove valuations v R S puq. The role of 9 φ of pG, 8 the following lemma will be similar to that of Lemma 6.4.1. Lemma 6.6.2. Theorem 2.2.1 holds for φ if r Ni
1,
¤ 3 and
1 ¤ i ¤ r.
r sim pGq. It is therefore a Proof. We are assuming that φ belongs to Φ 2 direct sum N inequivalent characters of order 1 or 2, where
1 N
r ¤ 3.
p is orthogonal, and G is simple. The secondary condition of Therefore G Theorem 2.2.1(a) consequently does not apply here. The primary condition of Theorem 2.2.1(a) follows from the global property
frN pφ9 q frG pφ9 q, 9
9
r pN q, fr P H 9
9
9
established in Lemma 5.4.2. For we can fix the component fru of fr away from 9
9
u so that frN pφ9 q is a nonzero multiple of its component frN pφq fru,N pφ9 u q at u. In other words, the linear form frN pφq in fr is the pullback of a stable r pG9 q. linear form f G pφq in f P H For the main assertion (b) of Theorem 2.2.1, we shall be content to treat p SOp3, Cq and G Spp2q. We may as well also the case N 3. Then G assume that the quadratic id`ele class character 9
9
ηφ9
φ1 φ2 φ3 9
9
9
equals 1, since we could otherwise replace each φ9 i by its product with p We shall ηφ9 η 9 1 . Then φ9 corresponds to a homorphism of WF9 to G. φ reduce the assertion to results of [LL], using the properties of Whittaker models implicit in Lemma 2.5.5. We do not know a priori that the stable distribution attached to φ in [LL] is the same as the linear form f G pφq. We therefore set fG pφq
¸
P
fG pπ q,
f
π Πφ
P H p G q,
where Πφ is the packet attached to the local parameter φ in [LL]. Since 9 the global results of [LL] φ is a local factor φ9 u of the global parameter φ, G G9 9 implicitly exhibit f pφq as a local factor of the φ-component Sdisc, pf9q of φ9 9 the stable trace formula for G. In particular, if
f9 f9u f9u
f f u, 9
6.6. LOCAL PACKETS FOR COMPOSITE φ
357
G we can fix f9u so that Sdisc, pf9q is a nonzero multiple of fGpφq. On the other φ9 9
SOp3, Cq that the formula
p hand, it follows easily from the fact that G (3.3.14) reduces to an identity
N G Idisc, pfrq rιpN, G9 q Spdisc, pfrGq, φ9 φ9 9
9
9
r pN q, fr P H 9
9
9
9 of which the left hand side is a nonzero multiple of frN pφ9 q. We choose the 9 9 variable component fr fru of fr at u so that it has the same image in SrpGq
as f . We can then fix the complementary component fru so that it has the 9
9 same image in SrpG9 u q as f9u , and so that frN pφ9 q is a nonzero multiple of f G pφq. It follows that
fG pφq e pφq f G pφq,
f
P H pG q ,
for a nonzero constant e pφq. The last equation is the analogue of the identity (6.4.4) from the proof of Lemma 6.4.1. If charpF q 2, we can combine it with a corresponding analogue of Lemma 2.5.5, which is very simple in this case and will appear in [A27]. It follows that e pφq 1. If charpF q 2, we can vary the global field F9 as in the proof of Lemma 6.4.1. It follows that e pφq 1 in this case as well. We see therefore that fG pφq f G pφq,
f
P H p G q,
in all cases. The possible ambiguity between [LL] and Theorem 2.2.1(a) thus resolved, we conclude that the packet Πφ of [LL] satsifies the required identity of Theorem 2.2.1(b). We shall now apply the general results from Chapter 5. Recall that Lemma 5.4.2 affirms the validity of Assumption 5.1.1 for the parameter φ9 P Fr2 pG9 q. As in the special case established in the last lemma, we need to convert this global property to the local assertion of Theorem 2.2.1(a). To do so, we temporarily allow G to be an element in the larger set Erell pN q. In other words, pG, φq pG9 u, φ9 uq is the local component of pair
pG, φq, 9
r sim pG9 q. G9 P Er ell pN q, φ9 P F 2 9
9
9
This is compatible with our understanding that the basic objects in the construction of §6.3 are really the simple parameters tφ1 , . . . , φr u and tφ9 1 , . . . , φ9 r u 9 q. 9 φ rather than the pairs pG, φq and pG, r pN q attached to the parameter Lemma 6.6.3. The linear form on H sim r φ P Φ2 pGq is the pullback of a stable linear form
f
ÝÑ
f G pφ q ,
f
P HrpGq,
358
6. THE LOCAL CLASSIFICATION
r pGq. If G G1 G2 is not simple, and on H
φ φ1 φ2 ,
then
φi
P Φr 2pGiq, i 1, 2,
f G pφq f1G pφ1 q f2G pφ2 q, 1
2
f
f1f2.
Proof. This is the local form of Lemma 5.4.2, which we shall apply 9 to the global parameter φ. Recall that φ φ9 u , for a valuation u of F9 . If v is an archimedean valuation, the analogue of the lemma for F9v will be assumed as part of the theory of twisted endoscopy in [Me]. Suppose that v u is not archimedean. If the corresponding centralizer S φ9 v is infinite, the analogue of the lemma for F9v follows by induction. In view of condition (ii) of Proposition 6.3.1, this leaves only the case that the integer N is less than 4, and φ9 v is a sum of N -inequivalent characters of order at most 2. In p must be orthogonal, and the analogue of the lemma follows these cases, G from Lemma 6.4.1. The assertions of the lemma for F F9u thus follow from the global assertion of Lemma 5.4.2, and their analogues for v u. The product formula from the lemma allows us to apply our induction hypotheses to any datum G P Erell pN q that is composite. We can therefore
assume henceforth that G P Ersim pN q and G9 P Er sim pN q, as before. Observe that the lemma includes the assertion φ is cuspidal lift, which will allow us to make use of the orthogonality relations of the last section. We are now ready to exploit the essential global properties of Chapter 5. Corollary 5.1.3, together with the application to Lemma 5.1.4 of its supplement Lemma 5.4.5 in case N is even and ηφ9 1, tells us that the stable 9
9 We thus have multiplicity formula of Theorem 4.1.2 is valid for G9 and φ.
G Sdisc, pf9q |ΦpG,9 φ9 q| |Sφ9 |1 f Gpφ9 q, φ9 9
(6.6.3)
9
r pG9 q, f9 P H
since εG pφ9 q 1 in the generic case at hand. We shall apply this to the 9 φ-component 9
(6.6.4)
G Idisc, pf9q φ9 9
¸
G9 1 Eell G9
P p q
9 G 9 1q S p1 ιpG, pf91q disc,φ9
of the stabilized trace formula given by (3.3.15). More precisely, we apply the corollary of Theorem 4.1.2 to the groups G9 1 that index the sum in (6.6.4). It thus follows from Corollary 4.1.3 that the right hand side of (6.6.4) equals ¸ ¸ 9 G 9 1q ιpG, |Sφ9 1 |1 f 1pφ9 1q. G9 1 PEell pG9 q φ1 PΦpG9 1 ,φ9 q
9 from the beginning of the elemenThis is the expression (4.7.7) (with ψ φ) tary proof of Lemma 4.7.1, which we recall was the conditional justification in §4.7 of the spectral multiplicity formula. The conditions from §4.7 included the local theorems, which of course are what we are now trying to
6.6. LOCAL PACKETS FOR COMPOSITE φ
359
establish (for generic parameters). However, the simple arguments from the proof of Lemma 4.7.1 are still available to us here. We shall revisit them very briefly, in order to obtain a modified global formula that will help us prove the local results. Arguing as in the proof of Lemma 4.7.1, we convert the last expression 9 to the φ-analogue ¸
pG1 ,φ1 q 9
9
|Sφ1 |1 |Z pGp1qΓ|1 |OutGpG1, φ1q|1 f 1pφ1q 9
9
9
9
9
9
9
9 of (4.7.8). The double sum is over the set of G-orbits of pairs pG9 1 , φ9 1 q in the 9 It can be replaced by a double family X pG9 q of §4.8 such that φ9 1 maps to φ.
p
9 sum over the set of G-orbits of pairs pφ9 G , x9 G q in the associated family Y pG9 q 9 The expression becomes such that φ9 G maps to φ.
p
¸
pφG ,xG q 9
9
|Sφ |1 f 1pφ1q, 9
9
9
G
since we can write
|Sφ1 |1 |Z pGp1qΓ|1 |OutGpG1, φ1q|1 |Sφ |1, 9
9
9
9
9
G
as again in the proof of Lemma 4.7.1. This summand depends only on the image x9 of x9 G in Sφ9 . The right hand side of (6.6.4) therefore equals
|ΦpG, φq| |Sφ|1 9
9
9
¸
P
f91 pφ9 1 q,
x9 Sφ9
9 x 9 q is 9 φ 9 q, and where we recall that ΦpG, where pG9 1 , φ9 1 q maps to the pair pφ, the preimage of φ9 in ΦpG9 q, a set of order 1 or 2. 9 Consider next the formula (4.1.1) (with ψ φ) for the left hand side 9 of (6.6.4). We claim that φ cannot contribute to the discrete spectrum of 9 any proper Levi subgroup M9 of G. For as we have argued in the past,
our induction hypothesis on the factor G9
P ErpNq 9
of M9 prevents any
P Ψr pN q that does contribute to the discrete spectrum of M r ell pN q that contains φ. Such assertions are of from lying in the subset Φ r pN q, which rests ultimately on course implicit in the definition of the set Φ
parameter ψ9
9
9
9
9
9
the general classification of Theorems 1.3.2 and 1.3.3. There is consequently no contribution from M9 to (4.1.1). The left hand side of (6.6.4) then equals G G Idisc, pf9q tr Rdisc, pf q φ9 φ9 9
9
¸
nφ9 pπ9 G q f9G9 pπ9 G q,
π9 G
where π9 G ranges over the set ΠpG9 q of irreducible unitary representations of G9 pA9 q, and nφ9 pπ9 G q are nonnegative integers. We are assuming that f9 lies in
360
6. THE LOCAL CLASSIFICATION
r pG9 q. This means that f 9 pπ9 G q depends only on the image π9 of π9 G in the H G r pG9 q of orbits in ΠpG9 q under the restricted direct product set Π ¹
r N pG9 q Out
r N pG9 v q. Out
v
We can therefore write the left hand side of (6.6.4) as ¸
|ΦpG, φq| 9
9
nφ9 pπ9 q f9G9 pπ9 q,
P p q
r G9 π9 Π
for a modified coefficient
¸
9 q|1 9 φ nφ9 pπ9 q |ΦpG,
(6.6.5)
P p
9 π π9 G Π G, 9
9 π 9 q. in which ΠpG, 9 q is the preimage of π 9 in ΠpG We have converted (6.6.4) to an identity
¸
(6.6.6)
P p q
r G9 π9 Π
¸
nφ9 pπ9 q f9G9 pπ9 q |Sφ9 |1
P
nφ9 pπ9 G q
q
f91 pφ9 1 q,
r pG9 q, f9 P H
x9 Sφ9
9 x 9 q. We shall apply this identity when f9 equals a where pG9 1 , φ9 1 q maps to pφ, product f9 f98 f9u f98,u , relative to the decomposition 9 q G 9 G9 pA 8 G9 u G9 8,u
GpF8q GpFuq GpA8,uq 9
9
9
9
9
9
of G9 pA9 q. Suppose that v is a valuation of F9 in the complement of S8 puq. The r 9 is then defined, and satisfies the conditions of Theorem 2.2.1. packet Π φv
r 2 pG9 v q, and therefore For if Sφ9 v is infinite, φ9 v lies in the complement of Φ represents the case we have already established. If Sφ9 v is finite, Proposition 6.3.1 tells us that N ¤ 3, in which case the local classification follows from Lemma 6.6.2. We can therefore write 9
¸
pf 8,uq1 pφ8,uq1
(6.6.7)
9
π9 8,u
xx, π8,uypf 8,uqG pπ8,uq, 9
9
9
9
9
where π9 8,u ranges over elements in the packet !
π9 8,u
â
R pq
v S8 u
π9 v : π9 v
of
and
P
φ9 8,u
r9 , Π φv
xx, π8,uy 9
9
x , πv y 1 for almost all v
¹
R pq
φ9 v ,
v S8 u
¹
R pq
v S8 u
xxv , πv y 9
9
)
6.6. LOCAL PACKETS FOR COMPOSITE φ
361
is the corresponding product of local pairings. If v is archimedean, we have
pfv1 qpφ1v q 9
¸
9
xxv , πv y fv,Gpπv q, 9
r9 π9 v PΠ
9
9
9
9
φv
by the results of Shelstad. We can therefore write ¸
pf81 qpφ18q xx, π8y pf8qG pπ8q, 9
(6.6.8)
9
9
9
9
9
9
π9 8
where π9 8 ranges over representations in the packet !
π9 8
â
P
v S8
of
φ9 8 and
xx, π8y 9
π9 v : π9 v ¹
P
v S8
¹
9
P
v S8
P
r9 Π
)
φv
φ9 v ,
xxv , πv y 9
9
is the associated product of local pairings. Consider the remaining valuation v u of F9 . Then F9u F , G9 u G and φ9 u φ. At this point we assume that the function f9u lies in the subspace r cusp pGq of H r pGq. We can then write H f9u1 pφ1u q
(6.6.9)
¸
P p q
r2 G π Π
cφ px, π q pf9u,G9 qpπ q,
in the notation of Lemma 6.5.3. We shall use the global identity (6.6.6) to extract information about the coefficients cφ px, π q. More precisely, we shall exploit the identity obtained by substituting the formulas (6.6.7), (6.6.8) and (6.6.9) for the three factors of the summand
1 pφ9 1 q f91 pφ9 1 q pf98,u q1 f91 pφ9 1 q f98 8 u u
pφ8,uq1 9
in (6.6.6). The right hand side of (6.6.6) becomes
|Sφ|1 9
¸ ¸
xx, π8y cφ,xpπq xx, π8,uy fGpπq, 9
P
9
9
9
9
9
9
x Sφ π9
where the inner sum is over products π9
π8 b π b π8,u 9
9
r 2 pGq and π9 8,u in the packet of representations π9 8 in the packet of φ9 8 , π in Π 8 ,u of φ9 , and x9 is the isomorphic image of x in Sφ9 . Suppose that ξ P Spφ is a character on the 2-group Sφ9 Sφ . It follows from the condition (iii)(a)
of Proposition 6.3.1 that there is a representation π9 8,ξ in the packet of φ9 8 such that xx,9 π9 8,ξ y ξpxq1, x P Sφ.
362
6. THE LOCAL CLASSIFICATION
For the places outside S8 puq, we take the representation π9 8,u p1q
â
R pq
v S8 u
πv p1q
in the packet of φ9 8,u such that for each v, the character
x , πv p1qy on Sφ 9
9
is 1. With an appropriate choice of the functions f8 and f 8,u , the formula 9
(6.6.6) then reduces to an identity (6.6.10)
¸
nφ pξ, π q fG pπ q
π
¸ π
|Sφ|1
¸
P
v
9
cφ px, π q ξ pxq1 fG pπ q
x Sφ
P HrcusppGq, where π is summed over Πr 2pGq on each side, and (6.6.11) nφ pξ, π q nφ π8,ξ b π b π 8,u p1q r 2 pGq are linearly independent Since the characters of representations π P Π r on Hcusp pGq, we conclude that ¸ cφ px, π q ξ pxq1 , (6.6.12) nφ pξ, π q |Sφ |1 for any f
9
9
9
P
x Sφ
r 2 pGq. In particular, the number (6.6.11) does depend only on for any π P Π the local data pφ, ξ, π q at u, as the notation suggests.
r 2 pGq, nφ pξ, π q is a nonnegative integer. Lemma 6.6.4. (a) For any π P Π (b) If mpφq and mpπ q are the orders of the preimages of φ and π in Φ2 pGq and Π2 pGq respectively, the product
n rφ pξ, π q mpπ q1 mpφq nφ pξ, π q
is also a nonnegative integer. Proof. By definition, nφ pξ, π q is the number nφ9 pπ9 q given by (6.6.5), with π9 π9 8,ξ b π b π9 8,u p1q. The summands nφ9 pπ9 G q on the right hand side of (6.6.5) are nonnegative integers. To prove (a), we need to show that the sum itself is divisible by 9 q|. We can therefore assume that G 9 φ p SOpN, Cq, the positive integer |ΦpG, 9 q is an orbit would r N pG9 q under which ΦpG, 9 φ with N even, since the group Out r N pG9 q is a group of order 2, the nontrivial otherwise be trivial. Then Out element being represented by a point in OpN, Cq with determinant equal to p1q. We can also assume that the rank Ni of each component φ9 i of φ9 is r N pG9 q itself. even, since the stabilizer of φ9 would otherwise be the group Out r 9 of With this condition, we consider the archimedean component π9 v P Π φv π9 at any v P S8 . The irreducible components of φ9 v are distinct and of degree 2, by the condition (iii)(a) of Proposition 6.3.1. It follows easily that r N pG9 q-orbit of irreducible representations of G9 pF9 v q, π9 v contains as an Out r N pG9 q acts freely on the indices of two elements π9 v,G . This implies that Out
6.6. LOCAL PACKETS FOR COMPOSITE φ
363
summation π9 G in (6.6.5). Since the original multiplicity nψ9 pπ9 G q is invariant 9 the sum in (6.6.5) is an even under the action of any F9 -automorphism of G, 9 q. This establishes 9 φ integer, and is hence divisible by the order 2 of ΦpG, (a). The proof of (b) is simpler. Since 9 q|, 9 φ mpφq mpφ9 q |ΦpG,
we need only show that the sum in (6.6.5) is in all cases divisible by the r N pGq-orbit of π. This number equals number of elements πG in the Out r N pG9 q acts freely on the indices of summation either 1 or 2. If it equals 2, Out π9 G in (6.6.5). It follows that the product ¸
rφ pξ, π q mpπ q1 n
nφ9 pπ9 G q,
π9 G
is also a nonnegative integer, as required.
We can now establish the essential properties of composite parameters for our group G P Ersim pN q over the local p-adic field F . Proposition 6.6.5. (a) For every φ r 2 pGq, together with a bijection of Π π
ÝÑ x , πy,
r φ onto Spφ , such that from Π
f 1 pφ 1 q
(6.6.13)
r P Φr sim 2 pGq, there is a finite subset Πφ
¸
π
P Πr φ,
xx, πy fGpπq,
f
P HrcusppGq, x P Sφ,
P 1 1 where pG , φ q is the pair (in Xell pGq) that maps to pφ, xq. rφ π Π
r sim pGq, the subsets Π r φ of Π r 2 pGq are disjoint. (b) As φ ranges over Φ 2
Proof. The key ingredient will be the orthogonality relations of Lemma 6.5.3(b). We shall combine them with the formula (6.6.12) for the nonnegative integer nφ pξ, π q. Suppose that ξ P Spφ and ξ1 P Spφ1 are characters of the finite groups r sim pGq. Applying (6.6.12) to each attached to two parameters φ and φ1 in Φ 2 of the integers nφ pξ, π q and nφ1 pξ1 , π q, we write ¸
r 2 pGq π PΠ
¸
n rφ pξ, π q nφ1 pξ1 , π q
mpπ q1 mpφq nφ pξ, π q nφ1 pξ1 , π q
π
mpφq |Sφ|1 |Sφ |1
¸
1
x,x1
ξ px q1 ξ 1 p x 1 q
¸
mpπ q1 cφ px, π q cφ1 px1 , π q ,
π
where x and x1 are summed over Sφ and Sφ1 respectively. The orthogonality relation (6.5.16) then tells us that the last sum over π vanishes unless pφ, xq
364
6. THE LOCAL CLASSIFICATION
equals pφ1 , x1 q, in which case it equals mpφq1 |Sφ |. The original sum over π therefore vanishes unless φ1 φ, in which case it equals
|Sφ|1 This in turn vanishes unless ξ1 established an identity (6.6.14)
¸
P p q
r2 G π Π
¸
P
ξ px q1 ξ 1 p x q.
x Sφ
ξ, in which case it equals 1.
n rφ pξ, π q nφ1 pξ1 , π q
#
We have
1, if pφ1 , ξ1 q pφ, ξ q, 0, otherwise.
Suppose first that pφ1 , ξ1 q pφ, ξ q. Then the nonnegative integers rφ pξ, π q are either both zero or both nonzero. Since the sum nφ pξ, π q and n over π of their product equals 1, we see that there is a unique element π pξ q r 2 pGq such that in Π (6.6.15)
rφ pξ, π q nφ pξ, π q n
#
1, if π π pξ q, 0, otherwise,
r 2 pGq. Then taking the case that pφ1 , ξ1 q pφ, ξ q, we see from for any π P Π (6.6.14) that the mapping
pφ, ξq ÝÑ πpξq is injective. In other words, if we define
tπpξq : ξ P Spφu, φ P Φr sim 2 p G q, r φ u of Π r 2 pGq are mutually disjoint. This is the assertion (b) the subsets tΠ rφ Π
of the proposition. To establish the remaining identity (6.6.13), we have only to invert (6.6.12). It follows from (6.6.15) that cφ px, π q equals ξ pxq if π π pξ q for some ξ P Spφ , and that cφ,x pπ q 0 otherwise. The required formula (6.6.13) then follows from (6.5.15), if we set
xx, πpξqy ξpxq cφpx, πξ q, This completes the proof of the proposition.
x P Sφ , ξ
P Spφ.
r φ , then Corollary 6.6.6. If π belongs to Π
mpπ q mpφq. Proof. The corollary follows from the equality (6.6.15) of n rφ pξ, π q with nφ pξ, π q.
6.7. LOCAL PACKETS FOR SIMPLE φ
365
6.7. Local packets for simple φ We continue with the discussion of §6.6. In this section we shall complete the local classification of tempered representations. We are taking r sim pN q to be a simple datum over the local p-adic field F , for the GPΦ fixed positive integer N . It remains to treat the simple generic parameters r sim pGq. However, to exploit the global construction of §6.2, we will φPΦ have to start with a representation π instead of a parameter φ. r sim pGq for the set of elements in Π r 2 pGq that do not We shall write Π sim r pGq of the disjoint sets Π r φ we constructed lie in the union over φ P Φ 2 in the last section. Let us also write Irsim pGq for the subspace of functions r sim pGq fG P Ircusp pGq such that fG pτ q 0 for every τ in the complement of Π in Trell pGq, and Srsim pGq for the subspace of functions f G P Srcusp pGq such that f G pφq 0 for every φ in Φsim 2 pG q.
Lemma 6.7.1. The isomorphism (6.5.10) maps Irsim pGq isomorphically onto Srsim pGq. Proof. In the statement, Srsim pGq is to be understood as a subspace of 1 collections of functions tf 1 f G u in à
G1 Eell G
P p q
Srcusp pG1 q
such that f 1 0 for G1 G. We observe from (6.5.6) that the orthogonal complement of Irsim pGq in Ircusp pGq may be identified with the space of r sim pGq. By functions on Trell pGq that are supported on the complement of Π (6.5.13), the orthogonal complement of Srsim pGq can be identified with the r sim pGq. It follows space of collections tf 1 u such that f G is supported on Φ 2 easily from the formulas (6.5.5) and (6.6.13) that the mapping (6.5.10) sends the first orthogonal complement onto the second. Since the mapping is an isometry, it does indeed send Irsim pGq isomorphically onto Srsim pGq.
r cusp pGq is a pseudocoefficient of a represenRemark. Suppose that fπ P H r tation π P Πsim pGq. This means that
fπ,G pτ q
#
1, if τ π, 0, otherwise,
for any τ P Trell pGq. Then fπ,G lies in Irsim pGq, according to the definition above. It follows that 1 fπ1 fπG 0, for any datum G1
P ErellpGq distinct from G.
We will now turn to the global arguments. We cannot use the discussion of §6.3, since it was based on the inductive application of Corollary 6.2.4. Corollary 6.2.4 is not available because it requires that the local theorems
366
6. THE LOCAL CLASSIFICATION
r pN q. We have therefore to go back to the be valid for parameters φ P Φ original Lemma 6.2.2, and the arguments from its proof. r sim pGq. Having chosen this object, we apply We fix a representation π P Π 9 π Lemmas 6.2.1 and 6.2.2. We obtain global objects pF9 , G, 9 q that satisfy the conditions of Lemma 6.2.2. In particular, pF, G, π q equals pF9u , G9 u , π9 u q, for a place u of F9 . If v belongs to S8 , π9 v belongs to the L-packet of a Langlands parameter φv in general position. If v lies in the complement of S8 puq, π9 v is a spherical representation, corresponding to a spherical Langlands parameter φv . The key property of π9 is of course that it occurs in the automorphic 9 discrete spectrum of G. This fact was established in the first half of the proof of Lemma 6.2.2. It then led naturally to the main point of the second half of the proof, the application of Corollary 3.4.3 to π. 9 This in turn allowed us
to attach a global parameter φ9
P Ψr pN q to π, which we then showed was in 9
9
r pN q of generic parameters. the subset of Φ In Corollary 6.2.3, we saw that for any valuation v u, the Langlands 9 parameter of the localization π9 v of π9 is the localization φ9 v of φ. Our task here will be to establish properties of the localization 9
φ φ9 u
r sim pGq, of φ9 at u. These will be consequences of the fact that π lies in Π together with the results we have established in sections following Lemma 6.2.2. r ell pN q. (See the We did see in proof of Lemma 6.2.2 that φ lies in Φ discussion following (6.2.8).) Moreover, it is a consequence of (6.2.9) (in the proof of Corollary 6.2.3) that there is a constant cpφq 0 such that
frπ,N pφq cpφq,
for any function frπ fru in Hcusp pN q with frπG fπG . It then follows that the linear form fr Ñ frN pφq on Hcusp pN q vanishes on the kernel of the r cusp pN q to Srcusp pGq. This in turn implies that transfer mapping from H r cusp pN q at all, the component fr pφq of G P Erell pN q in the for any fr P H decomposition (6.1.1) vanishes if G is distinct from G. Therefore, φ satisfies the condition (i) of the definition of a cuspidal lift in §6.5. We do not know a priori that φ is simple. Suppose however that it lies r sim pN q of Φ r sim pN q. Then Lemma 6.6.3 implies that in the complement Φ ell the conditions (i) and (ii) in the definition of a cuspidal lift are equivalent. More precisely, if we apply this lemma to the unique datum G P Erell pN q r sim pG q (the existence of which is guaranteed by our induction with φ P Φ s hypotheses), we see that G G. In other words, φ lies in the subset r sim pGq of Φ r sim pN q. It then follows from the formula (6.6.13) of Proposition Φ 2 ell 6.6.5 that ¸ fπG pφq fπ,G pπ q. rφ π PΠ 9
6.7. LOCAL PACKETS FOR SIMPLE φ
367
r sim pGq that Since fπ is a pseudocoefficient of π, a representation in a set Π r φ , the sum on the right vanishes. On the other hand, since is disjoint from Π φ satisfies the condition (i), we have
fπG pφq frπ,N pφq cpφq 0. This is a contradiction. r sim pN q of Φ r ell pN q. In this case, We have shown that φ lies in the subset Φ the first condition (i) for φ to be a cuspidal lift implies the second condition r sim pGq of Φ r sim pN q by virtue of the (ii). In other words, φ lies in the subset Φ fact that the definition in §6.1 is implied by the condition we have already r sim pN q is therefore a cuspidal lift. established. The local parameter φ P Φ 9 Consider the global parameter φ. It represents a generic automorphic representation of GLpN q whose component at u lies in the discrete series of GLpN, F q, since the corresponding component φ φ9 u of φ9 belongs to r sim pN q. The automorphic representation is therefore cuspidal, and φ9 conΦ r sim pN q of simple global parameters. This argument sequently lies in the set Φ is familiar from the proof of Corollary 6.2.4, where we were working with the benefit of the stronger hypothesis. As in the earlier case, φ9 also satisfies the condition 9
cpφ9 q ξφ9 cpπ9 q ,
π9
P A2 p G q , 9
of Theorem 1.4.1. It therefore meets the formal requirement for belonging
r sim pG9 q of Φ r sim pN q, according to the original definition of to the subset Φ §1.4. In the case here, however, we will have to work for a moment with the provisional definition of §5.1. 9
r9 sim pN q, we are now free to form the assoHaving shown that φ9 lies in Φ ciated family
(6.7.1)
r F 9
Frpφq t`φu 9
9
of global parameters. To make use of the results of Chapter 5, we need to know that the conditions of Assumption 5.4.1 hold, with V again being the set S8 of archimedean valuations. We cannot appeal to Proposition 6.3.1, as we did for the earlier compound parameter (6.6.1). However, the properties we need are already in the earlier Lemma 6.2.2. We know that φ9 v equals the original archimedean parameter φv . Conditions (5.4.1(a)) and (5.4.1(c)) then follow from the constraints we placed on φv in the construction of π9 in the first half of the proof of Lemma 6.2.2. Since the condition (5.4.1(b)) is r9 not relevant to this case, Assumption 5.4.1 is valid for F.
r We We can therefore apply the results of Chapter 5 to the family F. note that the local and global induction hypotheses of Chapter 5 are vacuous here, since the basic simple parameters φ and φ9 have degree N . In particular, we have to rely on the provisional definition of §5.1 for the set Frsim pG9 q. This 9
368
6. THE LOCAL CLASSIFICATION
poses no difficulty. For we established earlier that
Sdisc,φ9 pf9q tr Rdisc,φ9 pf9q ,
r pG9 q, chosen prior in (6.2.7) in the proof of Lemma 6.2.2, for a function f9 P H with the property that the right hand side is nonzero. Therefore, the left hand side is not identically 0. Therefore φ9 belongs to Fr2 pG9 q, according to any of the three equivalent conditions of Corollary 5.4.7. Among the results of Chapter 5 we can use is Lemma 5.4.2, which affirms that Assumption 5.1.1 is valid. Then, following the proof of Lemma 6.6.3, we can establish the local assertion that
frN pφq frG pφq,
r pN q, fr P H
r pGq. We note that we have already for a stable linear form f G pφq on H r cusp pN q, namely proved the weaker assertion for cuspidal functions fr P H that φ is a cuspidal lift. We had to do so independently, in order to show that the parameters φ and φ9 are simple. The main result from Chapter 5 is the stable multiplicity formula
(6.7.2)
G Sdisc, pf9q |ΦpG,9 φ9 q| f9Gpφ9 q, φ9 9
9
r pG9 q, f9 P H
9 for φ. It follows from Corollary 5.1.3, and the application to Lemma 5.1.4 of its supplement Lemma 5.4.6, in case N is even and ηφ9 1. We can now follow the arguments of §6.6, but with one essential difference. In §6.6, we started with a local parameter φ. Through a correspondence φ ÝÑ φ9 ÝÑ π9 ÝÑ π r φ of based on our global construction, we then obtained a local packet Π r representations π in Π2 pGq. In this section, the correspondence has had to go in the opposite direction. Starting with a representation π in the subset r sim pGq of Π r 2 pGq, the global construction has led us through a corresponΠ dence π ÝÑ π9 ÝÑ φ9 ÝÑ φ to a local parameter φ. Since π has been fixed from the beginning, we shall have to write π r 2 pGq that accompanies our attempt to for the variable representation in Π reverse the correspondence. We note that the group Sφ is now trivial. The sums over x P Sφ that were a part of the discussion of the last section will therefore not occur here. In particular, the analogue of the formula (6.6.6) is just
(6.7.3) where
¸
r G9 π9 Π
P p q
nφ9 pπ9 q f9G9 pπ9 q f9pφ9 q,
9 q|1 9 φ nφ9 pπ9 q |ΦpG,
r pG9 q, f9 P H
¸
Π G, 9 π π9 G 9
P p
q
q nφ9 pπ9 G
6.7. LOCAL PACKETS FOR SIMPLE φ
369
as in (6.6.5). It follows directly from the stable multiplicity formula above. We know that φ is a cuspidal lift. We can therefore consider the expansion (6.5.15) of r cusp pGq, f G pφ q , f PH in terms of elliptic values of discrete series characters fG pπ q,
f
P HrcusppGq,
We would like to describe the coefficients cφ pπ q cφ,1 pπ q,
π
π
P Πr 2pGq.
P Πr 2pGq.
Retracing the steps from §6.6, we arrive at a reduction nφ pπ q cφ pπ q
(6.7.4)
of the formula (6.6.12), for nonnegative numbers nφ pπ q nφ p1, π q, as in (6.6.11). Thus
π
P Πr 2pGq,
nφ pπ q nφ9 pπ9 q
is defined by the analogue of (6.6.5) above, in terms of the global representation π9 π9 8 p1q b π b π9 8,u p1q
9 q , and the global multiplicities n pπ r G9 pA in Π φ9 9 G q in the automorphic discrete
9 spectrum of G. The arguments 1 are of no significance in cφ,1 pπ q and nφ p1, π q; they are simply atavistic references to the character on the trivial group Sφ9 . We observe that the global representation
π9
π8 b π b π8,u 9
9
attached to the original representation π is of the required form. Since it 9 occurs in the discrete spectrum of G, the number nφ pπ q attached to π is strictly positive. Finally, the analogue of Lemma 6.6.4 is valid for any π . It tells us that both nφ pπ q and the product rφ pπ q mpπ q1 mpφq nφ pπ q n
are actually nonnegative integers. The next step is to apply the orthogonality relations (6.5.16), as at the beginning of the proof of Proposition 6.6.5. The properties of the coefficients cφ pπ q in Lemma 6.5.3 were stated with the understanding that φ is r sim pGq that is a cuspidal lift. Suppose that φ is an arbitrary parameter in Φ also a cuspidal lift. The corresponding expansion (6.5.15) supplies φ with its own set of coefficients cφ pπ q. It then follows from (6.5.16) that the sum (6.7.5)
¸
r2 G π Π
P p q
mpπ q1 cφ pπ q cφ pπ q
370
6. THE LOCAL CLASSIFICATION
vanishes unless φ equals φ, in which case it equals mpφq1 . Setting φ and making the substitution (6.7.4), we obtain ¸ π
φ,
rφ pπ q nφ pπ q 1. n
It then follows from the fact that n rφ pπ q and nφ pπ q are nonnegative integers that (6.7.6)
n r φ pπ q n φ pπ q
#
1, 0,
if π π, otherwise,
r 2 pG q . for any representation π P Π It follows from (6.7.4) and (6.7.6) that
cφ pπ q
(6.7.7)
#
1, π π, 0, otherwise,
r 2 pGq. The vanishing assertion in the last formula leads in for any π P Π turn to a collapse in the sum (6.7.5). The formula for (6.7.5) reduces simply to the relation mpπ q1 cφ pπ q cφ pπ q 0, in case φ φ. It follows that
cφ pπ q
(6.7.8)
#
1, if φ φ, 0, otherwise,
r sim pGq. In particular, the parameter φ is uniquely for any cuspidal lift φ P Φ determined by the representation π. We see now that the indirect global process by which we associated the parameter φ to π gives a canonical local construction. For it follows from (6.7.8) that the correspondence π Ñ φ is a well defined mapping from r sim pGq to Φ r sim pGq. We have next to show that this mapping is a bijection. Π We shall formulate the assertion in terms of the inverse mapping φ Ñ πφ , as a complement to the correspondence of Proposition 6.6.5. r sim pGq of Φ r sim pN q Proposition 6.7.2. (a) Every parameter φ in the subset Φ is a cuspidal lift. (b) There is a canonical bijection
ÝÑ πφ, φ P Φr simpGq, r sim pGq onto Π r sim pGq such that from Φ r cusp pGq, φ P Φ r sim pGq. (6.7.9) f G pφq fG pπφ q, f PH φ
Proof. The relation (6.7.9) is simply the general expansion (6.5.15) of Lemma 6.5.3, together with the identity (6.7.7) for the coefficients. To establish the proposition, however, we must prove the prior assertion that r sim pGq onto Φ r sim pGq. The the original mapping π Ñ φ is a bijection from Π
6.7. LOCAL PACKETS FOR SIMPLE φ
371
r c pG q condition (6.7.8) implies that the mapping is injective. Let us write Φ sim r c pGq equals Φ r sim pGq. The for its image. Our task is then to show that Φ sim r sim pGq satisfies the problem is that we do not know that every element φ P Φ first condition (i) of a cuspidal lift. It is essentially the question of showing that the union (6.1.2) is disjoint. Consider the subspaces Irsim pGq and Srsim pGq of Ircusp pGq and Srcusp pGq introduced at the beginning of the section. Lemma 6.7.1(a) asserts that the linear mapping fG ÝÑ f G , f P Ircusp pGq,
takes Irsim pGq isomorphically onto Srsim pGq. It is clear from the original definition that the invariant pseudocoefficients
P Πr simpGq, r sim pGq, set are a basis of the complex vector space Irsim pGq. For any π P Π f φ fπG , r c pGq is the image of π. Then f φ can be regarded as a stable where φ P Φ sim fπ,G ,
π
pseudocoefficient of φ, in the sense that #
1, if φ φ, f p φ q φ
0, otherwise,
for any φ
P Φr csimpGq. The stable pseudocoefficients r c p G q, f φ, φPΦ sim are then a basis of the complex vector space Srsim pGq.
We can augment this basis with objects from the last section. The r sim pGq are all cuspidal lifts. They also provide stable parameters φ in Φ 2 pseudocoefficients f φ , which can be expressed in terms of the corresponding packets by fφ
|Sφ|1
¸
P
fπG ,
rφ π Π
and which span the orthogonal complement of Srsim pGq in Srcusp pGq. The expanded family of pseudocoefficients f φ,
r c p G q, φPΦ 2
parametrized by the set r c p Gq Φ r sim pGq Φ 2 2
²
r c pG q , Φ sim
then forms a basis of Srcusp pGq. r sim pGq. By definition, φ is a paSuppose that φ is any element in Φ r sim pN q such that the linear form frG pφ q on Srcusp pGq defined by rameter in Φ r c pGq that is distinct (6.1.1) is nonzero. Suppose that φ is any parameter in Φ 2
372
6. THE LOCAL CLASSIFICATION
from φ . By Proposition 2.1.1, we can choose a function frφ is a preimage of φ, in the sense that
pfrφq
#
f φ, 0,
P HrcusppN q that
if G G, otherwise,
for any G P Erell pN q. Since φ is a cuspidal lift, we can regard frφ as a φ twisted pseudocoefficient of φ. In particular, pfrN qpφq vanishes, since φ and r ell pN q. It follows from (6.1.1) that φ represent distinct elements in Φ
pf φqpφq pfrφqGpφq frNφ pφq 0. On the other hand, there must be a φ such that f φ pφ q is nonzero, since Srcusp pGq is spanned by the functions f φ . There must therefore be a paramr c pGq that equals φ . Since φ P Φ r sim pGq, φ must lie in Φ r c p G q. eter φ P Φ 2 sim c r sim pGq equals Φ r pGq. This gives the assertion We have shown that Φ sim r c pGq are cuspidal lifts. It (a) of the proposition, since the elements in Φ sim r sim pGq to Φ r sim pGq is also tells us that the original mapping π Ñ φ from Π surjective, and hence a bijection. The inverse mapping φ Ñ πφ therefore exists, as asserted in (b). The proof of the proposition is complete.
Corollary 6.7.3. If π equals πφ , then mpπ q mpφq.
Proof. The corollary follows from the equality of n rφ pπ q with nφ pπ q in (6.7.6). We define the L-packet of any simple parameter φ singleton rφ Π
P Φr simpGq to be the
tπφu.
r φ to any Propositions 6.6.5 and 6.7.2 together thus attach an L-packet Π r 2 pGq. However, there is still square integrable Langlands parameter φ P Φ something more to be said about the corresponding character identities. We shall state it as a joint corollary of the two propositions. r 2 pGq. Then the Corollary 6.7.4. Suppose that φ is any parameter in Φ character identity
(6.7.10)
f 1 p φ1 q
¸
rφ π PΠ
xx, πy fGpπq,
x P Sφ ,
established in Propositions 6.6.5 and 6.7.2 for a cuspidal function f , remains r pG q . valid if f is an arbitrary function in H Proof. At first glance, the assertion might seem to be immediate. In the case that φ is not simple, for example, the global formula (6.6.6) from which we derived the cuspidal version (6.6.13) of the required character
6.7. LOCAL PACKETS FOR SIMPLE φ
P HrpGq.
identity is valid for any function f9 formula (6.7.11)
9
373
Likewise, the local transfer
frN pφq frG pφq
r pGq. However, we need to be of Lemma 6.6.3 holds for any function f P H careful. For we do not know a priori that the right hand side of (6.7.11) equals the value at x 1 of the right hand side of (6.6.13), if the function f r cusp pGq. Nor can we expect an immediate belongs to the complement of H rescue from (6.6.6). For we cannot a priori rule out the possible existence of global representations π9 on the left hand side for which π9 u is not elliptic. The simplest way to establish the corollary is to appeal to the two general theorems of [A11], and their twisted analogue for GLpN q (which we discussed briefly in the proof of Proposition 2.1.1). The proof of these theorems is not elementary, and has not yet been written down in the twisted case. However, we have already used them in the proof of Proposition 2.1.1. We may as well use the theorems again here, since it is exactly for this sort of problem that they were intended. In particular, they serve as a substitute for the fundamental lemma for the full spherical Hecke algebra [Hal]. The coefficients on the right hand side of (6.7.10), regarded as a sum over r temp pGq whose summands are supported on the subset Π r φ , are the full set Π already determined from the case that f is cuspidal. This is because the characters of discrete series are determined by their values on the strongly r pGq. Suppose elliptic set. Suppose then that f is a general function in H G also that we define f pφq alternatively as the value at x 1 of the right hand side of (6.7.10), rather than the left hand side of (6.7.11). Theorem 6.1 of [A11] then asserts that the resulting linear form in f is stable. Theorem 6.2 of [A11] asserts in addition that (6.7.10), with the left hand side being the analogue for G1 of the linear form just defined, is valid for f and for any element x in Sφ . Lastly, the twisted analogue for GLpN q of Theorem 6.2 of [A11] asserts that (6.7.11) holds for f . In other words, the alternative definition for f G pφq just given matches the one we have always used. We conclude that the character identity (6.7.10) does hold for the general function f .
We have established the general character identity of Theorem 2.2.1(b), r 2 pGq. Since Lemma 6.6.3 takes care of its first assertion for parameters φ P Φ (a), Theorem 2.2.1 itself is valid for any such parameter. We note that the character identity (6.7.10) also allows us to complete some of the global r Applied to the multiplicity formulas induction hypothesis for the family F. (6.6.6) and (6.7.3) we have deduced, it gives the contribution of φ9 to the discrete spectrum postulated in Theorem 1.5.2. Observe that Corollary 6.6.6 of Proposition 6.6.5 can be regarded as a proof of the assertion (b) of Theorem 2.2.4. Together with Corollary 6.7.3 of Proposition 6.7.2, it resolves an important induction hypothesis from Proposition 6.6.1. We are speaking of the hypothesis that for any parameter 9
374
6. THE LOCAL CLASSIFICATION
r 2 pGq in Φ r bdd pGq, the group W pπM q equals Wφ , in φ in the complement of Φ the notation of (6.5.3). This is what allowed us to conclude that the mapping (6.5.3) was surjective, which we recall was a starting point for Proposition 6.6.1. Proposition 6.6.1 itself tells us that Theorem 2.2.1 holds for any such φ. Theorem 2.2.1 is thus valid for any generic, bounded parameter r bdd pGq. Recall that this includes the assertion (a) of Theorem 1.5.1. φPΦ We shall state what remains of Theorem 1.5.1(b) as a joint corollary of the three Propositions 6.6.1, 6.6.5 and 6.7.2. r temp pGq of irreducible, tempered (Out r N pG q Corollary 6.7.5. The set Π r bdd pGq orbits of ) representations of GpF q is a disjoint union over φ P Φ r φ. of the packets Π
Proof. Proposition 6.6.1 includes the assertion that the complement of
p q
p q
r 2 G in Π r temp G is a disjoint union of packets Π r φ , in which φ ranges over Π r 2 G in Φ r bdd G . Propositions 6.6.5(b) and 6.7.2(b), the complement of Φ r sim G , tell us that Π r 2 G is the combined with the definition of the set Π r 2 G of the remaining packets Π r φ . The corollary disjoint union over φ Φ
follows.
p q
p q
P p q
p q
p q
We have now proved both of the local Theorems 1.5.1 and 2.2.1 for r bdd pGq. This gives the local Langlands correspondence parameters φ P Φ r N pGq) for any quasisplit orthogonal or (up to the action of the group Out symplective group G P Ersim pN q over the p-adic field F . Actually, we should really say that the local classification will be complete once we have resolved the induction assumptions on which the arguments of the chapter have been based. We shall do so now. 6.8. Resolution We have established Theorems 1.5.1, 2.2.1 and 2.4.1 for generic local parameters φ. These are the theorems that characterize the tempered representations of our classical group G P Ersim pN q over the p-adic field F . We are not quite done, however. As we have just noted, we have still to resolve the induction hypotheses on which the proof of the three theorems has been based. There are actually three kinds of hypotheses. First and foremost, we have the supplementary local Theorems 2.2.4 and 2.4.4 to prove. These theorems are of interest in their own right, since they give new information about characters and local harmonic analysis. But they were also forced on us for the statement of Theorem 2.4.1. We must prove them for the r pN q in order to complete the local hypothesis of this relevant parameters in Φ chapter. Secondly, we must also complete the global induction hypothesis of the chapter. That is, we must convince ourselves that the global theorems r9 pN q. Lastly, we need to resolve the temporary hold for parameters in F
6.8. RESOLUTION
375
r sim pGq in §6.1 in terms of the natural definition definition of the local set Φ provided by local Langlands parameters. This will take care of Theorem 6.1.1, the last of the local assertions. Consider then the supplementary local Theorems 2.2.4 and 2.4.4. They concern the case of a datum G P Ersim pN q, where N is even and p SOpN, Cq. As we recall, the standard outer automorphism of G gives a G r under G. The theorems apply in this case to a generic parameter bitorsor G r q of Φ r pGq. The proofs are similar to those of φ ψ in the subset of ΦpG Theorems 2.2.1 and 2.4.1. In particular, they rely on the global methods that originate with the construction of §6.3. However, the arguments now are considerably easier. For one thing, the parameter φ is never simple, so we have no need of the general discussion of §6.7. For another, we will be able to use a number of properties that were established ab initio in the earlier arguments. Suppose for a moment that
`1ψ1 ` ` `r ψr p r q. Recall that the centralizer S rψ in G r of is a general local parameter in ΨpG ψ
the image of ψ is an Sψ -torsor. It has a decomposition (6.8.1) Srψ
¹
P p q
Op`i , C
ψ
i Iψ G
¹
G i Iψ
P p q
Spp`i , Cq
¹
P
GLp`j , Cq
j Jψ
that is parallel to (1.4.8). We have written p q ψ here for the preimage of p1q under the sign character ξψ (noting that this minus sign is not related pGq). Similar remarks hold if the local pair pG, ψq to the superscript in IG 9 q over a global field F 9 ψ 9 . In the global case, the is replaced by a pair pG, 9 r In particular, we can form the definitions of §3.3 apply to the G-bitorsor G. 9 ψ-component 9
G Irdisc,ψ9 pfrq Idisc, pfrq, ψ9 9
9
r q, fr P HpG
9
9
9
r9 following (3.3.12). It of the discrete part of the twisted trace formula for G, satisfies the analogue
(6.8.2)
Irdisc,ψ9 pfrq 9
¸ 91 9 G Eell G
P p q
r G r1 q S r1 ιpG, pfr1q disc,ψ9 9
p
9
9
9 of (3.3.15). We can also form the ψ-component
G r r r R disc,ψ9 pf q Rdisc,ψ9 pf q, 9
9
9
9
r q, fr P HpG 9
9
G 9 q. of the canonical extension of the representation Rdisc, of G9 pA ψ9 9
376
6. THE LOCAL CLASSIFICATION
Returning to the focus of this chapter, we consider a generic local parameter φ `1 φ1
` ` `r φ r ,
P Φr simpGiq,
φi
Gi
P ErsimpNiq,
r q. We use Proposition 6.3.1 to construct simple global parameters in Φdisc pG r sim pG9 i q, and the associated family φ9 i P Φ
Frpφ1, . . . , φr q of compound global parameters. If pG, φq is the global pair attached to pG, φq, then φ lies in ΦpGrq. The corresponding centralizers Srφ and Srφ are r F 9
9
9
9
9
9
9
9
isomorphic. They are given by a product (6.8.1), but without the general linear factors, since the simple constituents φi of φ are self dual. The other conditions of Proposition 6.3.1, which were used to validate the results of Chapter 5, will not be needed here. Having described the underlying global machinery, we can be brief in r We will be content simply our treatment of the two local theorems for G. to sketch a proof of the main cases, those of square integrable parameters r q for Theorem 2.2.4, and elliptic parameters φ P Φell pG r q for Theoφ P Φ 2 pG rem 2.4.4. The various arguments of descent needed to deal then with the general cases are similar to those of the corresponding Theorems 2.2.1 and 2.4.1, and will be left to the reader. r q. In the We consider Theorem 2.2.4 first, for a parameter φ in Φ2 pG r r case F R, we treat G as we have GpN q, as an object that is part of the work in progress by Shelstad and Mezo. We shall therefore assume that F is a p-adic field, as in §6.6. We then have #
(6.8.3)
φ φ1 ` ` φr , Srφ Op1, Cqr q φ,
r sim pGi q and Gi P Ersim pNi q, with the requirement that some Ni be for φi P Φ odd. The proof of Theorem 2.2.4 will be a straightforward application of the discussion of §6.6.
9 r9 q. As in We first apply (6.8.2) and Corollary 4.1.3 to a function fr P HpG the remarks preceding (6.6.6), we obtain
Irdisc,φ9 pfrq 9
¸ 1 G
r G r1 q S r1 ιpG, pfr1q disc,φ9 9
p
9
9
9
|Srφ|1|ΦpG, φq| 9
9
9
¸
P
r Srφ x
fr1 pφ9 1 q, 9
9 x r 1 , φ9 1 q maps to the pair pφ, r q is r9 q. The condition that φ9 lies in Φ2 pG where pG equivalent to the property 9
9
|ΦpG, φq| 1. 9
9
6.8. RESOLUTION
377
The twisted analogue of (6.6.6) then becomes (6.8.4)
¸
r9 q |Sφ9 |1 nφ9 pπ9 q fr9 pπ 9
P p q
9 π9 Π G
G
¸
fr1 pφ9 1 q, 9
P
r Srφ x
G r9 is the canonical exten, and π where nφ9 pπ9 q is the multiplicity of π9 in Rdisc, φ9 9
9 q defined whenever n pπ r pA sion (4.2.7) of π9 to G φ9 9 q is nonzero. Recall that in the proof of the local results in §6.6 and §6.7, we evaluated nφ9 pπ9 q. We showed that it satisfies the formula of Theorem 1.5.2,with mφ9 being equal to 1, at least if the component π9 8,u of π9 outside S8 puq equals π9 8,u p1q. With 8,u r9 this condition on π , nφ9 pπ9 q vanishes unless the representation π9 8 lies in 9
9 the product Πφ9 8 of the archimedean packets Π φ9 v , the representation π
πu 9
Πφ at F Fu, and the character xx, π8y xx, πy, x P Sφ, is trivial, in which case nφ pπ q 1. We have only to investigate the form lies in the packet Πφ
9
9
u
9
9
9
9
taken by (6.8.4) for a suitable product fr fr8 f fr8,u , 9
9
9
f
fru. 9
Let ξ be a fixed character on Sφ . By the condition (iii)(a) of Proposition 6.3.1, we can choose a representation π9 8,ξ in Πφ9 8 that maps to the character ξ 1 in Sφ . The archimedean form of Theorem 2.2.4, which we are taking r9 8 determines an r9 8,ξ of π9 8 to G for granted, implies that any extension π
extension ξr of ξ to Srφ . We choose the first factor fr8 9
9 fr 9 pπ 8,G r8 q 9
#
1, if π9 8 π9 8,ξ , 0, otherwise,
P HpGr8q so that 9
for any representation π9 8 in the packet Πφ9 8 . We choose the third factor 9 r9 8,u q so that fr8,u P HpG #
1, if π9 8,u π9 8,u p1q, r9 8,u q fr8,u pπ 9
G 9
0, otherwise,
for any representation π9 8,u in the packet Πφ9 8,u . We recall that π9 8,u p1q is the product over v R S8 puq of those representations π9 v P Πφ9 v with x , π9 v y 1.
9 8,u q. At the remaining r pA r9 8,u p1q to G It therefore has a canonical extension π 9
r q H pG r u q. This repreplace u, we take f f9u to be any function in HpG sents a minor departure from this stage of the discussion in §6.6, where f was chosen to be cuspidal. 9
378
6. THE LOCAL CLASSIFICATION
Consider first the right hand side
|Sφ|1
¸ x r
¸
fr1 pφ9 1 q |Sφ |1 9
fr18 pφ9 18 q fr1 pφ1 q pfr8,u q1 9
x r
9
pφ8,uq1 9
of (6.8.4). Ruling out the trivial case that N 2, we shall assume that N ¥ 4, since N is even. It follows from the condition (ii) of Proposition
r v q for any v R S8 puq. It then follows from 6.3.1 that φ9 v does not lie in Φ2 pG Theorem 2.4.4 (which we will discuss in a moment), and the appropriate descent argument that 9
9
¸
pfr8,uq1 pφ8,uq1 9
xxr, πr8,uy fr8G,upπr8,uq 1. 9
π9 8,u PΠφ8,u
9
9
9
9
9
We see also from the definitions that fr18 pφ9 18 q
¸
9
P
π9 8 Πφ9 8
xxr, πr8y fr8,Gpπr8q ξrpxrq1. 9
9
9
9
9
The right hand side of (6.8.4) becomes
|Sφ|1
¸ r x
rq1 fr1 pφ1 q, ξrpx
r 1 , φ1 q is the preimage of pφ, x rq. To describe the other side, we where pG write π pξ q as in §6.6 for the representation of GpF q in the packet Πφ with x , πy ξ, and we take πrpξq πpξrq to be the extension of πpξq determined by the canonical extension (4.2.7) of
π8,ξ b πpξq b π8,up1q, the canonical extension of π 8,u p1q, and the extension we have chosen for π9
9
9
9
π9 8,ξ . The left hand side of (6.8.4) becomes ¸
r9 q nφ9 pπ9 q fr9 pπ 9
P p q
G
π9 Π G9
¸ π9
9 r r8,u 9 8,u r r nφ9 pπ9 q fr 9 pπ 8,G r8 q fGr pπrq f G9 pπr q fGr πpξ q . 9
9
The formula (6.8.4) therefore reduces to (6.8.5)
frGr π pξrq
|Sφ|1
¸
P
x r Srφ
ξrpx rq1 fr1 pφ1 q,
for the given character ξ P Spφ . r is any fixed point in Srφ , the product of ξrpx r q with either side of If x (6.8.5) is independent of the extension ξr of ξ. We sum each of the two
6.8. RESOLUTION
P Srφ. Since
products over ξ
|Sφ|1
¸
P
ξ Spφ
379
#
rx r , 1, if x r q ξrpx rq1 ξrpx
0,
the right hand simplifies. We conclude that fr1 pφ1 q
¸
P
ξ Spφ
otherwise,
xxr, ξry frGr πpξrq .
The extension ξr of ξ to Srφ is arbitrary. It determines the extension r pF q on the right hand side. The formula becomes π rpξ q π pξrq of π pξ q to G fr1 pφ1 q
¸
P
xxr, πry frGr pπrq,
π Πφ
r q. fr P HpG
This is the assertion (a) of Theorem 2.2.4. We recall that the assertion (b) of the theorem was Corollary 6.6.6. We have therefore established Theorem r q. 2.2.4, the first supplementary local theorem for parameters φ P Φ2 pG Consider next the assertion of Theorem 2.4.4, for a parameter φ in the r q in Φell pG r q. The conditions on φ are complement of Φ2 pG $ &φ
(6.8.6)
2φ1 ` ` 2φq ` φq 1 ` ` φr , %S rφ O p2, Cqq O p1, Cqrq , q ¥ 1, ψ
φ,reg in the with the requirement that there be a regular element w P W φ for pG, r φq. This is the twisted analogue for G r of the main case Weyl set W (6.4.1) treated in §6.4. There is no need to be concerned with the other two r follow exceptional cases (4.5.11) and (4.5.12). Their local analogues for G from Corollary 6.4.5, together with a natural descent argument. We are r of Lemma 2.4.2, which represents also taking for granted the analogue for G a reduction of M by descent. In other words, we shall assume as in §6.4 that M is minimal with respect to φ, in the sense that there is a parameter r 2 pM, φq. φM P Φ Theorem 2.4.4 can be proved in the same way as Theorem 2.4.1. Given 9 M 9 φ, 9 the local objects G, φ, M and φM , we again choose global objects G, 9 9 r 9 and φM according to Proposition 6.3.1. Then φ lies in the subset Φell pGq of r9 q over the global field F9 . The first step is to establish the G-analogue r9 Φell pG
(6.8.7)
¸
P
r Srφ,ell x
9 9 x 9 r r9 q fr fr19 pφ, q 9 pφ, x 9
G
G
0,
r q, fr P HpG 9
9
of the global identity (6.4.2). Recall that (6.4.2) was one of the assertions of the refinement Lemma 5.4.3 of Lemma 5.2.1. The original lemma was given by a very simple case of the standard model, which applies equally well to r Another consequence of Lemma 5.4.3 is that φ9 does not contribute to G. 9
380
6. THE LOCAL CLASSIFICATION
the discrete spectrum of any datum G
P ErsimpN q. 9
This implies that the
r of the sum (5.2.7) in Lemma 5.2.1 vanishes. The formula analogue for G 9
r of the lemma itself. (6.8.7) then follows from the analogue for G The next step is to remove the contributions to (6.8.7) of the valuations r of Lemma v in the complement of S8 puq. This reduces to the analogue for G 6.4.1, which amounts to the assertion of Theorem 2.4.4 in case 9
Ni
1,
1¤i¤r
¤ 3.
Since the Levi subgroup M of G can have no symplectic factors, we see from the first observation in the proof of Lemma 6.4.1 that M is actually abelian. It is then easy to see that G is a quasisplit form of either SOp4q or SOp2q. We leave to the reader the exercise of establishing Theorem 2.4.4 in either r of these cases. Once we have the G-analogue of the lemma, we can use the descent argument following its proof to treat the contributions to (6.8.7) of valuations v (6.8.8)
R S8puq. This reduces (6.8.7) to the Gr analogue ¹ ¸ ¹ 1 r r rv q rv q 0 f pφv , x f pφv , x v,G v,G 9
9
P pq
P
v S8 u
r Srφ,ell x
9
9
9
9
P pq
9
9
9
v S8 u
r v q. of (6.4.6), for functions frv P HpG The assertion of Theorem 2.4.4 for φ is the local identity 9
(6.8.9)
9
frG1r pφ, x rq fGr pφ, x rq,
r q. x r P Srφ,ell , fr P HpG
As in the proof of Theorem 2.4.1, the main point is to treat the special case that F equals R, and φ is in general position. To do so, we need to persuade r ourselves that Lemmas 6.4.2 and 6.4.3 remain valid if G is replaced by G. For Lemma 6.4.2, we use the simply transitive action x r
ÝÑ
x r xM ,
x r P Srφ,ell , xM
P Sφ
M
,
of the group SφM on Srφ,ell in place of its action on Sφ,ell . The proof for r For Lemma 6.4.3, we again introduce G then carries over directly to G. 7 supplementary degrees Nj . The original degrees Ni of φ of course equal 1 or 2, since F is archimedean. If Ni 2 for each i ¡ q, we set Nj7
#
Nj , Nq 1
as in the original proof. If Ni to be q 1, we set
Nr ,
1 for some index i ¡ q, which we can take
$ ' &Nj ,
Nj7 1, ' %
Nq
if i ¤ q, if i q 1,
1
Nr ,
if i ¤ q, if i q 1, if i q 2.
6.8. RESOLUTION
381
We leave the reader to verify that with this modification, the proof of Lemma r 6.4.3 for G carries over directly to G. Once we have established (6.8.9) for F R and φ in general position, we can apply the arguments Proposition 6.4.4. These arguments carry over r They yield the formula (6.8.9) for any F and φ. This directly from G to G. completes our discussion of Theorem 2.4.4, the remaining supplementary local theorem. r9 The second question to resolve is the global induction hypothesis for F. The global assertions are Theorems 1.5.2, 1.5.3, 4.1.2 and 4.2.2 (as well as Theorems 1.4.1 and 1.4.2, taken as implicit foundations for Theorem 1.5.2). r pN q. As We must check that they are valid for parameters φ9 in the set F 9
r satisfies Assumption we noted at the end of §6.3 and in §6.7, the family F 5.4.1. It consequently also satisfies Assumption 5.1.1, by Lemma 5.4.2. We 9
r can therefore apply any of the results of Chapter 5 to parameters φ9 P F. The central global assertion is the stable multiplicity formula of Theorem 4.1.2. The general global induction argument used in Proposition 4.5.1 can 9
r9 It yields the stable multiplicity formula for be restricted to the family F.
r pN q that do not fall into the critical cases studied in Chapparameters φ9 P F ter 5. For these remaining cases, the formula was established in Lemmas 9
r pN q. 5.4.3–5.4.6. Theorem 4.1.2 therefore holds for any parameter φ9 P F Theorem 1.5.2 is of course the actual multiplicity formula. It describes 9
representations in the discrete spectrum of a group G9 P Er sim pN q. For the packet attached to a parameter φ9 P Fr2 pG9 q, the formula follows from the local classification and the stable multiplicity formula, as we noted after Corollary 6.7.4, or as we can see from a direct appeal to the obvious variant 9
r pN q, Theorem of Lemma 4.7.1. If φ9 lies in the complement of Fr2 pG9 q in F 1.5.2 is a vanishing assertion. It follows from (4.4.12), (4.5.5) and the stable multiplicity formula. Theorem 1.5.3 has two parts. The assertion (a) follows from Lemma 5.4.6. The assertion (b) is not part of the induction argument for generic parameters, as we can see from the remarks at the beginning of §5.1. It need not concern us here. For Theorem 4.2.2, the essential assertion (b) applies 9
r q. It is a natural biproduct of the discussion of the to parameters φ9 P F2 pG local Theorem 2.2.4 above. Finally, Theorems 1.4.1 and 1.4.2 follow from our resolution of the definition of the global set Frsim pG9 q in Corollary 5.4.7. r sim pGq. We The last question concerns the definition of the local sets Φ need to resolve the temporary definition in §6.1 in terms of the natural definition in terms of local Langlands parameters. To do so, we introduce a third condition in terms of local L-functions. We assume for the rest of the section that the local field F is p-adic, although the discussion would also be valid in the archimedean case. Suppose 9
382
6. THE LOCAL CLASSIFICATION
r sim pN q is a simple datum. The question is whether φ lies in the that φ P Φ r sim pGq of Φ r sim pN q. Following the global construction from Lemma subset Φ r pN q. Then 5.3.2, we introduce the local parameter φ φ ` φ in the set Φ N 2N is even and ηψ 1, and there are two split groups G and G_ in Ersim pN q over F . They share a maximal Levi subgroup M GLpN q. As in the global case, G is distinguished from G_ by the condition that p contain the product G p G. p G To emphasize the analogy with the global resolution of Corollary 5.4.7, let us write δφ 1 if the local Langlands-Shahidi L-function Lps, φ, ρ_ q attached to a maximal parabolic subgroup P _ M N _ of G_ has a pole at p is orthogonal, s 0. This is what we expect if φ maps LF into L G. For if G _ x . The restriction of ρ_ to ρ is the symmetric square representation of M x then contains the trivial representation. Similarly, the image of L G in M _ p x , if G is symplectic, ρ is the skew-symmetric square representation of M L whose restriction to the image of G again contains the trivial representation. We can also write δφ 1 in case it is the L-function Lps, φ, ρ q attached to a maximal parabolic subgroup P M N that has a pole at s 0. This of course is what we have to rule out. r sim pN q and that G belongs to Ersim pN q. Corollary 6.8.1. Suppose that φ P Φ Then the following conditions on the pair pG, φq over our local p-adic field F are equivalent. (i) The linear form frG pφq on Srcusp pGq in (6.1.1) does not vanish. (ii) As a local Langlands parameter for GLpN q, φ factors through the image of L G in GLpN, Cq. (iii) The local quadratic characters ηG and ηψ are equal, and the local L-function condition δφ 1 holds.
Proof. The conditions (i) and (ii) represent our two possible definitions
p q
r sim G . They will be linked by the last condition (iii). of Φ
We recall that Henniart [He2] has established the identities Lps, φ, S 2 q Lps, S 2 φq
and
Lps, φ, Λ2 q Lps, Λ2 φq of local L-functions. The left hand sides are Langlands-Shahidi L-functions attached to the square integrable representation πφ of GLpN, F q, while the right hand sides are the L-functions attached to representations of LF . With these relations, the corollary becomes a consequence of the isomorphism between the representation theoretic and endoscopic R-groups of the parameter r pG q. For it follows from the remarks preceding the statement of φ P Φ the corollary that (ii) is equivalent to (iii), with L-functions interpreted as on the right. It follows from the local intertwining relation that (i) is also equivalent to (iii). More precisely, (i) is valid if and only if the induced representation IP pπφ q is reducible, since the left hand side of the analogue of
6.8. RESOLUTION
383
(2.4.7) for pG , φ q then represents the expected nonzero linear form φ φ on Srcusp pG Gq. By the definition of the representation theoretic R-group Rpπφ q, IP pπφ q is reducible if and only if the Plancherel density µP pπφ,λ q is nonzero at λ 0, or equivalently, the normalizing factor rP
|P pπφ,λ q rP |P pφλ q
has no pole at λ 0. The equivalence of (i) with (iii) then follows from the definition (2.3.3) of the normalizing factors in terms of L-functions. Corollary 6.8.1 is clearly parallel to its global counterpart Corollary 5.4.7, from Chapter 5. Together, they resolve the temporary definitions of the r sim pGq from §5.1 and §6.1 in terms of the original local and global sets Φ definitions. In each case, it is the condition (i) that gives the temporary definition in terms of harmonic analysis, and condition (ii) that gives the original definition. Notice, however, that while the global conditions (i) and (ii) of Corollary 5.4.7 are closely related, the local conditions (ii) and (iii) of Corollary 6.8.1 are the ones that are most closely related to each other. There is also another difference. Corollary 5.4.7 was proved, for the general family Fr with local constraints in §5.4, as part of a running induction
r argument on N . Corollary 6.8.1 was proved, with the help of the family F constructed in this chapter, only after the induction hypothesis on N had been resolved. With the proof of Corollary 6.8.1, we have reconciled the temporary defr sim pGq in terms of harmonic analysis with the natural definition inition of Φ in terms of local Langlands parameters. In particular, we have established Theorem 6.1.1. This completes our proof of the local Langlands corresponr N pG q dence for any group G P Ersim pN q (up to the action of the group Out of order 2, in case G is of type Dn ). 9
CHAPTER 7
Local Nontempered Representations 7.1. Local parameters and duality We come now to the local theorems for nongeneric parameters ψ. Our general strategy will be similar to that used in the last section to treat generic parameters φ. Given local objects pF, G, ψ q, we shall construct cor-
9 q, and a family of global parameters F 9 ψ r to responding global objects pF9 , G, which we can apply the results of Chapter 5. This section is parallel to §6.1. Its purpose is to examine local parameters that occur as completions ψ9 v of what will be our special global parameters 9 These completions will again fall into different categories. First of all, we ψ. will have the place v u at which ψ9 v equals the given local parameter ψ. To obtain information about ψ9 u , we will again need to control the behaviour of completions φ9 v at a set of places v P V distinct from u. In contrast to the last chapter, however, V will be a finite set of p-adic places rather than u of complementary archimedean places. the set S8 We are assuming in this chapter that F is a local field. Suppose r pGq is a general local parameter for a simple endoscopic datum that ψ P Ψ G P Ersim pN q over F . The main local result to establish is Theorem 2.2.1. The first assertion (a) of this theorem is now known. It was reduced in Lemma 2.2.2 to the generic case, which we resolved in Lemma 6.6.3. In r N pGq-symmetric linear form particular, the Out 9
(7.1.1)
f 1 pψ 1 q,
f
P HrpGq,
of the second assertion (b) of the theorem is well defined. The main point is to show that it has an expansion (2.2.6). We shall introduce a formal expansion of (7.1.1). Recall that the pair 1 pG , ψ1q is the preimage of pψ, sq, for a semisimple element s in Sψ . We first observe that f 1 pψ 1 q depends only on the image x of s in the quotient Sψ of Sψ . This would of course be a consequence of the local intertwining relation for ψ, which we have yet to prove. However, the claim is easy to establish directly. It follows from the local form of the discussion of the left hand side of (4.5.1), given prior to the statement of Proposition 4.5.1. Knowing that (7.1.1) depends only on x, we can expand it formally as a linear combination of irreducible characters on Sψ . Building a translation of r N pGq-symmetric Sψ by sψ into the definition, we obtain a general set of Out 385
386
7. LOCAL NONTEMPERED REPRESENTATIONS
linear forms f
ÝÑ
fG pσ q,
σ
P Σr ψ , f P HrpGq,
which is in bijection with the set of irreducible characters x on Sψ , such that (7.1.2)
ÝÑ xx, σy,
f 1 pψ 1 q
¸
P
rψ σ Σ
x P Sψ ,
xsψ x, σy fGpσq,
r pGq. The assertion (b) of Theorem 2.2.1 is that each for any x P Sψ and f P H r ψ is a nonnegative, integral linear combination of (Out r N pGq-orbits of) σPΣ irreducible unitary characters on GpF q. r ψ the packet of ψ, in anticipation We shall take the liberty of calling Σ r ψ and Π r ψ will then represent the same of what we expect to prove. For Σ object, differing only in their interpretations as packets over the respective r unit pGq. sets Spψ and Π We will use the expansion (7.1.2) to study the localization ψ9 u of ψ9 we want to understand. It will be necessary to allow the valuation v to be archimedean as well as p-adic. This is because the archimedean packets of [ABV], which exist for general groups, are not defined by twisted transfer to GLpN q. In fact, it will be valuations in the union
U
S8puq S8 Y tuu
of S8 and tuu that parametrize completions φ9 v for which we have to establish Theorem 2.2.1. The set U will in some sense be the analogue of the singleton tuu from Chapter 6. The rest of the section will be devoted to parameters that are to be localizations ψ9 v at places v in complement of U . Such valuations are of course nonarchimedean. We will break them into a disjoint union of a finite set V , and the complement S V,U of V and U . The localizations φ9 v at v P S V,U will play the role of the spherical parameters described in §6.1. As we have already said, the remaining set V will be the analogue of the set u in Chapter 6. We will take it to be any large finite set of valuations v of S8 F9 for which the degrees q9v of the corresponding residue fields are all large. We assume for the rest of the section that the local field F is nonarchimedean. Suppose for the moment that G is a general connected reductive group over F . An irreducible representation π of GpF q determines a connected component in the Bernstein center of G. This is given by a pair
pMπ , rπ q, where Mπ is a Levi subgroup of G, and rπ is a supercuspidal representation of the group ( Mπ pF q1 x P Mπ pF q : HMπ pxq 0
7.1. LOCAL PARAMETERS AND DUALITY
387
such that π is a subquotient of an induced representation IPπ prπ,λ q,
where rπ,λ ,
Pπ
P P p Mπ q ,
λ P aMπ ,C ,
is the aMπ ,C -orbit of supercuspidal representations of Mπ pF q attached to rπ . We write β pπ q p1qdimpAM0 {AMπ q ,
where M0 is a minimal Levi subgroup contained in Mπ . Since pMπ , rπ q is determined up to conjugacy, the sign β pπ q is an invariant of π. It plays an important role in the duality operator of Aubert-Schneider-Stuhler. The duality operator D DG is an involution on the Grothendieck group KpGq of the category of GpF q-modules of finite length. It is defined [Au, 1.5] by (7.1.3)
DG
¸
p1qdimpA
P0
{ AP q i G r G , P
P
P P0
where P M NP ranges over standard parabolic subgroups of G, and iG P and G rP are the functors of induction and restriction between KpGq and KpM q [BZ, §2.3]. A key property of DG is that it preserves irreducibility up to sign. More precisely, suppose that π is an irreducible representation of GpF q, with image rπ s in KpGq. Then (7.1.4)
ps, Drπ s β pπ q rπ
p of GpF q. (See [Au, Corollaire 3.9 and for an irreducible representation π Erratum] and [ScS, Proposition IV.5.1].) Duality also has the property of being compatible with endoscopy. Suppose that G1 is an endoscopic datum for G, equipped with auxiliary datum pGr1, ξr1q and corresponding transfer factors, as in §2.1. The transpose of the 1 of disassociated transfer mapping f Ñ f 1 of functions is a transfer S 1 Ñ SG r 1 pF q , tributions. Here S 1 is any stable, pζr1 q1 -equivariant distribution on G 1 1 and SG is the invariant, ζ -equivariant distribution on GpF q defined by
1 pf q Sp1 pf 1 q, SG
f
P HpG, ζ q.
Now the representation theoretic character provides an isomorphism from the complex vector space K pG q C
KpGq bZ C onto the space of invariant distributions on GpF q that are admissible, in the
sense that they are finite linear combinations of irreducible characters. The duality mapping DG can therefore be regarded as an involution on the space of finite invariant distributions on GpF q. The compatibility property is a formula (7.1.5)
pD1S 1qG αpG, G1q pDGSG1 q,
388
7. LOCAL NONTEMPERED REPRESENTATIONS
r 1 , and in which S 1 is assumed to be finite, D1 is the duality involution for G dimpAM0 {AM 1 q 0 . αpG, G1 q p1q
It comes with the implicit assertion that D1 maps the subspace of stable distributions to itself. (See [Hi], [A25].) More generally, suppose that G is a twisted triplet pG0 , θ, ω q, as in §2.1. One can define the complex vector space KpGq b C as the span of objects (2.1.1). The duality involution (7.1.3) makes sense in this generality, provided that the indices of summation P are treated as standard parabolic subsets of G [A4, §1]. It acts on the Grothendieck group KpGq of the subcategory of G pF q-modules of finite length whose irreducible constituents are of the form (2.1.1). The compatibility formula (7.1.5) then extends to this setting, with the appropriate definition of the sign αpG, G1 q. (For a proof of this and other properties discussed below, we refer the reader to the paper [A25] in preparation.) We note that we still have local parameters ψ P ΨpGq, if G represents a general triplet pG0 , θ, ω q. For any such ψ, we write ψppw, u1 , u2 q ψ pw, u2 , u1 q,
w
P WF ,
u1 , u2
P SU p2q,
for the dual parameter that interchanges the two SU p2q-factors in the group LF
SU p2q WF SU p2q SU p2q.
We also form the sign β pψ q p1q
p
{
dim AM0 AMψ
q,
where M0 is a minimal Levi subset of G, and Mψ is a minimal Levi subset for which the L-group L Mψ0 contains the image of ψ.
Assume now that G belongs to Ersim pN q. At this point, we are thinking of particular parameters that are to be localizations ψ9 v at places v outside p for generic parameters φ P Φ r p G q. of U . They will be of the form ψ φ, p r Any pair of local parameters φ P ΦpGq and ψ φ corresponds to a pair of irreducible, self-dual representations πφ and πψ of GLpN, F q. It follows from the conjecture of Zelevinsky [Z, 9.17], proved by Aubert [Au, Th´eor`eme 2.3] and Schneider and Stuhler [ScS], that π pφ equals πψ . However, our real interest is in the twisted characters rφ pfrq frG pφq tr π
and
rψ pfrq frG pψ q tr π
r pN q. These are finite invariant distributions on G r pN, F q, which we in fr P H r They behave in a simple way under the duality can denote by φr and ψ. r r operator D for GpN q. For it can be seen from twisted extensions of the arguments of [Au], applied to the very special case at hand, that r β pφ rq ψ. r rφ D
7.1. LOCAL PARAMETERS AND DUALITY
389
(See the general construction of [MW4, §3].) The parameters φ and ψ φp also provide finite, stable distributions on GpF q. We may as well denote r r N pGq-preimages of φ them again by φ and ψ, since they are the unique Out and ψr under the twisted transfer mapping of distributions S
ÝÑ
Sr SGr pN q .
To describe their behaviour under the duality operator D DG , we apply r above. We obtain r pN q of (7.1.5) and the formula for D rφ the analogue for G
r αpG, rq ψ. r r Gq D rφ r Gq β pφ pDφq αpG, r N pGq-invariant. The mapping S Ñ Sr is an injection if S is required to be Out
Since
rq β pφq r Gq β pφ αpG,
in this case, we conclude that Dφ β pφq ψ.
(7.1.6)
We shall apply these remarks to the case ψ φp of the general expansion (7.1.2). Since the groups Sφ and Sψ are the same, we can define a canonical r φ and Π r ψ , with matching characters bijection π Ñ σπ from Π
xx, σπ y xx, πy, x P Sφ, π P Πr φ. p for φ P Φ r pGq and G P Ersim pN q. Then Lemma 7.1.1. Suppose that ψ φ, (7.1.7) xsψ , πy σπ β pφq β pπq πp, r φ. for any π P Π Proof. The formula (2.2.6) is valid for the generic parameter φ. It can be written ¸ φ1G xx, πy π,
P
rφ π Π
r N pG q where pG1 , φ1 q corresponds to pφ, xq, and π is regarded as an Out invariant distribution on the right hand side. We shall apply the operator D DG to each side of this identity. It follows from (7.1.5) and the analogue for G1 of (7.1.6) that
DG φ1G Since we obtain
αpG, G1q pD1φ1qG αpG, G1q β pφ1q pψ1qG. αpG, G1 q β pφ1 q β pφq, DG φ1G
β pφqpψ1qG β pφq ψG1 .
390
7. LOCAL NONTEMPERED REPRESENTATIONS
The original identity becomes
1 ψG
¸
rφ π PΠ
¸
xx, πy β pφq Dπ
xx, πy β pφq β pπq πp
.
π
On the other hand, the expansion (7.1.2) can be written
1 ψG
¸
rψ σ PΣ
xsψ x, σy σ,
r N pGq-invariant distribution on the where σ is again understood as an Out right hand side. This identity then becomes
1 ψG
¸
P
rψ σ Σ
¸
P
rφ π Π
xx, σy xsψ , σy σ
xx, πy xsψ , πy σπ
,
since sψ can be regarded as an element in Sφ . It gives us a second expansion 1 , as a function of x into characters xx, πy. Identifying the coefficients of ψG of xx, π y, we obtain the required formula (7.1.7). rφ Theorem 2.2.1 asserts that the distribution σπ attached to any π P Π r N pGq). Combined with Lemma 7.1.1, is a character (up to the action of Out it implies the stronger assertion that σπ is actually equal to the irreducible character π p. It also implies the identity of signs
(7.1.8)
β pφq β pπ q xsψ , π y,
for every representation π P Πφ . Conversely, suppose that the identity p, so the assertion of Theorem (7.1.8) is valid for all π. Then σπ equals π 2.2.1 holds for π in this stronger form. r pGq, we write Π r G for the subset of Given the generic parameter φ P Φ φ r φ such that the identity (7.1.8) holds. We then define representations π P Π the corresponding subset
(7.1.9)
rG Π ψ
p: π π
P Πr Gφ
(
r ψ . The subset Σ rG Π r G of the packet Π rψ Σ r ψ thus consists of of Π ψ ψ p σπ . The assertion of (OutN pGq-orbits of) irreducible representations π p Theorem 2.2.1 for the dual parameter ψ φ amounts to the equality
(7.1.10)
rG Π ψ
Πr ψ
of the two sets. Lemma 7.1.1 also has implications for the function fG1 pψ, q on the left hand side of the nongeneric local intertwining relation (2.4.7) of Theorem
7.1. LOCAL PARAMETERS AND DUALITY
391
2.4.1. To describe it, we first transfer the duality operator to the function spaces IrpGq and SrpGq. r temp pGq. Any such function Recall that IrpGq is a space of functions on Π extends by analytic continuation to a function on the corresponding set of r pGq and standard representations, and hence by linearity, to a function on Π on the complex Grothendieck group KpGqC . We can therefore regard IrpGq r N pGq-symmetric as a space of linear functions on the vector space of Out invariant distributions on GpF q that are finite. Recall also that SrpGq is r bdd pGq. These objects can also be extended by a space of functions on Φ analytic continuation and linearity. The generic form of Theorem 2.2.1, established in Chapter 6, can be used to characterize the space of stable r N pGq-symmetric, invariant, finite distributions within the larger space of Out distributions. It allows us to regard SrpGq as a space of linear functions on r N pGq-symmetric stable distributions of GpF q that are finite. the space of Out These descriptions of IrpGq and SrpGq are compatible with the convention r pGq as a finite stable distribution (as in above of treating a parameter ψ P Ψ r (7.1.6)), and a representation π P ΠpGq as a finite invariant distribution (as in the proof of Lemma 7.1.1). With this understanding, we define duality operators D DG on both IrpGq and SrpGq by the adjoint actions
pDaqpπq apDπq,
and
a P IrpGq, π
pDbqpφq bpDφq,
P Πr temppGq,
r bdd pGq. b P SrpGq, φ P Φ
This gives continuous involutions on both IrpGq and SrpGq. Suppose that G, φ and ψ φp are as in Lemma 7.1.1. It is easy to deduce from the lemma that (7.1.11)
fG1 pψ, sψ xq β pφq pDfG q1 pφ, sψ xq,
f
P HrpGq,
Sφ. To do so, we first observe that xx, πy pDfGqpπq, x P Sφ,
for any element x in the group Sψ
pDfGq1pφ, xq
¸
P
rφ π Π
according to the definition (2.4.6) and the generic form of (2.2.6) we have established. We can therefore write ¸
pDfGq1pφ, xq xx, πy fGpDπq π
β p φq
¸ rφ π PΠ
xsψ x, πy fGpσπ q,
since the lemma tells us that Dπ
β pπq πp β pφq xsψ , πy σπ .
392
7. LOCAL NONTEMPERED REPRESENTATIONS
It then follows from (2.4.6) and (7.1.2) that
pDfGq1pφ, xq β pφq
¸
P
rψ σ Σ
xsψ x, σy fGpσq β pφq fG1 pψ, xq.
Replacing x by sψ x, we obtain (7.1.11). Now the generic form of Theorem 2.4.1 we have established implies that β pφq pDfG q1 pφ, sψ xq β pφq pDfG q pφ, sψ uq,
where u is any element in Nφ whose image xu in Sψ equals x. Given the reduction of Lemma 2.4.2, the assertion of Theorem 2.4.1 for the dual parameter ψ φp is consequently equivalent to the relation (7.1.12)
β pφq pDfG q pφ, sψ uq fG pψ, uq,
u P Nψ , f
P HrpGq.
This is a statement about the behaviour under duality of intertwining operators, equipped with the normalization (2.4.4) of Chapter 2. To be more precise, we fix a pair
pM, ψM q,
ψM
P Ψr 2pM, ψq,
as in the preamble to Proposition 2.4.3. Then ψM equals φpM , for a fixed r 2 pM, φq. We assume that M is proper in G, and that the parameter φM P Φ analogue of Theorem 2.2.1 holds inductively for pM, ψM q. According to the rψ Σ r ψ , of definition (2.4.5), fG pψ, uq is the sum, over σM in the set Π M M the linear forms (7.1.13)
fG pψ, u, σM q xu r, σ rM y tr RP pwu , σ rM , ψ q IP pσM , f q .
With this notation, it is not hard to see that Theorem 2.4.1 implies the identity of linear forms pM q, (7.1.14) β pφq pDfG q pφ, sψ u, πM q fG pψ, u, π
u P Nφ , f
P HrpGq,
pM q επM puqpDfG qpφ, u, πM q, fG pψ, u, π
u P Nφ , f
P HrpGq,
r φ . This is a straightforward consequence for every representation πM P Π M r φ of the of the fact that either side of (7.1.12) equals the sum over πM P Π M linear forms on the corresponding side of (7.1.14). Conversely, it is clear that if (7.1.14) is valid for all πM , then Theorem 2.4.1 holds for ψ. It is clear from the definition (7.1.13) that the putative identity (7.1.14) includes a condition on the representation theoretic R-groups of ψ and φ. D. Ban [Ban] has shown that for the groups G P Ersim pN q in question, the two R-groups are indeed the same. We recall that the endoscopic R-groups of ψ and φ are equal by definition. These are in turn isomorphic to any of the representation theoretic R-groups of φ, by the results of Chapter 6. Theorem 8.1 of [Ban], which relies on the results of [G], implies that the same is true of the representation theoretic R-groups of ψ. More precisely, r φ and the normalized intertwining operators for representations πM P Π M r pM P ΠψM correspond, up to a scalar multiple, under the action of D. π Stated in terms of the notation (7.1.13), this amounts to a weaker form
(7.1.15)
7.1. LOCAL PARAMETERS AND DUALITY
393
of (7.1.14), for a sign επM puq. The formula (7.1.15) is reminiscent of (6.1.5), the weaker form of the local intertwining relation for archimedean F and generic φ that follows from Shelstad’s work. In fact, the two formulas have almost identical roles. They apply to elements v in the set V of special valuations of F9 (archimedean in Chapter 6, p-adic here) by which the local intertwining relation (for generic parameters in Chapter 6, and general parameters here) is deduced by global means. In each case, επM represents a character on the subgroup Wψ of W pM q. Its inclusion in the formula has the effect of making the left hand side insensitive to the specific normalizations introduced in Chapter 2. The formulas would both hold (with different choices of characters επM ) for any normalizations Rpw, πM q of the intertwining operators that are multiplicative in w. They follow directly from the property that the operators Rpw, πM q are scalars for w P Wψ0 , and form a basis of the space of all intertwining operators of the induced representation IP pπM q as w ranges over Rψ . It is this property that follows from [Ban]. We note that Theorem pM be unitary. This condition is included in our 8.1 of [Ban] requires that π induction assumption that Theorem 2.2.1 is valid for ψM . (The results in [Ban] and [G] were actually stated for G split, but with our knowledge of the groups RpπM q from §6.6, they extend easily to the case that G P Ersim pN q is quasisplit.) r pGq, we write Π r G for the subset of Given the generic parameter φ P Φ φM r φ such that the identity (7.1.14) holds. We then representations πM P Π M define the corresponding subset
(7.1.16)
rG Π ψM
pM : πM π
P Πr Gφ
(
M
r ψ . The assertion of Theorem 2.4.1 for the dual parameter ψ of Π M amounts to the equality
(7.1.17)
rG Π ψM
φp
Πr ψ
M
of the two sets. Our discussion of duality has been designed for application to local completions ψ9 v at places v R U . The role of such completions will be similar to that of their analogues from Chapter 6. To emphasize this point, we can specialize the local parameters φ and ψ further. A finite dimensional representation of WF is said to be tamely ramified if its kernel contains the wildly ramified inertia group. In other words, the representation is trivial on the pro-p part of the inertia subgroup IF of WF , p being the residual characteristic of F . We shall say that the representation is quadratic if its irreducible constituents have dimension at most two. We thus observe a further symmetry with the discussion of §6.1. For it is clear r 2 p Gq that the tamely ramified, quadratic representations of WF that lie in Φ are in some sense analogues of the archimedean parameters described prior to Lemma 6.1.2.
394
7. LOCAL NONTEMPERED REPRESENTATIONS
We shall say that the local parameters φ and ψ φp are tamely ramified and quadratic if their restrictions to WF have these properties. r pGq are tamely ramified, Lemma 7.1.2. Suppose that ψ φp and φ P Φ quadratic parameters for our simple endoscopic datum G P Ersim pN q over the p-adic field F . r G in the dual group Spψ generates Spψ , and contains (a) The image of Π ψ the trivial character 1. r G in the dual group Spψ generates Spψ , and con(b) The image of Π M M ψM tains the trivial character 1.
Lemma 7.1.2 is intended as an analogue of Lemma 6.1.2. It provides the limited local information that will be used to deduce the general local theorems by global means. The two assertions of the lemma are obviously equivalent to their analogues for φ. It is really the first assertion (i) (taken for φ) that is parallel to Lemma 6.1.2. The second assertion (ii) is actually a stronger analogue of what was available in §6.1. It represents a refinement (for the p-adic field F here) of the formula (6.1.5), in which the characters επM pxq are all trivial. As we shall see in §7.3, this refinement will greatly simplify our proof of the local intertwining relation. We shall give a proof of Lemma 7.1.2 in [A25] that is based on the representations of Hecke algebras, r as well as a proof of a natural variant of the lemma for the bitorsor G r attached to an even orthogonal group G P Esim pN q. We should note that analogues of Lemma 7.1.2(i) have been established in complete generality by Moeglin [M1] [M2] [M3]. Using a remarkable family of partial duality involutions, she has established results that among r φ , for any generic parameter φ P Φ r pG q. r G equals Π other things imply that Π φ Her results are partially global, in that they depend on the generic form of Theorem 2.2.1, which was established by global means in Chapter 6. 7.2. Construction of global parameters ψ9 We will now introduce the families of global parameters needed to establish the general nontempered local theorems. The construction is essentially that of §6.2–§6.3. However, we shall have to impose local constraints at a finite set of nonarchimedean places. These will be used to apply the duality properties described in the last section. In particular, they will be used in the next section to modify the discussion of §5.4. Suppose that G P Ersim pN q is a fixed simple twisted endoscopic datum r pN q over the given local field F . Consider a nongeneric parameter for G (7.2.1)
ψ
`1ψ1 ` ` `r ψr ,
r pGq, with simple components in Ψ
ψi
P Ψr simpGiq,
Gi
P ErsimpNiq.
7.2. CONSTRUCTION OF GLOBAL PARAMETERS ψ9
395
Given the local pairs pGi , ψi q over F , we will want to choose global pairs
pGi, ψiq, 9
9
P ErsimpNiq, 9
G9 i
ψ9 i
P Ψr simpGiq, 9
over a suitable global field F9 such that
pFu, Gi,u, ψi,uq pF, Gi, ψiq, 9
1 ¤ i ¤ r,
9
9
for some valuation u of F9 . These will determine a global endoscopic datum
G9 P Er sim pN q such that the global parameter 9
ψ9
(7.2.2)
`1ψ1 9
`
`
`r ψ9 r
r pG9 q. The process will be similar to that of Proposition 6.3.1, belongs to Ψ except that supplementary constraints on the global data will have to be slightly different. There is also the minor difference that G was taken from the larger set Erell pN q in Proposition 6.3.1, rather than the set of simple data Ersim pN q we have specified here. As in the generic case in §6.3, we fix a Levi subgroup M of G such that r 2 pM, ψ q is nonempty. Given G, 9 the set Ψ we will also choose a global Levi r 2 pM, ψ9 q is nonempty. The analogues subgroup M9 of G9 such that Ψ 1 `1 (7.2.3) ψM ψ11 ψr`r ψ
and ψ9 M
(7.2.4)
1
ψ1` ψr`1 ψ 9 r
9 1
9
of (6.3.3) and (6.3.4) can then be identified with fixed elements in the re9 q. As before, we will have a special r 2 pM, ψ q and Ψ r 2 pM 9 ,ψ spective sets Ψ interest in the case that each `i equals 1. This of course is the case that M G and ψM ψ ψ , with M9 G9 and ψ9 M ψ9 ψ9 . Recall that there are decompositions ψi
µi b νi,
mini, 1 ¤ i ¤ r,
Ni
r sim pHi q is a simple generic parameter for a datum Hi P Ersim pmi q where µi P Φ over F , and νi is an irreducible representation of SU p2q of dimension ni . Set
I m,n
ti :
mi
m,
ni
n u,
for any positive integers m and n. We then have another decomposition (7.2.5)
ψ
where
n 1
8 à
p µ n b ν n q, à
p`i µ i q , and ν n is the irreducible representation of SU p2q of dimension n. The global µn
8 à
P
m 1 i I m,n
constituents of ψ9 will be of the form ψ9 i
µi 9
ν,
b 9i
1 ¤ i ¤ r,
396
7. LOCAL NONTEMPERED REPRESENTATIONS
r sim pH 9 q is a simple generic parameter for some H 9 r where µ9 i P Φ i i P E sim pmi q over F9 , and ν9 i is the extension of νi to the group SLp2, Cq. As we noted r sim pG9 i q. back in §1.4, these objects determine the simple data G9 i with ψ9 i P Ψ For the construction, we will apply a variant of Proposition 6.3.1 directly to the local generic parameter 9
µ
8 à
n 1
p µn q
r à
i 1
p`i µ i q.
There will be four possibilities for the global field F9 . The simplest case is when the local field F equals C. In this case we take F9 to be any imaginary quadratic extension of Q. If F R, and if the two dimensional generic constituents tµi : mi 2, 1 ¤ i ¤ ru
of ψ are in relative general position, we simply take F9 Q. If F R, but these two dimensional generic constituents are not in relative general position, we take F9 to be a totally real field with several archimedean places. Finally, if F is p-adic, we take F9 to be any totally real field of which F equals some completion. In all cases, we fix a valuation u of F9 such that F9u F . As agreed in the last section, we also fix a large finite set V of nonarchimedean places, which does not contain u, and for which q9v is large for any v P V . This set will assume the role played in Proposition 6.3.1 by the complementary set u S8 S8 tuu of archimedean places. According to our remarks in §7.1, it will be the set U
S8puq S8 Y tuu
that often takes the place of the individual valuation u in Chapter 6, while it will be the union S8 pu, V q S pU, V q U
that assumes the earlier role played by S8 puq.
YV
Proposition 7.2.1. Given the local objects G, ψ, M and ψM over F as 9 M 9 9 in (7.2.1) and (7.2.3), we can choose corresponding global objects G, ψ, 9 9 and ψM over F as in (7.2.2) and (7.2.4) such that the following conditions are satisfied. (i) There is a valuation u of F9 such that
pFu, Gu, ψu, Mu, ψM,uq pF, G, ψ, M, ψM q, 9
9
9
9
9
and such that the canonical maps Sψ9 M
ÝÑ
SψM
Sψ9
ÝÑ
Sψ
and are isomorphisms.
7.2. CONSTRUCTION OF GLOBAL PARAMETERS ψ9
397
(ii) For any valuation v outside the set S8 pu, V q, the local Langlands parameter µ9 v `1 µ9 1,v ` ` `r µ9 r,v ,
is a direct sum of tamely ramified quasicharacters of F9v , while the corresponding decomposition of the subparameter µ9 1,v
` ` µr,v 9
contains at most one ramified quasicharacter. r 2 pH 9 (iii)(a) For any v P V , the parameters µ9 i,v lie in Φ i,v q and are each a direct sum of tamely ramified representations of WF9v of dimension one and two, so in particular, ψ9 v equals the dual φpv of a generic parameter r pG9 v q. Moreover, the canonical mappings φv P Φ rM Π ψ9
M,V
and
rG Π ψ9
M,V
¹
pΠr Mψ q ÝÑ
Spψ9 M
Spψ
pΠr Gψ q ÝÑ
Spψ9 M
Spψ
9
P
v V
¹
P
9
v V
M,v
M,v
M
M
are surjective. (iii)(b) Suppose that each `i equals 1. Then there is a v r pG q for some G property that if ψ9 v lies in Ψ v v p equals G. p G
PV
with the
P Ersim,v pN q, the dual group 9
v
(iii)(c) Suppose that some `i is greater than 1. Then there is a v such that the kernel of the composition of mappings Sψ9
ÝÑ
Sψ9 v
ÝÑ
Rψ9 v
contains no element whose image in the global R-group Rψ9 to Rψ,reg . 9
PV
Rψ pGq belongs 9
9
Proof. The statement is clearly very close to that of Proposition 6.3.1. Notice that the condition (iii)(a) here applies to the two sets of p-adic representations from Lemma 7.1.2 (but with G replaced by M in the first set). The condition (iii)(a) of Proposition 6.3.1 applies to the L-packets over R from Lemma 6.1.2. We have included the possibility here that F C. In this case, however, each constituent is one-dimensional. Since it is also self-contragredient, µi must then be the trivial character on C . We simply take µ9 i to be the trivial id`ele class character for F9 . The various conditions then follow without difficulty. We can therefore assume that F C. Then F9 is a totally real field, chosen according to the description prior to the statement of the proposition. There will be some superficial differences from the setting of Proposition 6.3.1. For example, we are allowing a case F9 Q in which there is only one real valuation. Moreover, the constituents µi of µ need not be distinct, since they could be matched with different representations νi of SU p2q. Finally,
398
7. LOCAL NONTEMPERED REPRESENTATIONS
we will be applying the construction of Proposition 6.3.1 as it applies to the larger set of valuations u S8 pV q S8u Y V u . However, the general structure of the proof will be similar. in place of S8 The first step is to construct the primary global pairs
pHi, µiq, 9
P Ersimpmiq, 9
H9 i
9
µ9 i
P Φr simpHiq,
µi
P Φr simpHiq,
9
over F9 from the given local pairs
pHi, µiq,
P Ersimpmiq,
Hi
over F . We apply Corollary 6.2.4, supplemented by Remark 3 following its proof. For every i, the global pair then has the properties that
pFu, Hi,u, µi,uq pF, Hi, µiq, that µi,v is spherical for every v R S8 pu, V q, and that µi,v is an archimedean u . The new condition applies to parameter in general position for every v P S8 9
9
9
9
9
the finite set V of nonarchimedean places. According to the supplementary remark, we can arrange that for each v P V , the local Langlands parameter r 2 pH 9 µ9 i,v belongs to Φ 9 i,v is a direct sum i,v q. In fact, we can assume that µ of distinct, irreducible, self-dual, tamely ramified representations of WF9v of dimension two with unramified determinant, together with an unramified character of WF9v of order 1 or 2, in case mi is odd. Our hypothesis that the residual characteristics of valuations v P V are large insures that there are enough such two-dimensional representations of WF9v . As we noted above, the simple generic pairs pH9 i , µ9 i q can then be combined with the representations ν9 i of SLp2, Cq to give nongeneric pairs
pGi, ψiq, 9
9
P ErsimpNiq, 9
G9 i
ψ9 i
P EsimpGiq. 9
9
These objects in turn determine a global endoscopic datum G9 p together with a global parameter ψ9 with G9 G,
P ErsimpN q 9
P Ψr pGq as in (7.2.2). They x x, together with also determine a global Levi subgroup M of G with M M r pM , ψ q as in (7.2.4). The condition (i) follows. a global parameter ψM P Ψ p
9
9
9
9
9
9
9
To obtain the remaining conditions of the proposition, we need to impose more local constraints on the basic global pairs pH9 i , µ9 i q. We can by and large follow the prescription from the proof of Proposition 6.3.1, with V now being u . In particular, the chosen set of nonarchimedean valuations instead of S8 we will choose the global pairs successively with increasing i. Consider the quadratic id`ele class character
ηH ηµ detpµiq p i SOpmi , Cq. Its local values θi,v with H θ9i
9
i
9i
9
9 attached to any H9 i at places v P V are predetermined if mi is even. In this case, the conditions on µ9 i,v
7.2. CONSTRUCTION OF GLOBAL PARAMETERS ψ9
imply that θ9i,v
pεv qq ,
qi
i
399
12 mi,
where εv is the nontrivial unramified quadratic character on F9v . If mi is odd, we apply the criterion of Proposition 6.3.1 to V . Namely, we fix a set of distinct places V o tvi : mi oddu, and then require that #
q 1 ppεεv qqq , , v i
(7.2.6)
θ9i,v
i
if v if v
vi , vi ,
qi
12 pmi 1q,
for any v P V and i with mi odd. Recall that the local values θ9i,v are also determined at places v P S8 puq if mi is even, and at v u in general. To u and m be definite, we arbitrarily fix θ9i,v in the remaining case of v P S8 i odd. (We cannot use the criterion of Proposition 6.3.1, which we have just u is large. applied to our p-adic set V here, since we are not assuming that S8 There is no need to do so in any case, since V is now taking the earlier role u .) of S8 Having fixed the local values θ9i,v at any v P S8 pu, V q and i with p i SOpmi , Cq, we choose the global quadratic characters as in the proof H of Proposition 6.3.1. We require that for any v R S8 pu, V q there be at most one i such that θ9i,v ramifies, and that the ramification be tame. Recall that this is done by insisting that θ9i,v be unramified at any place v R S8 pu, V q, which divides 2, or at which θ9j,v ramifies for some j i. Once we have chosen the quadratic character θ9i , we obtain the global endoscopic datum p i . We then construct the global parameters H9 i from the dual group H9 i H r sim pH 9 q according to Lemma 6.2.2 and Corollary 6.2.4 (modified by µ9 i P Φ i u or V , we first the accompanying Remark 3). At the places v in either S8 9 specify the localizations φi,v in terms of their predecessors φj,v , with j i. This is how we arrange that as i varies, the constituents of φ9 i,v are mutually v. disjoint, and in fact, in relative general position if v P S8 The required property (ii) follows just as in the proof of Proposition 6.3.1. The same is also largely true of the remaining assertions in (iii)(a)– (iii)(c). They are for the most part trivial unless the subset I 1 of indices i P I o with mi 1 is not empty. As in the earlier proof, it is really only for this possibility that the property (7.2.6) was needed. The first assertion of (iii)(a) is an immediate consequence of the construction. In the other assertion of (iii)(a), the first mapping is analogous to that of Proposition 6.3.1(iii)(a). As before, it suffices to treat the case that M G. Working with the valuations vi as i varies over I 1 , we again show that the mapping x
Spψ9 V
ÝÑ
Spψ9
400
7. LOCAL NONTEMPERED REPRESENTATIONS
is surjective. We then establish this part of the assertion, using Lemma 7.1.2(a) in place of Lemma 6.1.2, from the fact that the set V is large. The second mapping in (iii)(a) has no counterpart in Proposition 6.3.1. However, the required surjectivity follows in the same way from Lemma 7.1.2(b). For the conditions (b) and (c) of (iii), we can again take v to be any valuation in V if I 1 is empty. Otherwise, we must take v v1 vi1 , for any fixed index i1 in I 1 . It might appear that (iii)(b) is complicated here by the existence of simple components ψi µi b νi with i P I 1 and ni ni1 . Indeed, the condition would fail on this account if we were allowing G to be a general element in Erell pN q, as in Proposition 6.3.1. However, we have deliberately taken G P Ersim pN q to be simple here. The condition (iii)(b) then follows easily. The proof of the last condition (iii)(c) is identical to that of Proposition 6.3.1. The discussion of this section has been parallel to that of §6.3. Having no need of a nontempered analogue of Lemma 6.3.2, we are now at the point we reached at the end of §6.3. Given simple local parameters ψi φi b νi over F , we have constructed global parameters ψ9 i φ9 i b ν9 i over F9 . We can then form the associated family (7.2.7)
r F 9
Frpψ1, . . . , ψr q t`1ψ1 9
9
9
9
`
`
`9r ψ9 r : `9i
¥ 0u
of compound global parameters. This represents another family of the sort treated in Chapter 5. In contrast with the earlier generic case, Assumption 5.1.1 requires no further effort to justify here. It follows from Lemma 2.2.2 and what we have established in the generic case. We recall that this assumption was the basic premise of Chapter 5. r that were in We now formally adopt the induction hypotheses for F force in Chapter 5. We fix the positive integer N , and assume that the r with degpψ q N . To local theorems all hold for general parameters ψ P Ψ this, we add the induction assumption that the global theorems all hold for 9
r with degpψ9 q N . parameters ψ9 in the given family F The global hypothesis is understood to include the conditions on Theorem 1.5.3, described near the beginning of §5.1. The assertion of Theorem 1.5.3(b) thus represents an induction assumption for the nontempered fam9
r which is resolved in Lemma 5.1.6. The assertion of Theorem 1.5.3(a) ily F, represents a condition on the simple generic constituents µ9 i of parameters 9
r which has also been treated. It is part of the resolution from Corolψ9 P F, lary 5.4.7, which among other things, is the foundation for the definition of 9
r We recall that the global theorems for simple generic the subsets FrpG9 q of F. parameters with local constraints were established in Lemma 5.4.6. The local constraints for the simple generic parameters µ9 i here are admittedly at u of the nonarchimedean places V , rather than the archimedean places S8 §5.4. But with the local theorems in place now for all of the constituents of 9
7.3. THE LOCAL INTERTWINING RELATION FOR ψ
401
µ9 i , the proof of Lemma 5.4.6 carries over in a straightforward way. Alternatively, we observe that the general proof of the global theorems, which we will complete in §8.2, can be carried out for generic parameters µ9 with no reference to the local results for nongeneric parameters of this chapter. We would like to apply the results of Chapter 5 to the nongeneric family (7.2.7) with local constraints. Assumption 5.1.1 is easy to verify in this case. It follows from the reduction in Lemma 2.2.2 of the local assertion of Theorem 2.2.1(a) and our induction assumption on the generic constituents µ9 i of parameters in (7.2.7). (The argument here is similar to that of the early part of the proof of Lemma 8.1.1, at the beginning of the next chapter. This places the family (7.2.7) in the general framework of §5.1. In particular, the general lemmas of §5.1–5.3 are valid (as applicable) for its parameters. However, their stronger refinements in §5.4, which we used to establish the local theorems of Chapter 6, hold only for the generic family of §5.4. We are going to have to extend them to the nongeneric family (7.2.7). To deal with Theorem 2.4.1, we will need nongeneric refinements of 2 pG 9q Lemmas 5.2.1 and 5.2.2. These apply to parameters ψ9 in the subset Frell r attached to a simple datum G9 P E r sim pN q. Any such ψ9 takes the form of F 9
9
#
(7.2.8)
ψ9 2ψ9 1 ` ` 2ψ9 q ` ψ9 q 1 ` Sψ9 Op2, Cqq Op1, Cqrq ψ9 ,
ψ9 r , q ¥ 1, `
as in (5.2.4). In contrast to the earlier generic refinements of §5.4, we will not need a separate argument for the case that the index r equals 1. To deal with Theorem 2.2.1, we will need nongeneric refinements of Lemmas 5.3.1 and 5.3.2. These apply to parameters
ψ1 ψr , ψi P Fr simpNiq, r We will again not require a separate argument in the subset Fr2 pGq of F.
(7.2.9)
ψ9 9
9
`
`
9
9
9
for the case that r equals 1. However, we will otherwise follow the general strategy of Chapter 5. In particular, we shall introduce the simple datum
G9
P ErsimpN q, for N N1 9
N , and the parameter ψ9
ψ1 9
` 9
ψ,
as in §5.3. We shall establish the necessary refinements in the next section. The argument will be similar to that of §5.4, except that the exceptional place v will be taken from the set of p-adic valuations V . In order to reduce the problem to Corollary 3.5.3, which was the underlying tool in §5.4, we shall apply the duality theory of the last section to the parameters ψ9 v and ψ9 ,v . 7.3. The local intertwining relation for ψ The nongeneric form of Theorem 2.4.1 will be easier than the generic case treated in §6.4. This is because we will have some special cases to work
402
7. LOCAL NONTEMPERED REPRESENTATIONS
with. They consist of a limited number of local intertwining relations that can be obtained by duality from the generic form of the theorem already established. We will be able to combine them with a simplified form of the global argument from §6.4. First, however, we will have to deal with the question raised at the end of the last section. We shall establish a pair of lemmas that extend the global results of §5.4. The first lemma gives the nongeneric refinements of Lemmas 5.2.1 and 2 pG 9 q of F. r It will be used later 5.2.2, for parameters (7.2.8) in the subset Frell in the section to derive the general local intertwining relation. 9
Lemma 7.3.1. Suppose that
pG, ψq,
G9 P Er sim pN q, ψ9 9
9
9
(7.3.1)
P Frell2 pGq, 9
r of the pair given in either Lemis an analogue for the nongeneric family F mas 5.2.1 or 5.2.2. Then the expressions 9
(7.3.2)
¸
c
P
px9 q f9G19 pψ,9 sψ9 x9 q f9G9 pψ,9 x9 q , εG ψ9 9
c ¡ 0,
x Sψ,ell
and
pf _qM pψ1q fG__ pψ, x1q ,
1, r pGq and f _ P H r p G _ q. from these lemmas vanish, for any functions f P H 1 8
(7.3.3)
9
9
9
9
9
9
9
if r
9
9
9
9
r of Lemmas 5.4.3 and Proof. This lemma represents an analogue for F 5.4.4, which we recall were the refinements of Lemmas 5.2.1 and 5.2.2 for generic parameters. In particular, (7.3.2) corresponds to one of the two expressions (5.2.8) or (5.2.13) (according to whether r ¡ 1 and r 1), while (7.3.3) corresponds to (5.2.12). As in the two refinements for generic parameters, we actually have to establish several vanishing assertions. In stating this lemma, we have included only those assertions that will be used directly in the coming proof of the nontempered local intertwining relation. Following the argument from the two lemmas of §5.4, we shall reduce the proof to the assertion of Corollary 3.5.3. The main step is to write the expression (7.3.2) in the form (5.4.10) obtained in the proof of Lemma 5.4.3. We choose v P V according to the condition (iii)(c) of Proposition r pG9 v q, as 7.2.1. In particular, ψ9 v is the dual φpv of a generic parameter φv P Φ prescribed by the condition (iii)(a) of the proposition. We take 9
f9 f9v f9v ,
f9v
P HrpGv q, 9
f9v
P HrpGv q, 9
to be decomposible. Then (7.3.2) equals c
¸
P
x Sψ,ell
εG px9 q f9v,1 G9 pψ9 v , sψ9 x9 v qpf9v q1G9 pψ9 v , sψ9 x9 v q f9v,G9 pψ9 v , x9 v q f9Gv9 pψ9 v , x9 v q , ψ9 9
7.3. THE LOCAL INTERTWINING RELATION FOR ψ
403
in the notation (5.4.7). There are two linear forms in f9v in each summand of this expression. To write them in terms of the generic parameter φv , we shall apply the duality operator Dv for the group G9 v . 1 pψ9 v , s 9 x9 v q. It equals The first linear form is f9v, ψ G9 β pφv qpDv f9v,G9 q1 pφv , sψ9 x9 v q β pφv qpDv f9v,G9 qpφv , sψ9 x9 v q,
by (7.1.11) and the local intertwining relation for the generic parameter φv . This in turn can be written as β pφ v q
¸
P
p q
r φv M v πv Π
pDv fv,Gq pφv , sψ xv , πv q 9
9
9
9
in the notation (7.1.13), or what is the same thing, the product of β pφv q with the analogue for Dv f9v,G9 , φv and sψ9 x9 v of the expression (5.4.8). (We are rφ r φ pMv q for the Mv -packet here, as in (5.4.8), rather than Π writing Π v v,Mv as in (7.1.13). The Levi subgroup Mv is chosen in terms of φv , and is not generally equal to the localization M9 v of Proposition 7.2.1.) As in the earlier 1 pψ9 v , s 9 x9 v q as a general sum discussion of (5.4.8), we can then express f9v, ψ G9 ¸
P
p q
r φv M v πv Π
α1 pπv , x9 v q pDv f9v,G9 q Mv , πv , rv pxq ,
following the definition (3.5.3). It is understood that α1 pπv , x9 v q is a complex coefficient, that rv pxq rv psψ xq
is the image of x in the local R-group Rφv , and that Mv , πv , rv pxq represents an element in the set T pG9 v q, as in the proof of Lemma 5.4.3. The other linear form is f9v,G9 pψ9 v , x9 v q. It can be written as ¸
r φv pMv q πv PΠ
f9v,G9 pψ9 v , x9 v , π pv q
in the notation (7.1.13), or equivalently, the analogue for f9v.G9 , ψ9 v and x9 v of (5.4.8). Our expectation (7.1.14) is that these expressions are equal to their analogues above. However, we do not need this finer property here. We can instead use the weaker formula (7.1.15) obtained from [Ban]. It asserts that pv q επv pxq pDv f9v,G9 q pφv , x9 v , πv q, f9v,G9 pψ9 v , x9 v , π r φ pMv q with associated sign επ pxq. We may therefore express for any πv P Π v v f9v,G9 pψ9 v , x9 v q as a general sum ¸
P
p q
r φv M v πv Π
αpπv , x9 v q pDv f9v,G9 q Mv , πv , rv pxq ,
according again to the definition (3.5.3), and for another complex coefficient αpπv , x9 v q.
404
7. LOCAL NONTEMPERED REPRESENTATIONS
We can now write the original expression (7.3.2) in the desired form. For we see from the formulas above that (7.3.2) is equal to ¸
(7.3.4)
P p q
δ pτv , f9v qpDv f9v,G9 qpτv q,
τv T G9 v
where δ pτv , f9v q equals the sum c
¸¸ x
εG px9 q α1pπv , x9 v qpf9v q1G9 pψ9 v , sψ9 x9 v q αpπv , x9 v q f9Gv9 pψ9 v , x9 v q , ψ9 9
πv
over elements x P Sψ,ell and πv 9
P Πr φ pMv q such that the triplet v
Mv , πv , rv pxq
belongs to the W0Gv -orbit represented by τv . We now apply Lemmas 5.2.1 and 5.2.2. Together, they assert that if tf u is any global compatible family of functions (5.2.6), the sum ¸
(7.3.5)
r sim N G E
P
9
p q
r ι N, G tr Rdisc, f , ψ9
p
p q
q
added to (7.3.3) in case r 1, equals (7.3.2). We are following the earlier convention of writing f9 or f9_ in place of f in case G equals G9 or G9 _ . The expression (7.3.5) is of course a nonnegative linear combination of irreducible 9 q. In the case r 1, the expression (7.3.3) characters on the groups G pA equals the analogue 1 8
(7.3.6)
rψ9 1 , ψ9 1 q IP _ pπ9 ψ9 1 , f9_ q tr 1 RP _ pw1 , π
r of the expression (5.4.11) obtained in the proof of Lemma 5.4.4. It too for F is a nonnegative linear combination of irreducible characters, in this case on the group G9 _ pA9 q. We can therefore write the sum of (7.3.5) with (7.3.3) as 9
¸¸ G π
cG pπ q fG pπ q,
G
P ErsimpN q, 9
π
P Πr pGq,
for nonnegative coefficients cG pπ q.
For any G P Er ell pN q, let Dv be the duality operator on Gv . At this point, we will actually be concerned with the “absolute value” 9
p p π q β p π q D pπ q π pv , D v v v v v
πv
P ΠpGv q,
pv is the dual (7.1.4) of the representation πv . This extends of Dv , where π to an involution p pπ q D p pπ q b π ,v D v v v
πpv b π,v ,
π
r pG q, and an adjoint involution on Π
πv b π,v ,
pDp v fG qpπq fG pDp v πq fG pπpv b π,v q
on IrpG q. Our sum of (7.3.5) and (7.3.3) can then be written as (7.3.7)
¸¸ G π
p cG π
p q pDp v fG qpπq,
G
P ErsimpN q, 9
π
P Πr pGq,
7.3. THE LOCAL INTERTWINING RELATION FOR ψ
405
for nonnegative coefficients p cG π
p q cG pDp v πq.
The earlier expression (7.3.4) takes the form ¸
(7.3.8)
P p q
p v f9 9 qpτv q, δppτv , f9v qpD v,G
τv T G9 v
where and
δppτv , f9v q β pτv q δ pτv , f9v q β pτv q β pπv q,
τv
Mv , πv , rv pxq .
The assertions of Lemmas 5.2.1 and 5.2.2 may thus be characterized as an identity between the expressions (7.3.7) and (7.3.8), for any compatible family of functions tf u. But it is easy to see from the definitions in §2.1 p u defines a correspondence of comand §3.4 that the family of operators tD v patible families. To be precise, suppose that tf u and tfp u are two families r pG q such that of functions on the spaces H p f D v G
fpG ,
G
P ErsimpN q. 9
Then if tf u is a compatible family, the same is true of tfp u. The identity between (7.3.7) and (7.3.8) therefore remains valid, with the functions p f replaced by fp , for any compatible family tfp u. D v G G We can now apply Corollary 3.5.3. It tells us that the coefficients p cG pπ q and δppτ9v , f9v q all vanish. Since (7.3.2) equals (7.3.8), it vanishes, as asserted. Corollary 3.5.3 likewise tells us that (7.3.6) vanishes, as do any of the summands in (7.3.5). Since (7.3.3) equals (7.3.6), it also vanishes. This is the second assertion of the lemma. The second lemma gives the nongeneric refinements of Lemmas 5.3.1
r It will be used and 5.3.2, for parameters (7.2.9) in the subset Fr2 pG9 q of F. in the next section to establish what is left of Theorem 2.2.1. 9
Lemma 7.3.2. Suppose that
pG, ψq, 9
9
G9 P Er sim pN q, ψ9 9
P Fr2pGq, 9
r9 of the pair given in either Lemis an analogue for the nongeneric family F mas 5.3.1 or 5.3.2. Then
(7.3.9)
G Sdisc, pf9q |ΨpG,9 ψ9 q| |Sψ9 |1 εGpψ9 q f Gpψ9 q, ψ9 9
9
9
r pG9 q. for any function f9 P H
r9 of Lemmas 5.4.5 and 5.4.6. Proof. This lemma is an analogue for F It is stated with the implicit assumption of the condition of Lemma 5.3.1,
406
7. LOCAL NONTEMPERED REPRESENTATIONS
namely that the global theorems are valid for parameters in the complement r p N q. of F9 ell pN q in F 9 We The required identity (7.3.9) is the stable multiplicity formula for ψ. can assume that N is even and ηψ9 1, since the formula is otherwise valid by Corollary 5.1.3. It then follows from Lemma 5.1.4 that the left hand side minus the right hand side is a multiple of the linear form 9
f9
ÝÑ
f9L pΛ9 q,
r pG9 q, f9 P H
9 q of G 9 q. (We r pG9 q obtained from the maximal Levi subgroup LpA 9 pA on H are writing L9 and Λ9 in place of L and Λ here, to be consistent with the surrounding notation.) We must show that Λ9 vanishes. As we agreed at the end of the last section, we introduce the simple
P ErsimpN q, for N N1 N , and the parameter ψ ψ1 ψ r ell pG q. We have then to modify the proofs of Lemmas 5.4.5 and 5.4.6 in in F datum G9
9
9
9
9
` 9
9
the manner of the last lemma. This entails working with compatible families of functions (5.3.3) on the groups G P Er sim pN q, and the correspondence p . We shall be brief. on such objects under the local duality operators D ,v 9 9_ We first introduce the composite endoscopic datum G9 _ 1 G1 G in 9
r pG9 _ q in the compatible family Er sim pN q, with corresponding function f91 P H 1 9 (5.3.3). We are including the case r 1 here, in which G9 1 G. It follows from the definitions that the supplementary summands in the expressions (5.3.4) and (5.3.21) of Lemmas 5.3.1 and 5.3.2 can be written 9
L9
pψ1 Λq
L9
p Γ Λq
b f91 and 1 2
f91
9
9
9
9
respectively. We then apply Corollary 3.5.3 as in Lemmas 5.4.5 and 5.4.6, p . This leads to the conclusion that in combination with the operators D v 9 attached to the maximal Levi subgroup the linear forms ψ9 1 Λ9 and Γ9 Λ, L9 G9 1 L9 of G9 _ 1 in the respective cases r ¡ 1 and r 1, vanish. If r ¡ 1, the linear form ψ9 1 attached to G9 1 is nonzero, according to our induction hypothesis. Therefore Λ9 vanishes in this case, as required. Suppose then that r 1. If Γ9 vanishes, the linear form f9pψ9 q f9G pΓ9 q 9
f9L pΛ9 q f9L pΛ9 q, 9
9
r pG9 q, f9 P H
is induced from the Levi subgroup L9 pAq. This contradicts the first assertion of Proposition 7.2.1(iii)(a), which is the analogue of the local generic condip . Therefore, it is tion (5.4.1)(a), as one sees easily from the properties of D v again the linear form Λ9 that vanishes, as required.
7.3. THE LOCAL INTERTWINING RELATION FOR ψ
407
We shall now establish Theorem 2.4.1 for nongeneric parameters r pGq. To recall the general strategy, we go back to the discussion for ψPΨ generic parameters at the beginning of §6.4. As we noted there, the reduction of the global intertwining relation in §4.5 can be made in the context r defined now by (7.2.7). The reduction also carries over to of the family F, the local intertwining relation of Theorem 2.4.1. In other words, the local r pN q, unless ψ belongs to intertwining relation holds for a parameter ψ P Ψ r ell pG q for some G P Esim pN q, or belongs to the local analogue for ψ of Ψ one of the two exceptional cases (4.5.11) or (4.5.12). As before, we prove the three exceptional cases together. Again, it is sufficient to establish the assertion of Theorem 2.4.1 for elements s and u that map to a point x in the set Sψ,ell . r ell pGq. The conditions In the first case, we fix G P Ersim pN q with ψ P Ψ are 9
#
(7.3.10)
2ψ1 ` ` 2ψq ` ψq 1 ` ` ψr , Sψ Op2, Cqq Op1, Cqrq ψ , q ¥ 1,
ψ
with the requirement that the Weyl group Wψ contain an element w in r together with Wψ,reg . Proposition 7.2.1 then gives us the global family F, 2 9 9 9 9 r a global pair pG, ψ q such that ψ P Fell pGq is of the form (7.2.8). The global pair satisfies the condition of Lemma 7.3.1. This lemma then gives us an identity 9
(7.3.11)
¸
P
εG px9 q f9G19 pψ,9 sψ9 x9 q f9G9 pψ,9 x9 q ψ9 9
0,
r pG9 q. fr P H 9
x Sψ,ell
r pGq, for The second and third cases apply to local parameters ψ P Ψ G P Ersim pN q, that are of the local forms of (4.5.11) and (4.5.12). In each
r together with of these two cases, Proposition 7.2.1 gives a global family F, 9 9 9 r 9 a global pair pG, ψ q such that ψ P F pGq has the corresponding global form (4.5.11) or (4.5.12). In both cases, Corollary 4.5.2 again tells us that (7.3.11) is valid as stated. Recall that if q r 1 in the original case (7.3.10), there r pG_ q. The corresponding is a second group G_ P Ersim pN q such that ψ P Ψ global pair pG9 _ , ψ9 q then belongs to the second exceptional case (4.5.11), and the global formula (7.3.11) in this case amounts to the vanishing of (7.3.3) established in Lemma 7.3.1. We can therefore include it in our discussion of (4.5.11), as we did for the generic parameters treated in §6.4. We have thus to extract the remaining local intertwining relation 9
fG1 pψ, sψ xq fG pψ, xq,
x P Sψ,ell , f
P HrpGq,
408
7. LOCAL NONTEMPERED REPRESENTATIONS
from the global identity (7.3.11), when ψ represents one of the three exceptional cases above. As in §6.4, we take f9
¹
f9v
v
to be a decomposable function. The summand of x in (7.3.11) then decomposes into a difference of products (7.3.12)
εG px9 q ψ9 9
¹
1 pψ9 v , s 9 x9 v q f9v, ψ G9
v
¹
f9v,G9 pψ9 v , x9 v q
v
over all valuations v. The first step is to remove the contributions to these products from valuations in the complement of the finite set S pU, V q. We shall establish a local lemma that can be applied to the places v P V , as well as those in the complement of S pU, V q. To simplify the notation, we shall formulate it in terms of the basic local pair pG, ψ q over F , specialized by r pGq. This of course implies that the requirement that ψ φp for some φ P Φ G r F is p-adic. For any such φ, we write Πφ,M for the subset of representations
r φ that lie in Π r φ pπM q, the set of irreducible constituents of the induced πPΠ r G defined prior to (7.1.16). representation IP pπM q, for some πM in the set Π φM
r pG q . Lemma 7.3.3. Assume that F is p-adic, and that ψ φp for some φ P Φ r pGq is such that the function fG pπ r φ is supported on pq of π P Π Then if f P H G r the subset Πφ,M , we have
fG1 pψ, sψ xq fG pψ, xq,
x P Sψ .
Moreover, if ψ is tamely ramified and quadratic, and f satisfies the stronger condition that fG pπ pq
(7.3.13) for any π
#
1, 0,
if x , π y 1, otherwise,
P Πr φ, then
fG1 pψ, sψ xq fG pψ, xq 1,
x P Sψ .
Proof. Using (7.1.11) and the generic form of Theorem 2.4.1, we write fG1 pψ, sψ xq β pφq pDfG q1 pφ, sψ xq
β pφq pDfGq pφ, sψ xq,
as in §7.1. Our condition on f implies that the function
pDfGq pπq fGpDπq β pφq fGpπpq,
π
P Πr φ,
r G of Π r φ . It then follows from the formula is supported on the subset Π φ,M r G , that the function (7.1.13), together with the definition of Π φ,M
pDfGq pφ, sψ u, πM q,
πM
P Πr φ
M
,
7.3. THE LOCAL INTERTWINING RELATION FOR ψ
409
r G of Π r φ . (We recall that u is any element in is supported on the subset Π M φM the group Nψ Nφ whose image in Sψ Sφ equals x.) We can therefore write
β pφq pDfG q pφ, sψ uq
β p φq β p φq
¸
M
P
M
¸
rG πM Π φ
¸
P
pDfGq pφ, sψ u, πM q
rφ πM PΠ
fG pψ, u, π pM q
rG πM Π φ
¸
pDφGq pφ, sψ u, πM q
M
P
rφ πM Π M
pM q fG pψ, u, π
fGpψ, uq, since the identity (7.1.14) holds by definition for representations πM We have established that
P Πr Gφ
M
.
fG1 pψ, sψ xq fG pψ, uq fG pψ, xq, as required. Suppose now that ψ is tamely ramified and quadratic. It then follows from Lemma 7.1.2(b) that (7.3.13) is indeed stronger than the first condition on f . In particular, the functions fG1 pψ, sψ xq and fG pψ, xq are equal. To see that they are equal to 1, we have only to observe that fG1 pψ, sψ xq
¸
P
rφ π Π
xx, πpy fGpπpq 1,
by (7.1.2), Lemma 7.1.1 and the fact that (7.1.8) holds by our induction hypothesis on M . This completes the proof of the lemma. r pGq, as Remark. Suppose that ψ φp for an arbitrary parameter φ P Φ in the lemma. Then for any complex valued function a on Spφ , there is a r pGq such that function f P H
a
x , πy fGpπpq,
π
P Πr φ.
This follows by duality from the assertion (b) of Theorem 1.5.1 (which we have now proved). In particular, we can always choose f so that the condition (7.3.13) holds. We will need another lemma for the unramified places v, which will again be stated in terms of pG, ψ q. Its proof does not rely on the induction hypothesis that M G.
410
7. LOCAL NONTEMPERED REPRESENTATIONS
Lemma 7.3.4. Suppose that pG, ψ q is unramified over F , and that f is the characteristic function of a hyperspecial maximal compact subgroup of GpF q. Then f satisfies the condition (7.3.13) of Lemma 7.3.3. Proof. The condition that ψ be unramified means that the restriction of ψ to LF is an unramified unitary representation of WF (and in particular, is trivial on the SU p2q factor of LF ). This implies that ψ is the dual of a r bdd pGq. tamely ramified, quadratic parameter in Φ We shall first establish that fG1 pψ, xq 1,
(7.3.14)
x P Sψ .
We recall that fG1 pψ, xq equals f 1 pψ 1 q, for any pair pG1 , ψ 1 q that maps to pψ, xq. The fundamental lemma for pG, G1q asserts that f 1 is the image in SrpG1 q of the characteristic function of a hyperspecial maximal compact subgroup of G1 pF q. Since we can assume inductively that f 1 pψ 1 q 1 whenever x 1, it will be enough to show that fG1 pψ, 1q f G pψ q 1. r where fr0 is the characteristic function of the standard maxiLet fr fr0 θ, r pN q , G mal compact subgroup of GLpN, F q. The fundamental lemma for G r pGq. It follows that asserts that frG equals the image f G of f in H
f G pψ q frG pψ q frN pψ q. But it is easy to see from the definition (2.2.1), and the properties of Whittaker functionals for GLpN, F q discussed in §2.5, that frN pψ q equals 1. The identity (7.3.14) follows. We need to be a little careful here, as I was reminded by the referee. The proof of the two fundamental lemmas by Ngo and Waldspurger applies only to the case that the residual characteristic of F is large. However, it is easily extended to the general case by using the generic forms of the local theorems we have now proved (results which of course would have been unavailable without the original two fundamental lemmas). We shall sketch a separate argument, even though it would also be possible to appeal directly to the unramified assertion of Theorem 1.5.1(a) we established at the beginning of §6.6. If we combine the identity (2.2.6) of Theorem 2.2.1 with the last assertion of Theorem 1.5.1(a), keeping in mind the remarks in §2.5 and §6.1 on spherical functions, we see that f
G
pφ q
#
1, if φ is unramified, 0, otherwise,
7.3. THE LOCAL INTERTWINING RELATION FOR ψ
411
r bdd pGq. This characterizes the image of f in SrpGq as a function for any φ P Φ r bdd pGq. We also see more generally that on Φ
f 1 pφ 1 q
#
1, 0,
if φ is unramified, otherwise,
if pG1 , φ1 q is a preimage of a given pair pφ, xq. Since the pair pG1 , φ1 q is unramified if and only if the same is true of φ, we obtain the formula f 1 p φ1 q
#
1, if φ1 is unramified, 0, otherwise,
r bdd pG1 q. Therefore, f 1 is indeed for G1 P E pGq unramified and any φ1 P Φ 1 r the image in S pG q of the characteristic function of a hyperspecial subgroup of G1 pF q, as asserted by the fundamental lemma for pG, G1 q. To treat the twisted fundamental lemma, we use the formula
frN pφq
#
1, if φ is unramified, 0, otherwise,
r bdd pN q. This is not hard to establish from the which holds for any φ P Φ unramified case of (i) from Corollary 2.5.2. It tells us that frG equals the r p N q, G . image f G of f in SrpGq, as asserted by the fundamental lemma for G The two fundamental lemmas thus hold in general, and the identity (7.3.14) is therefore valid with no restriction on the residual characteristic of F . To finish the proof of the lemma, we take φ to be the tamely ramified, p We must show that f r bdd pGq such that ψ φ. quadratic parameter in Φ satisfies (7.3.13). To this end, we write
f 1 pψ, xq f 1 pψ 1 q
¸
P
rφ π Π
xsψ x, πy fGpσπ q,
by (7.1.2). It then follows from (7.3.14) that fG pσπ q
#
1, if x , π y 1, 0, otherwise,
r φ . The required condition (7.3.13) is thus a consequence of for any π P Π Lemma 7.1.2(a), which tells us that σπ π p if x , π y 1.
We return to our proof of the local intertwining relation, in which r pGq falls into one of the three exceptional cases (7.3.10), (4.5.11) or ψPΨ (4.5.12). The global identity (7.3.11) asserts that the sum over x P Sψ,ell of (7.3.12) vanishes. We will use this identity to show that the two factors for v u in the two products in (7.3.12) are equal. The process will be considerably easier than its counterpart from §6.4, thanks to Lemma 7.3.3. If v is a valuation of F9 in the complement of U , Proposition 7.2.1 asserts that ψ9 v φv , for a tamely ramified, quadratic, generic parameter
412
7. LOCAL NONTEMPERED REPRESENTATIONS
r pG9 v q. Suppose first that v lies in the complement of both U and φv P Φ r pG9 v q so that it satisfies the V . In this case, we choose the function f9v P H analogue for φv of the condition (7.3.13). In the special case that ψ9 v is unramified, we can also assume that f9v is the characteristic function of a hyperspecial maximal compact subgroup of G9 pF9v q, by Lemma 7.3.4. It then follows from Lemma 7.3.3 that the corresponding two factors in (7.3.12) are each equal to 1. We may therefore remove them from the products. We have established that the two products in (7.3.12) can each be taken over the finite set of valuations v P S pU, V q. If v belongs to V , we choose f9v so that it satisfies the first condition of Lemma 7.3.3, namely that the r φ is supported on the subset Π r G . It then pv q of πv P Π function f9v,G9 pπ v φv ,M follows from the lemma that
1 pψ9 v , s 9 x9 v q f9 9 pψ9 v , x9 v q, f9v, v,G ψ G9
x P Sψ,ell .
The identity (7.3.11) reduces to ¸
P
εG px9 q f9U,1 G9 pψ9 U , sψ9 x9 q f9U,G9 pψ9 U , x9 q f9V1 pψ9 V , sψ9 x9 q, ψ9 9
x Sψ,ell 9
where
f9V1 pψ9 V , sψ9 x9 q
while f91
U,G9
¹
P
1 pψ9 v , s 9 x9 v q, f9v, ψ G9
v V
pψU , sψ xq and fU,GpψU , xq are defined by corresponding products 9
9
9
9
9
9
9
over v P U . It is a consequence of Proposition 7.2.1(iii)(a) that as f9V varies under the given constraints, the functions x
ÝÑ
f9V1 pψ9 V , sψ9 x9 q,
x P Sψ,ell ,
span the space of all functions on Sψ . It follows that (7.3.15)
1 pψ9 U , s 9 x9 q f 9 pψ9 U , x9 q, f9U, U,G ψ G9
x P Sψ,ell ,
r pG9 U q. for any function f9U P H It remains to separate the contribution of u to each side of (7.3.15) from u in U . To do so, we need only recall how we chose that of its complement S8 the global field F9 prior to Proposition 7.2.1. If F C, F9 is any imaginary quadratic field. Then U consists of the one valuation u, and (7.3.15) is simply the required identity (2.4.7) for the local parameter ψ ψ9 u . If F R, and the simple generic constituents
(7.3.16)
µi : m i
2, 1 ¤ i ¤ r
(
of ψ are in relative general position, F9 equals Q. Again U consists of the one valuation u, and (7.3.15) is the required identity for ψ ψ9 u . If F R, but the constituents (7.3.16) are not in relative general position, or if F is p-adic, we can take F9 to be a totally real field with several archimedean places. In these two cases, the corresponding global parameter ψ9 was chosen
7.4. LOCAL PACKETS FOR COMPOSITE AND SIMPLE ψ
413
v , the analogues for ψ9 in the proof of Proposition 7.2.1 so that for any v P S8 v of the generic constituents (7.3.16) are in general position. It follows from the case treated above that
f9v1 pψ9 v , sψ9 x9 v q f9v pψ9 v , x9 v q,
f9v
P HrpGv q, 9
u and x P S for any v P S8 ψ,ell . These linear forms are all nonzero, as one can see by relating the left hand side to GLpN, F9v q, according to the definitions (2.4.6), (2.2.3) and (2.2.1). We may therefore remove their contributions to (7.3.15). This leaves only the contribution of the complementary valuation v u. The formula (7.3.15) thus reduces to the required identity for ψ ψ9 u in these remaining two cases. We have now established the local intertwining relation for parameters r pGq in any of the three outstanding cases (7.3.10), (4.5.11) or (4.5.12). ψPΨ r pGq over This completes the proof of Theorem 2.4.1 for any parameter ψ P Ψ F. We also obtain the following corollary, which is proved in the same way as its generic analogue Corollary 6.4.5.
Corollary 7.3.5. The canonical self-intertwining operator attached to any r ψ satisfies r pGq, w P W 0 and πM P Π ψPΨ M ψ rM , ψ q 1. RP pw, π
7.4. Local packets for composite and simple ψ We are at last ready to complete our proof of the local theorems. They consist of five local theorems stated in Chapters 1 and 2. Having established everything for generic local parameters ψ φ in Chapter 6 (including the generic supplement Theorem 6.1.1), we are working now with nongeneric local parameters ψ. We dealt with the local intertwining relation of Theorem 2.4.1 in the last section. We shall sketch a proof of its supplement Theorem 2.4.4 at the end of this section. Since the nongeneric form of Theorem 1.5.1 is a special case of Theorem 2.2.1, we have then only to establish this last theorem, and its supplement Theorem 2.2.4. The general assertion (a) of Theorem 2.2.1 is now known. It was reduced in Lemma 2.2.2 to the generic case, which we resolved in Lemma 6.6.3. The assertion (b) of this theorem is the crux of the matter. For any pair
pG, ψq,
G P Ersim pN q, ψ
P Ψr pGq,
r ψ , together over the local field F , it postulates the existence of a packet Π r ψ that satisfies (2.2.6). Before we begin its proof, with a pairing on Sψ Π however, we shall derive a consequence of Theorem 2.2.1. We shall establish r ψ. a proposition that provides some elementary constituents of the packet Π A special case of the proposition will then be used in the general proof of Theorem 2.2.1.
414
7. LOCAL NONTEMPERED REPRESENTATIONS
r pGq is a general Langlands parameter We should first recall that if φ P Φ r φ is already defined. It consists of Langlands quotients for G, the L-packet Π πρ of elements ρ in a corresponding packet Prφ of standard representations. The packet Prφ in turn is defined by analytic continuation from the case that φ is bounded, as is the pairing
xx, ρy xx, πρy,
x P Sφ , ρ P Prφ .
However, the endoscopic identity (2.2.6) will be valid for general φ only if r φ is replaced by the packet Prφ of standard representations. Its failure for Π r φ of irreducible representations is of course one of the reasons the packet Π we have had to introduce other parameters ψ. This notation, incidentally, appears to be at odds with that of the original (nontempered) packet (1.5.1), in case the parameter φ ψ in (1.5.1) is generic. But since φ then belongs to r unit pGq of Φ r pGq, the representations in Prφ should be irreducible the subset Φ (according to Conjecture 8.3.1 in our next chapter). This would imply that r φ) r φ of (1.5.1) is the same as the packet Prφ (and the packet Π the packet Π defined here. Suppose for a moment that Theorem 2.2.1 is valid for the given pair pG, ψq, and that x is an element in Sψ . We can then choose a semisimple element s P Sψ with image x in Sψ , such that if pG1 , ψ 1 q is the preimage of the pair pψ, sq, the endoscopic datum G1 P E pGq is elliptic. The endoscopic identity (2.2.6) of Theorem 2.2.1(b) is f 1 pψ 1 q
¸
P
rψ π Π
xsψ x, πy fGpπq,
f
P HrpGq.
r ψ is a finite Π r unit pGq-packet, which is to say that it fibres over Recall that Π r r ψ pπ q for the fibre in Π r ψ of any element the set Πunit pGq. Let us write Π r unit pGq. The endoscopic identity can then be written πPΠ
(7.4.1)
f 1 pψ 1 q
¸
P
¸
P pq
p q
rψ π π r Π
r unit G π Π
xsψ x, πry
fG pπ q,
f
P HrpGq.
We shall combine it with the definition in Theorem 2.2.1(a) to give a formal r ψ. construction of the packet Π 1 1 Given pG , ψ q, we write G1
and where N
N11
ψ1
G11 G12,
ψ11 ψ21 ,
G1i
ψi1
P ErsimpNi1q,
P Ψr pG1iq, i 1, 2,
N21 . We can then write f 1 pψ 1 q
¸
r G1 φ1 Φ
P p q
npψ 1 , φ1 q f 1 pφ1 q,
7.4. LOCAL PACKETS FOR COMPOSITE AND SIMPLE ψ
415
for generic parameters φ1 r pG1 q, and integers in Φ
φ11 φ12,
φ1i
P Φr pG1iq,
rpψ 1 , φ1 q n n rpψ11 , φ11 q n rpψ21 , φ12 q,
by applying (2.2.12) separately to G11 and G12 . For any such φ1 , pG1 , φ1 q maps to a pair r pGq, spφ1 q P Sφpφ1 q . φpφ1 q, spφ1 q , φpφ1 q P Φ The analogue of (2.2.6) for standard representations is then f 1 pφ 1 q
¸
P
ρ Prφpφ1 q
xspφ1q, ρy fGpρq.
Lastly, we have the decomposition fG pρq
¸
r pGq π PΠ
mpρ, π q fG pπ q
of a standard representation ρ P Prφpφ1 q into irreducible representations π, for nonnegative integers mpρ, π q. Combining the three decompositions, we obtain a second endoscopic identity f 1 pψ 1 q
(7.4.2) where (7.4.3) λpψ 1 , π q
¸
P p q
r G π Π
¸
¸
r G1 ρ Prφpφ1 q φ1 Φ
P p q P
λpψ 1 , π q fG pπ q,
f
P HrpGq,
n rpψ 1 , φ1 q xspφ1 q, ρy mpρ, π q,
π
P Πr pGq.
It then follows from (7.4.1) that (7.4.4)
¸
P pq
rψ π r Π π
xsψ x, πry λpψ1, πq,
π
We recall that there is a generic parameter φψ is defined by
|w|
φψ pwq ψ w,
0
1 2
P Πr pGq. P Φr pGq attached to ψ. It
0
1 |w| 2
,
w
P LF ,
where |w| is the extension to LF of the absolute value on WF . The embedding of the centralizer Sψ into Sφψ factors to a surjective homomorphism from Sψ to Sφψ . This is dual to an injection of the character group Spφψ r φ take on an into Spψ . Our proposition asserts that the representations in Π ψ r ψ. endoscopic interpretation as elements in the larger packet Π
416
7. LOCAL NONTEMPERED REPRESENTATIONS
Proposition 7.4.1. Assume that Theorem 2.2.1 is valid for the pair pG, ψ q. r φ into the set of multiplicity free elements Then there is an injection from Π ψ r in the packet Πψ such that the diagram rφ Π ψ X
>
rψ Π
>
Spψ
_
Spφ ψ
_
rφ is commutative. In particular, the elements in the nontempered packet Π ψ are unitary.
Proof. As we observed in §2.2 and §3.5, an irreducible representation π of GLpN, F q comes with a linear form Λπ in the closure paB q of the associated maximal dual chamber. It is a measure of the failure of π to be tempered. Given parameters φ P ΦpN q and ψ P ΨpN q for GLpN q, we have been writing Λφ Λπφ and Λψ Λφψ Λπψ . In particular, we have linear r pGq and Ψ r p Gq forms Λφ and Λψ for parameters φ and ψ in the subsets Φ of ΦpN q and ΨpN q. The maximal closed chamber paP0 q for G embeds canonically in the chamber paB q for GLpN q. With this understanding, it r φ equals is clear that the linear form Λπ attached to a representation π P Π r pGq such that Π r φ contains π. Λφ , for the parameter φ P Φ The proposition will be a consequence of the identity (7.4.4) applied to r pGq, together with elementary properties of the factors a fixed element π P Π in the summands of (7.4.3). It might be a little simpler to argue in terms of the norm on paB q , as we did in the proof of Lemma 3.5.2, rather than the underlying partial order. The factor n rpψ 1 , φ1 q in (2.4.3) represents the r pG1 q in general coefficient in the expansion of a stable linear form ψ 1 on H terms of “standard” stable linear forms φ1 . If it is nonzero, we have
}Λφ1 } ¤ }Λψ1 } }Λψ }, with equality if and only if φ1 φψ1 . We also have Λφpφ1 q Λφ1 r pG1 q, and for any parameter φ1 P Φ r φ, Λφ Λρ Λπ , ρ P Prφ , π P Π r pGq. The factor mpρ, π q in (7.4.3) is the general for any parameter φ P Φ coefficient in the expansion of a standard representation ρ of GpF q in terms of irreducible representations π. If it is nonzero, we have
}Λπ } ¤ }Λρ}, with equality if and only if π is the Langlands quotient πρ of ρ. Taken together, these conditions place a similar restriction on the coefficient λpψ 1 , π q
7.4. LOCAL PACKETS FOR COMPOSITE AND SIMPLE ψ
417
defined by (7.4.3). Our conclusion is that if λpψ 1 , π q is nonzero, then
}Λπ } ¤ }Λψ1 },
r φ , for parameters φ φpφ1 q and with equality if and only if π lies in Π 1 1 φ φψ1 . We note that if φ does equal φψ1 , then φpφ1 q φψ and spφ1 q s. In this case, the original pair pψ, sq is transformed to pφψ , sq under the composition
pψ, sq ÝÑ pG1, ψ1q ÝÑ pG1, φψ1 q ÝÑ
φpφψ1 q, spφψ1 q
pφψ , sq.
r φ . The double sum in (7.4.3) then Suppose now that π belongs to Π ψ reduces to a sum over the single pair pφ1 , ρq pφψ1 , ρπ q, where ρπ is the representation in Prφψ corresponding to π. As the leading coefficients in rpψ 1 , φψ1 q and mpρπ , π q both equal 1. We see their respective expansions, n also that xspφ1q, ρy xs, ρπ y xx, πy.
The formula (7.4.3) thus reduces to
λpψ 1 , π q xx, π y.
It then follows from (7.4.4) that ¸
r ψ pπ q π r PΠ
xsψ x, πry xx, πy.
The point x P Sψ is arbitrary, and has been identified with its image in Sφψ on the right hand side of the formula. Since the image of sψ in Sφψ is trivial, the formula can be written ¸
(7.4.5)
P pq
rψ π π r Π
xx, πry xx, πy,
x P Sψ ,
1 after translation of x by sψ s ψ . If we set x 1 in (7.4.5), we see that the index of summation is trivial. r ψ pπ q of π in Π r ψ consists of one element, which In other words, the fibre Π we simply identify with π. This is the first assertion of the proposition. The other assertions follow from the resulting simplification of (7.4.5), and the r ψ are unitary. fact that the representations in Π
We can now go back to our main topic, the proof of Theorem 2.2.1. Our task is to establish assertion (b) of the theorem for the given pair pG, ψ q. r 2 pGq, the assertion follows from the local If ψ lies in the complement of Ψ intertwining relation we have now established, or rather its embodiment in Proposition 2.4.3. We have therefore only to consider the case that ψ r 2 p G q. belongs to Ψ We take (7.4.6)
ψ
ψ1 ` ` ψr ,
ψi
P Ψr simpGiq,
Gi
P ErsimpNiq,
418
7. LOCAL NONTEMPERED REPRESENTATIONS
r 2 pGq. Applying Proposition 7.2.1, we obtain to be a fixed parameter in Ψ an endoscopic datum G9 P Ersim pN q over the global field F9 and a global 9 q is 9 ψ parameter ψ9 P Fr2 pG9 q of the corresponding form (7.2.9). The pair pG, as in Lemma 7.3.2. It follows from this lemma that the stable multiplicity formula (7.3.9) holds for the pair. r9 (7.3.9) applies more generally By the global induction hypothesis on F, to the groups G9 1 that occur in
Idisc,ψ9 pf9q
(7.4.7)
¸
G9 1 Erell G9
P p q
9 G 9 1q S p1 ιpG, pf91q, disc,ψ9
9 the endoscopic decomposition (4.1.2) of the ψ-component of the discrete part of the trace formula. The specialization of (7.3.9) to any linear form on the right hand side was expressed in Corollary 4.1.3. Following the conditional proof of the spectral multiplicity formula in Lemma 4.7.1, as we did in the discussion of (6.6.4), we obtain a further decomposition
¸
G pf9q |ΨpG,9 ψ9 q| |Sψ9 |1 Idisc, ψ9 9
P
ε1 pψ9 1 q f91 pψ9 1 q,
x9 Sψ9
9 x 9 q. where pG9 1 , ψ9 1 q maps to the pair pψ, Continuing as in the relevant passage from §6.6, we turn next to the spectral decomposition (4.1.1) of Idisc,ψ9 pf9q. Since the parameter ψ9 belongs to
r 2 pG9 q, it cannot factor through any proper Levi subgroup M 9 of G. 9 Ψ Applying 9 9 our global induction hypothesis to the factor G of M as in the discussion of §6.6, we see that the summand of M9 in (4.1.1) vanishes. The left hand side of (7.4.7) therefore equals G G Idisc, pf9q tr Rdisc pf9q ψ9 9
9
¸
nψ9 pπ9 G q f9G9 pπ9 G q,
π9 G
9 q, where π9 G ranges over the set ΠpG9 q of irreducible representations of G9 pA 9 9 r 9 and nψ9 pπ9 G q are nonnegative integers. Since f lies in HpGq, fG9 pπG9 q depends
r pG9 q of orbits in ΠpG9 q under the only on the image π9 of π9 G in the set Π restricted direct product r N pG9 q Out
¹
r N pG9 v q. Out
v
We can therefore write
G Idisc, pf9q |ΨpG,9 ψ9 q| ψ9 9
for a modified coefficient
9 q|1 9 ψ nψ9 pπ9 q |ΨpG,
¸
P p q
r G9 π9 Π
nψ9 pπ9 q f9G9 pπ9 q,
¸
P p
9 π π9 G Π G, 9
q
nψ9 pπ9 G q,
9 π 9 q, as in (6.6.5). 9 q is the preimage of π 9 in ΠpG in which ΠpG,
7.4. LOCAL PACKETS FOR COMPOSITE AND SIMPLE ψ
419
We have converted (7.4.7) to an identity (7.4.8)
¸
¸
nψ9 pπ9 q f9G9 pπ9 q |Sψ9 |1
ε1 pψ9 1 q f91 pψ9 1 q,
r pG9 q, f9 P H
xPSψ P p q 9 x where pG9 1 , ψ9 1 q maps to pψ, 9 q. This formula is the analogue of the identity
r G9 π9 Π
9
9
(6.6.6) in §6.6. We shall apply it to a product
f9 f9V f9U f9V,U , relative to the decomposition 9 q G 9 G9 pA V
GU GV,U GpFV q GpFU q GpAV,U q. Recall that the set U S8 puq is composed of places at which we have little information, the analogue of the singleton tuu in §6.6. Recall also that the 9
9
9
9
9
9
9
9
u of archimedean places set V of p-adic places here plays the role of the set S8 in §6.6. It is what we will use to extract information about the places in U . We have been trying to follow the discussion of §6.6 as closely as possible, in order to clarify the structure of the general argument. At this point we come to a minor bifurcation. Our concern is the linear form
f 1 pψ 1 q f9u1 pψ9 u1 q,
f
fu, x P Sψ , 9
in which pG1 , ψ 1 q maps to pψ, xq. We want to express it in terms of irreducible characters. In §6.6, we proceeded directly. We decomposed the linear form (with φ in place of ψ) into a linear combination of irreducible characters π in f , with formal coefficients cφ,x pπ q in x that had then to be determined. It is convenient here to approach the question from the opposite direction. We shall decompose f 1 pψ 1 q into a linear combination of irreducible characters in x, with formal coefficients fG pσ q in f that have yet to be understood. In other words, we shall follow the general definition (7.1.2). In fact, we will apply it to each of the valuations v of U . For each v P U , (7.1.2) gives a sum f9v1 pψ9 v1 q
¸
xsψ xv , σv y fv,Gpσv q, 9
9
9
9
x9 v
9
σ9 v
P Sψ , 9
v
f9v
P HrpGv q, 9
r 9 of linear forms on H r pG9 v q. We write over σ9 v in the finite packet Σ ψv
(7.4.9)
¸
pfU1 qpψU1 q xsψ x, σU y pfU qGpσU q, 9
9
9
9
9
9
9
9
σ9 U
where σ9 U ranges over the linear forms in the packet !
σ9 U
â
P
σ9 v : σ9 v
P Σr ψ
v U
of ψ9 U
¹
P
v U
ψ9 v ,
9
v
)
420
7. LOCAL NONTEMPERED REPRESENTATIONS
and
¹
xx, σU y 9
9
xxv , σv y. 9
P
9
v U
We temporarily follow the same convention for the supplementary valuations v R U . We can then write ¸
pf V,U q1 pψV,U q1 9
(7.4.10)
9
xsψ x, σV,U y pf V,U qGpσV,U q, 9
9
9
9
9
9
σ9 U,V
where σ9 V,U ranges over elements in the packet !
σ9 V,U
â
R YU
σ9 v : σ9 v
P Σr ψ , x , σv y 1 for almost all v 9
)
9
v
v V
of ψ9 V,U
¹
ψ9 v ,
R YU
v V
and
¹
xx, σV,U y 9
9
R YU
xxv , σv y. 9
9
v V
We can also write (7.4.11)
¸
pfV1 qpφ1V q xsψ x, σV y pfV qGpσV q, 9
9
9
9
9
9
9
9
σ9 V
where σ9 V ranges over elements in the packet !
σ9 V
â
P
σ9 v : σ9 v
P Σr ψ 9
)
v
v V
of ψ9 V
¹
P
ψ9 v ,
v V
and
xx, σV y 9
9
¹
P
xxv , σv y. 9
9
v V
We will presently specialize the supplementary functions f9V and f9V,U , as in the argument of §6.6. We first substitute the formulas (7.4.9), (7.4.10) and (7.4.11) into the three factors of our linear form f91 pψ9 1 q f9V1 pψ9 V1 q f9U1 pψ9 U1 q pf9V,U q1 pψ9 V,U q1
in (7.4.8). According to Lemma 4.4.1, the coefficient ε1 pψ9 1 q in (7.4.8) equals 9 9 x 9 with pψ, 9 q being the usual the value of the character εψ9 εG on Sψ9 at sψ9 x, ψ9 image of pG9 1 , ψ9 1 q. The right hand side of (7.4.8) becomes
|Sψ |1 9
¸ ¸
P
x Sψ σ9
9 σ 9 y f9 9 pσ εψ9 psψ9 x9 q xsψ9 x, G 9 q,
7.4. LOCAL PACKETS FOR COMPOSITE AND SIMPLE ψ
an expression we write as
|Sψ |1
(7.4.12)
¸ ¸
9
421
εψ9 px9 q xx, 9 σ 9 y f9 9 pσ G 9 q,
P
x Sψ σ9
where the inner sums are over products σ9
σV b σU b σV,U 9
9
9
of elements in the packets of ψ9 V , ψ9 U and ψ9 V,U , and
xx, σy fGpσq 9
equals
9
9
9
9
xx, σV y fV,GpσV q xx, σU y fU,GpσU q xx, σV,U y fGV,U pσV,U q, 9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
while x9 is the isomorphic image of x in Sψ9 , as usual. Bear in mind that σ9 V , σ9 U and σ9 V,U at this point are still just linear forms. However, the two supplementary indices σ9 V and σ9 V,U both range over linearly independent sets of irreducible, signed characters. This follows from assertions (ii) and (iii)(a) of Proposition 7.2.1, which tell us that the localization ψ9 v at any v R U is dual to a tamely ramified, quadratic, generic parameter. We shall now fix elements σ9 V and σ9 V,U for which the sign is 1. Suppose that ξ P Spψ is a fixed character on the 2-group Sψ Sψ9 .
We apply condition (iii)(a) of Proposition 7.2.1 to the localization ψ9 V of r 2 pGq.) The ψ9 ψ9 M9 . (Bear in mind that M G here, since ψ belongs to Ψ condition implies that we can fix an element π9 V,ξ in the subset rG Π ψ9 9
V
¹
P
rG Π ψ9 9
v
v V
of the packet of ψ9 V such that the character x , π9 V,ξ y on Sψ9 equals the product ε9 1 ξ 1 . Away from V and U , we take the element ψ
π9 V,U p1q
â
R YU
π9 v p1q
v V
in the subset
rG Π ψ9 V,U 9
¹
R YU
v V
r Gv Π 9 9
ψv
of the packet of ψ such that for any v, the character x , π9 v p1qy on Sψ9 v is 1. We are relying here on Proposition 7.2.1(ii) and Lemma 7.1.2(a), which r G9 v of Π r 9 . It follows from together imply that π9 v p1q lies in the subset Π 9 ψv 9 V,U
ψv
the definition (7.1.9) that π9 v,ξ and π9 V,U p1q are both (orbits of) irreducible representations. We can now specialize the functions f9V and f9V,U . We have already noted that the two supplementary indices σ9 V and σ9 V,U in (7.4.12) are over linearly independent sets of irreducible signed characters. We choose the corresponding functions f9V,G9 pσ9 V q and f9V,U pσ9 V,U q, which represent factors of 9 G
422
7. LOCAL NONTEMPERED REPRESENTATIONS
the summands on the right hand side of (7.4.12), so that they are supported at σ9 V π9 V,ξ and σ9 V,U π9 V,U p1q. The original formula (7.4.8) then reduces to an identity ¸
nψ pξ, π9 U q f9U,G9 pπ9 U q
π9 U
¸
|Sψ |1 9
σ9 U
¸
xx, σU y ξpxq1fU,GpσU q, 9
P
9
9
9
9
x Sψ
r pG9 U q, σ9 U is summed over the packet of ψ9 U , and where π9 U is summed over Π
(7.4.13)
nψ pξ, π9 U q nψ9 π9 V,ξ b π9 U
b πV,U p1q . 9
This is the analogue of the formula (6.6.10) from the generic case in §6.6. 9 σ 9 U y of x P Sψ attached Since the groups Sψ9 v are all abelian, the function xx, r 9 pξ q is the set of σ9 U in the packet here to any σ9 U is a linear character. If Σ ψU r 9 of ψ9 U such that this character equals ξ, we conclude that Σ ψU
(7.4.14)
¸
P p q
r G9 U π9 U Π
nψ pξ, π9 U q f9U,G9 pπ9 U q
¸
P
r9 σ9 U Σ ψ
U
pξq
f9U,G9 pσ9 U q,
P Spψ and fU P HrpGU q. This is the analogue of (6.6.12) r pGU q, nψ pξ, πU q is a nonnegative integer. Lemma 7.4.2. For any πU P Π for any ξ
9
9
9
9
9
Proof. For this lemma, we need a slight embellishment of the proof of its generic counterpart Lemma 6.6.4(a). We recall from (7.4.13) that ¸
9 q|1 9 ψ nψ pξ, π9 U q |ΨpG,
P p
9
π9 G Π G,π9
where π9
q
nψ9 pπ9 G q,
πV,ξ b πU b πU,V p1q. 9
9
9
Since the summand nψ9 pπ9 G q is a nonnegative integer, the only possible source of difficulty is the inverse of the order
|ΨpG, ψq| |ΨpG, ψq| p is orthogonal and that of the preimage of ψ in ΨpGq. We can assume that G the degrees Ni mi ni of the components ψi µi b νi of ψ are all even, since the order |ΨpG, ψ q| would otherwise be 1, leaving us with nothing to 9
9
prove. Suppose that one of the numbers mi is odd. Then ni is even, and the corresponding representation νi of SU p2q is symplectic. Since the representation µi of LF is orthogonal in this case, ψi would then be symplectic. r 2 pGq is orthogonal. This contradicts the fact that ψ P Ψ We have shown that the degrees mi of the generic constituents µi of ψi are all even. Consider the construction of the corresponding global generic constituents µ9 i from Proposition 7.2.1. It has the property that for any
7.4. LOCAL PACKETS FOR COMPOSITE AND SIMPLE ψ
423
v P V , the completion µ9 i,v is a direct sum of 2-dimensional representations of WF9v . Moreover, for any n, the representation µ9 nv
à
µ9 i,v
ti: ni nu
of LF9v is multiplicity free. The completion ψ9 v
à
pµi,v b ν iq
à
9
i
pµnv b ν nq 9
n
pv , for some φv P Φ r 2 pG9 v q, and is of the form φ r 2 pG9 v q. of ψ9 therefore belongs to Ψ 9 Since the dual group of Gv is even orthogonal, there are two elements in each of the sets ΨpG9 v , ψ9 v q and ΦpG9 v , φv q. It then follows from Corollaries 6.6.6 r φ , regarded as an Out r N pG9 v q-orbit of irreand 6.7.3 that any element πv P Π v ducible representations, contains two representations. The same is therefore r 9 by duality, and hence for any component π9 v true for any element σ9 v P Σ ψv r N pG9 q acts freely of the factor π9 V,ξ of π9 at V . This implies that the group Out on the indices of summation π9 G above. Since the original multiplicity npπ9 G q 9 the sum itself is is invariant under the action of any F9 -automorphism of G, 9 q. This 9 ψ an even integer, and is therefore divisible by the order 2 of ΨpG, completes the proof of the lemma.
It remains to separate the contribution of u to each side of (7.4.14) from u in U . As in our treatment of (7.3.15) in the last that of its complement S8 section, we need to recall how we chose the global field F9 . If F C, or F R and the simple generic constituents (7.4.15)
t φi :
mi
2, 1 ¤ i ¤ ru
of ψ are in relative general position, U consists of the one valuation u. In these cases, there is nothing to do. In fact, in dealing with the remaining cases, we can assume inductively that the theorem has been established for pG, ψq in case U tuu. This is where we will use Proposition 7.4.1. The remaining cases are when F is real but the constituents (7.4.15) of u is not empty. ψ are not in relative general position, or F is p-adic. Then S8 9 It consists of real places v such that the analogues for ψv of the constituents (7.4.15) are in general position. By our induction assumption, we can apply Proposition 7.4.1 to the associated pairs pG9 v , ψ9 v q. We set u π9 8
â
P
u v S8
π9 v ,
r φ such that the character where for each v, π9 v is the element in the packet Π ψv x , π9 v y on Sφ9 v is trivial. The proposition asserts that π9 v belongs to the packet 9
r 9 , and has multiplicity 1. We set Π ψv u f98
¹
P
u v S8
f9v ,
424
7. LOCAL NONTEMPERED REPRESENTATIONS
r pG9 v q such that where for each v, f9v is a function in H #
f9v,G9
1 pπ9 1 q 1, if π9 v π9 v , v
0,
otherwise,
r 9 . We shall apply the identity (7.4.14) to the for any element π9 v1 in Π ψv product u f9U f f98 ,
r pG q H r pG9 u q. in which f is a variable function in the space H There will be only one term on the right hand side of (7.4.14) at the chosen function f9U . To describe it, let σ pξ q be the element in the packet rψ Σ r 9 with Σ ψu x , σpξqy ξ. It then follows from the definitions, and the fact that each π9 v above has r 9 , that the right hand side of (7.4.14) reduces simply to multiplicity 1 in Π ψv fG σ pξ q . The left hand side of (7.4.14) reduces to a finite sum of irreducible unitary characters in f , with nonnegative integral multiplicities u nψ pξ, π q nψ pξ, π9 8 b π q,
π
P Πr unitpGq.
The identity therefore takes the form (7.4.16)
¸
P
p q
r unit G π Π
nψ pξ, π q fG pπ q fG σ pξ q ,
f
P HrpGq.
Observe that in the earlier cases in which U contained the one element u, (7.4.16) is actually identical to (7.4.14). Our global argument has thus led to a local identity for pG, ψ q that holds in all cases. r ψ attached to Recall that σ pξ q is the element in the provisional packet Σ r pG q. the character ξ on Sψ . It was defined (7.1.2) only as a linear form on H r However, we see from (7.4.16) that it is actually a finite sum of (OutN pGqorbits of) irreducible characters. We have therefore obtained the following proposition. Proposition 7.4.3. For any ψ rψ Π
P Ψr 2pGq, define r ψ pξ q, Π ξ P Spψ ,
º
P
ξ Spψ
r ψ pξ q is the disjoint union over π where Π of npξ, π q-copies of π. Then
f 1 pψ 1 q
for any f
¸
¸
r ψ pξ q ξ PSpψ π PΠ
P Πr unitpGq of multisets consisting
ξ psψ xq fG pπ q
P HrpGq and x P Sψ , where xsψ x, πy ξpsψ xq,
π
¸ rψ π PΠ
xsψ x, πy fGpπq,
P Πr ψ pξq,
7.4. LOCAL PACKETS FOR COMPOSITE AND SIMPLE ψ
425
and as usual, pG1 , ψ 1 q maps to the pair pψ, xq.
We have now established the endoscopic identity (2.2.6). More prer ψ over Spψ determines cisely, we have shown that the provisional packet Σ r r the Πunit pGq-packet Πψ that satisfies the conditions of Theorem 2.2.1. This completes the proof of Theorem 2.2.1 and Theorem 1.5.1, for the pair pG, ψ q over F . In the last two sections, we have established Theorems 2.2.1 and 2.4.1 in general. These are the essential local theorems. We also know that Theorem 1.5.1 is valid. Any of its assertions that are not subsumed in Theorem 2.2.1 follow either from Lemma 7.3.4, or Corollary 6.7.5 and the general classification of tempered reprsentations established in §6.7. This leaves only the twisted local supplements, Theorems 2.2.4 and 2.4.4. They r q of Ψ r pGq, where G r is the bitorsor apply to parameters ψ in the subset ΨpG r attached to an orthogonal group G P Esim pN q with N even. In §6.8, we sketched a proof of the two supplementary theorems for r q of ΨpG r q. For any such φ, the dual parameters φ in the subset Φbdd pG p r r parameter ψ φ lies in ΨpGq. We shall first describe the analogues for G of the objects in Lemma 7.1.2. r q, there is a canonical mapping For any φ P ΦpG
ÝÑ
π r
σπr ,
π
P Πφ ,
r pF q and σπr is a GpF q-invariant linear form r is an extension of π to G where π r q, such that on HpG
fr1 pψ 1 q
¸
P
xsψ xr, πry frGr pσπr q,
r q. fr P HpG
π Πφ
p of (7.1.2) treated prior to the r of the case ψ φ This is the analogue for G r 1 , ψ 1 q is the preimage of a pair statement of Lemma 7.1.1. In particular, pG pψ, xrq attached to an element xr P Srψ . We define
(7.4.17)
r
ΠG ψ
tπp P Πψ :
σπr
πpru,
p r is an extension of the reprewhere π is the preimage of π p in Πφ , and π p r r is uniquely determined by σπr , and sentation π p to GpF q. It is clear that π r of the set Π rG r of π. This is the analogue for G hence by the extension π ψ
r 2 pGq in Φ r pGq, we can in Lemma 7.1.2(a). If φ lies in the complement of Φ r 2 pM, φq and choose a proper Levi subgroup M of G, with parameters φM in Φ r r ψM P Ψ2 pM, ψ q as in Lemma 7.1.2(b). The analogue for G of the associated r G is the set ΠGr of representations set Π ψM ψM
π pM ,
πM
P Πr φ
r ψ such that the analogue in the packet Π M
(7.4.18)
M
,
r frG qpφ, sψ u r, πM q frG pψ, u r, π pM q, βrpφqpD
r φ , fr P HpG r q, u rPN
426
7. LOCAL NONTEMPERED REPRESENTATIONS
r are taken with respect of the identity (7.1.14) holds. The objects βrpφq and D r r to G, which is to say that they are defined in terms of Levi subsets of G rather than Levi subgroups of G, according to the remarks prior to Lemma 7.1.1. p the parameter ψ P ΨpG rq Suppose that in addition to being of the form φ, r G is tamely ramified and quadratic. Then the image of Πψ in Spψ generates Spψ and contains the trivial character 1, while if M is proper in G, the image r p p of ΠG ψM in SψM generates SψM and contains the trivial character 1. This
r mentioned at the end of §7.1. assertion is the variant of Lemma 7.1.2 for G We shall prove it in [A25], along with the lemma itself. r of Lemma 7.1.2, we can establish the suppleGiven the analogue for G r q. In each mentary Theorems 2.2.4 and 2.4.4, for local parameters ψ P ΨpG case, we use Proposition 7.2.1 to construct a family (7.2.7) over the global 9 q is the global pair attached to pG, ψ q, ψ 9 lies in the subset 9 ψ field F9 . If pG, r q of Ψ r pG9 q. We have then simply to follow the steps of the proofs of the ΨpG corresponding Theorems 2.2.1 and 2.4.1. We shall spare the details, since r q of ΨpG r q, they contain no new arguments. If ψ belongs to the subset Ψ2 pG for example, the reduction of Theorem 2.2.4 to the proof of Theorem 2.2.1 in this section is essentially the same as the generic case ψ φ treated at r q in Ψell pG r q, the beginning of §6.8. If ψ belongs to the complement of Ψ2 pG 9
r of the expression (5.2.8) of Lemma we first observe that the analogue for G 5.2.1 vanishes. As in the generic case (6.8.6), this follows from the fact that 9
r of the other expression (5.2.7) in Lemma 5.2.1 vanishes. the analogue for G The proof of Theorem 2.4.4 for ψ can then be transcribed from the proof of Theorem 2.4.1 in §7.3. With these brief remarks, we have finished our discussion of Theorems 9
r the 2.2.4 and 2.4.4. We still have the global induction hypothesis for F, nongeneric family (7.2.7). To resolve it, we must persuade ourselves that 9
r9 pN q the relevant global theorems are valid for parameters ψ9 in the subset F
r9 of F. We shall again be brief. The stable multiplicity formula of Theorem 4.1.2 follows from the general reductions of Proposition 4.5.1 and the proofs of Lemmas 7.3.1 and 7.3.2. The actual multiplicity formula is the assertion of Theorem 1.5.2. If ψ9 lies in Fr2 pG9 q, it follows from the multiplicity formula (7.4.8) (and the local results we have now proved), or alternatively, a direct appeal to the relevant variant of Lemma 4.7.1. If ψ9 lies in the complement
r pN q, it follows as in generic case from (4.4.12), (4.5.5), and of Fr2 pG9 q in F the stable multiplicity formula. Theorem 1.5.3(a) is relevant only to generic parameters. Theorem 1.5.3(b) is also an assertion for generic parameters. However, its role in the induction argument is really for nongeneric parameters, insofar as they occur in proof of Lemma 4.3.1 from §4.6. The assertion 9
7.5. SOME REMARKS ON CHARACTERS
427
follows from an application of Lemma 5.1.6 to the nongeneric family (7.2.7). Finally, we have the supplementary global Theorem 4.2.2, for the relevant
r pN q. Its main assertion (b) is implicit in the proof of the parameters in F supplementary local Theorem 2.2.4. We passed over this proof as a variant of the generic case treated in §6.8. However, we will establish both assertions of Theorem 2.2.4, for general global parameters, in the first section of the next chapter. We have now resolved the various induction hypotheses for general local parameters ψ. With the end of the induction argument, we have established all of the local theorems. 9
7.5. Some remarks on characters We conclude this chapter by drawing attention to two character formulas for GLpN q. The first applies to the case F R. It is the formula of Adams and Huang for a Speh character on GLpN q. The second is a refinement by Tadic of results of Zelevinsky that are implicit in the discussion from §2.2 of p-adic infinitesimal characters on GLpN q. In each case, the formula is an explicit expansion of an irreducible character πψ on GLpN, F q in terms of standard characters. We will need the formulas in the next chapter. They will be used in §8.2 in the critical final stages of the global classification. The formulas might also be of interest in their own right. They apply to untwisted analogues rpψ 1 , φ1 q in the double sum (7.4.3), which we recall gives the of the integers n definition of the coefficients λpψ 1 , π q that determine the packets r ψ, Π
ψ
P Ψr pGq,
G P Ersim pN q.
We shall see that the two formulas are parallel. They hint at properties of r ψ that are common to the real and p-adic cases. the packets Π Suppose that N mn, and that θ is an irreducible, tempered character on GLpm, F q. Then θ corresponds to an irreducible, unitary, m-dimensional representation µ of LF under the Langlands correspondence for GLpmq. Let us write (7.5.1) θn pf q tr πψ pf q
tr πφ pf q , f P H GLpN q , ψ µ b ν n, for the irreducible character on GLpN, F q corresponding to µ and the irreducible, n-dimensional representation ν n of SU p2q. We will also consider
collections of pairs
ψ
pmi, θiq,
1 ¤ i ¤ r,
where
N m1 mr , and θi is a virtual character on GLpmi , F q. In other words, θi is a finite, integral, linear combination of irreducible characters on GLpmi , F q. Let us write (7.5.2)
θ1
`
`
θr
428
7. LOCAL NONTEMPERED REPRESENTATIONS
for the virtual character on GLpN, F q induced from the virtual character θ1 θr on the Levi subgroup GLpm1 , F q GLpmr , F q.
If the constituents θi are irreducible, (7.5.2) is of course a standard character. The formulas in [AH] and [Tad2] express a character (7.5.1) explicitly in terms of virtual characters (7.5.2). Suppose first that F R. Consider an irreducible representation of WF of dimension 2. It is then induced from a character
ÝÑ pz{|z|qk |z|λ zµ zν , z P C, k P N, λ P C, of the subgroup C of WF , where µ 21 pk λq and ν 12 pk λq, and 1 |z| pzzq 2 is the analytic absolute value. This representation of WF , which we denote by µpk, λq, parametrizes a representation in the relative discrete series of GLp2, F q, whose character we shall denote by θpk, λq. Following [AH, §5.10], we define θpk, λq by coherent continuation for all k P Z. For z
fixed λ,
θpk, λq : k
(
PZ
is then a coherent family of virtual characters based at λ [AH, §4]. In concrete terms, θpk, λq can be defined by the natural “continuation” in k from N to Z of Harish-Chandra’s explicit formula for θpk, λq. To describe it explicitly, let η p`, λq be the character x of R . Then while if k
ÝÑ
`
x{|x|
¥ 1, the sum
x P R , ` 0, 1, λ P C,
|x|λ,
θp0, λq η p0, λq θpk, λq
`
η p1, λq,
θpk, λq
equals η p0, k
λq ` η p1, k
λq
η p1, k
λq ` η p0, k
λq ,
λq ` η p0, k
λq
η p1, k
λq ` η p1, k
λq ,
if k is even, and η p0, k
if k is odd. These identities are easy consequences of Harish-Chandra’s formulas for the characters of representations in the relative discrete series of GLp2, Rq. Suppose that N 2n. For any k P N, we have the unitary, irreducible character θpk q θpk, 0q
7.5. SOME REMARKS ON CHARACTERS
429
in the relative discrete series for GLp2, Rq. The corresponding character θn pk q on GLpN, F q defined by (7.5.1) is known as the Speh character attached to k and n. According to [AH, (5.11)(c)], it has an expansion θ n pk q
¸
P
sgnpwq
w Sn
ð n
i 1
θ k pi wiq, pn
1q p i
wiq
in terms of virtual characters (7.5.2). Here, wi stands obviously for the image of i under the permutation w in the symmetric group Sn . Suppose next that F is p-adic. Let r be an irreducible unitary representation of WF of degree mr . If rλ pwq rpwq |w|λ ,
w
P WF ,
is the twist of r by a real number λ, the tensor product µr pk, λq rλ b ν k
1
,
k
¥ 0,
is an irreducible representation of the Langlands group LF WF SU p2q of degree m mr pk 1q. By the Langlands correspondence, it indexes a representation in the relative discrete series of GLpm, F q, whose character we denote by θr pk, λq. As in the case of R above, we need to define θr pk, λq for all integers k. The value k 0 is already part of the definition here. It corresponds to the case that the character θr pk, λq is supercuspidal. If k 1, we follow the convention of [Tad2] of setting θr pk, λq equal to 1, the trivial character on the trivial group t1u. If k 1, we simply set θr pk, λq 0. Suppose that N mn, m mr pk 1q.
The character θrn pk q on GLpN, F q associated to the unitary character θr pk q θr pk, 0q
by (7.5.1) is itself unitary. According to [Tad2, Theorem 5.4], it can be written explicitly as a signed sum θrn pk q
¸
P
sgnpwq
w Sn
ð n
i 1
θr k pi wiq, pn
1q pi
wiq
of virtual characters (7.5.2). We have changed the notation of [Tad2] slightly in order to make the p-adic formula parallel to the formula for F R. In [Tad2], d equals pk 1q, Wn equals Sn , and the sum is taken over the subset Wnpdq
(
P Wn : wpiq d ¥ i, 1 ¤ i ¤ n ( w P Sn : k pi wiq ¥ 1, 1 ¤ i ¤ n w
of elements in Wn whose summand in the p-adic formula is nonzero. Moreover, if ρ is the unitary supercuspidal representation corresponding to r, the k character of the representation denoted by δ prρ, ν ρsq on p. 342 of [Tad2] k equals θr k, 2 in our notation.
430
7. LOCAL NONTEMPERED REPRESENTATIONS
To write the two formulas together, we can always take r to be a unitary 1-dimensional representation of WF in case F R. The resulting character of GLp1, F q is of course the only irreducible unitary representation of a general linear group over F R that meets the formal definition of supercuspidal. The virtual characters θr pk, λq and θrn pk q then make sense for both real and p-adic F , as do the representations µr pk, λq and ψrn pk q µr pk q b ν n
of LF and LF SU p2q respectively. By including r in the notation, we can then restrict λ P C to be real, as we did for p-adic F . The case F R will still differ slightly from that of p-adic F , in that θr pk, λq will remain unchanged if r is multiplied by the sign character, but this is not serious. Our interest here will generally be confined to the original case that r is trivial. At any rate, the formulas for both real and p-adic F can now be written together as ¸
θrn pk q
(7.5.3)
P
sgnpwq θrw pk q,
w Sn
where (7.5.4)
θrw pk q
n ð
i 1
θr k pi wiq, pn
1q p i
wiq .
The outer automorphism x Ñ x_ of GLpN q gives an involution θ Ñ θ_ on the space of virtual characters on GLpN, F q. The virtual character in (7.5.3) satisfies θrn pk q_ θrn_ pk q,
since the associated unitary representation ψrn pk q of LF SU p2q has the same property. To describe the resulting mapping on the summands of (7.5.3), we write w_
w`w1w` w`1w1w`,
where w` : i
ÝÑ pn
1q i,
w
1 ¤ i ¤ n,
is the element of Sn of greatest length. Lemma 7.5.1. (i) The summand in (7.5.3) satisfies _ θrw pk q_ θrw_ pk q, w P Sn .
¥ n, and that 1 θrw pk q θrw pk q, for elements w, w1 P Sn . Then w w1 . (ii) Suppose that k
Proof. (i) It is clear that
θr p`, λq_
P Sn,
θr_ p`, λq,
7.5. SOME REMARKS ON CHARACTERS
431
for any ` P Z and λ P R. We shall check that this gives a bijection between the direct summands in the expansions (7.5.4) of each side of the proposed identity. Given w, we define a permutation i
ÝÑ
i_
w`wi,
1 ¤ i ¤ n,
of the indices in the formal direct sum (7.5.4). Then k pi_ w_ i_ q k pw` wi w` w1 w` w` wiq
k pw`wi w`iq k pn 1q wi pn
1q i
k pi wiq.
The permutation therefore stabilizes the first argument ` k pi wiq
of the corresponding two direct summands. Moreover,
pn
1q pi_
w_ i_ q pn
1q pw` wi
w` iq
pn 1q pn 1q wi pn 1q i pn 1q pi wiq . The permutation therefore acts as p1q on the second argument λ pn 1q pi wiq of the direct summands. The assertion (i) follows. (ii) The first condition k ¥ n implies that k pi wiq ¥ 1,
1 ¤ i ¤ n.
In particular, the linear forms pk q in (7.5.3) are standard characters, with direct summands that lie in the mutually inequivalent family θrw
θr p`, λq,
` P N, λ P R,
of essentially square integrable characters. The second condition implies that there is a bijection i Ñ i1 of the indices such that k pi wiq k pi1 w1 i1 q
and
1q pi wiq pn 1q pi1 w1 i1 q. These two equations imply that i1 i and w1 i1 wi, so that w1 claimed.
pn
w, as
This lemma is intended only for perspective. If k ¥ n, it tells us that pkq is self-dual if and only if w w_. If k n, θrw pkq is still self-dual if w w_ . However, the converse question seems to be more complicated, especially in case F R. The anti-involution w Ñ w_ of Sn is rather curious here. For it is the involution θrw
w
ÝÑ
w` w w` ,
w
P Sn,
432
7. LOCAL NONTEMPERED REPRESENTATIONS
whose kernel is the Weyl group of the relevant classical group. The proper roles of these two operations are unclear to me. There are two elements w P Sn whose summands in (7.5.3) are of special interest. The first is the identity permutation w 1. Then θrw pk q equals the standard character (7.5.5)
φnr pk q θrw1 pk q
n ð
i 1
θr k, pn
1q 2i ,
w1
1.
This case is characterized by the condition that the direct summands indexed by i are equal, up to twists by unramified quasicharacters. The other element is the longest permutation w w` . Then θrw pk q equals θrw` pk q
n ð
θr k
i 1
pn
1q 2i, 0 .
This case has the property that the factors indexed by i are often tempered. If k ¥ n as in Lemma 7.5.1(ii), they are all tempered, and θrw` pk q is again a standard character. With this restriction on k, we set w w` , and write w w` φn, r pk q θ r p k q θ r pk q.
The summand of w w` in (7.5.3) is more complicated if k n. For pkq is then only a virtual character, which could be 0 if F is p-adic. If F is real, and k, n ¥ 1 are arbitrary, we can set w w` as above, but we define θrw`
(7.5.6)
φn, r pk q
n ð
θr k
i 1
pn
1q 2i, 0 ,
where
θr p`, λq θr p|`|, λq. If F is p-adic, we have to define w differently. For any k, n ¥ 1, we set w i
#
pn 1q i, i p 1 k q,
for any i in the interval of integers
if i ¤ k if i ¥ k
1, 2, (
Nr1, ns i P N : 1 ¤ i ¤ n .
Since w restricts to an order reversing bijection from Nr1, k 1s onto Nrn k, ns, and an order preserving bijection from Nrk 2, ns onto Nr1, n k 1s, it is indeed a permutation in Sn . We then set n pkq θw pkq ð θr k pn 1q 2i, 0, (7.5.7) φn, r r
i1 where n mintn, k 1u. The factors of φn, r pk q are thus (irreducible) tem-
pered characters in all cases. In the p-adic case, we note that the condition k ¥ n, under which we gave a simpler definition for φn, r pk q above, could be relaxed to pk 1q ¥ n.
7.5. SOME REMARKS ON CHARACTERS
433
According to the definitions (7.5.5) and (7.5.6), φn, r pk q is induced from an irreducible, tempered, essentially square integrable character of a Levi subgroup of GLpN, F q. It is therefore an irreducible tempered character on GLpN, F q. The following lemma summarizes the properties that will be needed in §8.2.
Lemma 7.5.2. Suppose that k ¥ 0, n ¥ 1, r and N are as above, and that k ¥ 1 in case F is archimedean. (i) The irreducible tempered character φn, r pk q occurs with multiplicity p1q in the decomposition of the irreducible character θrnpkq into standard characters. (ii) The factor θr pk n 1, 0q occurs with multiplicity 1 in the general decomposition (7.5.2) of φrn, pk q into essentially square integrable characters.
Proof. (i) Suppose that F R. The character φn, r pk q is part of the n contribution to θr pk q of the summand of w w w` in (7.5.3). It rep resents the tempered part of the decomposition of θrw pk q into standard characters. This follows from (7.5.6), the definition (7.5.4) in case w w , and the decomposition of any direct factor θr p`, λq into standard characters of GLp2, F q in case ` ¤ 0. Similarly, if w w w` , the standard characters in the decomposition of θrw pk q are all nontempered, since
pn
1q pi
wiq 0,
1 ¤ i ¤ n.
pkq is the only tempered character in the decomposition of
φn, r
Therefore θrn pk q into standard characters. It actually occurs with multiplicity 1. Suppose that F is p-adic. The product w sgnpw q φn, r pk q sgnpw q θr pk q
is then equal to the summand of w w in (7.5.3). Since the other summands in (7.5.3) are all standard characters (up to a sign), it suffices to show that they are all nontempered. Assume that w P Sn indexes a summand in (7.5.3) that is nonzero and tempered. We consider the decomposition (7.5.4) of θrw pk q. The index i of any direct summand in (7.5.4) satisfies either wi i
(7.5.8)
k
1,
which means that the factor equals 1, or (7.5.9)
pn
1q pi
wiq 0,
which in the absence of (7.5.8) is necessary for the factor to be tempered. The condition (7.5.8) implies that i ¥ k 2, since wi ¥ 1. It then follows from (7.5.9) that the restriction of w to Nr1, k 1s equals that of w w` , and therefore maps Nr1, k 1s bijectively onto Nrn k, ns. This implies that w restricts to a bijection from Nrk 2, ns onto Nr1, n k 1s. But we must have wi ¥ i pk 1q, i P Nrk 2, ns, since the factor of i in (7.5.4) would otherwise vanish, contradicting the condition that θrw pk q is nonzero. This in turn can hold for every i P Nrk 2, ns
434
7. LOCAL NONTEMPERED REPRESENTATIONS
only if w maps Nrk 2, ns monotonically onto Nr1, n k 1s . In other words, the restriction of w to the remaining integer interval Nrk 2, ns also equals that of w . Therefore w equals w , as required. (ii) The general decomposition (7.5.2) for φn, r pk q is given explicitly by (7.5.6) or (7.5.7), according to whether F is real or p-adic. In either case, θr pk n 1, 0q is the factor indexed by i 1. In the case (7.5.6) of F R, it is clear from the definitions (and the fact that k ¥ 1) that the factor does occur with multiplicity 1. In the case (7.5.7) of p-adic F , the factors are all distinct, and therefore all have multiplicities 1. The lemma follows. In the course of proving (i), we established the following additional property.
Corollary 7.5.3. The character φn, r pk q is the only tempered constituent in the decomposition of θrn pk q into standard characters. The formula (7.5.3) is an explicit expansion of the irreducible character θrn pk q of GLpN, F q in terms of standard characters. Our real interest, however, is in the twisted characters for GLpN, F q. One would ultimately like a twisted analogue of (7.5.3) for any F , which is to say, an explicit description of the terms on the right hand side of the general formula (2.2.9). For p-adic F , Moeglin and Waldspurger [MW4] have established an inductive process, which is both subtle and complex, that leads to such a formula. The fact that (7.5.3) holds uniformly for any F suggests perhaps that a similar process might hold for F R. However, the twisted non-tempered characters of GLpN, Rq seem not to have been studied. In any case, it would be hard to establish an explicit twisted expansion from (7.5.3). The problem, pointed out to me by the referee, is that serious difficulties accumulate from the ambiguity of extensions of self-dual representations. Fortunately, we will need only partial information, which we will in fact be able to extract from (7.5.3). Roughly speaking, we shall show that rpψ, φq in (2.2.9) are congruent modulo 2 to their untwisted the coefficients n analogues. r pN q over the local field F . This was the setting We fix a parameter ψ P Ψ of the preamble to Lemma 2.2.2, from which we take our notation. We will also use similar notation for the more elementary untwisted analogues of objects introduced in §2.2. In particular, we have the ψ-subsets ΠpN, ψ q π
and
P ΠpN q : pµπ , ηπ q pµψ , ηψ q,
Λπ
¤ Λψ
P pN, ψ q ρ P P pN q : pµρ , ηρ q pµψ , ηψ q, Λρ
¤ Λψ
(
(
of the respective sets ΠpN q and P pN q of irreducible and standard representations of GLpN, F q. We also have untwisted analogues (7.5.10)
fN pρq
¸
P p
π Π N,ψ
q
mpρ, π q fN pπ q,
f
P HpN q, ρ P P pN, ψq,
7.5. SOME REMARKS ON CHARACTERS
and (7.5.11)
fN pπ q
¸
P p
ρ P N,ψ
q
npπ, ρq fN pρq,
f
435
P HpN q, π P ΠpN, ψq,
of the twisted expansions (2.2.7) and (2.2.8), or rather their restrictions to r pN, ψ q and Pr pN, ψ q of Π r pN q and Pr pN q. The expansion (7.5.3) the subsets Π we have just described is an explicit form of (7.5.11), in the special case that ψ is simple and π πψ . To exploit it, we need to find some relation between the general expansions we have just quoted with the twisted expansions of §2.2.
r pN q, assume that ρ pρrq0 and π pπ rq0 are Lemma 7.5.4. Given ψ P Ψ r pN, ψ q. restrictions to GLpN, F q of representations ρr P PrpN, ψ q and π r PΠ Then the coefficients in (2.2.7) and (7.5.10) and in (2.2.8) and (7.5.11) satisfy rq mpρ, π q (modulo 2) mpρr, π and r, ρrq npπ, ρq (modulo 2). n pπ
r pN, ψ q are sets Proof. According to the notation of §2.2, PrpN, ψ q and Π r r of representatives of orbits in the families tP pN qu and tΠpN qu from (2.2.7) r pN, F q that are defined and (2.2.8). They consist of representations of G by the Whittaker extensions from the beginning of §2.2. In particular, the corresponding coefficients mpρr, π rq and npπ r, ρrq are indeed integers. The same is of course true of the general coefficients mpρ, π q and npπ, ρq in (7.5.10) and (7.5.11), whether ρ and π are self-dual (which is to say, of the given form pρrq0 rq0 ) or not. The particular extensions ρr and π r are in fact irrelevant and pπ r pN, F q, to the lemma. So long as they are actually representations of G they are determined up to a sign, which does not affect the assertion of the lemma. The expansion (7.5.10) is just the decomposition of the standard character ρ of GLpN, F q into irreducible characters π. In particular, mpρ, π q is the multiplicity of π in the π-isotypical subspace V pρ, π q of the space on which ρ acts (or rather the image of this space in the Grothendieck group KpGqC ). The twisted expansion (2.2.7) is the decomposition of the standard characr pN, F q into irreducible characters π r, or rather the restriction of ter ρr of G r pN, F q of G r pN, F q. It is obtained from this decomposition to the subset G (7.5.10) by first removing the summands in (7.5.10) of representations π with π π _ , and then for the remaining π, keeping track of the restrictions of ρr to the subspaces V pρ, π q. For any π P ΠpN, ψ q with π π _ , we thus have rq m pρ, π q m pρ, π q, mpρr, π where m pρ, π q is the multiplicity of π in the ρr-isotypical subspace of V pρ, π q, and m pρ, π q is the multiplicity of π in the complementary subspace. Since
mpρ, π q m
pρ, πq
m pρ, π q,
436
7. LOCAL NONTEMPERED REPRESENTATIONS
we see that mpρr, π rq is congruent to mpρ, π q modulo 2. This is the first of the required congruence relations. The second congruence relation will be a little more complicated. We can regard P pN, ψ q and ΠpN, ψ q as partially ordered sets, defined by the partial order on the linear forms Λρ and Λπ in paB q . The mapping ρ Ñ πρ from P pN, ψ q to ΠpN, ψ q is then a canonical isomorphism of partially ordered sets. To simplify the notation, we introduce an abstract partially ordered set K pψ q, with corresponding isomorphisms k Ñ ρk , and k Ñ πk onto P pN, ψ q and ΠpN, ψ q. We then write
mpρk , πk1 q,
mkk1
k, k 1
P K p ψ q,
and
nkk1 npπk , ρk1 q, k, k 1 P K pψ q, for the integral coefficients in the expansions (7.5.10) and (7.5.11). The set K pψ q inherits an involution k Ñ k _ from the outer automorphism of GLpN q, and it follows from the expansions that mk_ pk1 q_
mpρ_k , πk_1 q mpρk , πk1 q mkk1
nk_ pk1 q_
npπk_, ρ_k1 q npπk , ρk1 q nkk1 ,
and
for any k and k 1 in K pψ q. Let U pψ q be the set of unipotent, integral K pψ q K pψ q -matrices k, k 1
a pakk1 q,
P K pψ q,
such that
1, (ii) ukk1 0, unless k 1 ¤ k,
(i) ukk and
(iii) uk_ pk1 q_
ukk1 .
It follows immediately that U pψ q is a group under matrix multiplication, in which our two coefficient matrices pmkk1 q and pnkk1 q represent inverse elements. To exploit the third condition, we borrow our earlier notation from the slightly different context of §1.4. We write K pψ q I pψ q
²
J pψ q
where
²
J _ pψ q, (
I pψ q i P K pψ q : i_ i , while J pψ q is a fixed set of representatives in K pψ q of orbits of order two under the involution and J _ pψ q is the complementary set of representatives. We can then introducea second group. Let UI pψ q be the group of unipotent, integral, I pψ q I pψ q -matrices aI
paii1 q,
i, i1
P I pψ q,
7.5. SOME REMARKS ON CHARACTERS
437
that satisfy the three conditions above (but with ii1 in place of kk 1 ). In this case, there are really only two conditions, since the adjoint condition (iii) is trivial. There is a natural restriction mapping a Ñ aI from U pψ q onto UI pψ q. However, this projection is not a group homomorphism. To describe the obstruction, consider the image in UI pψ q of a product of matrices a and b in U pψ q. For any indices i and i1 in I pψ q, we can write
pabqii1
¸
P p q
aik bki1
k K ψ
¸
Pp q
i I ψ
¸
Pp q
i I ψ
¸
aii bi i1
P
aij bji1
j J
aii bi i1
2
¸
P
¸ j_ J _
P
aij _ bj _ i1
aij bji1 ,
j J
since aij _ aij and bj _ i1 bji1 . In particular, the obstruction vanishes if the matrix coefficients are taken modulo 2. Let us therefore write U pψ q and U I pψ q for the two matrix groups defined in the same way as U pψ q and UI pψ q, but the coefficients in the field Z{2Z. The restriction mapping u Ñ uI is then a surjective homomorphism from U pψ q onto U I pψ q. To put the matter a little more strongly, we write x for the reduction modulo 2 of any integral matrix x. It then follows that the mapping
paI q, a P U pψq, is a surjective homomorphism from U pψ q onto U I pψ q. α: a
ÝÑ
aI
We now complete the proof of the lemma. We can of course identify the matrices m and n in U pψ q with the coefficient matrices of (7.5.10) and (7.5.11). In particular, n equals the inverse of m. It follows that nI
αpnq αpm1q αpmq1 pmI q1,
since as a homomorphism, α commutes with inverses. Having established the first congruence relation, we can identify mI with the reduction modulo 2 1 of the coefficient matrix in (2.2.7). Its inverse m I is therefore the reduction modulo 2 of the coefficient matrix in (2.2.8). Since this is in turn equal to nI , we have also established the second congruence relation. Our interest will be in the special case of (7.5.11) that π equals the Langlands quotient πψ πφψ attached to ψ. In this case, (7.5.11) can be written in the form (7.5.12)
fN pψ q
¸
P p
φ Φ N,ψ
q
npψ, φq fN pφq,
f
P H pN q ,
where ΦpN, ψ q φ P ΦpN q : pµφ , ηφ q pµψ , ηψ q, Λφ
¤ Λψ
(
438
7. LOCAL NONTEMPERED REPRESENTATIONS
and
npψ, φq npπψ , πφ q.
This is the untwisted analogue of the twisted expansion (2.2.9). It is the abstract setting for the explicit character identity (7.5.3). The formula (7.5.3) actually applies only to case that ψ is simple. However, it is easily extended to any parameter
`1ψ1 ` ` `r ψr . When ψ is simple, and corresponds to a pair pr, k q as in (7.5.3), we write θpψ q θrn pk q. ψ
for the right hand side of (7.5.3). For general ψ, we can then set (7.5.13)
θpψ q looooooooooomooooooooooon θpψ1 q ` ` θpψ1 q
`
`
p q
p q
θ ψr ` ` θ ψr . looooooooooomooooooooooon
`1
`r
As a character on GLpN, F q induced from an irreducible unitary character, θpψ q is itself irreducible. It is the character of the Langlands quotient πψ , whose value at a test function f P HpN q equals the left hand side of (7.5.12). The corresponding direct sum of the virtual characters on the right hand side of (7.5.3) is an explicit (though by now somewhat complicated) integral linear combination of standard characters on GLpN, F q. Its value at f equals the right hand side of (7.5.12). The two extremal standard characters in this explicit form of (7.5.12) are θpφψ q θloooooooooooomoooooooooooon pφψ1 q ` ` θpφψ1 q
`
`
`r
`1
and
θloooooooooooomoooooooooooon pφψr q ` ` θpφψr q
pφψ1 q ` ` θpφψ1 q θpφψ q θloooooooooooomoooooooooooon
`
`
`1
θloooooooooooomoooooooooooon pφψr q ` ` θpφψr q, `r
where if ψ is simple, θpφψ q φnr pk q and θpφψ q φn, r pk q, in the earlier notation. In the notation here, it is understood φψ and φψ stand for the Langlands parameters of the standard characters θpφψ q and θpφψ q. We know very well that npψ, φψ q 1. But from Lemma 7.5.2(i) we observe also that npψ, φψ q 1.
We recall that φψ indexes the constituent of (7.5.12) that is most highly nontempered, while it follows from Corollary 7.5.3 that φψ indexes the unique constituent of (7.5.12) that is tempered. With these remarks, we obtain the following corollary of Lemma 7.5.4.
7.5. SOME REMARKS ON CHARACTERS
439
r pN, ψ q, the coefficients in (2.2.9) and Corollary 7.5.5. For any φ P Φ (7.5.12) satisfy n rpψ, φq npψ, ψ q (modulo 2).
In particular,
rpψ, φψ q 1 n
(modulo 2),
so the parameter φψ has nonzero multiplicity in the twisted expansion (2.2.9). It indexes the unique constituent of (2.2.9) that is tempered. r pGq of Ψ r pN q attached to a simple Suppose that ψ lies in the subset Ψ r datum G P Esim pN q. The Langlands parameter
ÝÑ LG of the standard character θpφψ q on GLpN, F q is defined by φψ puq ψ u, ppuq , u P LF , φψ : LF
where
|u|
ppuq
1 2
u P LF .
0 1 ,
|u| 2
0
We can interpret the homomorphism p : LF
ÝÑ
SLp2, Cq
as the Langlands parameter for the trivial one-dimensional representation of P GLp2, F q. As such, it has a dual parameter p : LF
ÝÑ
SLp2, Cq,
which corresponds to the associated square integrable representation of P GLp2, F q. More precisely, p pw, sq s,
if F is p-adic, while if F
pw, sq P LF WF SU p2q,
R, we have
pz{zq
p p z q
and
0
1 2
0
1 pz{zq 2
p pσ F q
,
0 1 1 0 ,
z
P C ,
440
7. LOCAL NONTEMPERED REPRESENTATIONS
in the notation of §6.1. We can therefore associate a second Langlands parameter φψ : LF ÝÑ L G r pGq to ψ by defining in Φ
φψ puq ψ u, p puq ,
u P LF .
This is the Langlands parameter of the standard character θpφψ q on GLpN, F q. For p-adic F , it plays an important role in the work of Moeglin.
CHAPTER 8
The Global Classification 8.1. On the final step We are at last in a position to prove the global theorems. We have already had to treat some special cases in order to prove the local theorems. It is now time to work in the opposite direction. We shall use the local theorems we have just established to prove the global theorems in general. We will complete this argument in §8.1 and §8.2. In the last three sections of the chapter, we will discuss some ramifications of the results. Recall that there are four global theorems. They consist of Theorems 1.5.2 and 1.5.3 (which for their statements implicitly include Theorems 1.4.1 and 1.4.2), and the global supplements Theorems 4.1.2 and 4.2.2. The global intertwining relation of Corollary 4.2.1 is actually a corollary of the associated local intertwining relation, which we have established in all cases. On the other hand, the stable multiplicity formula of Theorem 4.1.2 is really the fundamental global assertion. It represents a foundation on which the others all depend. The underlying field F will be global throughout this chapter, unless r of selfspecified otherwise. The theorems apply to parameters ψ in the set Ψ dual representations that occur in the automorphic spectral decompositions of general linear groups over F . We shall combine the local results we have established with the global formulas of §5.1–5.3. To this end, we will take r To be consistent the global family Fr of Chapter 5 to be the entire family Ψ. with the conventions of §5.1, we must consider the case that Fr equals the r of generic parameters in Ψ, r as well as that of F r Ψ. r However, we subset Φ r in place of F, r distinguishing when necessary between will generally write Ψ the generic and nongeneric cases. As usual, when working with nongeneric r we will assume implicitly that we have already established parameters in Ψ, the global theorems for generic parameters. We fix a positive integer N , and as in §5.1, assume inductively that the r with degpψ q N . We do not need global theorems all hold for any ψ P Ψ to add a local hypothesis, of course, since we have now established all the local theorems. We do however have to check that the family satisfies the additional hypothesis of §5.1. r 2 pGq. Then Lemma 8.1.1. Suppose that G P Eell pN q and that ψ belongs to Ψ the conditions of Assumption 5.1.1 hold for the pair pG, ψ q. 441
442
8. THE GLOBAL CLASSIFICATION
Proof. This is the analogue of Lemma 5.4.2, for the maximal family r However, it will be a little easier for us to prove, now that we have Fr Ψ. the local theorems in hand. Suppose for example that G GS GO is composite. From our induction hypothesis, we observe that any localization r pGv q. The condition of Assumpψv ψS,v ψO,v belongs to the local set Ψ tion 5.1.1 in this case then follows from its local version (2.2.4) in Theorem 2.2.1. We therefore assume that G is simple. If we can show that any localr pGv q, we can appeal again to the relevant ization ψv of ψ still belongs to Ψ local assertion (2.2.3) of Theorem 2.2.1. Suppose that ψ is composite. An application of our induction hypothesis to the simple constituents ψi of ψ, r pGq and Ψ r pGv q, then combined with the inductive definition of the sets Ψ r confirms that ψv lies in ΨpGv q, and hence that ψ satisfies Assumption 5.1.1. If ψ is simple but not generic, we can apply our induction hypothesis to r pGv q, and that ψ its simple factor µ. We see again that ψv belongs to Ψ satisfies Assumption 5.1.1. We can therefore assume that φ ψ is simple and generic. We have reduced the proof of the lemma to the case of a simple generic pair pG, φq, G P ErsimpN q, φ P Φr simpGq. r p G v q. The problem is to show that any localization φv of φ belongs to Φ The reader will recognize in this the assertion of Theorem 1.4.2, the second seed theorem from Chapter 1. However, we are now using the temporary r sim pGq from §5.1, rather than the original definition from definition of Φ Theorem 1.4.1 in terms of which Theorem 1.4.2 was stated. The logic of the r sim pGq temporary definition is actually somewhat convoluted. For we took Φ r to be the subset of elements φ P Φsim pN q that satisfied the condition of G 0. Assumption 5.1.1, in addition to the natural global condition Sdisc,φ We then had to reinterpret Assumption 5.1.1 for simple generic pairs as the condition (5.1.6). Our problem, thus modified, is to show that for any r sim pN q, we can find a simple datum G P Ersim pN q such that S G φPΦ disc,φ 0
and such that the linear form frN pφq transfers to G. r sim pN q, we write For the given φ P Φ N N Irdisc,φ pfrq tr Rrdisc,φ pfrq
frN pφq,
r pN q. fr P H
r pN q in place of G) This is one of the various reductions of (4.1.1) (with G r pN q-analogue with which we are now very familiar. It then follows from the G (3.3.14) of (4.1.2) that
frN pφq
¸
P
p q
G Ersim N
p
q
p q
G r ι N, G Spdisc,φ frG ,
r pN q . fr P H
Here we are using the fact, obtained by the usual induction argument, that G Sdisc,φ vanishes for any G in the complement of Ersim pN q in Erell pN q. There
8.1. ON THE FINAL STEP
443
must be at least one G P Ersim pN q whose summand does not vanish identically in fr. If there is exactly one, we have G ιpN, Gq Spdisc,φ frN pφq r pfrGq,
r pN q, fr P H
r pGq, as required. and frN pφq is the pullback of a unique stable linear form on H This is the case for example if N is odd or the character ηψ is nontrivial. We can therefore assume that N is even and ηψ 1. The summand of G then vanishes unless G is one of the two split groups in Ersim pN q, which we denote as usual by G and G_ . The identity becomes G G_ frN pφq r ιpN, Gq Spdisc,φ pfrGq rιpN, G_q Spdisc,φ pfrG_ q.
We can assume that neither summand vanishes identically in fr, since we would otherwise be in the case settled above. We can also assume that the linear form frN pφq does not transfer to either G or G_ , since we would otherwise again be done. We must show that these conditions lead to a contradiction of the identity. The second condition implies that there are valuations v and v _ such r pGv q and Φ r pG__ q. that φv and φv_ do not lie in the respective local sets Φ v r v pN q such Suppose first that v v _ . One can then choose a function frv P H _ that frvG frvG 0, but such that frv,N pφv q 0. We leave the reader to check that this is possible, using the characterization of Proposition 2.1.1 r pG q attached to and the fact that φv must still belong to the local set Φ v some Gv P Erv pN q. Once frv is chosen, we set fr frv frv , for any function r v pN q such that frv pφv q 0. Then frG frG_ 0, while frN pφq 0. frv P H N In other words, the right hand side of the last identity vanishes, while the left hand side is nonzero. This is a contradiction. Suppose next that v v _ . r v pN q and frv_ P H r v_ pN q such that We can then choose functions frv P H _ G G r r r r fv 0 and fv_ 0, but such that fv,N pφv q and fv_ ,N pφv_ q are both _ nonzero. We then set fr frv frv_ frv,v , for a complementary function _ _ v,v frv,v with frN pφv,v_ q 0. Once again, we have frG frG_ 0 and frN pφq 0, and therefore a violation of the last identity. We thus have obtained a contradiction in both cases. Our conclusion is that the linear form frN pφq does indeed transfer to one of the two groups. Consequently, r sim pGq there is a simple datum G P Ersim pN q such that φ lies in the subset Φ r of Φsim pN q. This completes the proof of the lemma, in the last case of a simple generic parameter. Remark. We observed that Assumption 5.1.1, when restricted to the case of r sim pN q, is essentially the assertion of Theosimple generic parameters φ P Φ rem 1.4.1. The only difference is that it is expressed in terms of our alternate r sim pN q and Φ r v pN q. In other words, Assumption 5.1.1 definitions of the sets Φ (for simple generic φ) is tied to the conditions (i) in Corollaries 5.4.7 and
444
8. THE GLOBAL CLASSIFICATION
6.8.1, while Theorem 1.4.2 is tied to the equivalent conditions (ii) in these corollaries. r (or F r Φ) r within the We have now placed the maximal family Fr Ψ general framework of §5.1. In particular, the family comes with a set of induction hypotheses on the positive integer N . Our task is to complete the inductive proof by establishing the global theorems for parameters ψ in the r pN q (or F r pN q Φ r pN q). set FrpN q Ψ Suppose for a moment that ψ belongs to the complement of the set r ell pN q. Recall that this means that ψ does not belong to Ψ r 2 pGq, for any Ψ G P Erell pN q. The global theorems reduce in this case to the two vanishing assertions of Proposition 4.5.1. However, the proposition actually imposes a r ell pGq, further requirement that ψ also not belong to the larger elliptic set Ψ if G P Ersim pN q is simple. This second condition does not subsume the first, r 2 pG q for some G P Erell pN q that is not simple. since it permits ψ to lie in Ψ And indeed, there really was something to establish in this case. We did it in Lemma 5.1.6, which was our setting for a general proof of Theorem 1.5.3(b), and where we also completed the proof of the two vanishing assertions for pairs
pG, ψq,
G P Ersim pN q, ψ
P Ψr 2pGq.
(The corresponding two assertions do not have to be proved for the pair pG, ψq. They follow immediately from our induction hypothesis on N , G G and the fact that the linear forms trpRdisc,ψv q and Sdisc,ψ for composite G decompose into the associated products. We note, incidentally, that the roles of G and G here have been interchanged from Lemma 5.1.6.) Proposition 4.5.1 and Lemma 5.1.6, taken together, thus yield the global theorems for any ψ that does not belong to the elliptic set attached to any simple G. r ell pGq, for We can therefore assume from this point on that ψ lies in Ψ some G P Ersim pN q. One sees easily from the general form (4.5.9) of Sψ that G is uniquely determined by ψ. r 2 pGq of Ψ r 2 pGq in Ψ r ell pGq. Suppose first that ψ lies in the complement Ψ ell r Then ψ lies in the complement of Ψell pN q. The global theorems therefore reduce to the two vanishing assertions of Proposition 4.5.1, even though the proposition itself does not apply to this case. We shall instead apply Lemma 5.2.1 or Lemma 5.2.2, according to whether the index r in the decomposition (5.2.4) of ψ is greater than 1 or equal to 1. The global intertwining relation (4.2.6) of Corollary 4.2.1, which we now know is valid, tells us that the given expressions (5.2.8), (5.2.12) and (5.2.13) in these lemmas all vanish. It follows that the remaining expressions (5.2.7) and (5.2.11) in the two cases also vanish. In other words, we have ¸ G Ersim N
P
p q
r ι N, G tr Rdisc,ψ f
p
q
p q 0,
8.1. ON THE FINAL STEP
445
for any r, and any compatible family of functions (5.2.6). It then follows from Proposition 3.5.1 that
pf q tr Rdisc,ψ
(8.1.1)
G
0,
P ErsimpN q,
f
This is one of the two vanishing assertions. The second, (8.1.2)
0
pf q 0, Sdisc,ψ
G
P ErsimpN q,
f
P HrpGq.
P HrpGq,
follows from the first. Indeed, the difference (5.2.9) of the left hand sides of (8.1.1) and (8.1.2) was seen in the proofs of Lemmas 5.2.1 and 5.2.2 to equal one of the expressions (5.2.8), (5.2.12) and (5.2.13) we now know are all equal to 0. We have thus established the global theorems when ψ does not lie in r Ψ2 pGq, for any G P Ersim pN q. This is a major step, even though it amounts only to a pair of vanishing formulas. Theorem 1.5.2 asserts that the contribur 2 pGq exhaust the automorphic discrete spectrum tions from parameters in Ψ of G. The first formula (8.1.1), applied to G G (and augmented by Corollary 3.4.3), asserts that this is indeed the case. In particular, G has no embedded eigenvalues, in the sense described at the beginning of §4.3. This result has been hard won. The proof we have just completed calls upon much of what we have done since Chapter 1. At its heart are the comparisons from the standard model in §4.3 and §4.4, in which the two vanishing formulas are treated side by side. The second formula (8.1.2) is equivalent to the stable multiplicity formula (4.1.11) for the pair pG, ψ q. We have now established that it holds when the group Sψ attached to pG, ψ q is infinite. 0 This of course is the case in which the coefficients σ pS ψ q in (4.1.11) vanish, according to the property (4.1.9). We come now to the last case, in which ψ is “square integrable”. We r 2 pGq, for some G P Ersim pN q. In particular, assume henceforth that ψ lies in Ψ ψ
ψ1
`
`
ψr
is multiplicity free. For this case, there will be assertions to be established from all of the global theorems. If ψ is compound, in the sense that r ¡ 1, the techniques we have been using will be quite sufficient. The case r 1 that ψ is simple is considerably harder. It requires something further, which we will discuss in the next section. In both cases, the stable multiplicity formula remains the key result to be proved. Lemma 5.1.4 gives a criterion for the stable multiplicity formula to be valid for any pair pG , ψ q, with G P Ersim pN q. It tells us that we need only show that the linear form Λ on Sr0 pLq vanishes. We recall that L and Λ are defined only if N is even and ηψ 1, or equivalently, if there is a second group G_ P Ersim pN q with ηG_ 1. Then L GLp 21 N q represents a maximal Levi subgroup of both G and G_ , and Λ is the obstruction for _ S G_ on G_ to vanish as expected. Recall the stable linear form Sdisc,ψ disc,ψ also that if N is odd or nψ 1, we have already established the stable multiplicity formula for each pG , ψ q in Corollary 5.1.3.
446
8. THE GLOBAL CLASSIFICATION
Assume now that r ¡ 1. We apply Lemma 5.3.1 to the supplementary parameter ψ . The global intertwining relation, which we now know holds in general, tells us that the expression (5.3.5) vanishes. It then follows from the lemma that the sum ¸
G Ersim N
P
p q
r ι N , G tr Rdisc,ψ
p
q
pf q
b f L pψ 1 Λ q
vanishes, for a positive constant b , and any compatible family of functions (5.3.3). We can now argue exactly as in the proof of Lemma 5.4.5. That is, we replace the supplemental summand on the right by the linear form
P HrpG_1 q, _ r for the composite endoscopic datum G_ 1 G1 G in Eell pN q, with corL
b f1
pψ1 Λq,
f1
responding function f1 in (5.3.3). This is a nonnegative linear combination of irreducible characters in f1 , by Corollary 5.1.5 and the application of our induction hypothesis to ψ1 . We can therefore apply Proposition 3.5.1 to the modified sum. This allows us to say that the linear form ψ1 Λ attached to the maximal Levi subgroup L of G_ 1 vanishes. Since the linear form attached to ψ1 is nonzero by our induction hypothesis, we conclude that Λ vanishes. The stable multiplicity formula thus holds for any pair pG , ψ q, and for pG, ψ q in particular, whenever the index r for ψ is greater than 1. Now that we have the stable multiplicity formula for parameters r 2 pGq with r ¡ 1, we shall establish Theorem 1.5.2 for any such ψ. ψPΨ More precisely, for any G P Ersim pN q, we shall show that the contribution of ψ to the discrete spectrum of G is the summand of ψ on the right hand side of the analogue for G of the decomposition (1.5.5) of Theorem 1.5.2. r pG q, by If G P Ersim pN q is distinct from G, ψ does not belong to Ψ definition. There is consequently no contribution of ψ to the analogue for G of (1.5.5). According to the stable multiplicity formula for pG , ψ q, attached to pG , ψ q vanishes. It follows from the stable linear form Sdisc,ψ (4.4.12) that
pf q tr Rdisc,ψ
pf q 0, Sdisc,ψ
f
P HrpGq,
so there is also no contribution of ψ to the discrete spectrum of G . It therefore suffices to consider the case that G G. Since ψ belongs to r 2 pGq, its contribution to (1.5.5) is a nontrivial multiplicity formula, rather Ψ than zero. But we have already seen how to prove this formula. The proof is of course the content of our conditional Lemma 4.7.1, from the end of §4.7. The conditions of this lemma (both explicit and implicit) have now all been met. The conclusion (4.7.10) of the lemma, namely that the contribution of ψ to the discrete spectrum of G is indeed the summand of ψ in (1.5.5), is therefore valid. Theorem 1.5.2 therefore holds for the compound parameter r 2 p G q. ψPΨ The only other global theorem that is relevant to ψ is Theorem 4.2.2. r attached to a datum G P Ersim pN q, in case N is It concerns the bitorsor G
8.1. ON THE FINAL STEP
447
p SOpN, Cq, and ψ lies in Ψ r 2 pGq. Recall that (4.2.7) provides a even, G canonical extension rdisc,ψ RGr R disc,ψ G r pAq. The theorem has of the representation Rdisc,ψ to the GpAq-bitorsor G
r q of Ψ r 2 pGq or two assertions, according to whether ψ lies in the subset Ψ2 pG not. The argument is similar to the special case discussed, somewhat hastily, in §6.8. We form the ψ-component G Irdisc,ψ pfrq Idisc,ψ pfrq,
r q, fr P HpG
r
r following (3.3.12). of the discrete part of the twisted trace formula for G, According to Hypothesis 3.2.1, it satisfies the analogue
Irdisc,ψ pfrq
¸ r G1 Eell G
P p q
r1 r G r1 q S r1 ιpG, disc,ψ pf q p
r of (4.1.2). It also satisfies the analogue for G r of (4.1.1). As we have for G observed many times in the case of a “square integrable” parameter ψ, the terms with M G in (4.1.1) all vanish. The remaining term with M G rdisc,ψ pfrq, by the general definitions at the end is the trace of the operator R of §3.1. It follows that
(8.1.3)
rdisc,ψ pfrq tr R
¸
r 1 Eell G r G
P p q
r1 r G r1 q S r1 ιpG, disc,ψ pf q. p
r q in Ψ r 2 pGq. Then ψ does Suppose that ψ lies in the complement of Ψ2 pG r 1 P Eell pG r q. It follows r pG r 1 q, for any twisted endoscopic datum G not lie in Ψ 1 r , or in fact the application to G r1 from the stable multiplicity formula for G 1 vanishes. of the more elementary assertion of Proposition 3.4.1, that Srdisc,ψ The left hand side of (8.1.3) therefore also vanishes. This is the assertion (a) of Theorem 4.2.2. r q of Ψ r 2 pGq. The set Srψ Now suppose that ψ belongs to the subset Ψ2 pG is then nonempty, and is a bitorsor over Sψ . To establish the corresponding assertion (b) of Theorem 4.2.2, we can follow the argument from §6.8 for a generic global parameter with local constraints. Starting with (8.1.3), we obtain a generalization
(8.1.4)
¸
P p q
r π Π G
rq |Sψ |1 nψ pπ q frGr pπ
¸
P
r Srψ x
ε1 pψ 1 q fr1 pψ 1 q,
r 1 , ψ 1 q maps to pψ, x r q is the subset of represenof (6.8.4), in which pG rq, ΠpG r r tations in ΠpGq that extend to GpAq, and nψ pπ q is the multiplicity of π in G r is the canonical extension of π defined Rdisc,ψ . On the left hand side, π whenever nψ pπ q 0.
448
8. THE GLOBAL CLASSIFICATION
We noted in §4.8 that the proof of Lemma 4.4.1 in §6.6 is quite general. r We can In particular, it carries over without change to a twisted group G. therefore write ¸
P
r Srψ x
ε1 pψ 1 q fr1 pψ 1 q
¸
P
r Srψ x
¸
rq εψ p s ψ x
P
xsψ xr, πry frGr pπrq,
π Πψ
according to the general argument of §4.7. With a change of variables in the sum over x r, we see that the right hand of (8.1.4) equals
|Sψ |1 This in turn equals
¸
P
x r Srψ
¸
rq εG ψ px
P
xxr, πry frGr pπrq.
π Πψ
¸
P p q
r q, frGr pπ
π Πψ ε ψ
where Πψ pεψ q is the set of representations π in the global packet Πψ such 1 G r is the that the linear character x , π y on Sψ equals ε ψ εψ εψ , while π canonical extension of π determined by εψ . To be more precise, we recall r pAq to any r of π from GpAq to G that Theorem 2.2.4 assigns an extension π r extension x , π ry of the linear form x , π y from Sψ to Sψ . In the last sum, π r r corresponds to the canonical extension of εψ x , π y to Sψ . The formula (8.1.4) thus takes the form ¸
P p q
r π Π G
rq nψ pπ q frGr pπ
¸
P p q
π Πψ εψ
r q, frGr pπ
r q. fr P HpG
rG r represents the extension of π determined by R On the left hand side, π disc,ψ , while on the right hand side, it stands for the extension determined by εψ . The formula tells us that the two extensions match, as asserted in Theorem 4.2.2(b). This completes the proof of Theorem 4.2.2, and hence our study r 2 pGq with r ¡ 1. of global parameters ψ P Ψ r sim pGq is a simple Suppose now that r 1. In other words, ψ P Ψ r parameter for the group G P Esim pN q. As we have noted, this case will be more of a challenge. We shall begin the discussion here, leaving the main burden of proof for the next section. There are two general problems in this case. One remains the stable multiplicity formula, which entails proving that the linear form Λ vanishes. We shall soon see how to reduce this to the case that ψ is not generic. The other r sim pGq. In is the assertion of Theorem 1.5.3(a) for generic parameters ψ P Ψ §5.3, we formulated it in terms of a sign δψ , which equals 1 or 1 according to whether the assertion is valid for ψ or not. This of course applies only to the case that ψ is generic. There will thus be one problem to solve if ψ is not generic, and another if it is. We remind ourselves that we are working in the general framework of Chapter 5. In particular, we are following the convention at the beginning
8.1. ON THE FINAL STEP
449
r sim pGq of Ψ r sim pGq. Once we of Chapter 5 for the definition of the subset Φ have established the stable multiplicity formula in case ψ is generic, we will r sim pN q be able to say that G is uniquely determined by φ, and hence that Φ r sim pGq. This leads directly is a disjoint union over G P Ersim pN q of the sets Φ to a proof of Theorem 1.4.1. It will therefore tell us that the temporary r sim pGq is equivalent to the original definition in terms of this definition of Φ theorem. We recall that these matters were treated in §5.4, when Fr was the family with local constraints introduced there. We are of course now taking r or its subset Φ r of generic parameters. Fr to be either the full set Ψ r sim pGq as we We begin our investigation of the simple parameter ψ P Ψ did in the compound case above. That is, we turn to the only result available to us, the relevant lemma from §5.3. This is Lemma 5.3.2, which we apply to the supplementary parameter
ψ
ψ
`
ψ,
and a compatible family of functions (5.3.3). Consider the terms in Lemma 5.3.2. We first note that the global intertwining relation holds for the pairs pG , ψ q and pG_ , ψ q. It gives identities fG1 pψ , sψ x1 q fG pψ , x1 q
and
pf _qM pψq pf _q1G_ pψ
, x1 q fG__ pψ , x1 q
between the terms in the two expressions (5.3.22) and (5.3.23) of the lemma. The difference between (5.3.22) and (5.3.23) therefore equals 1 8
p1 δψ qpf _qM pψq 18 p1 δψ qfG pψ
, x1 q,
an expression we can write as 1 8
p1 δψ q f M pψq fG pψ
since
, x1 q ,
pf _ qM pψ q f M p ψ q .
Now f M pψ q represents a unitary character on the group G pAq. It is induced from the Langlands quotient representation πψ of the maximal Levi subgroup M pAq GLpN, Aq of G pAq. By definition, fG pψ , x1 q is a difference of two unitary characters on G pAq whose sum equals f M pψ q. Since δψ equals 1 or 1, the last expression is therefore a nonnegative linear combination of irreducible characters. Lemma 5.3.2 tells us that its sum with ¸
G Ersim N
P
p q
r ι N , G tr Rdisc,ψ
p
q
pf q
1 2
f L p Γ Λq ,
450
8. THE GLOBAL CLASSIFICATION
the expression we labelled (5.3.21) in the statement of the lemma, vanishes. Arguing as in the proof of Lemma 5.4.6, we replace the supplemental summand in (5.3.21) by the linear form
P HrpG_1 q, _ r for the composite endoscopic datum G_ 1 G G in Esim pN q, with cor1 L 2 f1
p Γ Λ q,
f1
responding function f1 from the family (5.3.3). By Corollary 5.1.5 and the definition of Γ, this is a nonnegative linear combination of irreducible characters in f1 . The same is of course true of the other summands in (5.3.21). The terms in Lemma 5.3.2 thus combine to give an expression of the general form (3.5.1). The lemma itself asserts that the expression so obtained vanishes. It then follows from Proposition 3.5.1 that the corresponding coefficients vanish. In particular, we see that (8.1.5)
Rdisc,ψ pf q 0,
that and that
G
P ErsimpN q,
f
P HrpGq,
Γ Λ 0,
p1 δψ q
f M pψ q fG pψ , x1 q
0. The second of these equations tells us that either Λ 0, which is what we want, or Γ 0. In other words, we have (8.1.6) Γ 0, if Λ 0. The third equation tells us that either δψ 1, which is what we want, or the complementary factor on the left hand side vanishes. In other words, we have (8.1.7)
f M pψ q fG pψ , x1 q,
if δψ
1,
r pG q. What is one to make of these conditions? for any function f P H G Recall that Sdisc,ψ pf q equals a scalar multiple of f G pΓq, according to the G definition prior to Lemma 5.3.2. If ψ is generic, Sdisc,ψ is nonzero, by our
r sim pGq. It follows that Γ 0, and therefore that Λ 0. We definition of Φ note in passing that this argument could also have been used in the proof of Lemma 5.4.6. The local condition (5.4.1)(a) we used instead is parallel to its nontempered analogue needed for the proof of Lemma 7.3.2. In any case, we have now established the stable multiplicity formula for generic ψ. This is the result we anticipated above. It resolves the temporary definition r sim pGq from §5.1 in terms of the original definition by Theorem 1.4.1, of Φ r p G q. and thus completes our inductive definition of the full set Ψ This leaves only the two problems mentioned above. We must show that δψ 1 if ψ is generic, and that Λ 0 if ψ is not generic. At the beginning of the next section, we shall see whether we can solve the two problems by applying the conditions (8.1.7) and (8.1.6).
8.2. PROOF BY CONTRADICTION
451
8.2. Proof by contradiction The task now is to finish proving the most intractable case of the global theorems. In the last section, we fixed a positive integer N , and imposed the usual induction hypothesis that the theorems all hold for global parameters r with degpψ q N . We then described how our earlier results yield the ψPΨ r sim pN q. global theorems for parameters of degree N in the complement of Ψ r It remains to treat the simple global parameters ψ P Ψsim pN q. In the last section we also obtained information from the application of Lemma 5.3.2. In particular, we proved the stable multiplicity formula in the r sim pGq of generic parameters in Ψ r sim pGq. It case that ψ lies in the subset Φ remains only to establish the condition δψ 1 of Theorem 1.5.3(a) in this r sim pGq is not generic, δψ 1 case. In the complementary case that ψ P Ψ by definition. In this case, it is only the stable multiplicity formula for ψ that needs to be established. We thus have the two separate properties to establish, one when ψ is generic, and the other when it is not. In each case, there is a corresponding condition (8.1.7) or (8.1.6) we can bring to bear on the proof. Are the conditions (8.1.7) and (8.1.6) we have obtained from the application of Lemma 5.3.2 sufficient to establish the final two properties? Our initial answer to this question is not likely to be encouraging. However, with patience, we will see that it eventually leads to a happy conclusion. We shall study the question for a general pair
pG, ψq,
G P Ersim pN q, ψ
P Ψr simpGq.
The most critical case is obviously that of N even and nψ 1. This is the case in which G is split, and where we have the auxiliary objects G_ , L and Λ. We shall discuss these objects in treating the general case, with the implicit understanding that any terms in which they occur vanish if N is odd or nψ 1. Consider the condition (8.1.6) that either Λ or Γ must vanish. If it is Λ that vanishes, as required, we see from Lemma 5.1.4 that G Sdisc,ψ pf q |ΨpG, ψq| |Sψ |1 εGpψq f Gpψq orpGq f LpΛq
orpGq 1 1 f Gpψq orpGq f Gpψq,
and that G_ Sdisc,ψ
orpG_q f _,LpΛq 0, P HrpGq and f _ P HrpG_q
for functions f that we may as well assume come from a compatible family. These are of course the stable multiplicity formulas for pG, ψ q and pG_ , ψ q. In this case, we also obtain the local identity f G pψ q f G pΓq
f L pΛq f G pΓq,
452
8. THE GLOBAL CLASSIFICATION
from the formula obtained in §5.3 prior to Lemma 5.3.2. On the other hand, if it is Γ that vanishes, we have the local identity f G pψ q f G pΓ q
f L p Λ q f L pΛ q.
In particular, the stable distribution on GpAq attached to the original parameter ψ transfers locally from the Levi subgroup LpAq. In this case, it follows from Lemma 5.1.4 that G Sdisc,ψ pf q |ΨpG, ψq| |Sψ |1 εGpψq f Gpψq orpGq f LpΛq
orpGq 1 1 f LpΛq orpGq f LpΛq 0,
and that G_ Sdisc,ψ pf _q orpG_q f _,LpΛq orpG_q f Gpψq.
Consider the case that ψ is generic. Then Λ 0, as we have seen. The problem here is to show that δψ 1. Assume the contrary, namely that r pG q, we obtain identities δψ 1. For any function f P H f M pψ q fG pψ , x1 q fG1 pψ , x1 q
f GGpψ ψq f GGpΓ Γq,
from (8.1.7), the global intertwining relation, and the definition of fG1 in terms of the endoscopic transfer f GG of f to the endoscopic datum G1 G G for G . It then follows from Corollary 5.3.3 and (8.1.5) that Sdisc,φ pf q 12 orpG G
q f M p ψ q.
r pG_ q lies in a compatible family that contains f , we In addition, if f _ P H obtain an identity
f M pψ q fG__ pψ , x1 q,
as above. It then follows from Corollary 5.3.3, (8.1.5), and the vanishing of Λ that G_ Sdisc,φ pf _ q 14 orpG_ q f M pψ q. To summarize what we have obtained so far, we take two pairs of functions
tf1 P HrpGq,
f1_
P HrpG_qu
tf2 P HrpG q,
f2_
P Hr pG_qu
and
8.2. PROOF BY CONTRADICTION
453
that are compatible, in the sense that they belong to compatible families for Erell pN q and Erell pN q respectively. We then have $ G ' Sdisc,ψ f1 ' ' ' ' _ ' &S G f_
(8.2.1)
p q orpGq f1Gpψq, disc,ψ p 1 q 0, M G ' Sdisc,ψ pf2 q 21 orpG q f2 pψ q, ' ' ' _ ' ' M %S G 1 _ rpG_ q f2 pψ q, disc,ψ pf2 q 4 o
under the assumption that δψ 1 (so that ψ is generic and Λ 0). Suppose next that ψ is not generic. Then δψ 1 by definition. However, in this case we will still have to show that Λ 0. Assume the contrary, namely that Λ 0, and consequently that Γ 0. The first condition implies that N is even and G is split. In particular, we can work with the p in place of L G, if we choose. The second condition implies dual group G that the invariant linear form on GpAq attached to ψ is the pullback of the invariant linear form Λ on the Levi subgroup LpAq. From this property, we shall establish the following consequence of the results of §7.5. Lemma 8.2.1. With our assumption that Γ 0, the localization ψv of the r sim pN q at any valuation v factors through the Levi subgroup parameter ψ P Ψ p p L of G. Proof. We are working with a simple, self-dual parameter ψ in the r sim pGq of Ψ r sim pN q. It has the usual decomposition subset Ψ ψ
µ
b
ν,
r sim pmq, ν µPΦ
ν n,
for positive integers m and n with N mn. The unipotent component ν has degree n ¡ 1, by our assumption that ψ is not generic. The generic component µ has a determinant ηµ , which is an id`ele class character with ηµ2 1, since µ is self dual. It will not be hard to show that the generic degree m is even, and that the character ηµ equals 1. These properties follow from a global argument we shall give at the end of the proof. In the meantime, we shall establish the lemma under the assumption that they hold. This part of the argument is purely local. We shall describe it in local notation that is compatible with the discussion of §7.5. We therefore take F to be local until near the end of the proof, at which point we will revert to the global notation. We take G P Ersim pN q to be a r pGq to be a local parameter. We split endoscopic datum over F , and ψ P Ψ are assuming that N mn, where n ¡ 1 and m is even, and that ψ is of the form p pmq, ν ν n , ψ µ b ν n, µPΨ
454
8. THE GLOBAL CLASSIFICATION
where ν n is the irreducible representation of SU p2q of degree n and µ has determinant ηµ equal to 1. The generic factor µ is a self-dual representation µ
r à
`i µ i
i 1
of LF , for positive integers `i and distinct simple elements µi P Φsim pmi q such that m `i mi `r mr . The split group G has a Siegel maximal Levi subgroup L GLpN {2q. Our primary assumption on ψ is that the stable character f G p ψ q,
f
P HrpGq,
is induced from a linear form on the Levi subgroup LpF q of GpF q. We must use this to deduce that the homomorphism ψ : LF
SU p2q ÝÑ
p G
p of G. p factors through the subgroup L p p The embedding of L into G can of course be taken to be
x
ÝÑ px, x_q,
p GLpN {2, Cq. xPL
Since µ is self-dual, there is an involution i Ñ i_ on the set of indices i such that µi_ µ_ i and `i_ `i . An orbit of order 2 contributes a subrepresentation n `i pµi b ν n q ` `i pµ_ i bν q
p Similarly, for any i with µ_ of ψ that factors through L. i of `i contributes a subrepresentation
2`1i pµi b ν n q,
`1i
µi, the even part
r`i{2s,
p If `i is odd, however, there will be a supplementhat also factors through L. tary subrepresentation pµi b ν n q of ψ whose image cannot also be embedded p We have therefore to show that the set of indices in L.
I
ti :
µ_ i
µi ,
`i oddu
is empty. Suppose first that F C. A component µi is then a quasicharacter on C , which is self dual if and only if it equals 1. If there is an i with µi 1, the corresponding multiplicity `i must be even, since m is even, and the other components occur in pairs. Therefore I is empty and ψ factors p as required. through L, Suppose then that F is real or p-adic. We assume that the subset of indices I is not empty. Our task is to obtain a contradiction of the condition on the stable character f G pψ q. By the remarks above, it suffices to show that if à ψ pµ i b ν n q iPI
8.2. PROOF BY CONTRADICTION
and
N
¸
P
455
mi n,
i I
r pG q, then and if G is the datum in Ersim pN q such that ψ belongs to Ψ the twisted character G r p G q, f pψ q, f P H
is not induced from the maximal Levi subgroup L of G . The answer lies in the decomposition (2.2.12) (with pG , ψ q in place of pG, ψ q) of this stable character in terms of standard stable characters. Since standard stable characters on G are defined directly by the twisted transfer of standard characters from GLpN q, it suffices to consider the summands in the decomposition (2.2.9) (with pN , ψ q) in place of pN, ψ q of the twisted irreducible character fr
ÝÑ
fr,N pψ q tr πψ pfr q ,
fr
P HrpNq,
into twisted standard characters. We shall apply the character formulas of §7.5 to the direct summands ψ,i
µi b ν n ,
i P I ,
of ψ . As an irreducible representation of LF of degree mi , µi is self-dual and therefore unitary. If F is p-adic, LF is of course a product WF SU p2q. We write µi r b ν k 1 , mi mr pk 1q, where r is an irreducible unitary representation of LF of degree mr , and k is a nonnegative integer. The irreducible character on GLpmi n, F q attached to pµi b ν nq then equals θrnpkq, in the notation of §7.5. If F R, mi equals 1 or 2. In case mi 2, µi parametrizes a self-dual representation of GLp2, Rq in the relative discrete series. This in turn is parametrized by a positive integer k P N. The irreducible character on GLp2n, F q attached to pµi b ν n q equals θn pk q, in the notation of §7.5. In case mi 1, there are two possibilities for the self-dual linear character µi of R , either the trivial character 1 or the sign character εR . But it follows from the fact that m is even that there is a second index i1 P I such that µi1 εR µi . The irreducible character on GLp2n, F q attached to pµi ` µi1 q b ν n then equals θn p0q, again in the notation of §7.5. We thus obtain a decomposition θpψ q
ð
P
i I
θpψ,i q
ð
pr,kq
θrn pk q
from (7.5.13) and the surjective correspondence of indices i
ÝÑ pr, kq,
i P I , k
¥ 0,
where r is understood to be trivial if F R. The correspondence is almost a bijection, in that the fibre of pr, k q in I consists of one element unless F R and k 0, in which case it is a pair ti, i1 u. We fix an index pr1 , k1 q
456
8. THE GLOBAL CLASSIFICATION
with k1 maximal. If F R, pr1 , k1 q k1 is uniquely determined. In this case, we have k1 ¥ 1. For if k1 0, we obtain an identity ηµ
r ¹
p η µ q` i
i
i 1
¹
P
i I
ηµi
1 εR εR ,
which contradicts the condition ηµ 1. If F is p-adic, there could be several r paired with k1 . We may as well choose r1 from among these so that mr1 is maximal. We shall apply Lemma 7.5.2 and Corollary 7.5.5 to the direct summand n θr1 pk1 q of θpψ q, or rather the tempered constituent φn, r1 pk1 q of this summand. This gives an extremal factor θr1 pk1 n 1, 0q of φr1 pk1 n 1, 0q, which can be compared with the other factors θr p`, λq from the decompositions (7.5.3) and (7.5.4) of any direct summand θrk pk q of θpψ q. It would not be hard to see directly that θn pk1 n 1, 0q is distinct from all of these other factors. This would lead us to the required contradiction. However, it is perhaps more natural (if a little less direct) simply to restrict our attention to the tempered constituent θpφψ q
ð
P
i I
θpφψ,i q
ð
pr,kq
φn, r pk q
of θpψ q. That this suffices is a consequence of the parametrization of standard characters (ordinary or twisted) for GLpN q by the associated r pN q, and the fact that set of Langlands parameters (either ΦpN q or Φ these objects form a basis of the space of all virtual characters (ordinary or twisted) for GLpN q. For it can then be seen, using the germ expansions of twisted characters around singular points for example, that if a given finite linear combination of twisted standard characters tθru is induced from r on the maximal Levi subset a distribution Λ r pF q GLpN {2, F q GLpN {2, F q L
θrpNq r (The symbol θrpN q here denotes r pN , F q, the same is true of each θ. of G the standard outer automorphism of GLpN q, or rather, its restriction to r 0 . It is not to be confused with any of the given the Levi subgroup L L
r In other words, the Langlands parameter of each θr twisted characters θ.) factors through the L-group of L . We are assuming that the twisted character
θrpψ q fr,N
ÝÑ
fr,N pψ q
r . We know from Lemma 7.5.2(i) and Corollary is induced from such a Λ 7.5.5 that the twisted tempered character θrpφψ q attached to θpφψ q occurs in the decomposition (2.2.9) of θrpψ q into twisted standard characters. To obtain a contradiction, we need only show that the self-dual Langlands r0 . parameter φψ of θpφψ q does not factor through the L-group of L
8.2. PROOF BY CONTRADICTION
457
Lemma 7.5.2(ii) tells us that the factor θr1 pk1 n 1, 0q occurs with multiplicity 1 in the direct summand φn, r1 pk1 q of θ pφψ q. Here we are relying on the fact that k1 ¥ 1 if F R (a condition that need not hold if F is p-adic, but which is then superfluous to Lemma 7.5.2(ii)). The integral argument pn k1 1q in this factor is maximal. It is in fact strictly larger that the arguments in any of the other factors of direct summands ψrn, pk q of θpφψ q, except for the maximal arguments from pairs
pr, k1q,
r
r1,
in the p-adic case. But the standard characters θr pn k1 1, 0q and θr1 pn k1 1, 0q for GL mr pk1 1q are distinct, since the same is true of the corresponding Langlands parameters. It follows that there is a self-dual factor in the decomposition of θpφψ q that occurs with multiplicity 1. In other words, there is an irreducible self-dual subrepresentation of the Langlands r pN q that occurs with multiplicity 1. This precludes parameter φψ P Φ r 0 , and gives us the the possibility that φψ factor through the L-group of L required contradiction. We have shown that the existence of indices in I leads to a contradiction. The set I is therefore empty. This implies that the parameter r pGq factors through L, p as we have seen. We have thus established the ψPΨ assertion of the lemma, in local notation, under the assumption that m is even and ηµ 1. This assumption was actually used only in the case that F is archimedean, but of course, we still have to justify it. We return to the global notation we have been using throughout Chapter 8. Then F , G and ψ µ b ν are the global objects from the beginning of the proof. It remains to show that m degpµq is even, and that the global id`ele class character ηµ equals 1. Let v be any valuation of F at which G and ψ are unramified. We are free to apply the local argument above to the completion
ψv
µv b ν n
of ψ. Since the assumption on m and ηµv was not needed in this case, the argument tells us that the given condition on the stable character fvG pψv q,
fv
P HrpGv q,
namely that it is induced from LpFv q, implies that the local parameter ψv p in G. p This in turn implies that m is even, as factors through the image of L required, and that the local determinant ηµv equals 1. It then follows from Tchebotarev density theorem that the global determinant ηµ equals 1, as required. In particular, ηµv 1 for any valuation v of F , the condition we used in the archimedean case of the local argument above. This completes the global argument, and hence the proof of the lemma.
458
8. THE GLOBAL CLASSIFICATION
The lemma implies that any localization ψ p in the Levi subgroup diagonal image of L
,v
of ψ factors through the
pL p GLpN {2, Cq GLpN {2, Cq L p . The centralizer of the image of L p in G p of G connected group
can be identified with the
GLp2, Cq GLp2, Cq, which contains the global centralizer Op2, Cq Sψ . It follows that Sψ maps to the identity component of Sψ ,v . Applying this to the nontrivial point x1 in Sψ , we see from the global intertwining relation r pG q. In other words, the that fG pψ , x1 q equals f M pψ q, for any f P H condition (8.1.7) holds in the present case, even though δψ 1. It then follows from Corollary 5.3.3 and (8.1.5) that Sdisc,ψ pf q G
1 4
orpG
q f M p ψ q.
r pG_ q belongs to some compatible family that contains f , it again If f _ P H follows from Corollary 5.3.3 and (8.1.5), in combination with the global intertwining relation and the fact that the group Sψ_ Spp2, Cq is connected, that G_ Sdisc,ψ pf _ q 14 orpG_ q fG__ pψ , x1 q f _,LL pΛ Λq
41 orpG_q f M pψq 12 orpG_q f M pψq.
f M pψ q
We thus have a second family of identities
(8.2.2)
$ G Sdisc,ψ f1 ' ' ' ' ' &S G_ f1_
p q 0, q orpG_q f1Gpψq, disc,ψ p G M ' Sdisc,ψ pf2 q 14 orpG q f2 pψ q, ' ' ' ' % G_ M Sdisc,ψ pf2_ q 12 orpG_ q f2 pψ q,
under the assumption that Λ 0 (so that ψ is not generic and δψ 1), and for functions f1 , f1_ , f2 and f2_ as in (8.2.1). Notice the similarity with (8.2.1). The two families (8.2.1) and (8.2.2) are completely parallel, but with G and G interchanged with their counterparts G_ and G_ .
8.2. PROOF BY CONTRADICTION
459
For comparison, we can also write down what the equations in (8.2.1) and (8.2.2) would be if δψ 1 and Λ 0, as expected. We obtain $ G Sdisc,ψ f1 ' ' ' ' ' &S G_ f1_
p q orpGq f1Gpψq, q 0, disc,ψ p (8.2.3) G ' Sdisc,ψ pf2 q 0, ' ' ' ' % G_ M Sdisc,ψ pf2_ q 14 orpG_ q f2 pψ q, under the assumption δψ 1 and Λ 0 (which imposes no further condi-
tions on ψ), and for functions f1 , f1_ , f2 and f2_ as in (8.2.1) and (8.2.2). These follow from Corollary 5.3.3 and discussion above, particularly (8.1.5) and the preceding applications of the global intertwining relation. They would also follow easily from the (as yet unproven) stable multiplicity formula. We are of course trying to establish the conditions δψ 1 and Λ 0. Our task is therefore to focus on the equations (8.2.1) and (8.2.2) that would follow if the conditions are not met. However, the equations in (8.2.1) and (8.2.2) are vaguely disturbing. They contain nothing that is at odds with anything we have established so far. As a matter of fact, their symmetry seems to give them an air of authority, which is reinforced by their relative simplicity. But the equations are based on premises that we must unequivocally refute! We are left with the uncomfortable feeling that our methods may have run their natural course. Given the symmetry between (8.2.1) and (8.2.2), we appear to have no choice but to look for a uniform argument that applies to both problems. We shall introduce a second supplementary parameter, which at first sight seems to be unnatural and without promise. The surprise is that it actually works! The new parameter will be seen in fact to be quite natural, and to be exactly what is required to contradict the assumptions δψ 1 of (8.2.1) and Λ 0 of (8.2.2). For the given pair pG, ψ q, we define our second supplementary parameter by
3ψ ψ ψ ψ. r pN q, for N Then ψ belongs to Ψ 3N . Remember that we are assuming that N is even and ηψ 1, so that G and G_ are distinct split groups in Ersim pN q. Let G and G_ be the split groups in Ersim pN q whose dual pG pG p and p groups G and pG_ q^ contain the respective products G _ ^ _ ^ _ ^ pG q pG q pG q . We then have the maximal Levi subgroups M M G GLpN q G ψ
and
M_
`
`
M _ G_ GLpN q G_
460
8. THE GLOBAL CLASSIFICATION
of G and G_ respectively. The product ψ r pN as a parameter for M whose image in Ψ associated compatible family of functions (8.2.4)
ψ can then be treated q equals ψ . Given an
f
(
P HrpGq : G P ErellpN q , f and f _ f as usual in the cases G G
we shall write f and G G_ . The stage is now set for the lemma that is the key to the final classification. This lemma is founded on the techniques of §5.3, but relies also on the character formulas of §7.5 embodied in Lemma 8.2.1. Lemma 8.2.2. For the given pair
pG, ψq,
G P Ersim pN q, ψ
P Ψr simpN q,
assume that N is even and ηψ 1. In addition, assume that either δψ as in (8.2.1), or Λ 0 as in (8.2.2). Then the sum (8.2.5)
¸
G Ersim N
P
p
q
tr Rdisc,ψ
pf q
1 2
fM
1
pψ ψ q
vanishes for any compatible family of functions (8.2.4). Proof. Once again, we have to modify the arguments from Chapter 4. The proof will be loosely modeled on that of Lemmas 5.3.1 and 5.3.2 for ψ . However, our exposition here will be slightly different, since we now have the local theorems at our disposal. In particular, analogues of the terms (5.3.22) and (5.3.23) need not be an explicit part of the discussion here, thanks to the local and global intertwining relations. We fix a compatible family of functions (8.2.4). The arguments of Chapter 4 would lead ultimately to the vanishing of the sum (8.2.6)
¸
G Ersim N
P
p
q
p
r ιN
, G q tr Rdisc,ψ
pf q ,
if we could assume inductively that the global theorems were valid for proper subparameters of ψ . We cannot do this, of course, because we have assumed that one of the contrary conditions δψ 1 or Λ 0 holds. However, we can use this condition, as it is reflected in either (8.2.1) or (8.2.2), to modify the expected formula for (8.2.6). The modifications of (8.2.6) are of three sorts. They arise from the endoscopic contributions of proper products G G1 G2 in the set Erell pN q, from endoscopic contributions of proper products G1 G1 G2 in the sets Eell pG q attached to simple data G P Ersim pN q, and from spectral contributions from proper Levi subgroups M of the groups G P Ersim pN q. As we have seen, explicitly in the proof of Lemma 5.3.1 and implicitly in that of Lemma 5.3.2, most such contributions reduce immediately to zero. We shall have to consider only one pair of contributions of each of the three
8.2. PROOF BY CONTRADICTION
461
kinds. In each case, we will have to subtract the expected value of the contribution from the actual value. The difference will then have to be added or subtracted, as a correction term for (8.2.6), in order that the resulting expression vanish. As in the earlier proofs, the sign of this last operation (addition or subtraction) is dictated by the role of the given contribution in the standard model. Since these signs are critical, it would be a good idea to review again the process by which they are determined. We know that we can write (8.2.6) as the sum of (8.2.7)
¸
G Ersim N
P
and
p
(8.2.8)
¸ G
q
p
r ιN
p
r ιN
, G q trpRdisc,ψ
, G q 0 Sdisc,ψ
pf q,
pf q 0Sdisc,ψ pf q G
P Erell0 pN q,
0 pN q is the complement of ErsimpN q in ErellpN q, since the where Erell condition (5.2.15) of Lemma 5.2.3 holds for ψ . This is the identity, with (8.2.6) on the left hand side and the sum of (8.2.7) and (8.2.8) on the right, that is to be our starting point. The first pair of modifications describes the obstruction to the expected vanishing of the summands in (8.2.8). To simplify the right hand side of the identity, we will want to move the obstructions to the left hand side. In other words, we add them to (8.2.6), since the summands come with minus signs. For the remaining terms on the right hand side, we will take what we can from Corollaries 4.3.3 and 4.4.3. We can certainly write
tr Rdisc,ψ
1 pf q I 1 pf q, pf q Iend pf q Sdisc,ψ spec
G P Ersim pN
q,
1 and I 1 stand for the proper parts of the respective expansions where Ispec end (4.1.1) and (4.1.2) (namely, the sums taken only over M G and G1 G respectively, but of course with pG , ψ q in place of pG, ψ q). Corollaries 4.3.3 and 4.4.3 (and their comparison by Proposition 4.1.1 that in the case of pG, ψ q culminated in (4.5.4)) pertain to the expected values of the terms on the two sides of this last formula. Since the expected value of the trace rdisc,ψ
pf q tr Rdisc,ψ pf q
is 0, they tell us that
tr Rdisc,ψ
pf q 0Sdisc,ψ pf q 0rdisc,ψ pf q 0sdisc,ψ pf q 1 pf q 0 I 1 pf q, 0Iend G P Ersim pN q, spec 1 pf q and 0 I 1 pf q stand for the variance of I 1 pf q and I 1 pf q where 0 Iend spec spec end respectively from their expected values. The remaining modifications describe the obstruction to the expected vanishing of these last two terms. 1 pf q. To move the resulting The second of the three pairs applies to 0 Iend obstruction to the left hand side, we must subtract it from (8.2.6). The 1 pf q. To transfer the resulting obstruction in this third pair applies to 0 Ispec
462
8. THE GLOBAL CLASSIFICATION
last case, we must add it to (8.2.6). The expression (8.2.6), thus modified by the three kinds of correction, will then vanish. The arguments for the two possible conditions δψ 1 or Λ 0 are essentially parallel, so we shall only treat one of them in detail. We assume until near the end of the proof that the condition Λ 0 is in force. Then ψ is not generic, δψ equals 1, and the equations (8.2.2) hold. The first pair of corrections applies to the linear forms 1 2 pf q Sdisc,ψ pf1q Sdisc,ψ pf2q, G P Erell0 pN q, 0 pN where G G1 G2 is either of the groups G G_ or G_ G in Erell q, and f f1 f2 is the corresponding function in (8.2.4). If G equals G G_ , the actual value of (8.2.9) vanishes, according to the first equation
(8.2.9)
Sdisc,ψ
in (8.2.2), while the expected value equals
14 orpGq orpG_q f1Gpψq f2M pψq,
by the first and fourth equations in (8.2.3). If G equals G_ G , the actual value of (8.2.9) is 1 4
orpG_ q orpG
q f1Gpψq f2M pψq,
by the second and third equations in (8.2.2), while the expected value vanishes by the second (or third) equation in (8.2.3). In each case, we subtract the expected value from the actual value, and then multiply the difference by the coefficient (8.2.10)
p
r ιN
, G q
1 4
orpG q1
14 orpG1q1 orpG2q1.
We obtain two equal contributions, whose sum (8.2.11)
1 8
fM
pψ ψ q ,
must be added as a correction term to (8.2.6). The second pair of corrections applies to two terms in the analogue for ψ of (4.1.2). These are the linear forms (8.2.12) 1 1 2 Sdisc,ψ pf 1q Sdisc,ψ pf1q Sdisc,ψ pf2q, G1 P EsimpGq, G P ErsimpN q, where G1 G1 G2 is either of the groups G G P Erell pG q or r pG1 q with the G_ G_ P Eell pG_ q, and f 1 f1 f2 is a function in H 1 1 same image in SrpG q as the chosen compatible family. If G equals G G , the actual value and the expected value of (8.2.12) both vanish, by the first equation in (8.2.2) and the third equation in (8.2.3). If G1 equals G_ G_ , the actual value of (8.2.12) is
12 orpG_q orpG_q f1Gpψq f2M pψ q, by the second and fourth equations in (8.2.2), while the expected value vanishes, by the second equation in (8.2.3). In each case, we subtract the
8.2. PROOF BY CONTRADICTION
463
expected value from the actual value, and then multiply the difference by the coefficient (8.2.13) r ιpN , G q ιpG , G1 q 41 orpG q1 OutpG , G1 q 14 orpG1 q1 orpG2 q1 . This time we must subtract the result from (8.2.6). The two minus signs cancel, and we obtain another copy of (8.2.11) to be added as a correction term to (8.2.6). The third pair of corrections applies to two terms in the analogues for ψ of (4.1.1). These are the linear forms (8.2.14)
tr MP ,ψ
pwq IP ,ψ pf q ,
G
P ErsimpN q,
where G is either G or G_ , M is the corresponding Levi subgroup M or M _ , and w is the associated element in Wψ ,reg pM q. This requires a little more discussion. Suppose first that G equals G . Recall that M is isomorphic to GLpN q G. The actual discrete spectrum for G and ψ vanishes under our assumption Λ 0, since the first equation in (8.2.2) tells us that the corresponding stable linear form vanishes. The actual value of (8.2.14) therefore vanishes. To describe the expected value, we note that the centralizer group is isomorphic to SOp3, Cq. Because ψ is not generic, the induction Sψ hypothesis underlying the proof of Lemma 4.3.1 in §4.6 is valid. By a very elementary case of the discussion in §4.6, the global normalizing factor implicit in (8.2.14) therefore equals p1q. If we then combine the first equation in (8.2.3) with the global intertwining relation, we see that the expected value of (8.2.14) is
orpGq fG pψ
, x1 q orpGq f M
p ψ ψ q,
where x1 stands for the only element in the trivial group Sψ . We must subtract the expected value from the actual value, multiply the difference by the coefficient (8.2.15) r ιpN , G q |W pM q|1 | detpw 1q|1 12 orpG q 14 18 orpGq1 , and add the result to (8.2.6). This gives us a third copy of (8.2.11) to be added to (8.2.6). Suppose finally that G equals G_ . In this case, we cannot quite apply the discussion of §4.6 to the global normalizing factor in (8.2.14), because our assumption Λ 0 violates the earlier premises. However, we can still represent the normalizing factor as a product of quotients of two automorphic L-functions. One is the Rankin-Selberg L-function Lps, ψ ψ q. Indeed, it is built out of local Rankin-Selberg factors Lps, ψv ψv q attached to localizations ψv ψv : LFv
SU p2q ÝÑ pM _ q^ GLpN, Cq pG_q^ GLpN, Cq GLpN, Cq
464
8. THE GLOBAL CLASSIFICATION
of ψ ψ. (We shall recall before how ψv can be treated as a local parameter for the group G_ .) Since ψ is self-dual, it has a pole at s 1. The other is the Langlands-Shahidi L-function attached to the parameter ψ for G_ . Since Sψ_ is isomorphic to SLp2, Cq, this L-function also has a pole at s 1. We are of course using our induction hypothesis here that Theorem 1.5.3(a) is valid for the generic component of ψ, as we did in discussing the case G G above. The two L-functions each contribute a factor p1q to the product, so the global normalizing factor equals 1 in this case. (By contrast, in the case G G above, Sψ is isomorphic to Op2, Cq, and the corresponding L-function does not have a pole, and therefore does not contribute a second factor p1q.) Continuing our analysis of the case G G_ , we consider the local normalized intertwining operators in (8.2.14). According to Lemma 8.2.1 and our assumption Λ 0, any localization ψv of ψ factors through the p of G. p Since L p also represents a Levi subgroup of pG_ q^ , Levi subgroup L we can treat ψv as a local parameter for G_ as well as for G. The image in pG_ q^ of the corresponding localization ψ
,v
ψv
`
ψv ` ψv
of ψ is then contained in the threefold diagonal image of pG_ q^ . The centralizer of this diagonal image, taken modulo the center of pG_ q^ , is isomorphic to the connected group SOp3, Cq. Since w w_ represents the nontrivial element in the Weyl group of SOp3, Cq, we can choose a local representative of w_ in the connected centralizer pSψ_ ,v q0 . The local normalized intertwining operator therefore equals the identity. We have shown that if G G_ , the local and global factors of the operator MP ,ψ pw q in (8.2.14) are all equal to 1. The operator is therefore trivial. In fact, from the global intertwining relation and the second equation in (8.2.2), we see that (8.2.14) itself equals the linear form _ orpG_ q fG__ pψ , x1 q orpG_ q pf _ qM pψ ψ q orpG_ q f M pψ ψ q. On the other hand, the expected value of (8.2.14) is equal to 0, by the second equation in (8.2.3). Subtracting the expected value from the actual value, and multiplying by the coefficient (8.2.16) r ιpN , G_ q |W pM _ q|1 | detpw_ 1q|1 21 orpG_ q 14 18 orpG_ q1 , we obtain a fourth copy of (8.2.11) to be added to (8.2.6). We have obtained four correction terms in all, each of which is equal to (8.2.6). The sum of (8.2.6) with the total correction term 4
1 8
fM
pψ ψq 12 f M pψ ψq
therefore vanishes. Since the sum is equal to the given expression (8.2.2), we have obtained a proof of the lemma in case Λ 0. Assume now that δψ 1. Then ψ is generic, Λ 0, and the equations (8.2.1) hold. The structure of the proof is the same in this case, with the only
8.2. PROOF BY CONTRADICTION
465
minor differences in detail resulting from our use of the equations (8.2.1) in place of (8.2.2). We can be brief. Suppose that the group G in (8.2.9) equals G G_ . Then the actual value of (8.2.9) equals 1 4
orpGq orpG_ q f M
p ψ ψ q,
while the expected value is
14 orpGq orpG_q f M pψ ψq,
by the first and fourth equations in (8.2.1) and (8.2.3). If G equals G_ G , the actual and expected values of (8.2.9) both vanish, by the second equations in (8.2.1) and (8.2.3). The product of the total difference with the coefficients (8.2.10) is equal to the linear form (8.2.11) we obtained before. Suppose that the group G1 in (8.2.12) equals G G . Then the actual value of (8.2.12) equals
12 orpGq orpG q f M pψ ψq, while the expected value vanishes, by the first and third equations in (8.2.1) and (8.2.3). If G1 equals G_ G_ , the actual and expected values of (8.2.12) both vanish, by the second equations in (8.2.1) and (8.2.3). The product of the total difference with the coefficient (8.2.13), when subtracted from (8.2.6), again gives a second copy of the correction term (8.2.9). Suppose that the groups G and M in (8.2.14) equal G and M . The actual discrete spectrum for M and ψ ψ is equal to what is expected, by the first equations in (8.2.1) and (8.2.3). Since the group Sψ equals Op3, Cq, the normalized intertwining operator in (8.2.14) equals 1, as expected. The global normalizing factor implicit in (8.2.14) is therefore the only quantity that differs from its expected value. The expected value equals the sign of w w in SOp3, Cq, which is p1q. The actual value equals 1, as in our discussion of G G_ for the case Λ 0 above, this time using the definition of δψ 1. Appealing again to the first equations of (8.2.1) and (8.2.3), we see that the actual value of (8.2.12) is orpGq fG
pψ
, x1 q orpGq f M
pψ ψ q,
while the expected value equals p1q times this quantity. The product of the difference with the coefficient (8.2.15) gives two more copies to (8.2.11) to be added as correction terms to (8.2.6). Finally, suppose that the groups G and M in (8.2.14) equal G_ and _ M . The discrete spectrum for M _ and ψ ψ vanishes, as expected, by the second equations in (8.2.1) and (8.2.3). The actual and expected values of (8.2.14) therefore both vanish in this case, and give no further correction terms for (8.2.6). The total correction term in this case thus equals 2
1 8
fM
pψ ψ q
1 4
fM
pψ ψq 12 f M pψ ψq,
466
8. THE GLOBAL CLASSIFICATION
as it did in the earlier case. The given expression (8.2.2) therefore again vanishes. We have completed the proof of the lemma in the remaining case δψ 1. With the proof of Lemma 8.2.2, we have finally obtained the result we want. It remains only to appeal one last time to Proposition 3.5.1. The linear form (8.2.17)
fM
pψ ψ q ,
f
P HrpG q,
in (8.2.5) is an induced character. In fact, it is induced from the nonzero character on the Levi subgroup M pAq obtained from the local constituents ψv ψv of ψ ψ. In particular, it is a nonnegative integral combination of irreducible characters. Since the same is true of the linear forms in f in (8.2.5), the entire expression (8.2.5) is a nonnegative linear combination of irreducible characters of the general form (3.5.1). If the expression vanishes, we can indeed apply Proposition 3.5.1. It will tell us that the corresponding coefficients all vanish. Since there are nonzero coefficients in (8.2.17), this cannot happen. The expression (8.2.5) therefore does not vanish. We have arrived at a contradiction to the premise of Lemma 8.2.2. In other words, we have established that δψ 1 and Λ 0. It follows that Theorem 1.5.3(a) holds if φ ψ is generic, and that the stable multiplicity formula of Theorem 4.1.2 is valid in the remaining case that ψ is not generic. These are the two assertions left over from the last section. r sim pN q are The remaining global properties for simple parameters ψ P Ψ in the assertions of Theorems 1.5.2 and 4.2.2. They follow immediately. Indeed, given the stable multiplicity formula for ψ, we appeal to (4.4.12) and Lemma 4.7.1 as above to establish the required contribution (4.7.10) to the spectral multiplicity of Theorem 1.5.2. (In case ψ is generic, we are also implicitly relying on the fact that Lemma 8.1.1 includes a proof of Theorem 1.4.2.) If pG, ψ q is as in Theorem 4.2.2, the simple parameter ψ r q of Ψ r 2 pGq. The assertion (b) of this theorem lies in the complement of Ψ2 pG is therefore irrelevant, while assertion (a) again follows from the fact that the right hand side of (8.1.3) vanishes. These are the last assertions to be r sim pN q. Since the simple parameters proved for the simple parameter ψ P Ψ were all that remained after the last section, we have now established the r pN q. This completes the induction global theorems for any parameter ψ P Ψ argument on N begun in the previous section, and therefore completes our proof of the global theorems. In summary, we have now resolved all of the induction hypotheses. We have consequently proved all of the theorems. These are the generic local theorems established in Chapter 6, the nongeneric local theorems treated in Chapter 7, and the global theorems proved in this chapter. In particular, r sim pN q of simple the analogue of Corollary 5.4.7 for the full set Frsim pN q Φ global generic parameters is valid. We shall restate it as a final corollary of this discussion, for easy reference.
8.3. REFLECTIONS ON THE RESULTS
467
r sim pN q is any simple, self-dual, generic Corollary 8.2.3. Suppose that φ P Φ parameter of rank N , and that G belongs to Ersim pN q. Then the following three conditions on the pair pG, φq over our global field F are equivalent. G r pGq does not vanish. (i) The linear form Sdisc,φ on H (ii) Theorem 1.4.1 holds for φ, with Gφ G. (iii) The global quadratic characters ηG and ηφ are equal, and the global L-function condition δφ 1 of Theorem 1.5.3(a) holds.
Proof. The proof follows from the global theorems we have now established. In particular, it is identical to that of Corollary 5.4.7, the special case for the global family Fr of §5.4. Corollary 8.2.3 tells us that any of the three equivalent conditions (i)– r sim pGq of simple, generic global (iii) can serve as a definition for the set Φ parameters for G. Thus ends the last ambiguity from the construction of r pGq and the the§1.4, which led to the complete set of global parameters Φ orems they satisfy. Once again we observe the local-global symmetry, here between the global corollary we have just proved, and its local counterpart Corollary 6.8.1. We can now go on to other things. In the rest of this chapter, we shall discuss a few ramifications of what we have done. We will then study the representations of inner twists of groups G P Ersim pN q in the final Chapter 9. 8.3. Reflections on the results Having finally proved the theorems, we shall pause for a few moments of reflection. We will begin the section with a brief overview of the results, taken from the general perspective of characters and multiplicities. We will then pose some questions that are particular to the even orthogonal groups SOp2nq. We will complete the section with a proof that local and global L-packets contain generic representations. We fix one of the quasisplit groups G P Ersim pN q over our field F . We recall that the integer N equals 2n, 2n 1 or 2n, according to whether G equals a simple group SOp2n 1q, Spp2nq or SOp2nq from the infinite p equal to Spp2n, Cq, family Bn , Cn or Dn , with corresponding dual group G SOp2n 1, Cq or SOp2n, Cq. In the second and third cases, G comes with an arithmetic character η (a character on F if F is local or A {F if F is global) of order 1 or 2. In the third case, η determines G as a quasisplit inner twist of the corresponding split group. The group G is of course split in the other two cases. The role of η in the second case G Spp2nq is to specify the L-embedding of L G into the L-group of GLpN q that determines G as twisted endoscopic datum for GLpN q. Suppose for the moment that F is local. The local Langlands group LF is then defined as the split extension (1.1.1) of WF . We have classified the r φ indexed by the irreducible representations of GpF q in terms of packets Π
468
8. THE GLOBAL CLASSIFICATION
r pGq of equivalence classes of Langlands parameters set Φ
φ : LF
ÝÑ
L
G.
p p equals Spp2n, Cq Equivalence is defined by G-conjugacy as usual when G r or SOp2n 1, Cq. In these cases, Πφ is the L-packet conjectured by Langlands for any connected reductive group over F . When G equals SOp2n, Cq, equivalence is defined by conjugacy under the exension Op2n, Cq of Z{2Z by SOp2n, Cq. In this case, the packet of φ is still an L-packet if the Op2n, Cqorbit of φ equals its SOp2n, Cq-orbit . If the Op2n, Cq-orbit contains two r φ is a set of pairs of irreducible representaSOp2n, Cq orbits, however, Π tions. We have thus established the full local Langlands correspondence for the split groups SOp2n 1q and Spp2nq, and a slightly weaker form of the correspondence for the quasisplit groups SOp2nq. It is worth emphasizing that the endoscopic classification of representar φ are represented tions is by characters. That is, the elements in a packet Π by their characters. These are defined in terms of the finite 2-group
Sφ
Sφ{Sφ0 Z pGpqΓ,
Γ ΓF
GalpF {F q,
and twisted characters for general linear groups. Let us be more explicit. The distributional character
fG pπ q tr π pf q ,
P H p G q, of any irreducible representation π P ΠpGq of GpF q is a locally integrable function. In other words, fG pπ q
»
p q
Greg F
ΘG pπ, xq f pxqdx
f
»
p q
Γreg G
IG pπ, γ q fG pγ q dγ,
where Γreg pGq ΓG-reg pGq is the space of strongly G-regular conjugacy classes in GpF q, equipped with its natural measure, and IG pπ, γ q |Dpγ q| 2 ΘG pπ, γ q, 1
γ
P Γreg pGq,
is the normalized character. This assertion is a fundamental theorem of Harish-Chandra, which is at the heart of harmonic analysis on real and p-adic groups. The endoscopic classification is in terms of tempered representations, or equivalently, their (reducible) analytic continuation to standard representations. For real groups, tempered (and standard) characters have natural formulas, which are based on Harish-Chandra’s explicit formulas [Ha1] [Ha2] for the characters of discrete series. Characters for p-adic groups are more complex, but they have many interesting and well understood qualitative properties. It is not hard to describe endoscopic transfer in terms of normalized characters. Recall that the tempered representations of GpF q correspond to r bdd pGq of parameters φ P Φ r pGq of bounded image. We will need the subset Φ a slightly different convention for treating the constituents of the associated
8.3. REFLECTIONS ON THE RESULTS
469
r φ , since they are elements in the set Π r pGq of orbits in ΠpGq under packets Π the group r pGq Out r N pGq Aut r N pGq{rIntN pGq. O
For any π in this set, the sum IrG pπ, γ q
¸ π
IG pπ , γ q,
π
P ΠpG, πq,
r pGq-orbit ΠpG, π q of π in ΠpGq depends only on the image of γ over the O in the quotient r reg pGq Γreg pGq { O r p G q. Γ
r reg pGq inherited from Γreg pGq. We then We take the quotient measure on Γ observe that
(8.3.1)
fG pπ q
»
p q
r reg G Γ
IrG pπ, γ q fG pγ q dγ,
f
P HrpGq,
r pGq, and in particular, for π in any packet Π r φ . The stable anafor any π P Π r bdd pGq. logue of this distribution is the linear form f G pφq attached to φ P Φ It can be expanded as an integral
f G pφ q
»
r reg pGq ∆
SrG pφ, δ q f G pδ q dδ,
f
P HrpGq,
r reg pGq ∆ r G-reg pGq of O r pGq-orbits of strongly G-regular over the space ∆ stable conjugacy classes in GpF q, equipped with the measure inherited from Γreg pGq. The normalized stable character SrG pφ, δ q in the integral is obtained by transfer from GLpN q, as we shall see explicitly in a moment. The function SrG pφ, δ q of course has an analogue 1 r G-reg pG1 q, Sr1 pφ1 , δ 1 q SrG pφ1 , δ 1 q, δ1 P ∆
for the preimage pG1 , φ1 q of any given pair pφ, sq. For any f then write f 1 pφ 1 q
»
» »
r G-reg G1 ∆
p q
Sr1 pφ1 , δ 1 q f 1 pδ 1 q dδ 1 ¸
r G-reg G1 ∆
p q γ PΓr reg pGq ¸
r reg pGq Γ
r G-reg G1 δ1 ∆
P
P HrpGq, we can
p q
Sr1 pφ, δ 1 q ∆pδ 1 , γ q fG pγ q dδ 1 Sr1 pφ1 , δ 1 q ∆pδ 1 , γ q fG pγ q dγ,
1 by the definition of f 1 f G , and a change of variables as, for example, in [A11, Lemma 2.3]. On the other hand, if x is the image in Sφ of the given
470
8. THE GLOBAL CLASSIFICATION
point s P S π,ss , we can also write f 1 pφ 1 q
¸
rφ π PΠ
xx, πy fGpπq
»
¸
p q πPΠr φ
r reg G Γ
xx, πy IrGpπ, γ q fGpγ q dγ,
by (2.2.6) and (8.3.1). We thus obtain an identity ¸ δ1
Sr1 pφ1 , δ 1 q ∆pδ 1 , γ q
¸
xx, πy IrGpπ, γ q
π
of the two integrands. An inversion on the group Sφ then yields the general formula (8.3.2)
IrG pπ, γ q
¸
¸
P
x Sφ δ 1 ∆ r G-reg G1
P
p q
xx, πy1Sr1pφ1, δ1q ∆pδ1, γ q,
r φ and Γ r reg pGq respectively, and pG1 , δ 1 q corresponds in which π and γ lie in Π to x according to the usual mappings
pG1, φ1q ÝÑ pφ, sq ÝÑ pφ, xq, s P S φ,ss. If pG1 , φ1 q equals pG, φq and δ is the stable conjugacy class of γ,
second last formula reduces to SrG pφ, δ q
¸
P
rφ π Π
IrG pπ, γ q,
γ
the
P Γrreg pGq,
as might be expected. This is not to be regarded as a means to compute stable characters, however, since stable characters are really the primary objects. We instead use the twisted character (8.3.3)
frN pφq
»
r reg pN q Γ
rφ , γ rq frN pγ rq dγ r, IrN pπ
r pN q, fr P H
r bdd pN q for GLpN q, and in particattached to any self-dual parameter φ P Φ r ular, to any φ P Φbdd pGq. The left hand side here equals the trace of π rφ pfrq, r reg pN q on the right hand by definition (2.2.1). The domain of integration Γ side is the set of strongly regular, twisted conjugacy classes in GLpN, F q. The kernel function in the integrand is the normalized twisted character r N pπ rφ , γ r q, IrN pπ rφ , γ rq |D pγ rq| 2 Θ 1
r reg pN q, rPΓ γ
whose existence has been established in [Clo1]. An analysis similar to that of (8.3.2), using (2.2.1) and (8.3.3) in place of (2.2.6) and (8.3.1), yields the general formula (8.3.4)
SrG pφ, δ q
r bdd pGq and δ for any φ P Φ
¸
P
p q
r reg N γ r Γ
rq , rφ , γ rq ∆pδ, γ IrN pπ
P ∆r N -reg pGq.
8.3. REFLECTIONS ON THE RESULTS
471
r pGq-orbits of) irreducible The formulas (8.3.2) and (8.3.4) classify the (O tempered representations of GpF q in terms of twisted, tempered characters for general linear groups. Suppose for example that π lies in the subset r 2 pGq of square integrable (orbits of) representations. Then the parameter Π r φ of π lies in the subset Φ r 2 pGq of Φ r bdd pGq. The endoscopic φ for the packet Π 1 data G that index the terms in the formula (8.3.2) for the character of π consequently lie in the subset Erell pGq of E pGq, and are products
G1
G11 G12,
G1i
P ErsimpNi1q,
for a partition of N into two even integers Ni1 . The stable characters Sr1 pφ1 , δ 1 q in the corresponding summands decompose accordingly into products Sr1 pφ1 , δ 1 q Sr11 pφ11 , δ11 q Sr21 pφ12 , δ21 q, whose factors can be written in terms of twisted characters on the groups GLpNi1 , F q by the formula (8.3.4). The supplementary coefficients on the right hand sides of (8.3.2) and (8.3.4) are concrete functions. The factor xx, πy1 in (8.3.2) is just the value at x1 of the character on the 2-group rq are transfer Sφ attached to π. The other coefficients ∆pδ 1 , γ q and ∆pδ, γ factors, which we know are subtle objects, but which are nonetheless defined ([LS1], [KS]) by explicit formulas. The local classification embodied in the character formulas (8.3.1)–(8.3.4) was proved by global means. We recall that the proof also had a purely local component. It was the collection of orthogonality relations from §6.5, for elliptic tempered characters on GpF q, and twisted, elliptic tempered characters on GLpN, F q. r pGq of equivaThe local theorems apply also to the more general set Ψ lence classes of nongeneric parameters ψ : LF
SU p2q ÝÑ
L
G.
r ψ are These are complicated by the fact that elements in the packets Π generally reducible. However, the character formulas (8.3.1)–(8.3.4) remain valid for ψ, with the point x in (8.3.2) being replaced by its translate sψ x. They lead to explicit character formulas for the reducible elements π (which r ψ (as the set over Srψ we denoted by Σ r ψ in §7.1), we denoted by σ in §7.1) in Π in terms of twisted, nontempered characters for general linear groups. The r pGq are not part of the local classification nongeneric local parameters ψ P Ψ of admissible representations. They serve rather in support of a global goal, the description of nontempered automorphic representations. But since the constituents of their packets are all unitary, the parameters ψ might still have a role in the local classification of unitary representations. We do have to account for the possible failure of the analogue of Ramanujan’s conjecture for GLpN q. In other words, we have to work with the larger set r r r Ψ unit pGq Ψ pGq X Ψunit pN q
472
8. THE GLOBAL CLASSIFICATION
r pGq. This is the intermediate set of local parameters in place of Ψ r pG q Ψ r r Ψ unit pGq Ψ
p G q,
defined in §1.5 in terms of the irreducible unitary representations of GLpN, F q. r r For any ψ P Ψ unit pGq, the elements in the packet Πψ are the induced representations (1.5.1). Their characters are obtained by analytic continuation of characters (2.2.6) that are provided by our local Theorem 2.2.1. In particular, the discussion above applies to the more general local parameters r unit pGq. We would expect the induced representations in the packets ψPΨ r ψ to be irreducible. We shall state this as a conjecture, even though its Π proof might be quite straightforward. Conjecture 8.3.1. Assume that F is local, that G P Ersim pN q, and that ψ r is a parameter in the set Ψ unit pGq. Then the induced representations IP pπM,λ q,
πM
P Πr ψ
M
,
r ψ are irreducible and unitary. in the packet Π
Rather than attempt a proof of the assertion, let me just append a couple r of brief remarks. There is a bijection ψ Ñ πψ from the set Ψ unit pN q attached r r unit pN q to GLpN q onto Πunit pN q. The irreducible representation πψ P Π of GLpN, F q can be regarded either as the Langlands quotient attached to the Langlands parameter φψ , or an irreducible induced representation IP pN q pπM pN q,λ q, for a Levi subgroup M pN q of GLpN q and the Langlands quotient πM pN q of a parameter ψM pN q P Ψ M pN q . The classification of the unitary dual Πunit pN q in [Tad1] and [V2] is in terms of such induced representations. In particular, the irreducibility is built into the construction. The problem is to show that the irreducibility property is preserved under the transfer from GLpN q to G. One would have to show that the singularities of the relevant intertwining operators for GpF q are dominated by those for GLpN, F q. Such singularities are governed by exponents of representations [BoW, p. 113] for p-adic F , and by some variant of the conditions of [SpehV], when F is archimedean. In the case that ψ lies in the subset r unit pGq Ψ r r Φ unit pGq X ΦpGq
r of generic parameters in Ψ unit pGq, the two assertions of the conjecture are not hard to check.
Suppose now that F is global. If the fundamental local objects are irreducible characters, the fundamental global objects are Hecke eigenfamilies c tcv u. The global theorems classify the Hecke eigenfamilies attached to automorphic representations of G in terms of those for general linear groups. But they do more. Just as the local theorems sort irreducible characters into local packets for G in relating them to GLpN q, so the global theorems characterize the automorphic representations for G attached to eigenfamilies in terms of a global packet, and hence in terms of tensor products of local
8.3. REFLECTIONS ON THE RESULTS
473
characters, even as they relate the eigenfamilies to those of GLpN q. The global process could be regarded as a description of the ramification properties of automorphic representations that are hidden in the unramified data of Hecke eigenfamilies. This is all summarized compactly in the multiplicity formula of Theorem 1.5.2. Recall that Acusp pN q denotes the set of equivalence classes of unitary, cuspidal automorphic representations of GLpN q. When we began in §1.3, we agreed that this was the basic global object. It contains the family Arcusp pN q r sim pN q of self-dual such representations, which we have also denoted by Φ when thinking in terms of global parameters. The disjoint unions Arcusp
º N
Arcusp pN q
Arcusp
º N
Acusp pN q
are then the foundation for all of the global results. Theorem 1.3.2 implies that these sets map bijectively onto the corresponding sets of Hecke eigenfamilies in the disjoint unions (8.3.5) where and
Crsim
º N
Crsim pN q
Csim
º N
Crsim pN q c P Csim pN q : c_
Csim pN q, (
c
,
Csim pN q Ccusp pN q is our fundamental set of simple Hecke eigenfamilies from (1.3.11). The more concrete data in (8.3.5) can thus also serve as a foundation for the global results. Ideally, one would like to be able to formulate general global properties explicitly in terms of data from (8.3.5). Sometimes one can do so in an elementary way. For example, Crsim pN q is the subset of families c tcv u in Csim pN q that are equal to their corresponding dual families c_ tc_ v u. The same goes for the description in §1.3 of the larger sets of Hecke eigenfamilies n 1 n 1 ( cpψ q cpµq b cpν q cv pψ q cv pµq qv 2 ` ` cv pµq qv 2 ,
which are attached to elements in the set Ψsim pN q A2 pN q that parametrizes the automorphic discrete spectrum of GLpN q. These are obtained in an elementary way from families cpµq in the basic sets Csim pmq. Sometimes one must use transcendental means. This is the case for the description of the subset Csim pGq of families c P Crsim pN q attached to the given G, according to which of the partial L-functions LS ps, c, S 2 q
¹
R
det I
S 2pcv q qvs
det I
Λ2pcv q qvs
v S
or
LS ps, c, Λ2 q
¹
R
v S
474
8. THE GLOBAL CLASSIFICATION
has a pole at s 1 (since the partial L-functions have the same behaviour at s 1 as the completed L-functions of Theorem 1.5.3). Sometimes, however, the ties are less direct. For example, it would be hard to claim that the local data at ramified places v can be obtained from the unramified data in this explicit way. Theorems 1.5.1 and 1.5.2 do reduce the question to the case of GLpN q. One might then argue that the answer is in the theory of Rankin-Selberg L-functions, specifically the completion of a partial Lfunction LS ps, c1 c2 q to a full L-function Lps, π1 π2 q with the appropriate functional equation. At any rate, the global theorems are ultimately statements about data from (8.3.5). In particular, Theorem 1.5.2 describes the automorphic disr 2 pGq attached to comcrete spectrum of G in terms of parameters ψ P Ψ pound Hecke eigenfamilies cpψ q. These in turn are derived from simple r sim pGq above. families µi P Crsim pmi q, as in the special case of the subset Ψ It is the multiplicity formula (1.5.3) that provides the quantitative link ber 2 p G q, tween global data and local properties. Through the parameters ψ P Ψ r pGv q, and the corresponding mappings their completions ψv P Ψ Sψ
ÝÑ
¹ v
Sψv
of centralizers, it characterizes the automorphic discrete spectrum explicitly in terms of local characters for G. These are then classified in terms of twisted characters for general linear groups by the formulas (8.3.1)–(8.3.4), and their generalizations for nongeneric parameters ψv . The global theorems of course have other consequences. Since some of these were discussed as they came up in the text, we will not spend more time on them here. We recall only that they include a general theory of Rankin-Selberg L-functions and ε-factors for pairs of representations of G that is inherited from GLpN q, the special properties of L-functions and ε-factors given by Theorem 1.5.3, the relation between multiplicities and symplectic ε-factors implicit in the sign character εψ of Theorem 1.5.2, and the absence of embedded eigenvalues for G established in Chapter 4. Our proof of the local and global theorems is reminiscent of a fundamental theorem from the beginnings of the subject. This is the analytic classification by Hermann Weyl of the representations of compact, simply connected Lie groups [We1]. Weyl’s proof, which includes the Weyl character formula, is by invariant harmonic analysis. It entails a felicitous interplay of characters and multiplicities, linked by the orthogonality relations satisfied by irreducible characters. For a proof in the elementary case of U pN q (which we could consider the analogue of our reductive group GLpN q), see [We2, 377–385]. These general observations on characters and multiplicities are all implicit in the statements of the various theorems. They have served as a pretext for us to collect our thoughts. They also lead us naturally to the
8.3. REFLECTIONS ON THE RESULTS
475
next topic of discussion, that of even orthogonal groups. This will be divided into two parts, according to whether the integer mpψ q attached to a parameter ψ equals 1 or 2. We shall see that each part comes also with a broader philosophical question. We fix an even orthogonal group G P Ersim pN q,
p SOpN, Cq, N G
2n,
r pGq is of order 2 in this case, and G is either split or over F . The group O r pGq. Recall that a quasisplit outer twist by the nontrivial element θr in O r pGq, mpψ q denotes the number of G-orbits p for any ψ P Ψ in the associated r r pGq is the disAutN pGq-orbit of L-homomorphisms attached to ψ. Then Ψ joint union of the subset rq ψ ΨpG
(
P Ψr pGq : mpψq 1
r r G θ, G
,
and its complement (
rq ψ Ψc pG
P Ψr pGq : mpψq 2 , r 1 pGq. We write Ψ2 pG r q, Φbdd pG r q, Φ2 pG rq which we will usually denote by Ψ r 1 p G q, Φ r 1 p G q, Φ r 1 pGq, etc., for the obvious subsets ΨpG r q and Ψ r 1 pG q. and Ψ 2 2 bdd Consider, for example, a parameter
ψ1 ψr , ψi P Ψr simpGiq, Gi P ErsimpNiq, r 2 pGq if r 2 pGq of Ψ r pGq. Then ψ lies in the subset Ψ r 1 pGq of Ψ in the subset Ψ 2 ψ
`
`
and only if each of the integers Ni is even. This is the most important case. r pGq is a self-dual sum (1.4.1) of simple parameters A general parameter ψ P Ψ with higher multiplicities `i . It is then easy to see that ψ belongs to the r 1 pGq if and only if Ni is even for each i in the subset Iψ of indices subset Ψ that correspond to self dual simple parameters. The condition can also be r sim pG q we have expressed in terms of the general subparameter ψ P Ψ attached to ψ, in which the associated endoscopic datum G P Ersim pN q is now an even orthogonal group. We write ψ
ψ
`
ψ ,
ψ
P Ψr pGq,
G
P ErsimpN q,
where G is also an even orthogonal group, and ψ factors through a Levi subgroup of G that is a product of general linear groups. Then ψ lies in r 1 pGq if and only if ψ and ψ lie in the corresponding subsets Ψ r 1 pG q and Ψ r 1 pG q of Ψ r pG q and Ψ r 2 pG q (with the convention that Ψ r 1 p Gq Ψ r pGq in Ψ 2 case N 0). The remarks here, with the accompanying notation, obviously r pG q. extend to the larger family of parameters Ψ r bdd pGq of generic paWe shall limit our discussion here to the subset Φ r pGq. (If F is global, the subscript bdd here is a little misleading. rameters in Ψ It anticipates our being able at some point to replace the complex algebraic
476
8. THE GLOBAL CLASSIFICATION
group Lψ by a locally compact group.) The first part of the discussion conr q of Φ r bdd pGq. It is an extension of cerns parameters φ in the subset Φbdd pG our remarks on characters and multiplicities to twisted endoscopy for G. r q, Theorem Suppose that F is local. For a local parameter φ P Φbdd pG r 2.2.4(b) tells us that the elements in Πφ are actually irreducible represenr pGq-orbits. In other words, the packet tations, rather than nontrivial O r φ is an actual L-packet, as we noted at the beginning of the section. Πφ Π The character formulas (8.3.1)–(8.3.4) are of course still in force. However, there are supplementary character formulas that come from Theorem r 2.2.4(a). They apply to the G-twisted characters (8.3.6)
rq frGr pπ
»
p q
r ΓG-reg G
r, γ rq frGr pγ rq dγ r, IGr pπ
r q, fr P HpG
r pF q of representations π P Πφ . An analysis similar to r to G of extensions π that of (8.3.2), with (2.2.17) and (8.3.5) in place of (2.2.6) and (8.3.1), yields the general formula
(8.3.7)
IGr pπ r, γ rq
¸
¸
r1 x r Srφ δr1 ∆G-reg G
P
P
p q
xxr, πry1 Sr1pφr1, δr1q ∆pδr1, γrq,
r1 q r q, π P Πφ and γ r q and for π r1 , φ r P ΓG-reg pG r, Srφ and pG for any φ P Φbdd pG as in Theorem 2.2.4(a). Given that the L-packet Πφ has already been defined by the original formula, one might ask whether (8.3.7) is superfluous. It is not. Suppose for r q of Φbdd pG r q. The endoscopic data example that φ lies in the subset Φ2 pG 1 r that index the terms in (8.3.7) then lie in the subset Eell pG r q of E pGq, and G are products r1 G r1 G r1 , r 1 P Ersim pN r 1 q, G G 1 2 i i
r 1 . The stable characters in the for a partition of N into two odd integers N i corresponding summands decompose accordingly into products
Sr1 pφr1 , δr1 q Sr1 pφr11 , δr11 q Sr1 pφr12 , δr21 q
to which we can apply (8.3.4). The formula (8.3.6) thus expresses twisted r pF q in terms of twisted characters of general linear groups characters on G r 1 . The original formula (8.3.2) expresses ordinary characters of odd rank N i on GpF q in terms of twisted characters of general linear groups of even rank Ni1 . Therefore, (8.3.7) represents a separate reciprocity law, which gives new local information. r q has a Suppose now that F is global. A global parameter φ P Φ2 pG global L-packet Πφ , whose contribution to the discrete spectrum is given explicitly by Theorem 1.5.2. Since its completions φv lie in the corresponding r v q, the representations in the local packets Πφ satisfy the local subsets ΦpG v supplementary twisted character formulas (8.3.7). What more is there to say? The extra piece of global information is given by Theorem 4.2.2(b). It
8.3. REFLECTIONS ON THE RESULTS
477
r pAq of any π P Πφ in the discrete asserts that two canonical extensions to G spectrum, one obtained from (4.2.7), the other from Theorem 1.5.2 and the extensions π rv in (8.3.7), are the same. Viewed as a signed multiplicity formula for the extension (4.2.7), it tells us that the sign in question equals 1. r v q, but that φ Suppose again that each completion φv of φ lies in ΦpG 1 c r r q. This is the itself belongs to the complementary global set Φ2 pGq Φ2 pG r φ that contributes to the discrete case in which any representation π P Π spectrum does so with multiplicity 2. Theorem 4.2.2(a) implies that for r r q, the restriction of the operator RGr any fr P HpG disc,φ pf q to the π-isotypical 2 subspace Lπ of the discrete spectrum has vanishing trace. The eigenspaces r G pθrq then give a canonical decomposition of the operator Rdisc,φ
(8.3.8)
L2π
L2π, ` L2π,,
π
P Πφ,
G of L2π into invariant subspaces on which the restriction Rdisc,φ is equivalent to π. This phenomenon is related to a general philosophical question posed by A. Beilinson. Guided by his work on the geometric Langlands program, Beilinson has asked whether spaces of automorphic forms of higher multiplicity might have natural bases. Examples from [LL] lend support to his suggestion. The decomposition (8.3.8), which we have obtained from twisted endoscopy for G, can be regarded as further evidence. It would obviously be interesting to investigate this question heuristically in terms of parameters for general groups. The other half of our discussion of the even orthogonal group G is for the r q of Φbdd pG r q. This is not implicit r 1 pG q Φ c p G complementary subset Φ bdd bdd in the earlier theorems. It concerns the question of constructing genuine Lr φ , and will be the setting for two new packets Πφ from the coarser packets Π theorems in the next section. Our purpose here is to lay the groundwork for these theorems. r 1 pGq as an Aut r N pGq-orbit Suppose that F is local. Regarding φ P Φ bdd p of L-homomorphisms, we write Φpφq for the corresponding pair of G-orbits r of L-homomorphisms φ . We also write Πφ for the preimage of Πφ in ΠpGq r pGq. Then Πφ is a disjoint union of under the projection of ΠpGq onto Π r pGq-torsors in ΠpGq, which is to say, transitive O r pGq-orbits (of order 2) O of irreducible representations of GpF q. We would like to separate it into r pGq-orbits, which are two disjoint subsets Πφ, of representatives of the O compatible with endoscopic transfer, and are canonically parametrized by r pGq-torsor Φpφq. We will eventually realize the two elements φ in the O this goal, apart from the canonical parametrization by Φpφq. The idea is to r pGq-torsor T pφq constructed directly in terms of replace Φpφq by a second O representations.
478
8. THE GLOBAL CLASSIFICATION
For the formal definition of T pφq, consider first a simple parameter r sim pGq. Notice that this automatically implies that φ lies in Φ r 1 p G q, φPΦ bdd since N is even. According to our weaker version of the local Langlands r pGq-orbit πφ P Π r pGq of irrecorrespondence for G, φ corresponds to an O ducible representations in Πunit pGq. By Corollary 6.7.3, the order mpπφ q of r pGq-torsor πφ . the orbit equals 2. We define T pφq in this case to be the O Consider next a general element φ φ1 ` ` φr ,
φi
P Φr simpGiq,
r 1 pGq. In this case, we take in Φ 2
(8.3.9)
Gi
T pφq t t1 tr : ti
P EsimpNiq,
Ni even,
(
P T pφ i q {
to be a set of equivalence classes in the product over i of the sets T pφi q. The equivalence relation is defined simply by writing t1 t if the set i : t1i
ti
(
r 1 pGq in Φ r 1 pG q. is even. Finally, suppose that φ lies in the complement of Φ 2 bdd If φ φ , in the notation above, φ is the image of a product of local Langlands parameters for general linear groups, and there is no ambiguity in identifying the two Langlands parameters in Φpφq with two L-packets. In this case, we simply set T pφq Φpφq. We can therefore assume that φ is a proper direct sum φ ` φ . Having already defined T pφ q for the parameter r 2 pG q, we set φ P Φ
(8.3.10)
T pφ q t t t
(
P T p φ q T pφ q { ,
where t1 t if the components of t1 and t are either the same or both distinct from each other. With our reference to Corollary 6.7.3 in the definition of T pφq, we have implicitly taken F to be p-adic. The same definition of course holds if F is archimedean. In fact, by the general form of the Langlands correspondence r pGq-isomorphism between the torsors for real groups, there is a canonical O Φpφq and T pφq. For p-adic F , we do not have a canonical bijection. Our goal, which we will pursue in the next section, will be to construct the two L-packets for φ using T pφq in place of Φpφq. r 1 pG q . Suppose that F is global, and that φ is a global parameter in Φ bdd r pGq-torsor in a rather similar fashion. Suppose We can then define an O r sim pGq of Φ r 1 pGq. For any valuation v on F , first that φ lies in the subset Φ 2 r φ corresponding to the we take πv to be the representation in the packet Π v trivial character on Sφv . In the case of archimedean v, where the mapping r φ to Spφ is not surjective, we appeal to Shelstad’s characterization from Π v v r φ with Whittaker model for the [S6, Theorem 11.5] of the elements in Π v  existence of πv . The global product π πv , which is to be regarded as
8.3. REFLECTIONS ON THE RESULTS
an orbit under the group r N pG A q Out
â v
479
r N pG v q , Out
then occurs in the global discrete spectrum. More precisely, there are exactly two representations π in the orbit of π that occur in the discrete spectrum of G. They each occur with multiplicity 1, and together, form a transitive r pGq of Out r N pGA q. This follows directly orbit under the diagonal subgroup O from the proof of Proposition 6.7.2, or alternatively, the general assertion of r pGq-torsor tπ u, in Theorem 1.5.2. We can therefore define T pφq to be the O case φ is simple. Suppose next that φ is a general parameter in the subset r 1 pGq, with simple components tφi u. We then define the global O r pGq-torsor Φ 2 T pφq by the local construction (8.3.9). Finally, if φ lies in the complement r 1 pGq in Φ r 1 pGq, we follow the local prescription (8.3.10) to define the of Φ 2 bdd global torsor T pφq. r sim pGq is simple, and φv lies in the subset ΦpG r v q of We note that if φ P Φ r pGv q for every v, the two representations π above are actually equivalent. Φ r we But as we saw in the decomposition (8.3.8) obtained from the action of θ, can indeed treat them as two separate representations in the definition. At any rate, our interest in global torsor T pφq attached to φ P Φbdd pGq will be r 1 p G v q, confined to the case that some φv lies in the corresponding local set Φ since we have treated the other case above. In the next section, we will use the local torsor T pφq to construct the r 1 pGq. We shall use the two L-packets attached to a local parameter φ P Φ bdd global torsor T pφq to establish corresponding multiplicity formulas for global r 1 pGq. This will give the refined endoscopic classification, parameters φ P Φ 2 both local and global, for generic parameters for our even orthogonal group G. These results represent a different perspective, in at least one sense. The characters of individual representations π in a local packet Πφ, cannot be defined in terms of twisted characters for general linear groups. They do satisfy internal endoscopic relations analogous to (8.3.2), but nothing akin to (8.3.4). From this point of view, the representations π are less accessible r pGq-orbits π. than their O Of interest also is what we do not obtain. We will not be able to define a canonical parametrization of the two L-packets Πφ, by the two Langlands parameters φ P Φpφq. The difficulty seems to be deeply entrenched, at least insofar as it pertains to the methods of this volume. The obstruction can be likened to what in physics might be called a pZ{2Zq-symmetry. The philosophical question is whether there might be any endoscopic way to resolve it. Is there any “experiment” within the conjectural theory of endoscopy that r pGq-torsors? would determine the predicted bijection between the two O The third topic for this section is that of Whittaker models. We shall prove a strong form of the generic packet conjecture, for any group G P Ersim pN q. To do so, we have only to combine our classification of local
480
8. THE GLOBAL CLASSIFICATION
and global L-packets for G with the results of Cogdell, Kim, PiatetskiiShapiro and Shahidi, and Ginzburg, Rallis and Soudry on the transfer of automorphic representations with global Whittaker models. The basic idea of applying the work that culminated in [CKPS2] and [GRS] to the general existence of Whittaker models is not new. I learned of it from Rallis, but the idea was also known to Shahidi, and no doubt others as well. We should recall the underlying global definition. Assume for a moment that F is global, and that G is a general quasisplit, connected group over F . Suppose also that pB, T, tXk uq is a ΓF -stable splitting of G over F , and that ψF is a nontrivial additive character on A{F . The function χpuq ψF pu1
un q,
u P NB pAq,
where tuk u are the coordinates of u with respect to the simple root vectors tXk u, is a nondegenerate, left NB pF q-invariant character on NB pAq. It represents a global Whittaker datum pB, χq for G. Given χ, one defines the Whittaker functional ω phq
»
p qz p q
NB F NB A
hpnq χpnq1 dn
on the space of smooth functions h on GpF qzGpAq. Suppose that π is an irreducible cuspidal automorphic representation of GpAq, acting on a closed invariant subspace V of L2cusp GpF qzGpAq, χG (where χG denotes a central character datum). We say that pπ, V q is (globally) pB, χq-generic if the restriction of ω to the space of smooth functions in V is nonzero. If this is so, the restriction ωv of ω to any local constituent πv of π is a local pBv , χv q-Whittaker functional. We assume that F is either local or global, and that G is a fixed group in r Esim pN q over F . We can then fix pB, χq by fixing ψF , and taking pB, T, tXk uq to be the standard splitting. If F is global, and pGv , Fv q is a localization of pG, F q, we can take pBv , χv q to be the corresponding localization of pB, χq.
r bdd pGq is a Proposition 8.3.2. (a) Suppose that F is local, that φ P Φ r bounded generic parmeter, and that π represents the OutN pGq-orbit in the r φ such that the linear character x, π y on Sφ is trivial. Then π local packet Π is locally pB, χq-generic. (b) Suppose that F is global, and that φ is a generic parameter in the r 2 pGq. Then there is a globally pB, χq-generic element π in larger global set Ψ r φ such that the linear character x, π y on Sφ is trivial. the global packet Π
Proof. (a) If F is archimedean, the result is known for any quasisplit G, and for the general L-packet attached to a Langlands parameter φ P Φbdd pGq. For it follows from [V2, Theorem 6.2] and [Kos, p. 105] that there is exactly one pB, χq-generic representation π in the packet Πφ . It then follows from [S6, Theorem 11.5] that with the pB, χq-Whittaker normalization of transfer factors we are using, π is indeed the representation in r φ such that x , π y equals 1. Π
8.3. REFLECTIONS ON THE RESULTS
481
We can therefore assume that F is p-adic. Suppose first that φ belongs r 2 pGq. The argument in this case is purely local. We to the complement of Φ choose a local parameter φM P Φ2 pM, φq, for a proper Levi subgroup M of r φ such that the linear form x , πM y G. Let πM be the representation in Π M on SφM is trivial. We can assume inductively that πM is pM, χM q-generic, since M is proper in G. Let π be the unique pB, χq-generic component of IP pπM q. It then follows from (2.5.3), together with the definition (2.4.17) of rM y for πM and the the character x , π y in terms of its twisted analogue x , π self-intertwining operators RP pw, π rM q, that the character is trivial. This gives the proposition for φ. r 2 pGq. In this case we use a global arSuppose next that φ belongs to Φ gument, specifically the global construction of §6.3. From Proposition 6.3.1, together with the discussion at the beginning of this section in case φ is sim9 q from the given local objects pF, G, φq. 9 φ ple, we obtain global objects pF9 , G, r ell pN q. It corresponds r 2 pG9 q of Φ Then φ9 is a generic parameter in the subset Φ to an automorphic representation πφ9 of GLpN q, which is induced from a selfdual, cuspidal automorphic representation of a Levi subgroup. It follows by induction, or by Lemma 5.4.6 in case φ9 is simple, that Theorem 1.5.3 is 9 r sim pG9 i q of φ. valid for the simple components φ9 i P Φ This is the required condition for the global descent theorem of [GRS] and [So] (which depends for its proof on [CKPS2]). The theorem states that there is a globally pB,9 χ9 q-generic cuspidal automorphic representation π9 of G9 pA9 q whose near equivalence class (in the language of [So]) maps to that of πφ9 . In other words, π9 satisfies the condition 9
ξφ9 cpπ9 q
cpπφq cpφq, 9
9
and thus contributes to the subspace of the automorphic discrete spectrum 9 of G9 determined by φ. As we noted following the proof of Corollary 6.7.4, r 9 we now have at π9 therefore represents an element in the global packet Π φ our disposal, for which associated linear character (8.3.11)
xx, πy 9
¹
9
xxv , πv y, 9
v
9
x9 P Sφ9 ,
equals 1. 9 χ 9 q-generic, each of its components π 9 v is locally Since π9 is globally pB, pB9 v , χ9 v q-generic. If v R S8puq, we know that π9 v is characterized uniquely as r 9 such that the local linear character x , π9 v y the element in the local packet Π φv on Sφ9 v equals 1. This follows from the discussion based on the group Spp2q at the end of §2.5, which led to the proof of Lemmas 6.4.1 and 6.6.2. We have just seen that the same condition holds for the archimedean places v P S8 . This leaves only the p-adic place u of F9 , which we recall has the property that pF9u , G9 u , φ9 u q equals pF, G, φq. The component π9 u of π9 at u lies in the rφ Π r 9 . It follows from the product formula (8.3.11) that local packet Π φu
482
8. THE GLOBAL CLASSIFICATION
the corresponding linear character x , π9 u y on the group Sφ9 u Sφ equals 1, and consequently that π9 u equals the given representation π of GpF q. But we also know that π9 u is pB9 u , χ9 u q-generic. The given representation is therefore pB, χq-generic, as required. (b) The existence of π is proved in the same way as that of π9 above. That is to say, it follows directly from the global descent theorem of [GRS] and [So]. The assertion on the character x, π y is just the condition that π occurs in the discrete spectrum. Remarks. 1. We have identified the irreducible representation π with the r N pGq-orbit of representations in the packet Π r φ . This is corresponding Out just our usual convention in the local case (a), and applies also to the global r N pGq-orbit of representations in the case (b) if π is taken to be the Out r N pGA q-orbit that actually occur in the discrete spectrum. associated Out r N pGq in The group of F -automorphisms of G with which we identified Out Chapter 1 stabilizes pB, χq. We can therefore regard a pB, χq-Whittaker r N pGq-orbit of functionals for functional for the representation π as an Out r N pGq-orbit of representations. the corresponding Out 2. In the local case (a), it is expected that π is the unique pB, χq-generic r φ . In fact, we had to use the known analogue of this element in the packet Π property for valuations v u in the proof. The property for G follows from r φ ), in the case G SOp2n 1q. [JiS1] (and the existence of the packet Π (See also [JiS2], for a proof of the local Langlands correspondence for generic representations of the p-adic group SOp2n 1q.) Extensions of [JiS1] to the other groups G were sketched in [JiS3], while the group G Spp2nq was recently treated in detail in [Liu]. Taking these for granted, we see that the automorphic representation π in (b) is the (unique) element in the global r φ whose local components are the elements πv P Π r φ whose linear packet Π v characters x, πv y on the corresponding groups Sφv are all trivial. 3. Consider again the local case (a). As we noted in §2.5, Konno [Kon] r φ , once constructed, would contain a pB, χq-generic proved that the packet Π element. His methods are purely local. Their refinement, based on a study of the Whittaker normalization of transfer factors in [KS, (5.3)], would presumably lead to a local proof of the proposition in case charpF q 2, as well as a proof of the uniqueness of π. 8.4. Refinements for even orthogonal groups We shall now continue the discussion of even orthogonal groups begun in the last section. Our goal is to construct genuine L-packets from the r φ obtained in the main theorems. To see that this might cruder packets Π be possible, we need only observe that the full force of the stabilized trace formula has yet to be exploited. We have used it so far just for test functions in the symmetric global Hecke algebra. We will now apply it as it was stated
8.4. REFINEMENTS FOR EVEN ORTHOGONAL GROUPS
483
in the general formula (3.3.6) of Lemma 3.3.1, for arbitrary functions in the Hecke algebra. As in the last section, we fix an even orthogonal group G P Ersim pN q,
p SOpN, Cq, N G
2n,
over the field F . We are interested in parameters φ that lie in the subset r 1 pG q φ P Φ r pGq : mpφq 2 Φ
(
r pGq. For any such φ, the associated pair of Φ
Φpφq ΦpG, φq
of parameters is a torsor under the group r pGq Out r N pGq Aut r N pGq{rIntN pGq O
r pGq-torsor T pφq in terms of order 2. In the last section, we defined another O r φ will be of irreducible representations. The L-packets we construct from Π parametrized by T pφq instead of Φpφq. r q of Φ r 1 pGq in Φ r pGq, the set Φpφq consists If φ lies in the complement ΦpG of φ alone. A formal application of the earlier definition of T pφq to this case yields a set that also contains only one element. With this understanding, r pGq, the set T pφq behaves well under endoscopic we note that for any φ P Φ transfer. Suppose for example that s belongs to the subset S φ,ell of elliptic elements in S φ . The endoscopic preimage of the pair pφ, sq is then a pair pG1, φ1q, where G1 G11 G12 , G1i P Ersim pNi1 q,
belongs to Eell pGq, and
φ1
φ11 φ12,
r pG1 q. The finite group belongs to Φ
φ1i
P Φr pG1iq,
r pG 1 q O r pG 1 q O r pG 1 q O 1 2
is an extension 1
ÝÑ
OutG pG1 q
ÝÑ OrpG1q ÝÑ OrpGq ÝÑ
1,
which acts transitively on the product
T pφ1 q T pφ11 q T pφ12 q.
It is clear that there is a surjective mapping T pφ1 q Ñ T pφq, which is comr pG1 q and O r pGq. In particular, the mapping patible with the actions of O 1 1 pG , φ q Ñ pφ, sq extends to a surjective mapping
pG1, φ1, t1q ÝÑ pφ, s, tq,
t P T pφ q, t 1
P T pφ 1 q . r 1 pGq (or ever the We are really only interested in the case that φ lies in Φ r 1 pGq), so that T pφq has order 2 and is indeed an O r pGq-torsor. smaller set Φ bdd
484
8. THE GLOBAL CLASSIFICATION
r pG1 q-torsor. In this The set T pφ1 q then typically has order 4, and is an O 1 case, the fibres of the mapping T pφ q Ñ T pφq are OutG pG1 q-torsors. r represents a subset of Π r unit pGq, we shall Suppose that F is local. If Π write Π for its preimage in Πunit pGq. In particular, Πφ is the preimage in r φ of O r pGq-orbits in Π r unit pGq we have attached to Πunit pGq of the packet Π r 1 pGq of Φ r pGq. It follows from any φ. Assume that φ lies in the subset Φ bdd r pGq-orbit of any π P Π r φ has order 2. The the results of Chapter 6 that the O fibres of the mapping rφ Πφ ÝÑ Π r pGq-torsors. The following theorem posits among other things a are thus O pair of canonical sections r φ ÝÑ Πφ Π parametrized by T pφq. Their images in Πunit pGq represent the true L-packets attached to the two Langlands parameters in the set Φpφq. r 1 pGq. Then Theorem 8.4.1. Suppose that F is local, and that φ lies in Φ bdd r pGq-equivariant bijection there is an O
t
ÝÑ
t P T pφ q,
φt ,
from T pφq onto a pair of stable linear forms
ÝÑ f Gpφtq, f P HpGq, t P T pφq, r pGq-equivariant bijection on HpGq, and an O pπ, tq ÝÑ πt, π P Πr φ, t P T pφq, r φ T pφq onto Πφ , such that from Π r p G q, t P T pφ q , (8.4.1) f G p φ q f G p φ t q, f PH f
and
(8.4.2)
fG pπ q fG pπt q,
f
P HrpGq, t P T pφq,
and such that for any x P Sφ , the identity
(8.4.3)
f 1 pφ1t1 q
¸
P
rφ π Π
xx, πy fGpπtq,
f
P H pG q, t P T pφ q,
is valid for any endoscopic preimage pG1 , φ1 , t1 q of pφ, s, tq.
Remarks. 1. The formula (8.4.1) tells us that the action φt of T pφq on the r pGq on stable distributions in question is compatible with the action of O corresponding parameters. The formula (8.4.2) asserts that the action πt of r pGq. The T pφq on the relevant characters is also compatible with that of O formula (8.4.3) is a refinement (for generic parameters) of the main assertion (2.2.6) of the local Theorem 2.2.1. 2. Given (8.4.1) and (8.4.2), one sees without difficulty that the proposed mappings t Ñ φt and t Ñ πt are uniquely determined by (8.4.3).
8.4. REFINEMENTS FOR EVEN ORTHOGONAL GROUPS
485
3. Suppose that the theorem holds for F . It then extends formally to r 1 pGq. In standard representations attached to parameters in the larger set Φ r φ attached to any parameter particular, the theorem is valid for the packet Π in the subset r 1 pG q Φ r 1 p Gq X Ψ r Φ unit unit pGq r 1 pG q . of Φ
We shall use global methods to establish this local theorem. Before doing so, we may as well first state the associated global theorem. The burden of proof for both theorems will then fall on the stabilized trace formula for G. r bdd pGq is a generic global Suppose that F is global, and that φ P Φ parameter. We can then write (8.4.4)
Idisc,φ pf q
¸
G1 Eell G
P p q
1 pf 1 q, ιpG, G1 q Spdisc,φ
f
P H pG q ,
in the notation (3.3.15). The essential point here is that the formula holds for general functions in HpGq, rather than just the functions in the symmetric r pGq. To make the argument work, we will need a nonsymmetric subalgebra H version of the stable multiplicity formula, at least for some φ, to express the terms on the right. r unit pGv q of Ψ r Any localization φv of φ lies in the generic subset Φ unit pGv q. Assume that Theorem 8.4.1 is valid for Fv , and that φv lies in the subset r 1 pGq of Φ r unit pGq. The assertions of Theorem 8.4.1 are then valid for φv , Φ unit as we noted above in Remark 3. This allows us to define an isomorphism
ÝÑ T p φ v q. t
tv
between the torsors T pφq and To describe it, suppose first that r φ P Φsim pGq is simple. According to the definition in §8.3, an element t P T pφq is represented by an automorphic representation π πt attached r pGq-orbit π in Π r φ . We use the main assertion (8.4.3) of the theorem to the O to define tv as the element in T pφv q such that πv,tv
πt,v .
We extend the construction from this basic case to more general φ, first in r 1 pGq and then in the complement of Φ r 1 pGq in Φ r 1 pGq, directly from the Φ 2 2 bdd definitions. Suppose now that Theorem 8.4.1 holds for every completion Fv of F . The mappings t Ñ tv then allow us to globalize the two constructions of the r pGq-equivariant mapping local theorem. We define a global O t
ÝÑ
t P T p φ q,
φt ,
from T pφq to the space of linear forms on the global space S pGq by setting φt
â
pφv,t q. v
v
486
8. THE GLOBAL CLASSIFICATION
r 1 pGv q, tv of course represents the lone If φv lies in the complement of Φ unit r pG v qelement in the set T pφv q. In this case, φv,tv is understood to be the O invariant linear form φv we identified in (2.2.2) with the corresponding parameter. To describe the second global construction, we extend its accompanying local notation in the natural way. Namely, we write Π for the r of the restricted tensor product preimage in Πunit pGq of any subset Π r unit pGq Π
â
r unit pGv q . Π
v
r pGq-equivariant mapping We then define an O
pπ, tq ÝÑ
πt ,
π
P Πr φ, t P T pφq,
π
P Πr φ, t P T pφq.
r φ T pφq to Πφ by setting from Π
πt
â
πv,tv ,
v
This is compatible with our definition of tv in the special case that φ lies in r sim pGq. Φ r 1 pGq. Then Theorem 8.4.2. Suppose that F is global, and that φ lies in Φ 2
(8.4.5)
Sdisc,φ pf q |Sφ |1
and (8.4.6)
tr Rdisc,φ pf q
¸
P pq
f G pφ t q ,
f
t T φ
¸
¸
r φ p1q tPT pφq π PΠ
fG pπt q,
P H pG q, f
P H p G q.
Remarks. 1. The assertions are refinements (for generic, square-integrable parameters) of the global Theorems 4.1.2 and 1.5.2. By not requiring f to r pGq, they place the group G of type lie in the symmetric Hecke algebra H Dn on an equal footing with our groups of type Bn and Cn . 2. We recall that the left hand side of (8.4.6) was defined in (3.4.5). The r φ p1q on the right hand side is the set of representations π in indexing set Π r φ such that the character x, π y on Sφ is trivial. the global packet Π The proofs of Theorems 8.4.1 and 8.4.2 have a common core. It begins with the preliminary interpretation of the stabilized trace formula (8.4.4). To describe this, we assume inductively that both theorems hold if N is replaced by any positive integer N 1 N . Suppose that F and φ are as in the global Theorem 8.4.2. The analogue of (8.4.5) is then valid for the summand of any G1 G on the right hand side of (8.4.4). Making the appropriate substitution, and recalling how we treated these terms in (6.6.4) for example, we find that the right hand side
8.4. REFINEMENTS FOR EVEN ORTHOGONAL GROUPS
of (8.4.4) equals (8.4.7)
G Sdisc,φ pf q
|Sφ|1
¸
¸
P
t T φ
x Sφ x 1
P pq
487
f 1 pφ1t1 q,
where pG1 , φ1 , t1 q is a preimage of pφ, x, tq. The left hand side of (8.4.4) reduces to the left hand side of (8.4.6). For the present, we write it simply as (8.4.8)
¸
¸
r unit pGq π PΠpπ q π PΠ
nφ pπ q fG pπ q,
where
Πpπ q ΠpG, π q is the preimage of π in Πunit pGq, and nφ pπ q is the multiplicity of π in the discrete spectrum. The identity of (8.4.7) and (8.4.8) will be a starting point for the proofs of both theorems. We shall establish the two theorems in succession. Proof of Theorem 8.4.1. We are assuming that F is local, and that r 1 pGq. If F is archimedean, the theorem follows from the endoφ lies in Φ bdd scopic Langlands classification of Shelstad. In fact, the Langlands classifir pGq-bijection between the torsors cation in this case provides a canonical O Φpφq and T pφq, which is compatible with the assertions of the theorem. It is the lack of this bijection in the p-adic case that forces us to introduce T pφq as a substitute for Φpφq. We therefore assume that the local field F is nonarchimedean. To deal with this case, we shall use the global methods of Chapter 6. In particular, we shall apply Proposition 6.3.1 to the local parameter φ. Having established the local theorems of Chapter 6, we do not need the induction hypothesis Ni N from Proposition 6.3.1 that allowed us to apply Corollary 6.2.4. In other words, we do not have to rule out the case that φ is simple. r 1 p Gq The essential case is that of a parameter (6.6.1) in the subset Φ 2 r 1 pGq. With this assumption on φ, we obtain global objects F9 , of Φ bdd r 1 pG9 q from the local objects F , G and φ, which satisfy G9 P Ersim pN q and φ9 P Φ 2 the conditions of Proposition 6.3.1. In particular, we have
pFu, Gu, φuq pF, G, φq, 9
9
9
for a p-adic valuation u on the global field F9 . We identify the local group r p Gq O r pG9 u q (of order 2) with its global counterpart O r pG9 q. From the O construction of Proposition 6.3.1, together with the fact that the simple r 1 pG9 v q degrees Ni degpφ9 i q degpφi q are even, we see that φ9 v belongs to Φ 2 for each archimedean valuation v P S8 . In particular, the set Φpφ9 v q is r pG9 v q-torsor. This allows us to identify the group O r pG q O r pG9 q also an O r pG9 v q for any v P S8 , and thereby treat Φpφ9 v q as an with its analogue O r OpGq-torsor.
488
8. THE GLOBAL CLASSIFICATION
There is a canonical OpGq-bijection t Ñ t9 from T pφq to T pφ9 q. In the case that φ is simple, this follows from the fact that a global representation π9 in T pφ9 q is uniquely determined by its component π π9 ,u at u. For arbitrary φ, the bijection is then a consequence of the definition of the two parallel equivalence relations for T pφq and T pφ9 q. Its construction thus does not require the validity of Theorem 8.4.1 for the completion φ φ9 u . We do know that the theorem holds for the completion φ9 v of φ9 at any archimedean valuation v P S8 . The earlier construction then gives us an OpGq-bijection t Ñ t9v from T pφq to T pφ9 v q. This in turn can be composed r pG9 v q-torsors with the isomorphism between T pφ9 v q and any of the other O 9 9 r attached to φv . We thus obtain an OpGq-isomorphism t Ñ φv,t from T pφq r pGq-isomorphism t Ñ π9 v,t from T pφq to T pπ9 v q, for any to Φpφ9 v q, and an O r9 . π9 v P Π φv We are carrying the earlier induction hypothesis that both theorems hold if N is replaced by a positive integer N 1 N . The expressions (8.4.7) and 9q 9 φ (8.4.8), obtained from the stabilized trace formula (8.4.4) but with pF9 , G, in place of the earlier triplet pF, G, φq, are therefore equal. To be consistent with the notation here, we need also to write f9 for the global test function in HpG9 q that was earlier denoted by f . We fix its component f98,u
¹
R pq
f9v
v S8 u
r pG9 v q to be a symmetric, spheriaway from S8 puq by choosing each f9v P H 1 1 9 9 cal function so that fv pφv q 1, where pG9 1v , φ9 1v q is the v-component of the preimage pG9 1 , φ9 1 q of the pair attached to any x P Sφ . We can then regard the equality of (8.4.7) and (8.4.8) as an identity of linear forms in the complementary component
f98,u
f8 fu f8 f, 9
9
f98
9
P H p G 8 q, f P H pG q , 9
of f9. The summand of x and t in (8.4.7) equals
1 pφ9 1 1 q f 1 pφ1 1 q, f91 pφ9 1t1 q f98 8,t t since
pf 8,uq1 pφ8,uq 9
¹
9
R pq
v S8 u
f9v1 pφ1v q 1.
The expression (8.4.7) therefore equals (8.4.9)
G pf9q Sdisc, φ9 9
|Sφ|1
¸
¸
P
t T φ
x Sφ x 1
P pq
1 pφ9 1 1 qf 1 pφ1 1 q. f98 8,t t
The description of (8.4.8) requires a little more discussion.
8.4. REFINEMENTS FOR EVEN ORTHOGONAL GROUPS
489
r pGq-orbit of representations For any character ξ P Spφ , we have the O r φ such that x, π pξ qy ξ. We also have the set Π r9 π pξ q P Π φ8 ,ξ of representations â r9 , π9 v , π9 v P Π π9 8 φv v PS8 r 9 such that the character in the packet Π φ8 ¹
xx, π8y 9
xxv , πv y, 9
9
x P Sφ ,
9
v PS8 1 on Sφ equals ξ pxq . It follows from Proposition 6.3.1(iii)(a) that this set is r not empty. In general, the elements π9 8 in Π φ8 represent simply transitive 9 orbits in Πunit pG8 q under the product 9
r pG9 8 q O
¹
P
v S8
r pG9 v q. O
r9 For any π9 8 P Π φ8 ,ξ and any local representation π pξ q a unique global representation
P Π πpξq , there is
π8, b πpξq b π8,u,
π9
9
9
where π9 8, belongs to the set
ΠpG9 8 , π9 8 q
â
P
v S8
ΠpG9 v , π9 v q
and π9 8,u P ΠpG9 8,u q is spherical, such that the coefficient nφ9 pπ9 q on (8.4.8) is nonzero. This follows from Proposition 6.3.1(ii) and either the proof of Propositions 6.6.5 and 6.7.2 or simply the general assertion of Theorem 1.5.2. In particular, the representation π9 8, is uniquely determined by π9 8 and π pξ q. We can then write π9 8,
â
π8,t8pq 9
for a direct product t98 pq t8 π9 8 , π pξ q
P
v S8
¹
P
v S8
π9 v,t9v pq , tv π9 8 , π pξ q
of uniquely determined elements t9v pq tv π9 8 , π pξ q in T pφq. Since the nonzero coefficient nφ9 pπ9 q actually equals 1, the expression (8.4.8) becomes (8.4.10)
¸
¸
P
P
¸
p qPΠpπpξqq
r9 π ξ ξ Spφ π9 8 Π φ8 ,ξ
f98,G9 pπ9 8, q fG pπ pξ q .
r φ , there is an O r pGq-isomorphism t Ñ πt from Lemma 8.4.3. For any π P Π r 9 , we have T pφq onto Πpπ q such that for any ξ P Spφ and π9 8 P Π φ8 ,ξ
tv π9 8 , πt pξ q
t,
t P T p φ q, v
P S8 .
490
8. THE GLOBAL CLASSIFICATION
Proof. We have to show that for any ξ and π pξ q, the T pφq-components
tv π9 8 , π pξ q ,
P S8,
v
of the representations π9 8, π9 8,t98 pq in (8.4.10) are independent of π9 8 and v. The main point is to show that tv π9 8 , π pξ q is independent of the first argument π9 8 . The proof of this will be based on the identity of (8.4.9) and (8.4.10) that we have obtained from the stabilized trace formula. We first impose a constraint on the function f98 P HpG9 8 q that forces the summands on the right hand side of (8.4.9) to vanish. We require that for any x 1 in Sφ and any t P T pφq, the linear form
1 pφ9 1 1 q f98 8,t
¸
xx, π8y f8,Gpπ8,tq 9
P
r9 π9 8 Π φ8
9
9
9
9
vanishes. In other words, for any t P T pφq, the sum f98,G9 pξ, tq
¸
P
r9 π9 8 Π φ8 ,ξ
f98,G9 pπ9 8,t q,
ξ
P Spφ,
is independent of ξ. This forces the sum over x in (8.4.9) to vanish. We then fix the indices ξ and π pξ q in (8.4.10), and take f P HpGq to be a pseudo-coefficient f pξ q of the representation π pξ q P Π2 pGq. This simplifies r 9 . The equality of (8.4.9) the triple sum in (8.4.10) to a simple sum over Π φ8 ,ξ and (8.4.10) becomes (8.4.11)
G Sdisc, pf9 fξ, f98,uq φ9 8
¸
9
P
r9 π9 8 Π φ8 ,ξ
f98,G9 pπ9 8 q.
Consider the abstract spectral expansion of the left hand side of (8.4.11), as a stable linear form in f98 . Using the right hand to bound its spectral support, and keeping in mind the spectral criterion for stability in terms of archimedean Langlands parameters, we obtain a linear combination (8.4.12)
G Sdisc, f9 f pξ q f98,u φ9 8 9
over indices t98
¹
P
v S8
t9v ,
tv
¸ t98
G 9 cpt98 q f98 pφ8,t98 q 9
P T pφ q ,
with complex coefficients cpt98 q c t98 , f pξ q . The formulas (8.4.11) and (8.4.12) were derived with the condition that 9 f8 pξ, tq be independent of t. We can actually treat the two right hand sides r9 as linear forms on the space of functions in the preimage Πφ9 8 ,ξ of Π φ8 ,ξ in ΠpG9 8 q. To be precise, we claim that if the coefficients cpt98 q are rescaled by the factor |Πφ9 8 | |Πφ9 8,ξ |1 |Πr φ9 8 | |Πr φ9 8,ξ |1,
8.4. REFINEMENTS FOR EVEN ORTHOGONAL GROUPS
491
the two formulas remain valid if the condition in f98 is replaced by the requirement that the function f98,G9 on ΠpG9 8 q be supported on Πφ9 8 pξ q. This is because the restriction mapping, from the subspace of functions f98 P HpG9 8 q that satisfy the first condition onto the space of functions f98,G9 on Πφ9 8 ,ξ , is surjective. The claim follows easily from this fact, together with the form of the terms on the two right hand sides. The two formulas can then be combined as an identity (8.4.13)
¸
P
r9 π9 8 Π φ8 ,ξ
f98,G9 pπ9 8, q
¸ t98
G 9 cpt98 q f98 pφ8,t98 q, 9
for complex coefficients cpt98 q c t98 , f pξ q , and any f98 P HpG9 8 q such that the function f98,G9 on ΠpG9 8 q is supported on Πφ9 8 ,ξ . The representation π9 8,
r9 on the left hand side is the image of π9 8 under the section from Π φ8 ,ξ to 9 Πφ9 8 ,ξ determined by π pξ q, while the index of summation t8 on the right hand is over elements in the direct product of |S8 |-copies of T pφq. The left hand side of (8.4.13) does not vanish, since we can always take f98 to be a pseudocoefficient of some π9 8, . We can therefore choose an index t98 so that the corresponding coefficient cpt98 q on the right is nonzero. 9 r9 Suppose that π9 8 P Π φ8 ,ξ is arbitrary, and that f8 is a pseudocoefficient of the representation π9 8,t98 . The left hand side then equals 1 if t8 π9 8 , π pξ q equals t98 , and vanishes otherwise. But the right hand side reduces in this case to the nonzero coefficient cpt98 q c t98 , f pξ q . It follows that the product ¹ t8 π9 8 , π pξ q tv π9 8 , π pξ q v PS8 equals the index ¹ t98 t9v . v PS8 r 9 . It follows that for any v P S8 , But π9 8 was taken to be any element in Π φ8 ,ξ tv π9 8 , π pξ q equals t9v , and is therefore independent of π9 8 . We now have to check that t9v tv π9 8 , π pξ q is independent of v. If 9 φ is simple, ξ equals 1, and Π φ consists of one element π. In this case, the definitions give isomorphisms
t πt
ÝÑ
πv,t ÝÑ φv,t r pGq-torsors. (We recall that for simple φ, T pφq is just defined to be the of O r OpGq-torsor Πφ .) The global representation πt is determined by πt and a r r particularly chosen factor π8 in the set Π φ8 Πφ8 ,1 . It follows that tv pπ8 , πt q t, by the definition of the left hand side. Having already shown that tv pπ8 , πt q π9 t
ÝÑ
π9 t,v
9
9
9
9
9
9
9
9
is independent of π9 8 , we conclude that it is also independent of v. For
492
8. THE GLOBAL CLASSIFICATION
r pGq-torsors T pφq and Φpφ9 v q. general φ, we have an isomorphismt Ñ φ9 v,t of O The independence of tv π9 8 , π pξ q on v then follows from the definitions, and the case we have just established for the simple constituents of φ. r φ. We can now construct the required isomorphism. Suppose that π P Π p Then π π pξq, for a unique character ξ P Sφ . For any representation π pξ q P Π π pξ q , we now know that the point
t tv π9 8 , π pξ q ,
π9 8
P Πr φ8,ξ , v P S8, 9
in T pφq is independent of π9 8 and v, and therefore depends only on πξ, . We r pGq-torsors Π π pξ q thus obtain an isomorphism π pξ q Ñ t between the O and T pφq. Its inverse t Ñ πt then satisfies the conditions of the lemma. We return to the proof of Theorem 8.4.1. There are two mappings to r pGq-equivariant isomorphism pπ, tq Ñ πt is provided by the consider. The O last lemma. We define the other mapping by setting f G pφ t q
(8.4.14)
¸
rφ π PΠ
fG pπt q,
t P T p φ q, f
P H pG q.
The required identities (8.4.1) and (8.4.2) then follow from the definitions. We must still show that the right hand side of (8.4.14) is stable in f in order to justify the notation on the left hand side, and of course also because it is one of the requirements of the theorem. We have also to establish (8.4.3). The mapping pπ, tq Ñ πt is defined by the formula t8 π9 8 , πt pξ q
t,
π9 8
P Πr φ8,ξ , t P T pφq, 9
according to the discussion at the end of the proof of Lemma 8.4.3. The expression (8.4.10) can then be written as ¸
(8.4.15)
P
¸
P pq
ξ Spφ t T φ
¸
P
r9 π9 8 Π φ8 ,ξ
f98,G9 pπ9 8,t q fG πt pξ q .
The inner sum in brackets is what we denoted by f98,G9 pξ, tq near the begin-
ning of the proof of the lemma. As before, let us assume that f98 is such that this is independent of ξ. The inner sum can then be written as the product of |Sφ |1 with ¸
P
ξ Spφ
f98,G9 pξ, tq
¸
P
r9 π9 8 Π φ8
G 9 f98,G9 pπ9 8,t q f98 pφ8,tq. 9
Our expression (8.4.15) for (8.4.10) reduces to the simpler expression
|Sφ|1
¸
P pq
t T φ
G 9 f98 pφ8,tq 9
¸
P
ξ Spφ
fG πt pξ q
,
8.4. REFINEMENTS FOR EVEN ORTHOGONAL GROUPS
which in turn equals
|Sφ|1
¸
493
G 9 f98 pφ8,tq f Gpφtq, 9
P pq
t T φ
according to the definition (8.4.14). On the other hand, our condition on f98 implies that the summands in (8.4.9) with x 1 all vanish. The equality of (8.4.9) and (8.4.10) becomes (8.4.16)
G Sdisc, pf9q |Sφ9 |1 φ9
¸
9
G 9 f98 pφ8,tq f Gpφtq. 9
P pq
t T φ
The left hand side of this formula is stable in the variable function f P HpGq. G9 pφ9 The coefficients f98 8,t q may be chosen arbitrarily by varying f98 . It follows that the linear form f G pφt q defined by (8.4.14) is stable, as required. It remains to prove (8.4.3). The formula (8.4.16) is still valid without the condition on f98 under which it was derived. This follows from the stability of each side and the fact that the subspace of functions f98 P HpG9 8 q that satisfy the condition maps onto S pG9 8 q. We substitute it for the leading term on the right hand side (8.4.9) of the trace formula (8.4.4). The right hand side of (8.4.4) becomes
|Sφ|1
for any f98 (8.4.17)
¸
¸
P
P pq
x Sφ t T φ
1 pφ9 1 1 q f 1 pφ1 1 q, f98 8,t t
P HpG8q and f P HpGq, an expression we write as ¸¸ ¸ |Sφ|1 xx, π8y f8,Gpπ8,tq f 1pφ1t1 q. 9
9
x
t
9
9
9
P
9
r9 π9 8 Π φ8
This consequently equals the expression (8.4.15) we have obtained for the left hand side (8.4.10) of (8.4.4). We choose the function f98 to be a pseudocoefficient of a representation π9 8,t P Πφ9 8 pξ q, for fixed ξ and t. The expres sion (8.4.15) reduces to fG πt pξ q . The other expression (8.4.17) becomes
|Sφ|1
¸
P
ξ pxq1 f 1 pφ1t1 q,
x Sφ
1 . Applying Fourier inversion on Sφ to each side of the 9 π 9 8 y ξ px q since xx, resulting identity, we obtain f 1 pφ1t1 q
¸
P
ξ Spφ
ξ pxq fG πt pξ q
¸
P
rφ π Π
xx, πy fGpπtq.
This is the required formula (8.4.3). We have completed the proof of Theorem 8.4.1 in case φ is a parameter r 1 pGq. It remains to consider a parameter φ in the complement (6.6.1) in Φ 2 r 1 pGq in Φ r 1 pGq. Then φ is of the general form (6.3.1), and is the image of Φ 2 bdd r 2 pM q, for a proper Levi subgroup M of G. We shall of a parameter φM P Φ say only a few words about this case since it is much simpler.
494
8. THE GLOBAL CLASSIFICATION
We define the mappings t Ñ φt and pπ, tq Ñ πt directly from their analogues for M , guided by the requirement that they be compatible with induction. In this way, we also obtain general analogues fG1 pφt , xq and fG pφt , xq of the two sides of the local intertwining relation (2.4.7). The required identities (8.4.1) and (8.4.2) follow directly from the definitions. It remains to establish the transfer identity (8.4.3). As we saw in the proof of Proposition 2.4.3, this in turn will follow from the local intertwining relation. In other words, it suffices to establish the following analogue of Theorem 2.4.1. Proposition 8.4.4. Suppose that F is local, and that φ lies in the compler 1 pGq in Φ r 1 pGq. Then ment of Φ 2 bdd fG1 pφt , sq fG pφt , uq,
(8.4.18)
f
P H p G q, t P T pφ q ,
for u and s as in Theorem 2.4.1. Proof. Recalling the discussion at the beginning of §6.4, and noting that there is nothing new to be established in the two exceptional cases (4.5.11) and (4.5.12), we see that it is enough to prove (8.4.18) for a parameter φ of the special form (6.4.1), with r ¡ q, and Ni even whenever q i ¤ r. We can then take u s to be an element x in the set we denoted by Sφ,ell . We shall be brief. Suppose first that F R. By the results of Shelstad and the multiplicativity of intertwining operators, the generalization fG pφt , xq
¸
P
rφ π Π
επM,t pxq xx, π y fG pπt q,
t P T pφ q , f
P H pG q ,
P HrpGq, this equals ¸ fG pφ, xq fG1 pφ, xq xx, πy fGpπq,
of (6.1.5) remains valid. If f
P
rφ π Π
by the special case of (8.4.18) given by Theorem 2.4.1. We can choose r pGq so that f PH #
1, if π π, fG pπ q fG pπ q t
for any π
0, otherwise,
P Πr φ. It then follows that επ pxq 1, x P Sφ,ell , t P T pφq. M,t
Therefore
fG pφt , xq
for any t, x and f
¸
P
rφ π Π
xx, πy fGpπtq fG1 pφt, xq,
P HpGq, as required.
8.4. REFINEMENTS FOR EVEN ORTHOGONAL GROUPS
495
Suppose now that F is p-adic. In this case, we use Proposition 6.3.1 to 9 M 9 9 φ, 9 and φ attach global objects F9 , G, M to F , G, φ, M and φM , as at the beginning of §6.4. We then have the analogue ¸
(8.4.19)
¸
f9G19 pφ9 t , x9 q f9G9 pφ9 t , x9 q
P pq P
0,
f9 P HpG9 q,
t T φ x Sφ,ell
9 q, 9 φ of (6.4.2). This follows from an application of the standard model to pG, as in the relevant part of the proof of Lemma 5.2.1, and the fact that G9 pf9q vanishes, as we have shown for symmetric functions f9 P HrpG9 q R9 disc, φ9
(and consequently any f9). We can fix a symmetric function f98,u away from S8 puq so that if f9 f98 f f98,u ,
f98
P HrpG8,uq
P HrpG8q, f P HpGq, 9
the left hand side of (8.4.19) decomposes ¸
¸
P pq P
t T φ x Sφ,ell
9
1 pφ9 8,t , x9 q f 1 pφt , xq f9 9 pφ9 8,t , x9 q fG pφt , xq . f98 G 8 ,G ,G9
It then follows from the archimedean case we have proved that ¸¸ t
x
f98,G9 pφ9 8,t , x9 q fG1 pφt , xq fG pφt , xq
0.
Since the linear forms f98,G9 pφ9 8,t , x9 q,
f98
P HpG8q, t P T pφq, x P Sφ,ell, 9
are linearly independent, we conclude that fG1 pφt , xq fG pφt , xq,
f
P HpGq, t P T pφq, x P Sφ,ell,
as required. This completes our sketch of the proof of Proposition 8.4.4, and hence also of the remaining part of the proof of Theorem 8.4.1.
We write (8.4.20)
Πφ,t
tπt : π P Πr φu,
t P T p φ q,
r 1 pGq, as in Theorem 8.4.1. The sets Πφ,t parametrized by for any φ P Φ bdd the two elements t P T pφq are then the two L-packets attached to φ. If φ r q, T pφq consists of one element, belongs to the complementary set Φbdd pG and the assertions of Theorem 8.4.1 hold trivially with φt φ and πt π. If we extend the definition (8.4.20) to this case, Πφ,t is simply the L-packet r φ we have already attached to φ. Πφ Π
Corollary 8.4.5. (a) The set Πtemp pGq of irreducible, tempered representations of GpF q is a disjoint union of the L-packets Πφ,t ,
r bdd pGq, t P T pφq. φPΦ
496
8. THE GLOBAL CLASSIFICATION
(b) The distributions r bdd pGq, t P T pφq, φPΦ
φt ,
form a basis of the space of stable linear forms on HpGq.
r bdd pGq, the preimage Πφ of Π r φ in Πtemp pGq Proof. (a) For any φ P Φ is the disjoint union over t P T pφq of the packets Πφ,t . The assertion follows r temp pGq, which was stated in Theorem 1.5.1(a) and from its analogue for Π proved in Corollary 6.7.5. (b) We can assume that F is nonarchimedean, as we have done implicitly in (a), since the assertions for archimedean F follow from the general endoscopic classification of Shelstad. In particular, Shelstad’s archimedean results imply that the spectral characterization of stability (in terms of Langlands parameters) matches the geometric characterization (in terms of stable conjugacy classes), a property we have already used in the derivation of (8.4.12). This is what must be verified for p-adic F . Theorem 8.4.1 tells us that the distributions φt are stable. Part (a) above implies that they are linearly independent. What remains is to show that they span the space of all stable linear forms on HpGq. This is the r pN q, alluded to analogue for G of the assertion of Proposition 2.1.1 for G at the beginning of the proof of the proposition. It is implicit in the main results [A11] (namely, Theorems 6.1 and 6.2, together with Proposition 3.5), and will be made explicit in [A24].
Proof of Theorem 8.4.2. We assume now that F is global. We have established that Theorem 8.4.1 holds for the localizations φv of the given r 1 pGq. However, we must still retain what is left of global parameter φ P Φ 2 the original induction hypothesis, namely that Theorem 8.4.2 holds if N is replaced by any integer N 1 N . The expressions (8.4.7) and (8.4.8) are therefore equal. The main point is to prove the stable multiplicity formula (8.4.5). The argument will be similar to a part of the proof of Theorem 8.4.1, specifically Lemma 8.4.3, so we can be brief. We shall first take care of the other assertion, the actual multiplicity formula (8.4.6). One advantage we have here is that the right hand side of (8.4.5) is actually defined. It equals the contribution of x 1 that is missing from G (8.4.7). Following a convention from §4.4, we write 0 Sdisc,φ pf q for the difference between the two sides of (8.4.5). Since (8.4.7) equals the left hand G side of (8.4.6), namely the trace of Rdisc,φ pf q, we see that the difference G tr Rdisc,φ pf q
equals
G 0Sdisc,φ pf q ,
|Sφ|1
¸
¸
P
P pq
x Sφ t T φ
f
f 1 pφ1t1 q.
P H pG q ,
8.4. REFINEMENTS FOR EVEN ORTHOGONAL GROUPS
497
Another advantage is that we can now apply the formula (8.4.3) to the local components of the global summands f 1 pφ1t1 q. From a simplified variant of the remarks at the end of §4.7, we deduce that the last expression equals ¸
¸
P pq
P
r φ,1 t T φ π Π
fG pπt q.
G This is just the right hand side of (8.4.6). Writing 0 rdisc,φ pf q for the difference of the two sides of (8.4.6), we conclude that 0 G rdisc,φ
(8.4.21)
G pf q 0Sdisc,φ p f q,
f
P H pG q.
In particular, the stable multiplicity formula (8.4.5) implies the actual multiplicity formula (8.4.6). It remains then to establish (8.4.5). If the localization φv of φ lies in the r v q of Φ r unit pGv q for each v, the formula (8.3.6) reduces to what subset Φunit pG r pGq. The formula (8.4.5) we have established for symmetric functions f P H is then also valid, and there is nothing further to do. We can therefore r 1 pGv q for some v. This assume that φv lies in the complementary set Φ unit property will allow us to put together a proof of (8.4.5) parallel to that of its analogue (8.4.16) from Theorem 8.4.1. Suppose that V is some finite set of valuations of F such that φv lies in r 1 pGv q of Φ r unit pGv q for each v P V . We fix a character ξ on the subset Φ unit V Sφ , and an element π pξ q in the packet r V Π φ
!â
R
πv : πv
P Πr φ , x, πv y 1 for almost all v
)
v
v V
with the property that the character
xx, πV pξqy
¹
R
xxv , πv y,
v V
x P Sφ ,
on Sφ equals ξ. We also fix a locally symmetric function f V pξ q such that # 1, if π V π V pξ q, f V p ξ q G pπ V q 0, otherwise, for every π V
P Πr φ
V
P HrpGV q
. We then set f
fV f V pξq,
fV
P H p G V q,
and regard (8.4.21) as an identity in the complementary function fV . G The left hand side 0 rdisc,φ pf q of (8.4.21) is defined as the difference of the two sides of the putative identity (8.4.6). The left hand side of (8.4.6) in turn equals the double sum (8.4.8). Arguing as in the derivation of (8.4.10) from (8.4.8) in the proof of Theorem 8.4.1, we see that this equals ¸
¸
rφ π V PΠ
πV,
V ,ξ
fV,G pπV, q,
498
8. THE GLOBAL CLASSIFICATION
r φ ,ξ is the subset of elements πV P Π r φ such that the character where Π V V x, πV y on Sφ equals ξ1, and πV, ranges over some OrpGq-orbit of order 2 in the set ΠpπV q. In particular, we have
πV,
πV,t
V
pq
for a direct product tV pq tV πV , π V pξ q
â
πv,tV pq ,
P
v V
¹
P
t v π V , π V pξ q
v V
r pGq. The of points in T pφq that is determined up to the diagonal action of O right hand side of (8.4.6) is just equal to ¸
¸
P pq
P
r φ ,ξ t T φ πV Π V
where πV,t
â
P
πv,t ,
fV,G pπV,t q, t P T pφ q .
v V
The difference of these last two expressions therefore equals the right hand G side 0 Sdisc fV f V pξ q of (8.4.21). Supplementing the argument used to derive (8.4.12) in the proof of Lemma 8.4.3 with the criterion for stability at places v P V in Corollary 8.4.5(b), we write the stable linear form 0SG V disc fV f pξ q as a linear combination ¸
cptV q fVG pφV,tV q
tV
over indices tV
¹
P
tv ,
tv
P T pφ q ,
v SV
with complex coefficients cptV q cptV , fξV q. The identity (8.4.21) becomes (8.4.22)
¸
P
pq
rφ ξ πV Π V
where
f V,G pπV,tV pq q f V,G pπV,t q f V px q
¸
P p q
r G α O
¸
cptV q fVG pφV,tV q,
tV
fV αpxq .
r φ pξ q, the point The problem is to show that for each πV P Π V V tV pq tV πV , π pξ q is diagonal, in the sense that it is the image of a fixed point t P T pφq. We shall assume that this is false, and then derive a contradiction to (8.4.22). Our assumption implies that the left hand side of (8.4.22) does not vanish. There is then a corresponding contribution to the right hand side. That is, there is a nondiagonal index tV such that the coefficient cptV q is r φ ,ξ . We can then choose fV P HpGV q so nonzero. Suppose that πV P Π V
8.4. REFINEMENTS FOR EVEN ORTHOGONAL GROUPS
499
that the function fV,G on ΠφV ,ξ equals 1 at πV,tV , and vanishes on the complement of πV,tV . The left hand side of (8.4.22) then equals 1 if the index tV pq tV πV , π V pξ q equals tV , and vanishes otherwise. Since the right hand side of (8.4.22) equals the nonzero coefficient cptV q, we deduce that tV pq equals tV . Since πV is arbitrary, we conclude that tV tV pq is independent of πV . Armed with this property, we obtain the contradiction from r pGq-torsors the definition of the isomorphism of O t
ÝÑ
πv,t
ÝÑ
φv,t ,
t P T pφ q, v
P V,
πv
P Πr φ , v
as in the closing arguments from the proof of Lemma 8.4.3. That is, we show that tV is in fact diagonal, first in case φ is simple, and then in general by an application of the definition of T pφq to the simple components of φ. We have established the necessary contradiction. It follows that the r (OpGq-orbits of) indices tV pq and t in any summand on the left hand side of (8.4.22) are equal. The sum itself therefore vanishes. But the left hand G pf q of (8.4.21), if f V equals side of (8.4.22) equals the left hand side 0 rdisc,φ the chosen function f V pξ q. Letting ξ
P Spφ vary, we conclude that G 0 G pf q 0, (8.4.23) rdisc,φ pf q 0 Sdisc,φ r 1 pGv q, and any function for any finite set of valuations v P V with φv P Φ unit V V r pG V q. (8.4.24) f fV f , fV P HpGV q, f P H It remains to establish (8.4.23) for ± any function f P HpGq. Assuming fv is decomposable, we fix a finite without loss of generality that f v
set S of valuations outside of which fv is unramified. We then take V to r 1 pGv q. For any v in be the subset of places v P S such that φv lies in Φ unit the complement of S, fv is the characteristic function of our fixed maximal r pGv q-symmetric subspace compact subgroup Kv of GpFv q, which lies in the O r pGv q of HpGv q. If v lies in the complement of V in S, φv lies in Φunit pG r v q, H r pGv q-symmetric. One sees readily from the discussion above and is itself O G pf q of (8.4.22) remains unchanged if fv P HpGv q is that the left hand 0 rdisc,φ replaced by the symmetric function 21 f v . It then follows from (8.4.22), and the case of (8.4.23) we have established for functions f of the form (8.4.24), that (8.4.23) does indeed hold for any function f P HpGq. According to our definitions, the required global formulas (8.4.5) and (8.4.6) are just the two vanishing assertions of (8.4.23). Having established these assertions, we therefore completed the proof of Theorem 8.4.2. We have completed our study of local and global L-packets for the even orthogonal group G. In particular, we have obtained a refined local endoscopic classification of admissible representations, and a refined global endoscopic classification of the representations in the discrete spectrum of Ramanujan type. We have not tried to treat the packets attached to nongeneric parameters ψ. The global methods of this section undoubtedly apply
500
8. THE GLOBAL CLASSIFICATION
to such ψ. However, they would have to be combined with a finer understanding of the constituents of their packets. It would be interesting to apply the p-adic results of Moeglin to this question. As we noted in §8.3, the refined endoscopic classification we have established for p-adic F does not quite imply a refined Langlands correspondence. The impediment is the internal pZ{2Zq-symmetry implicit in the lack of a canonical bijection between two sets of order 2. In principle, there will be r 1 pGq. The one copy of the group pZ{2Zq for each local parameter φ P Φ bdd obstruction is actually a little more modest. Suppose that for each even N , and each even orthogonal group G P Ersim pN q over the p-adic field F , the pZ{2Zq-symmetry for any simple parameter φ P Φr simpGq has been “broken”. r pGq-torsors Φpφq and T pφq preIn other words, the bijection between the O dicted by the local Langlands conjecture has been defined. The same then r 1 pGq, which are atholds for any of the more general parameters φ P Φ bdd tached to any N and G. This is a direct consequence of the definitions. Notice that the question here is purely local. Our makeshift global group Lψ is too coarse at this point even to consider a global analogue of the problem.
8.5. An approximation of the Langlands group We would like to be able to streamline the definitions of §1.4. The goal would be to replace the complex groups Lψ in terms of which we formulated the global theorems by a locally compact group that is independent of ψ. We shall conclude Chapter 8 with a step in this direction. We shall introduce a group that characterizes most of the global parameters we have studied. The new group will be closer in spirit to the hypothetical Langlands group LF . As an approximation it is still pretty crude. It serves as a substitute for LF only so far as to describe the global endoscopic problems we have now solved. In fact, the group treats just a part of the problem, in that it pertains only to global parameters we shall call regular. Its missing complement depends on the solution of two supplementary endoscopic problems. There is no point in attempting to formulate it here, even though these problems are undoubtedly easier than the ones we solved. Our aim for now is merely to give a partial description, which illustrates what we can hope is a general construction in the simplest of situations. There are two steps. The first will be to introduce a group that characterizes automorphic representations of general linear groups. We will then describe a subgroup that accounts also for automorphic representations of orthogonal and symplectic groups. For any N ¥ 1, we have the basic set Csim pN q, defined in §1.3 and discussed further in §8.3. We recall that it consists of simple families c tcv : v
R Su
8.5. AN APPROXIMATION OF THE LANGLANDS GROUP
501
of semisimple conjugacy classes in GLpN, Cq, taken up to the equivalence relation c1 c if c1v cv for almost all v. As usual, S ValpF q represents a finite set of valuations of F that contains the set S8 of archimedean places. The special case ∆ ∆F C p1q Csim p1q is an abelian group under pointwise multiplication. It has an action δ: c
ÝÑ
δc,
δ
P ∆, c P CsimpN q,
on Csim pN q, given by pointwise scalar multiplication on families of pN N qmatrices. The complex determinant GLpN, Cq
ÝÑ
GLp1, Cq,
which is of course dual to the central embedding GLp1q
ÝÑ
GLpN q,
determines a second operation. Its pointwise application gives a mapping εp : Csim pN q
such that
ÝÑ
∆
εppδcq δ N εppcq. We shall use these two operations to define an endoscopic Langlands group that treats “most” automorphic representations of general linear groups. We first recall that the two operations have natural analogues for automorphic representations. We know that there is a canonical bijection π
ÝÑ cpπq,
π
P AcusppN q,
from the set of unitary cuspidal automorphic representations of GLpN q onto Csim pN q. This is the theorem of strong multiplicity one, which can be regarded as the specialization of Theorem 1.3.2 to Acusp pN q, and is the basis for our definition of the set Csim pN q. The abelian group X
XF Acuspp1q
of id`ele class characters acts in the usual way χ: π
ÝÑ
χπ
π b pχ detq,
χ P X, π
P AcusppN q,
on Acusp pN q. The central character then gives a second operation, a mapping ε : Acusp pN q ÝÑ X such that εpχπ q χN εpπ q. These are the automorphic analogues of the two operations for Csim pN q. We shall write c ÝÑ πc , c P Csim pN q, for the inverse of the bijection π Ñ cpπ q, and also δ
ÝÑ
χδ ,
δ
P ∆,
502
8. THE GLOBAL CLASSIFICATION
for its specialization to the case N πδc
χδ πc,
and
1. We then have δ P ∆, c P Csim pN q,
εp π c q χ c , c P Csim pN q, for the id`ele class character χc χεppcq . For any c P Csim pN q, we have isomorphic subgroups
tδ P ∆ :
∆c
δc cu
and
Xc tχ P X : χ πc πc u tχδ : δ P ∆c u of ∆ and X. They consist of elements of order dividing N , and are expected to be cyclic. We shall say that c and πc are regular if these two subgroups are trivial, which is to say that the associated representations πc b pχ detq,
χ P X,
are mutually inequivalent. We then write Csim,reg pN q for the set of regular elements in Csim pN q. The complement of Csim,reg pN q in Csim pN q should then be sparse in Csim pN q. We shall introduce a locally compact variant Lc of the L-group of GLpN q for every c P Csim,reg pN q. The unitary group U pN q is a compact real form of the complex dual group GLpN, Cq. For any c P Csim,reg pN q, we define an extension (8.5.1) of WF
ÝÑ WF ÝÑ 1 by the special unitary group Kc SU pN q by setting ( Lc g w P U pN q WF : detpg q χc pwq . 1
ÝÑ
Kc
ÝÑ
Lc
We have identified the central character χc of πc here with a character on the Weil group. The group Lc is a locally compact extension of WF by the compact, simply connected group SU pN q, which need not be split. By itself, Lc depends only on χc . It has the property that for any unitary, N -dimensional representation of WF with determinant equal to χc , the corresponding global Langlands parameter factors through the image of the L-embedding Lc
ÝÑ
L
GLpN q
GLpN, Cq WF .
We shall make Lc more rigid by attaching the local Langlands parameters associated to c. The group SU pN q is of course a compact real form of the complex special linear group SLpN, Cq. To account for the possible failure of the generalized Ramanujan conjecture for GLpN q, we will need to work with the complexification Lc,C of Lc . This is the extension (8.5.2) of WF
ÝÑ Kc,C ÝÑ Lc,C ÝÑ by Kc,C SLpN, Cq defined by Lc,C g w P GLpN, Cq WF : 1
WF
ÝÑ
1 (
detpg q χc pwq .
8.5. AN APPROXIMATION OF THE LANGLANDS GROUP
503
It has the property that any L-homomorphism φ : Lc
ÝÑ
L
G,
for a connected reductive group G over F , extends analytically to Lc,C . Our interest in Lc,C is in the local Langlands parameters LFv
ÝÑ
L
GLpN qv
GLpN, Cq WF
v
attached to the local components πc,v of the cuspidal automorphic representation πc of c. They provide the left hand vertical arrows in the diagrams LFv ÝÝÝÝÑ WFv (8.5.3)
,
Lc,C
ÝÝÝÝÑ
v
P ValpF q,
WF
and are defined up to conjugacy by the subgroup Kc,C of Lc,C . We would like to treat Lc as an object that depends only on the ∆-orbit of c. Since ∆ acts freely on Csim,reg pN q, any point c1 in the orbit of c equals δc, for a unique element δ P ∆. The mapping gw
ÝÑ
gχδ pwq w,
gw
P Lc,
is then a canonical L-isomorphism from Lc to Lc1 , which is compatible with the associated diagrams (8.5.3). Before we formalize the last property in our definitions, we should build r pN q of outer automorphisms of GLpN q. This is the group of in the group O order 2 generated by our standard automorphism θrpN q. Its canonical actions r pN q and X O r p N q. on ∆ and X give isomorphic semidirect products ∆ O 1 r pN q acts in the obvious way on Csim pN q. If c is any The first group ∆ O point in the corresponding orbit of c, we can still choose a canonical element r pN q such that c1 αc. We do need to be a little careful if the α in ∆ O orbit of c is self-dual, in the sense that the family c_ θrpN qpcq equals for r pN q then has order 2. In some δ P ∆, since the stability group of c in ∆ O this case, we simply take α to be the unique element in the normal subgroup r pN q. The second group X O r pN q acts on the L-group of GLpN q ∆ of ∆ O according to the automorphisms L χθ
L θ χ , 1
χ P X, θ
P OrpN q,
defined at the beginning of §3.2. In other words,
pχ θq : g w ÝÑ θpgq χpwq w, g P GLpN, Cq, w P WF . In particular, if c1 equals pδ θqc, the restriction of the automorphism defined r pN q by pχδ θq maps Lc to Lc1 . It follows that for any point c1 in the ∆ O orbit of c, we have a canonical L-isomorphism from Lc to Lc1 , which is compatible with the associated diagrams (8.5.3). We can therefore treat c as an element in the set Csim,reg pN q of r ∆ OpN q -orbits in Csim,reg pN q. With this interpretation of c, we can still
504
8. THE GLOBAL CLASSIFICATION
form a canonical extension (8.5.1), with complexification (8.5.2) and local diagrams (8.5.3). It comes with a canonical isomorphism onto the associated concrete object we have attached to any c1 in the class c, and is defined as a formal inverse limit over the points c1 , as in Kottwitz’s definitions at the beginning of [K3]. We are now working with the family
Csim,reg
of orbits of the group
º
Csim,reg pN q
¥
N 1
à
r pN q ∆ O
¥
N 1
in the subset Csim,reg
Csim
º
Csim,reg pN q
¥
N 1
of
º
¥
N 1
Csim pN q .
We amalgamate the groups we have attached to elements of the family into a fibre product LF,reg
(8.5.4)
¹
pLc ÝÑ WF q
cPC
sim,reg
over WF . We thus obtain a locally compact extension (8.5.5)
1
ÝÑ
KF,reg
ÝÑ
LF,reg
ÝÑ
WF
ÝÑ
1
of WF by a product KF,reg of compact, simply connected groups. The extension comes with a “complexification” 1
ÝÑ
KF,reg,C
ÝÑ
LF,reg,C
ÝÑ
WF
ÝÑ
1,
where KF,reg,C is an (infinite) topological product of complex special linear groups, and an associated family of conjugacy classes of local embeddings LFv ÝÝÝÝÑ WFv LF,reg,C
Ñ
ã
Ñ
ã
(8.5.6)
ÝÝÝÝÑ
,
v
P ValpF q.
WF
Consider a (continuous, semisimple) representation of L
F,reg .
φ : LF,reg
ÝÑ
GLpN, Cq
In case φ is irreducible, we shall call it standard if it factors to an irreducible, unitary representation of a quotient Lc of LF,reg , whose restriction to the subgroup Kc SU pN q is standard, in the extended sense that it is equivalent to either the standard embedding of SU pN q into GLpN, Cq or its dual. An irreducible standard representation thus amounts to a tensor product of a standard representation of SU pN q with a unitary character of
8.5. AN APPROXIMATION OF THE LANGLANDS GROUP
505
WF . In particular, it is trivial on the commutator subgroup WFc of WF . In the general case, we shall say that φ is standard if it is a (finite) direct sum of irreducible standard representations. We then write Φreg,bdd pN q for the set of standard representations of LF , and Ψreg pN q for the set of equivalence classes of unitary representations ψ : LF,reg SU p2q
ÝÑ
GLpN, Cq
whose restriction to LF,reg belongs to Φreg,bdd pN q. We have constructed a formal analogue LF,reg of the global Langlands group LF . It governs the basic representation theory of GLpN q, which is to say, the theory encompassed by the three theorems stated in §1.3. Of course, we have really to preface this statement with the adjective regular. Let Areg pN q be the subset of representations in ApN q, the set of automorphic representations of GLpN q introduced in §1.3 that occur in the spectral decomposition, whose cuspidal components are regular. It then follows from the definitions, together with the three theorems, that there is a canonical bijection ψ ÝÑ πψ , ψ P Ψreg pN q,
from Ψreg pN q onto Areg pN q such that πψ,v
πψ ,
v
v
P ValpF q.
Here, ψv P Ψv pN q is the local parameter attached to the complexification of ψ by (8.5.6), while πψ,v is obviously the local component of the automorphic representation πψ . There is no need to be concerned that the “complexification” LF,reg,C of LF,reg is not locally compact. Its only role is to extend the domain of parameters φ P Φreg,bdd pN q. Any such φ factors to a representation of an , which in turn extension of WF by a finite dimensional quotient of KF,reg extends analytically to the extension of WF by the corresponding quotient of KF,reg,C . We recall again that the complexification LF,reg,C is needed only because we do not know the generalized Ramanujan conjecture for cuspidal automorphic representations of GLpN q. The group LF,reg is adapted to the theory of (standard) endoscopy for general linear groups, which amounts to the three theorems stated in §1.3, together with the theory of Eisenstein series for GLpN q. We shall now introduce a subgroup of LF,reg that is adapted to the more sophisticated theory of endoscopy for orthogonal and symplectic groups. Consider the subset º º ( Crsim pN q c P Csim pN q : c_ c Crsim
¥
¥
N 1
N 1
of self-dual families in Csim . For any c P Crsim , we write rc ∆
tδ P ∆ :
δc P Crsim u.
506
8. THE GLOBAL CLASSIFICATION
Since
pδcq_ δ_c_ δ1c,
we see that
rc ∆
tδ P ∆ :
δ2
P ∆c u .
We shall again restrict our attention to the corresponding subset Crsim,reg
º
¥
N 1
Crsim,reg pN q
º
¥
N 1
Crsim pN q X Csim,reg pN q
of regular elements. For any c in this subset, the associated subset r ∆ rc ∆
tδ P ∆ :
δ2
1u
of ∆ is the subgroup of elements of order dividing 2, which is of course independent of c. We form the corresponding family
Crsim,reg
º
Crsim,reg pN q
¥
N 1
r of ∆-orbits in Crsim,reg . There is then an injection
Crsim,reg
ÝÑ Csim,reg , r pN q -orbits in C ∆ O sim,reg that have represen-
whose image is the set of tatives in Crsim,reg . It follows from Theorem 1.4.1 that Crsim,reg pN q
º
P
p q
G Ersim N
Crsim,reg pGq ,
where Crsim,reg pGq is the subset of families c P Crsim,reg pN q such that c cpπ q, r on Crsim,reg pN q for some π in the set A2 pGq. If N is even, the free action of ∆ stabilizes the associated subsets Crsim,reg pGq, as one sees from the criterion r of Theorem 1.5.3. If N is odd, it is the opposite that holds. In this case ∆ r acts simply transitively on the data G P Esim pN q. Recall however that these different endoscopic data all have the same endoscopic group G SppN 1q, p SOpN, Cq. with dual group G Suppose that c belongs to the subset Crsim,reg pGq of Crsim,reg pN q attached r c for the standard maximal compact to G P Ersim pN q, for some N . We write K p Thus, K r c equals the special unitary quaternion subgroup subgroup of G. p SU pn, Hq of G Spp2n, Cq if G SOp2n 1q, the compact orthogonal p SOp2n 1, Cq if G Spp2nq, and the compact subgroup SOp2n 1q of G p SOp2n, Cq if G SOp2nq. In the first orthogonal subgroup SOp2nq of G r c , while in the third case, it acts according two cases, WF acts trivially on K to the mapping of WF to the group of outer automorphisms defined by χc . We define an extension (8.5.7)
1
ÝÑ
rc K
ÝÑ
rc L
ÝÑ
WF
ÝÑ
1
8.5. AN APPROXIMATION OF THE LANGLANDS GROUP
r c simply by setting of WF by K rc L
Kr c WF .
r c as a compact real form of the group We can think of L words, its complexification is the extension
(8.5.8)
1
ÝÑ
ÝÑ
r c,C K
507
r c,C L
ÝÑ
WF
ÝÑ
L G.
In other
1
p c,C G p defined by the L-group L r c,C L G. of WF by the complex group K Our notation here is obviously designed to be compatible with that of the groups Lc and Lc,C above. In particular, we have local diagrams
ÝÝÝÝÑ
rF L v
(8.5.9)
r c,C L
ÝÝÝÝÑ
WF v
,
v
P ValpF q,
WF
in which the left hand vertical arrows are defined up to conjugacy by the r c,C . This follows from Theorem 1.4.2, but with one r c,C of L subgroup K r c,C G p SOp2n 1, Cq, we must take c to caveat. In the case that K r be the element in its ∆-orbit that χc 1 in order that the images of the r c,C . mappings of the groups LFv actually be contained in L r c as an Following our convention for the group Lc above, we can treat L r object that depends only on the image of c in the set Crsim,reg pGq of ∆-orbits p SOp2n 1, Cq we have just flagged, this is in Crsim,reg pGq. In the case G part of the definition. In either of the other two cases, suppose that c1 δc r The mapping is the element in the orbit of c attached to δ P ∆. gw
ÝÑ
gχδ pwq w,
gw
P Lrc,
r c to L r c1 , which is compatible with is then a canonical L-ismorphism from L r the associated diagrams (8.5.9). We can therefore regard c as a ∆-orbit in all three cases. With this interpretation, we still have a canonical extension (8.5.7), with complexification (8.5.8) and local diagrams (8.5.9). It again comes with a canonical isomorphism onto the concrete object we have attached to any c1 in the class c. We can now define our subgroup of LF,reg . We have assigned a group
r c with complexification L r c,C to each c P Cr r L sim,reg . Let us identify Csim,reg with its injective image in Csim,reg . For any c in its complement, we just r c Lc and L r c,C Lc,C . With these conventions, we amalgamate the set L groups as a fibre product r L F,reg
(8.5.10)
¹
c Csim,reg
P
pLrc ÝÑ
WF q
over F . We thus obtain a locally compact extension (8.5.11)
1
ÝÑ
r K F,reg
ÝÑ
r L F,reg
ÝÑ
WF
ÝÑ
1
508
8. THE GLOBAL CLASSIFICATION
r of WF by a product K F,reg of compact, connected groups. It comes with a “complexification”
1
ÝÑ
r K F,reg,C
ÝÑ
r L F,reg,C
ÝÑ
WF
ÝÑ
1,
r where K F,reg,C is an (infinite) topological product of complex dual groups, and an associated family of conjugacy classes of embeddings LFv ÝÝÝÝÑ WFv r L F,reg,C
Ñ
ã
Ñ
ã
(8.5.12)
ÝÝÝÝÑ
,
v
P ValpF q.
WF
r To embed L F,reg as a subgroup of LF,reg , consider a class c
P Crsim,reg .
By choosing suitable representatives of c in Crsim,reg and Csim,reg , we obtain r c into Lc from the standard embedding of K rc a canonical L-embedding of L r into Kc . If c belongs to the complement of Csim,reg in Csim,reg , the groups r c,reg and Lc,reg are equal by definition. It follows from the definitions (8.5.4) L and (8.5.10) that there is a canonical L-embedding r L F,reg
(8.5.13)
Ñ
ã
LF,reg ,
which is compatible with the local diagrams (8.5.6) and (8.5.12). Consider an L-homomorphism r φr : L F,reg
ÝÑ
L
G,
G P Ersim pN q.
We shall say that φr is standard if it can be embedded into a commutative diagram r L F,reg
r
φ ÝÝÝÝ Ñ
LG
,
LF,reg ÝÝÝÝÑ GLpN, Cq where the left hand vertical arrow is the L-embedding we have just constructed, the right hand vertical arrow is the representation determined by G as a twisted endoscopic datum, and φ is a standard representation r of LF,reg defined as above. We write Φ reg,bdd pGq for the set of standard φ
r taken up to the equivalence relation defined by conL-homomorphisms φ, L r N pGq. We define Ψ r pGq to be the set of jugation of G by the group Aut reg L-homomorphisms r ψr : L F,reg SU p2q
ÝÑ
L
G
r r whose restriction to L F,reg belongs to Φreg,bdd pN q, taken up to the same equivalence relation. r The point of course is that one could use the single group L F,reg in r reg pGq of place of the family of ad hoc groups Lψ attached to the subset Ψ
8.5. AN APPROXIMATION OF THE LANGLANDS GROUP
509
r pGq. A parameter (1.4.1) in Ψ r p Gq “regular” parameters in the earlier set Ψ would again be called regular if the cuspidal automorphic representations µi P Arcusp pmi q in its simple constituents ψi µi b νi are regular. One sees r pGq onto Ψ r reg pGq that from the definitions that there is a bijection from Ψ reg is compatible with the local diagrams (8.5.12) and (1.4.14), and such that
ψ pLF,reg,C q Lψ ,
ψ
P Ψr reg pGq,
for suitable representatives of the corresponding parameters within their r N pGq-orbits. Moreover, the bijection preserves the global centralizers Out Sψ , their localizations Sψv , and the associated homomorphisms Sψ
ÝÑ
¹ v
Sψv .
Therefore our theorems, insofar as they apply to regular parameters, can r indeed be formulated in terms of the group L F,reg .
r for the locally compact exWe have reserved the symbols LF and L F r tensions of LF,reg and L that could be expected if one included the F,reg complementary, “nonregular” representations in the construction. If c lies in the complement of Csim,reg pN q in Csim pN q, ∆c should be a cyclic group, whose order d then divides N . In this case, c would be obtained by automorphic induction from the subgroup WK of WF attached to Xc by class field theory. One would define Lc as an extension (8.5.1) of WF by the smaller simply connected group
Kc
SU pkq SU pkq, loooooooooooomoooooooooooon
dk
N,
d
on whose factors WF acts by permutation through its cyclic quotient WF {WK . For any point c1 δc in the ∆-orbit of c, the L-isomorphism from Lc to Lc1 would then depend only on the image of δ in ∆{∆c , and would therefore be canonical. This would lead to a canonical group Lc that depends only on pN q of ∆ OpN q-orbits in Csim pN q. Similar the image of c in the set Csim properties could be expected of any point c in the complement of Crsim,reg pN q r c that depends only on in Crsim pN q. The would lead to a canonical group L r r r the image of c in the set Csim pN q of ∆-orbits in Csim pN q. We would then r as fibre products (8.5.4) and be able to define the larger groups LF and L F (8.5.10), but taken over the larger set
Csim
º N
pN q. Csim
r should not be The detailed construction of the larger groups LF and L F difficult, given the global results of §8.1 and §8.2. However, it would take us beyond the methods of this volume. We add here only the remark that the two groups will come with an L-embedding r L F
Ñ
ã
LF ,
510
8. THE GLOBAL CLASSIFICATION
r pGq. These would allow and two larger families of parameters Ψ pN q and Ψ us to formulate our global results entirely in terms of the natural approxir of the Langlands group LF . We hope to return to these mations LF and L F matters at some point in the future, in the general context of quasiclassical groups.
CHAPTER 9
Inner Forms 9.1. Inner twists This last chapter might be more elementary than some of the others. My original intention was to extend the classification to general orthogonal and symplectic groups. However, it became clear that the complete proofs would take considerably more than one chapter. For there are interesting new phenomena that arise from the inner forms of quasisplit orthogonal and symplectic groups. These all require further discussion, and sometimes more complex proofs. Our goal for the last chapter has therefore to be more modest. We will be content to look at nonquasisplit inner forms from the perspective of earlier chapters, and then simply to state analogues of the results for quasisplit groups. In this sense, the last chapter is parallel to the first. The difference is that the assertions we make here will have to be proved elsewhere [A28]. The first three sections of the chapter will each be built around a distinctive property of inner forms that bears on their representation theory. The first section is devoted to a review of the classification of inner forms. The new phenomenon here is the possible failure of an outer automorphism to have a representative (in its outer class) that is defined over the ground field. Among other things, this is responsible for the failure of the Hasse principle for groups of type Dn . In the second section, we will discuss the nonabelian groups that must be used in place of the abelian 2-groups Sψ attached to local parameters ψ for quasisplit groups. They lead ultimately to higher global multiplicities for automorphic representations. The third section concerns the noncanonical nature of the local transfer factors in [LS1]. We shall describe how to rectify the problem, according to ideas of Kaletha and Kottwitz. We will then be able to state the local results in §9.4, and the global results in §9.5. A high point in the theory of algebraic groups is the classification of reductive groups over F (our local or global field of characteristic 0). We shall review this part of the theory, as it applies to orthogonal and symplectic groups. We described the quasisplit orthogonal and symplectic groups G in §1.2. (From this point on, we will be writing G for the groups that were earlier denoted by G, since they now assume only a secondary role.) What remains to discuss are the inner forms of these groups. This part of the
511
512
9. INNER FORMS
classification is best formulated in terms of the finer classification of inner twists of G . An inner twist of the quasisplit group G P Ersim pN q over our field F is a pair pG, ψ q, where G is a connected reductive group over F , and ψ is an isomorphism from G to G such that for every element σ in the Galois group ΓF , the automorphism αpσ q ψσ pψ q1
of G is inner. An isomorphism from pG, ψ q to a second inner twist pG1 , ψ1 q is an isomorphism θ1 : G Ñ G1 over F such that the automorphism ψ1 θ1 ψ 1 of G is inner. (See [KS, p. 141].) The correspondence that sends pG, ψ q to the 1-cocycle αpσ q P Z 1 pF, Gad q then descends to a bijection between the set of isomorphism classes of inner twists of G and the Galois cohomology set H 1 pF, Gad q. An inner form of G can be defined simply as a connected reductive group over F that is the first component of some inner twist of G . There is consequently a surjective mapping from the set of isomorphism classes of inner twists onto the set of isomorphism classes of inner forms. The fibres of the mapping correspond to the orbits in H 1 pF, Gad q of the group OutpG q of outer automorphisms of G , under an action we can denote by θ : α pσ q
ÝÑ
θ α p σ qσ pθ q1 ,
θ
P OutpGq, αpσq P Z 1pF, Gadq,
or equivalently, by θ˜ : αpσ q
ÝÑ
θ˜ αpσ q
θ˜ αpσq pθ˜q1,
where θ˜ is the F -automorphism of G in the F -outer class θ that preserves a given F -splitting. We will return to these matters in the discussion of Lemma 9.1.1 below, once we have described the classification. We note in passing that our remarks here remain valid if G is replaced by any quasisplit group over F . In the case at hand, the group OutpG q is trivial unless G is of the form SOp2nq, in which case it equals Z{2Z. Thus, although the classification of inner twists of G is finer than that of inner forms, its fibres are quite transparent. Accordingly, we will often let G stand for a given inner twist pG, ψ q of G (in the spirit of our earlier convention for endoscopy), in addition to the orthogonal or symplectic group that is its first component. There are really three classifications of inner twists. One is by Galois cohomology, which can be based on the elegant formulation of Kottwitz [K5, §1–2], with the results in [Se, III.35–III.37] taken as an archimedean supplement. This is the best suited to our purposes. A second classification is by the Tits indices in [Ti]. It is actually a classification of inner forms G (rather than inner twists), which comes with a transparent description of the Levi subgroups M of G. The third is the explicit classification in terms of bilinear forms, about which we will say little.
9.1. INNER TWISTS
513
p q Given a group G P Ersim pN q over F , we may as well write Zpsc Z pG sc ) for the centre of the simply connected dual group Gp . (rather than Zpsc sc Then we have $ Z 2Z, ' ' ' &
(9.1.1)
{ Z{2Z, Zpsc ' pZ{2Zq pZ{2Zq, ' ' % Z{4Z,
if if if if
G G G G
SOp2n 1q, Spp2nq, SOp2nq, with n ¡ 1 even, SOp2nq, with n ¡ 1 odd.
Γ of elements in Z psc fixed by the Galois group We also have the subgroup Zpsc Γ Z psc . Γ ΓF . If G is split, Γ of course acts trivially on Zpsc , and Zpsc If G is not split, we recall that G is of the form SOp2nq, and Γ acts on Zpsc through the quadratic extension E {F . The nontrivial element in the corresponding quotient of Γ acts by permutation of the two factors of Zpsc if Γ n is even, and by the automorphism z Ñ z 1 if n is odd. It follows that Zpsc equals Z{2Z if G is not split, a property we can recall from earlier chapters. Suppose first that F is local. Then there is a canonical morphism of pointed sets
Kad
(9.1.2)
KG
ad
: H 1 pF, Gad q
ÝÑ ΠpZpscΓ q,
Γ q is the group of (linear) characters on Z pΓ . If F is p-adic, the where ΠpZpsc sc map is bijective. If F is archimedean, we have
q im H 1 pF, G q kerpKad sc
and
q ker ΠpZpΓ q impKad sc
ÝÑ
H 1 pF, Gad q
ÝÑ ΠpZpscq ,
where the kernel on the right is induced from the norm homomorphism from Γ given by the action of Γ in Z psc . In particular, K is neither Zpsc to Zpsc F ad injective nor surjective if F R. The set H 1 pF, Gad q is of course trivial if F C. (See the general local result [K5, Theorem 1.2], in which Gad is Γq replaced by an arbitrary connected reductive group G over F , and ΠpZpsc is replaced by the finite character group p q Γ { Z pG p qΓ ApG q Π Z pG
0
.q
Suppose next that F is global. Then the canonical mapping H 1 pF, Gad q
à
ÝÑ
v
H 1 pFv , Gad q
is injective. Its image is the kernel of the composition à v
H 1 pFv , Gad q
ÝÑ
à v
Γv ΠpZpsc q
ÝÑ ΠpZpscΓ q.
(See the general global results [K5, Theorem 2.2 and Proposition 2.6]. If Gad is replaced by a general group G, the mapping need not be injective, though it does have an easily described kernel [K3, §4]. In addition, the
514
9. INNER FORMS
Γv q and ΠpZ pΓ q in the composition above have to be replaced by groups ΠpZpsc sc ApGv q and ApGq respectively.) We are generally letting G stand for an isomorphism class of inner twists pG, ψq over F (as well as the associated connected reductive group). If F is local, we can write
(9.1.3)
ζpG pz q xz, Gy,
z
P ZpscΓ ,
Γ obtained from K and the image of G in H 1 pF, G q. for the character on Zpsc ad ad Γ to the group Z pΓv attached If F is global, we have a mapping z Ñ zv from Zpsc sc to any completion Fv of F . The isomorphism classes of inner twists over F are then bijective with the set of families
G tGv : v
P valpF qu
of (isomorphism classes of) local inner twists such that Gv all v, and such that the character (9.1.4)
xz, Gy
¹
xzv , Gv y,
z
Gv for almost
P ZpscΓ ,
v Γ is trivial. The analogy with the deeper pairings of Theorems 1.5.1 on Zpsc and 1.5.2 is clear. This classification of inner twists by Galois cohomology is easy to work with. However, the description we have just given is incomplete. It does not explicitly characterize the inner twists over the local field F R. A related gap is the lack of an immediate description of the Levi subgroup of G over F , especially in case F equals R or is global. when F R. If G equals It is easy to describe the image of Kad SOp2n 1q or Spp2nq, G is split. The norm mapping from the group Γ Z psc Z{2Z to itself is trivial, and K is surjective. The third case Zpsc ad that G SOp2nq depends on whether n is even or odd. If n is even, one is surjective if G is split, and trivial otherwise. If n is checks that Kad odd, Kad is surjective if G is not split, and maps onto the subgroup (of Γ q of elements of order 2 if G is split. In particular, we index 2) in ΠpZpsc is surjective if and only if GpRq has square observe in all cases that Kad integrable representations. The problem still is that our description does not account for the fibres of the mapping. There are two ways we could refine the discussion so as to fill the gap. One modification would be to let G be what we called a K-group in [A12, §2] (for F local) and [A14, §4] (for F global), following ideas of Kottwitz and Vogan. This would have the effect of compressing the fibres into single objects. It would also account for the Levi subgroups of Kad of G [A12, Lemma 2.1], [A14, Lemma 4.1] (by redefining the problem as much as actually solving it). However, there is no point letting G stand here for anything other than the connected reductive group it has been up until now. The other refinement would be to include Serre’s formulation of the archimedean cohomology sets H 1 pR, Gad q [Se, III.35–III.37]. This would
9.1. INNER TWISTS
515
also lead to a description of the Levi subgroups of the connected group G. However, we will not give the details, since they would take us too far afield. We shall instead simply list the relevant Tits indices from [Ti]. More precisely, we list the Tits indices for our groups of type Bn , Cn and Dn , specialized to the local or global field F . They are taken directly from Table II of [Ti], but adjusted slightly so that they represent inner twists G of the groups G P Ersim pN q rather than inner forms. (The distinction is only relevant to indices of type 1 Dn , below, specifically the subdiagrams (9.1.5) and (9.1.6).) Together with a few supplementary details we include without proof, the indices allow one to interpolate the missing information. We refer the reader to the explanations in [Ti, p. 39,54]. Each index is built on the Coxeter-Dynkin diagram of the group G . It comes with two supplementary integers, in addition to the number n of vertices in the diagram: the F -rank r P t0, 1, . . . , nu of G and a degree d P t1, 2u. Vertices that lie in the same Galois orbit (which occur only for the quasisplit, nonsplit groups G of type 2 Dn below) are written side by side. Vertex orbits that correspond to isotropic simple roots of G are then circled. At the suggestion of Bill Casselman, we have also included broken loops to indicate vertex orbits of order two that are anisotropic (and that are hence not enclosed in a solid loop). The minimal Levi subgroup M0 of G (or rather its derived group, which is called the anisotropic kernel in [Ti]) is represented by the diagram obtained by removing the isotropic vertex orbits. In particular, the split rank r of G over F equals the number of circled vertex orbits. The standard Levi subgroups M of G are in bijection with subsets of isotropic vertex orbits. They are represented by the diagrams obtained by deleting these subsets. If F is local, the inner twist G is determined by its index (with the interpretations below for the subdiagrams (9.1.5) and (9.1.6)). In this case, Γ in terms of we state without proof a description of the character ζpG on Zpsc its index. If F is global, the inner twist G is not determined by the index. It is characterized instead by its completions pGv q, which is to say their local indices, subject to the condition (9.1.4). In this case, the set of isotropic vertex orbits in the global index can be seen to be the intersection over v of the sets of local isotropic vertex orbits (suitably interpreted in case G is of the type 2 Dn below, and some Gv of type 1 Dn has diagram of the form (9.1.5)). We can thus construct a global index explicitly from the associated family of local indices. Type Bn
r
n−r
,
I thank Casselman for his composition of all of these diagrams.
d 1.
516
9. INNER FORMS
(The degree d 2 does not occur in this case.) Further conditions on r and d F R: none. F p-adic: n r P t0, 1u. F global: none. The group G G
SOp2n SOp2n
G r over F . p G
1q is split.
1, qr q, where qr is a (nondegenerate) quadratic form of index
Spp2n, Cq Gp.
The character ζpG (F local) ζpG
pεpscqnr ,
Γ where εpsc is the nontrivial character on the group Zpsc
Z{2Z Zpsc.
Type Cn ,
r=n
2r
n − 2r
d 1.
,
d 2.
Further conditions on r and d F R: n r 0 if d 1 (as in the diagram). No further conditions if d 2. F p-adic: n r 0 if d 1 (as in the diagram). n 2r P t0, 1u, if d 2. F global: n r 0 if d 1 (as in the diagram). No further conditions if d 2. The group G G
Spp2nq is split. G G Spp2nq, if d 1; G SU pn, hr q, where hr is a (nondegenerate) Hermitian form of index r over a quaternion algebra Q over F , if d 2. p G p SOp2n 1, Cq. G
9.1. INNER TWISTS
517
The character ζpG (F local) ζpG
pεpscqd1,
Γ where εpsc is again the nontrivial character on the group Zpsc
Z{2Z Zpsc.
Type 1 Dn
, r
d 1.
n−r
, n − 2r
2r
If d 2 and n 2r
d 2.
0, the right hand end is one of the two diagrams
(9.1.5(i))
or
,
if n ¡ 2, and one of the two diagrams (9.1.5(ii))
or
,
if n 2 (in which case these are of course the full diagrams). If d 2 and n 2r ¥ 3, we take the right hand end somewhat artificially to be one of the two formal copies of the diagram
(9.1.6)
518
9. INNER FORMS
which we identify with one of the two anisotropic, central simple algebras of degree 4 that are implicit in G as a group over F . In each of these cases, the associated two diagrams represent the two inner twists G in the fibre of an inner form. (Compare with the diagrams at the bottom of p. 56 in [Ti].) Further conditions on r and d F R: n r is even if d 1. n 2r 0 if d 2 (so in particular, n is even). F p-adic: n r P t0, 2u if d 1. n 2r P t0, 3u if d 2 (so n can be even or odd). F global: n r is even if d 1. n 2r is even or equal to 3 if d 2. The group G
G
SOp2nq is split. G SOp2n, qr q, where qr is a (nondegenerate) quadratic form of discriminant 1 and index r over F , if d 1; G SU pn, hr q, where hr is a non
degenerate anti-Hermitian form of discriminant 1 and index r over a quaternion algebra Q over F , if d 2.
p G
SOp2n, Cq Gp.
The character ζpG (F local)
#
ζpG
pnrq pεpscq 2 , 1
psc , χ
if d 1, if d 2,
where εpsc is the nontrivial character on the group #
Γ Zpsc
q pZ{2Zq, Zpsc ppZZ{{2Z 4Zq,
n even, n odd,
psc is one of the remaining two nonthat is fixed by the automorphism, and χ trivial characters. The latter is determined by the diagram (9.1.5) attached to G if n is even, and the anisotropic, central simple algebra of degree 6 attached to G and the diagram (9.1.6) if n is odd (and F is p-adic).
Type 2 Dn , r
d 1.
n−r
, 2r
n − 2r
d 2.
9.1. INNER TWISTS
If d 1 and n r
519
1, the right hand end is the diagram ,
and G equals G . If d 2 and n 2d 1, the right hand end is the diagram .
Further conditions on r and d F R: n r is odd if d 1. n 2r 1 if d 2 (so in particular, n is odd). F p-adic: n r 1 if d 1. n 2r P t1, 2u if d 2 (so n can be even or odd). F global: none. The group G
G SOp2n, ηE q is the nonsplit, quasisplit group attached to a quadratic extension E {F . G is as in 1 Dn , except that the forms qr and ? hr in question now have discrimimant α P F {pF q2 such that E F p αq. p SOp2n, Cq G ΓE {F .
Gp, with the nontrivial L-action of the Galois group
The character ζpG (F local)
pεpscqd1, Γ Z{2Z Z psc . where εpsc is the nontrivial character on Zpsc ζpG
To focus the discussion, we have agreed to consider one property in each of the first three sections that makes inner twists qualitatively different from quasisplit groups, and that complicates the classification of their representations. The question for this section is the possible nonexistence of rational automorphisms. Suppose that G is isomorphic to one of the quasisplit groups SOp2nq over F . We shall now write θr for the F -automorphism of rpN q in (1.2.5). This of course repreG that was denoted by θr Int w sents the nontrivial element in OutpG q. Given an inner twist pG, ψ q with associated 1-cocycle αpσ q, we can ask whether there is a corresponding F automorphism of G. In other words, is there an F -automorphism of G, r that transforms under ψ to an automorphism in which we now denote by θ, r the inner class of θ ?
520
9. INNER FORMS
Lemma 9.1.1. The following conditions on the objects pG, ψ q and θr attached to G P Ersim pN q, the given quasisplit group of type Dn over our local or global field F , are equivalent. (i) There is an F -automorphism θr of the group G in the inner class of the automorphism ψ 1 θr ψ. (ii) The inner twist pG, θr ψ q is isomorphic to pG, ψ q. (iii) The cocycles αpσ q and θr αpσ q
θr αpσq pθrq1, σ P ΓF , have the same image in H 1 pF, Gad q. (iv) There is an element γ P IntpG q such that θr αpσ q pγ q1 αpσ qσ pγ q, σ P ΓF .
Proof. The lemma is a direct consequence of the definitions. For example, we can regard ψ as an F -isomorphism from G to G if we equip G with the twisted Galois action σG
αpσqσ ψσpψq1σ,
σ
P ΓF .
The first condition (i) becomes the existence of an element γ such that σG pγ θr q γ θr .
P IntpGq
This amounts in turn to the existence of an F -automorphism θr ψ 1 γ θr ψ
(9.1.7)
of G such that the automorphism γ
pψθrqpθrψq1
of G is inner. In other words, θr is an isomorphism from pG, θr ψ q to pG, ψ q, the existence of which is the second condition (ii). Conditions (i) and (ii) are therefore equivalent. The conditions (iii) and (iv) are just alternate ways of stating (ii), so they are also equivalent to the first two conditions. Remarks. 1. What happens when the conditions of the lemma are not met? Given pG, ψ q and θr as in the lemma, let pG_ , ψ _ q be any inner twist of G such that α_ pσ q ψ _ σ pψ _ q1 def
θr αpσq ,
σ
P ΓF .
The isomorphism (9.1.8)
θr_
pψ_q1θrψ :
G
ÝÑ
G_
is then defined over F , since we have σ pθr_ q σ pψ _ q1 σ pθr qσ pψ q pψ _ q1 α_ pσ qθr αpσ q1 ψ
pψ_q1θrαpσqαpσq1ψ pψ_q1θrψ θr_,
9.1. INNER TWISTS
for any σ
521
P ΓF . For example, we could take pG_, ψ_q pG, θrψq, and θr_ I.
Alternatively, we could define
pG_, ψ_q pθrG, ψq,
and θr_
ψ1θrψ,
where θr G is the group G with the twisted Galois action for which the isomorphism ψ 1 θr ψ from G to θr G is defined over F . Suppose that the 1-cocycles αpσ q and α_ pσ q have distinct images in H 1 pF, Gad q. Then θr_ is not an isomorphism of the inner twists pG, ψ q and pG_, ψ_q, as we see directly from the fact that the automorphism θr
ψ_θr_ψ1
of G is not inner. Thus, θr_ is an isomorphism between G and G_ as groups over F , but not as inner twists. A second observation is that θr does not exist as an F -automorphism of G. For there is no canonical way to identify G and G_ over F other than by θr_ . These two related points are both important, obvious as they may be. 2. The lemma of course deals with the case that αpσ q has the same image in H 1 pF, Gad q as the 1-cocycle α_ pσ q θr αpσ q (condition (iii)). We can then choose a corresponding “intertwining operator” γ P IntpG q (condition (iv)), from which we define θr as a product (9.1.7). Consequently θr exists, as an isomorphism from the inner twist pG_ , ψ _ q pG, θr ψ q to pG, ψq (condition (iii)), and as a nontrivial F -automorphism of G (condition (i)). We return to the classification, as illustrated in the table of indices above. We are assuming for the rest of this section that G is of type Dn , so that θr exists as an automorphism of G of order 2. It acts by permutation on the indices of G , which are by definition the family of markings on its Coxeter-Dynkin diagram in the list above, by interchanging the two right hand vertices in the diagram. We see by inspection that a θr -orbit of indices for G has order 1 unless G is of type 1 Dn and the integer d of its index equals 2, in which case the orbit is of order 2. Suppose that F is local. The inner twist G is then completely determined by its index. Therefore θr transfers to an F -automorphism θr of G in this case unless G is of type 1 Dn and d 2, or equivalently, unless the index of G contains a subdiagram (9.1.5) or (9.1.6). This is to be regarded as the exceptional case. It occurs only when G is split (of type Dn ), and then only for one θr -orbit of indices if F is p-adic. If F R, it occurs for one orbit of indices if n is even, and none if n is odd. We see from these remarks that θr transfers to an F -automorphism of G (as either a group or an inner twist) if and only if (9.1.9)
θr ζpG
ζpG,
522
9. INNER FORMS
Γ Z psc attached to G (as an inner for the character ζpG on the group Zpsc twist). The exceptional case thus occurs when θr ζpG is distinct from ζpG . In this case, we can attach the complementary inner twist pG_ , ψ _ q to the given inner twist pG, ψ q, as in Remark 1 above. Then θr ζpG ζpG_ , and we have the F isomorphism θr_ from G to G_ as groups, but not as inner twists. Suppose that F is global. Then the inner twist G is determined by the indices of its localizations pGv q, subject to the condition (9.1.4) on the characters ζpGv . In this case, θr transfers to an F -automorphism of G (as a group as an inner twist) if and only if
θrv ζpGv
(9.1.10)
ζpG , v
for all v. This follows, for example, from the application of any of the conditions of Lemma 9.1.1 to G and to each Gv . The exceptional case occurs when θrv ζpGv is distinct from ζpGv , for some v. The complementary inner twist pG_ , ψ _ q over F is determined by the property that θrv ζpGv ζpG_ v for all v. We again have an F -isomorphism θr_ from G to G_ as groups, but not as inner twists. But in the global case, there are other inner twists pG, ψq over F that are related to pG, ψq. Suppose that pG, ψq is such that its localizations pGv q satisfy (9.1.4), and such that one of the identities ζpGv ζpGv or ζpGv θr ζpGv holds for any v, but such that neither of the identities holds for all v. Then there is an Fv -isomorphism between the groups Gv and Gv , but no F -isomorphism between the groups G and G . In other words, the groups G and G over F are locally isomorphic, but not globally isomorphic. In this sense, they violate the Hasse principle. However, the only inner twist pG , ψ q over F that is locally isomorphic to pG, ψq (at all places v) is pG, ψq itself. In other words, the Hasse principle, in its usual interpretation for the Galois cohomology set H 1 pF, G q, remains valid. Although it is perhaps redundant, let us illustrate these ideas with a simple example of type 1 D2 . The underlying quasisplit group G is split, and equals G Spp2q Spp2q{t1u, where the group t1u is embedded diagonally. Suppose first that F is local, and that the integer d equals 2. This is the case of the general classification that corresponds to the θr -orbit of indices (9.1.5(ii)). The two indices parametrize the two groups G1
Spp2q Q1{t1u
G2
Q1 Spp2q{t1u,
and
where Q1 pF q is the group of elements of norm 1 in a quaternion algebra over F . We assume implicitly that we have identified Q1 pF q with Spp2, F q.
9.1. INNER TWISTS
523
Then G1 and G2 are defined as inner twists by the isomorphisms ψ1 It follows that
ψ2 1 1.
pG2, ψ2q pG_1 , ψ1_q pθrG, ψq,
in the notation of Remark 1 above, while the F -isomorphism θr_ in (9.1.8) is equal to the permutation θr of the two factors. This is obviously the only way to identify G1 and G2 , since the identity isomorphism is not defined over F . Suppose next that F is global. If S is a finite, nonempty, even set of valuations of F , we write Q1 pS q for the group represented by the elements of norm 1 in a quaternion algebra over F that is ramified at S and unramified outside of S. We fix S, and take G1
Q1pS1q Q1pS11 q
G2
Q1pS2q Q1pS21 q,
and for two partitions
S1 > S11 S2 > S21 of S into nonempty, even subsets. For any v P S, the mapping # I, if v P pS1 X S2 q Y pS11 X S21 q, θrv_ r θ, if v P pS1 X S21 q Y pS11 X S2 q, is then an Fv -isomorphism from G1 to G2 . If v R S, we obviously have two S
Fv -isomorphisms from G1 to G2 , either the identity I or the permutation r Therefore G1 and G2 are locally isomorphic as groups over mapping θ. F . On the other hand they are not globally isomorphic unless S1 S2 or S1 S21 . They are of course neither locally or globally isomorphic as inner twists, except if S1 S2 . We thus have a clear illustration of the failure of the Hasse principle for inner forms of type 1 Dn , even though it remains valid for inner twists. This completes our discussion of inner twists. For the transfer of automorphisms, our special topic for the section, we will keep in mind the summary from Remark 1 following Lemma 9.1.1, especially the two points at the end. We have concentrated on the first of these, the difference between isomorphisms of groups and isomorphisms of inner twists, and its relation to the Hasse principle. The other is the difference between F -isomorphisms and F -automorphisms. Its main implication is that there are inner forms G r pGq. This means for which we cannot define the symmetric Hecke algebra H that the formulations of the theorems for quasisplit groups in §1.5 do not carry over as stated to the inner twist G.
524
9. INNER FORMS
9.2. Parameters and centralizers In this section we consider a second point of departure for general orthogonal and symplectic groups. If G G is quasisplit, our description of r pGq and centralizits representations has been in terms of parameters ψ P Ψ ers Sψ . If G is not quasisplit, we need to modify these objects. In particular, we need to consider a finite extension of the abelian quotient Sψ of Sψ that is often nonabelian. We shall look at the structure of these groups, and describe their irreducible characters. We fix G. which is to say the inner twist pG, ψ q of G P Ersim pN q over F that G represents. Following a convention from [LS1, (1.2)], we will usually use ψ to identify the L-groups of G and G . In case G is itself quasisplit, we might as well simply take G G and ψ 1. In general, it makes sense to suppress ψ from the notation whenever we can, since we will again be using this symbol to represent a local or global parameter. r pGq for the Having identified the L-groups of G and G , we write Ψ r subset of parameters ψ P ΨpG q that are locally G-relevant. For F local, this means that ψ has a representative with the usual property that if its image is contained in a parabolic subgroup L P of L G, then L P corresponds to a parabolic subgroup P of G that is defined over F . For F global, it means that any localization ψv of ψ is Gv relevant. In other words, ψ belongs to the r pGq of Ψ r pG q if and only if each localization ψv belongs to global subset Ψ r pGv q of Ψ r pG q. We remark in passing that this definition the local subset Ψ v is not entirely standard. The usual definition applies (at least for local F ) r pGq represents the set of Out r N pG q-orbits. to the finer set ΨpGq, in which Ψ r pGq here would be more satisfactory if we treated G itself Our definition of Ψ r as an OutN pG q-oribt of inner twists. We shall touch on this point again in §9.4 in discussing difficulties raised by the questions of §9.1. We are interested in the irreducible representations of GpF q if F is local, and the automorphic representations of G if F is global. Our first eight chapters were devoted to quasisplit G. We saw that in this case, the contribution of a parameter ψ to the representation theory of G is closely related to the (nonconnected) complex reductive group Sψ
Sψ {Z pGpqΓ.
For general G, we shall attach a slightly larger group to any (locally Grelevant) parameter. p ad of G. p We r pGq, S ψ is contained in the adjoint group G For any ψ P Ψ p sc shall write Sψ,sc for the preimage of S ψ in the simply connected cover G p ad . Then Sψ,sc contains the group Z psc , and is a central extension of G (9.2.1)
1
ÝÑ
Zpsc
ÝÑ
Sψ,sc
ÝÑ
Sψ
ÝÑ
1
of S ψ by Zpsc . For quasisplit G, the component group Sψ S ψ {S ψ governs the endoscopic packet of representations attached to ψ. For general G, the 0
9.2. PARAMETERS AND CENTRALIZERS
525
larger component group
0 Sψ,sc Sψ,sc {Sψ,sc is what is needed for the associated endoscopic packet. It is a central extension
(9.2.2)
1
of Sψ by the group In general, Sψ,sc for its centre, and
ÝÑ
Zpψ,sc
ÝÑ
Sψ,sc
ÝÑ
Sψ
ÝÑ
1
0 Zpψ,sc Zpsc {Zpsc X Srψ,sc . is a nonabelian finite group. We shall write
Zψ,sc
Z pSψ,scq
Zψ Zψ,sc {Zpψ,sc for the image of Zψ,sc in the abelian quotient Sψ of Sψ,sc . The fact that the irreducible characters of Sψ,sc need not be one-dimensional obviously adds interest to the representation theory of G. The group Sψ,sc is of Heisenberg type, so its representation theory is still quite simple. In this section, we shall describe the group explicitly in terms of ψ, and review how to classify its irreducible characters. The subgroup Zψ plays an essential role in this classification. It is really the local case that is of interest here, since we will see that the global theory descends to the original group Sψ . We may as well therefore assume that F is local. The groups Sψ,sc and Sψ,sc were introduced for p-adic F and parameters φ in the subset ΦpGq ΨpGq X ΦpG q
in [A19] (where Sφ,sc was denoted by Srψ ). The purpose of [A19] was to state a conjectural parametrization of the local packet attached to φ in terms of representations of the associated nonabelian group Sψ,sc . The stated parametrization [A19, §3] depends on a noncanonical choice of an extension Γ to the larger group Z psc . With this in mind, we of the character ζpG on Zpsc fix a section
Ñ ΠpZpscq Γ q that depends only on the for the restriction mapping from ΠpZpsc q to ΠpZpsc Γ q and ΠpZ psc q for quasisplit group G P Ersim pN q. (We are now writing ΠpZpsc (9.2.3)
Γ ΠpZpsc q
ã
Γ and Z psc .) For our given the groups of characters on the abelian groups Zpsc p inner twist G, the section then gives an extension to Zsc of the associated Γ q, which we continue to denote by ζ pG . character ζpG P ΠpZpsc The group G belongs to one of the three general families. If it is of type Bn , G is isomorphic to the split group SOp2n 1q, and its dual group is p G p and Sψ,sc Sψ , and we have isomorphic to Spp2n, Cq. In this case, G sc nothing to do. We suppose therefore that G is of type Cn or Dn . Then G is isomorphic to either a split group Spp2nq or a quasisplit group SOp2nq,
526
9. INNER FORMS
and its dual group is isomorphic to either SOp2n 1, Cq or SOp2n, Cq. We r pGq, for an inner twist G of a have therefore to deal with parameters ψ P Ψ r p p group G P Esim pN q with G G SOpN, Cq. The essential case is that of a parameter ψ in the subset r 2 p Gq Ψ r p Gq X Ψ r 2 pG q Ψ
of square integrable parameters. The group Sψ,sc is then finite, and the short exact sequence (9.2.2) reduces to (9.2.1). For any such ψ, we would like to describe the structure of the finite group Sψ,sc Sψ,sc . This reduces to a lemma on complex spin groups, which was suggested to me by Steve Kudla. It will be convenient to modify the earlier convention from §1.2 slightly by taking the more familiar embedding of OpN, Cq into GLpN, Cq. In other words, we treat OpN, Cq as the group of complex pN N q-matrices that preserve the dot product on Cn . For any partition Π pN1 , . . . , Nr q,
of N , the product
N
N1
Nr ,
OpΠq OpN1 , Cq OpNr , Cq
then represents the corresponding group of block diagonal matrices in OpN, Cq. Set ApN q looooooooooooooomooooooooooooooon Op1, Cq Op1, Cq X SOpN, Cq, N
and let B pN q be the preimage of ApN q in the extension SpinpN, Cq of SOpN, Cq. Then B pN q is an extension
ÝÑ t1u ÝÑ B pN q ÝÑ ApN q ÝÑ 1 of ApN q by the multiplicative group t1u. We can write ApN q as the group of elements as , parametrized by even subsets S of t1, . . . , N u, with (9.2.4)
1
multiplication
aS aT defined by the divided difference
aS4T
pS Y T q pS X T q. Indeed, we identify aS with an element in SOpN, Cq by setting # ej , if j P S, aS ej ej , if j R S, for the standard orthonormal basis te1 , . . . , eN u of Cn . Given aS , we introS4T
duce a formal product bS
ei ei , 1
s
S
ti1 i2 isu.
We then define a group law on the set (9.2.5)
bS , bS : S
t1, . . . , N u even
(
9.2. PARAMETERS AND CENTRALIZERS
527
by writing (9.2.6)
bS bT
for the 2-cocycle εpS, T q
¹
P
εpS, T qbS4T ,
p1q|T |, i
Ti
tj P T : j iu.
i S
It is of course understood that the element p1q bφ lies in the centre of the group (9.2.5). It is then clear that (9.2.5) represents a central extension of the group ApN q by t1u. Lemma 9.2.1. There is a canonical isomorphism from the group (9.2.5), as an extension of ApN q by t1u, onto the group B pN q in (9.2.4). Moreover, the elements in (9.2.5) satisfy the supplementary relations $ &b2 p1q |S |p|S2 |1q , S (9.2.7) %b b p1q|S XT | b b , S T T S for even subsets S and T of t1, . . . , N u.
Proof. The basis vectors ei embed into the Clifford algebra C pQq attached to the standard quadratic form Qpv q v v on the complex vector space V CN . We can then regard the products bS as elements in the even part C 0 pQq of C pQq. The lemma is a consequence of the construction of C pQq, and its relation to the complex spin group SpinpN, Cq. Let us review these notions, for the sake of the author, if not the reader. The Clifford algebra C pQq is the associative C-algebra generated by the orthonormal basis vectors tei u, with relations #
(9.2.8)
e2i pei ei q 1, ei ej ej ei 2ei ej
0,
if i j.
It has a pZ{2Zq-grading, defined by the algebra involution generated by the isomorphism v Ñ pv q of V , and hence a decomposition C pQq C 0 pQq ` C 1 pQq
into even and odd parts. This is a C-algebra grading, which is not to be confused with the natural Z-grading of C pQq as a vector space. The Clifford algebra also has a transpose anti-involution generated by the mapping
pv1 vk q ÝÑ pv1 vk qt vk v1,
vi
P V.
This is used to define the Clifford scalar product Qpxq pxt xq0 ,
x P C pQq,
I will follow the elementary discussion of an article Clifford algebra in Wikipedia, as it appeared on June 10, 2011.
528
9. INNER FORMS
where pq0 denotes the projection of C pQq onto C given by 0-components of its Z-grading. The Clifford scalar product reduces to the original quadratic form on the subspace V of C pQq. The spin group SpinpN, Cq lies in a chain of embedded subgroups SpinpN, Cq G SpinpN, Cq C 0 pQq
C pQq,
where as usual, C 0 pQq and C pQq denote the groups of units in the underlying algebras. The general spin group G SpinpN, Cq is identified with the subgroup of elements x P C 0 pQq such that the conjugation mapping y
xyx1 ,
y
G SpinpN, Cq
ÝÑ
ÝÑ
P C pQq,
stabilizes the subspace V of C pQq. The resulting representation of G SpinpN, Cq on V maps this group onto the subgroup SOpN, Cq of GLpN, Cq, and defines a short exact sequence 1
ÝÑ
The spinor norm x
C
ÝÑ
ÝÑ
xt x pxt xq0 ,
SOpN, Cq
ÝÑ
1.
x P G SpinpN, Cq,
is a homomorphism from G SpinpN, Cq onto C . Its kernel is the subgroup SpinpN, Cq, while the kernel of its restriction to the central subgroup C of G SpinpN, Cq is the subgroup t1u of C . We thus have a canonical embedding of the extension
ÝÑ t1u ÝÑ SpinpN, Cq ÝÑ SOpN, Cq ÝÑ into the group of units C 0 pQq in the even Clifford algebra. 1
1
The lemma is now a direct consequence of the embedding of SpinpN, Cq into C 0 pQq . The multiplication law (9.2.6) for elements bS in (9.2.5) reduces to its analogue (9.2.8) for general elements in C pQq. The supplementary relations (9.2.7) follow in turn from (9.2.6), as the reader can easily check. We return to our inner twist G of the quasisplit group G with pG p SOpN, Cq. G Suppose that r sim pNi q, ψ ψ1 ` ` ψr , ψi P Ψ
P ErsimpN q,
r 2 pGq. Given ψ, we have a decomposition is a parameter in Ψ
I
Ie > Io Iψ,e > Iψ,o
of its set of indices
I Iψ t1, . . . , ru into two disjoint subsets, consisting of those indices k whose associated degrees Nk are either even or odd. Let Aψ be the group of elements as , parametrized by subsets s I of indices such that the intersection so
s X Io
9.2. PARAMETERS AND CENTRALIZERS
529
is even, with multiplication
as4t. We identify Aψ with a subgroup of ApN q under the injective homomorphism as Ñ aS , where Ss is the (disjoint) union over k in s of the subsets tN1 Nk1 1, . . . , N1 Nk u t1, . . . , N u of the original indexing set. In particular, Aψ is a subgroup of SOpN, Cq. Its preimage in SpinpN, Cq is the subgroup Bψ tbs : s I, so evenu of B pN q, where bs bS . We thus obtain a subextension 1 ÝÑ t1u ÝÑ Bψ ÝÑ Aψ ÝÑ 1 as at
s
s
of (9.2.4). We will need to know when Bψ is abelian. More generally, we consider the quotient Bψ {Zψ B ψ {Z ψ ,
where Zψ Z pBψ q is the centre of Bψ , and Z ψ B ψ Bψ {Zpsc . Observe that
|Bψ | 2|Aψ | 2r
where εψ
εψ
Zψ {Zpsc is its image in
,
#
1, if |Io | 0, 0, if |Io | 0.
If N is odd, the set Io has an odd number of elements. In particular, it is nonempty, and |Bψ | 2r . In this case, the centre Zpsc of SpinpN, Cq equals t1u. If N is even, the set Io has an even number of elements. It can be empty, so that |Bψ | can equal either 2r 1 or 2r . In this case, we have elements aI P AI and bI P Bψ attached to the maximal set s I, and the center of SpinpN, Cq equals Zpsc It follows that (9.2.9)
t1, bI u.
|B ψ | |Bψ |{|Zpsc| 2r
where δψ
δN
#
1, 2,
,
ε ψ δψ
if N is odd, if N is even.
As for the center of Bψ , we claim that (9.2.10)
Zψ
tbs : s Ieu Zpsc.
To check this, we appeal to the specialization (9.2.11)
1 b t bs bt
p1q|sXt|bs p1q|s Xt |bs o
o
530
9. INNER FORMS
of the second relation in (9.2.7), which holds for any subsets s and t of I with so and to even. It follows that Zψ contains the right hand side of (9.2.10). Conversely, suppose that bs lies in Zψ , but that s is not contained in Ie . It is then easy to verify from (9.2.10) and the various definitions that so Io , and that |Io | 2. This implies that N is even, and that bs is the product of an element pbI q P Zpsc with an element bt P Bψ such that t is contained in Ie . The claim follows. It follows from (9.2.10) that
|Zψ | 2|I |
,
δψ ε ψ
e
and that (9.2.12)
|Z ψ | |Zψ |{|Zpsc| 2|I |ε e
ψ
We have now only to combine (9.2.12) with (9.2.9). This gives a formula (9.2.13)
|Bψ {Zψ | 2|I | o
2εψ δψ
for the order of the quotient in question. In particular, we see from the definitions of εψ and δψ that Bψ is abelian if and only if |Io | ¤ 2. In the case that Bψ is not abelian, (9.2.11) implies that its derived group satisfies (9.2.14)
Bψ,der
t1u
R, and that φ ψ lies in the subset r 2 p Gq Ψ r 2 pG q X Φ r pG q Φ
Suppose for example that F
r 2 pGq. The simple components φk of φ must be distinct and self-dual, of Ψ and their degrees Nk are either 2 or 1. There can be at most two k with Nk 1, the trivial character 1R on R and the sign character εR . It follows that |Io | ¤ 2 for any such φ, and hence that the group Bφ is abelian. This property, which is valid for any G, is a basic tenet of the work of Shelstad [S6]. We note that if F C, the only simple, self-dual, generic parameter φk is the trivial character 1C . In this case, we have
|Io| |I | 1 N,
and the group Bφ t1u is trivial. We shall use the relations we have established among the various groups
t1u Zpsc Zψ Bψ to characterize the set ΠpBψ q of equivalence classes of irreducible characters of Bψ . It suffices to treat the subset ΠpBψ , ζpψ q of characters in ΠpBψ q whose
central character on Zpsc equals a fixed character ζpψ . We can assume that Bψ is nonabelian, since we would otherwise be dealing with a set of onedimensional characters. We can also assume that the value of ζpψ on the element p1q in Zpsc is p1q, since the quotient of Bψ by t1u is the abelian group Aψ .
9.2. PARAMETERS AND CENTRALIZERS
531
r 2 pGq is nonLemma 9.2.2. Assume that the group Bψ attached to ψ P Ψ abelian, and that the given character ζpψ on Zpsc satisfies ζpG p1q 1. Then there is a bijective correspondence
P ΠpZψ , ζpψ q, from the set of (abelian) characters ΠpZψ , ζpψ q on Zψ whose restriction to Zpsc equals ζpG , onto the corresponding nonabelian character set ΠpBψ , ζpψ q ξ
ÝÑ xb, ξy,
on Bψ , such that
xb, ξy
(9.2.15) where
dψ
#
b P Bψ , ξ
#
dψ ξ pbq, 0,
2p|Io |1q{2 , 2p|Io |2q{2 ,
if b P Zψ , if b R Zψ , if N is odd, if N is even,
is the degree of any character in ΠpBψ , ζpψ q. Proof. As a finite nonabelian group, Bψ is particularly simple. It is a 2-stage nilpotent group of Heisenberg type, whose representation theory is elementary and well known. We shall just quote what we need. The quotient Bψ {Zψ is a finite abelian 2-group. It has a well defined pairing rx, ys xyx1y1, x, y P Bψ {Zψ , which takes values in t1u by (9.2.14), and is nondegenerate by (9.2.11). With this structure, we can regard Bψ {Zψ as a 2-dimensional vector space over the field of 2-elements, in which addition (in both the vector space and the field) is written multiplicatively. As a vector space, Bψ {Zψ has dimension equal to
|Io|
2εψ δψ
#
|Io| δψ ||IIo|| 1, 2, o
if N is odd, if N is even,
by (9.2.13), and our condition that Bψ is nonabelian. Since |Io | has the same parity as N , this integer is even, which of course is also a consequence of the existence of the symplectic form r, s. Let U {Zψ be a maximal isotropic subspace of Bψ {Zψ . This is a subspace of dimension one-half that of Bψ {Zψ . Its preimage in Bψ is a maximal abelian subgroup U , which embeds into the chain t1u Zψ U Bψ . The degree, defined in the statement of the theorem as a power of 2, then satisfies 1 dψ 2 2 dimpBψ {Zψ q 2dimpU {Zψ q |Bψ {U |.
Suppose that ξ belongs to ΠpZψ , ζpψ q. We first extend ξ as an abelian character to the subgroup U , and then form the induced representation ρξ
IndBU pξq ψ
532
9. INNER FORMS
of Bψ . It is an elementary consequence of the theory of induced representations that the representation ρξ is irreducible, that its character vanishes on the complement of Zψ , and that tr ρξ pz q
xz, ξy |Bψ {U | ξpzq dψ ξpzq,
for any z P Zψ . In particular, the equivalence class of ρξ is independent of the choice of U and the extension of ξ. The lemma follows.
r 2 pG q. We have been working with a square integrable parameter ψ P Ψ It is clear that the definitions of the groups in the lemma match those of the centralizer groups defined at the beginning of the section. We have Sψ Aψ , Sψ,sc Bψ , S ψ Sψ Bψ {Zpsc , Zψ,sc Zψ and Zψ Z ψ . Lemma 9.2.2 therefore gives us a description of the set ΠpSψ,sc , ζpG q of irreducible characters on Sψ,sc with central character ζpG ζpψ on Zpsc . r pGq. We recall More generally, suppose that ψ is a general parameter in Ψ that the original centralizer group Sψ comes with a short exact sequence
1
ÝÑ
Sψ1
ÝÑ
Sψ
ÝÑ
Rψ
ÝÑ
1.
What is the analogue of this for the extension Sψ,sc of Sψ in (9.2.2)? Since ψ is relevant to G, we can choose a Levi subgroup M of G with respect to which ψ is square integrable. In other words, we can choose a r 2 pM, ψ q. The group Sψ is then pair pM, ψM q such that ψM lies in the set Ψ M canonically isomorphic to the subgroup Sψ1 of Sψ , a property that we have used repeatedly since it first arose in §2.4 (and in the global case of §4.2), and which follows from the well known fact that xqΓ Z pM
xqΓ Z pGpqΓ Z pM
0
.
(See [A12, Lemma 1.1], for example.) The last formula holds also for the xsc of M x in G p sc , where we interpret it as an identity preimage M xsc qΓ π0 Z pM
xsc qΓ ZpscΓ {ZpscΓ X Z pM
0
Γ between two finite groups. On the other hand, the character ζpG on Zpsc Γ attached to G is trivial on the intersection of Zpsc with the identity component xsc qΓ . (See for example the proof of [A12, Lemma 2.1], specifically of Z pM the remarks at the end of the second paragraph on p. 219 of [A12].) We have agreed to write ζpG also for its extension (9.2.3) to a character on Zpsc , xsc q. Since a larger group which is still contained in Z pM
xsc qΓ Zpsc X Z pM
0
xsc qΓ ZpscΓ X Z pM
0
,
we can identify ζpG with a character on the quotient Zpψ,sc of Zpsc by this intersection. According to (9.2.2), Zpψ,sc is the kernel of the projection of Sψ,sc onto Sψ . Since elements in Zpsc commute with the connected group xsc qΓ Z pM
0
0 Sψ,sc ,
9.3. ON THE NORMALIZATION OF TRANSFER FACTORS
533
1 . It is in fact the kernel of the projection the quotient Zpψ,sc embeds into Sψ,sc 1 of Sψ,sc onto Sψ1 . As a consequence of these various remarks, we obtain a commutative diagram
1 ∨
Zpψ,sc
1 ∨
Zpψ,sc
∨
(9.2.16)
1
1
>
1 Sψ,sc
∨ > S1 ψ
∨ >
Sψ,sc
>
Sψ
∨
∨
1
1
>
1
>
1
}
∨ >
Rψ
>
Rψ
of short exact sequences. The upper horizontal sequence is the analogue for Sψ,sc of the lower horizontal sequence for Sψ . We can regard M as an inner twist of a Levi subgroup M of G that is a product (2.3.4) of general linear groups and a group G of the same type as G . From the list of indices in the last section, we observe that M is a product of general linear groups, all over either F or a fixed F -quaternion 1 in (9.2.16) algebra, with an inner form G of the group G . The group Sψ,sc satisfies 1 Sψ,sc SψM ,sc Sψ ,
r 2 pG q is the subparameter of ψ attached to G . Its irreducible where ψ P Ψ characters are given by Lemma 9.2.2. The irreducible characters of the larger 1 , and the characters on group Sψ,sc are then defined in terms of those of Sψ,sc the abelian R-group Rψ , by a simple process analogous to that in the proof of Proposition 2.4.3. We will establish the analogue for G of Proposition 2.4.3 in [A28], as part of the local intertwining relation for G.
9.3. On the normalization of transfer factors In the earlier chapters, we classified the representations of a quasisplit orthogonal or symplectic group G in terms of those of general linear groups. The purpose of this chapter is to describe the representations of a general orthogonal or symplectic group G in terms of those of groups G . In the first section, we distinguished between inner forms and inner twists of a given G P Ersim pN q. The reason for working with the latter is that the parametrization of representations of an inner form G really depends on
534
9. INNER FORMS
the extra structure of a quasisplit inner twist ψ. In this sense, ψ behaves somewhat like a “co-ordinate system” for G. The representation theory of a given group depends on something else as well. Its quasisplit inner twist ψ must be supplemented by a system of transfer factors for its endoscopic groups G1 [LS1, (3.7)]. (We are assuming at this point, and until further notice, that the underlying field F is local.) If G is not quasisplit, its transfer factors cannot be normalized by Whittaker data as in [KS, §5.4] (or by F -splittings as in [LS1, (3.7)]). According to the definitions in [LS1, (3.7)], a transfer factor ∆ for pG, G1 q is determined only up to a multiplicative constant u P U p1q of absolute value 1. This is our topic for the third section. We shall first discuss the abstract problem of normalizing transfer factors. We will then describe the natural normalizations established in recent work of Kaletha [Kal1], [Kal2], [Kal4]. We fix G, which according to our convention represents an inner twist pG, ψ q of a (connected) quasisplit orthogonal or symplectic group G P Ersim pN q over F . We still have the notion of an isomorphism α1 of endoscopic data G11 and G12 for G (which in this case represent triplets pG11 , s11 , ξ11 q and pG12 , s12 , ξ21 q). As earlier, we follow the definition of [KS]. Then α1 is p such that Intpg 1 q1 restricts to an identified with an element g 1 g pα1 q in G L-isomorphism between the L-subgroups ξ21 pL G12 q and ξ11 pL G11 q of L G, and such that g 1 ps11 qpg 1 q1 belongs to the set p qΓ Cent ξ 1 pL G1 q, G p Z pG 2 2
0
s12.
The symbol α1 serves also to represent a dual F -isomorphism between the quasisplit groups G11 and G12 . The logic might be a little clearer here if we define G11 and G12 to be strongly isomorphic if there is an isomorphism (of endoscopic data) G11 and G12 with g pα1 q 1. A general α1 can be treated as an isomorphism of strong isomorphism classes of endoscopic data. It is p and in fact by the image then uniquely determined by the element g 1 P G, 1 1 p ad . Intpg q of g in G We are in the practice of writing E pGq for the set of isomorphism classes of endoscopic data for G. For any G1 P E pGq, let us also write E pG, G1 q for the set of strong isomorphism classes in the general isomorphism class G1 . With this understanding, E pG, G1 q is a complex algebraic variety on which p acts transitively. The stabilizer in G p of any G1 P E pG, G1 q is a the group G 1 p whose group of connected components is the finite reductive subgroup of G, group we have denoted by OutG pG11 q. We can also write E p Gq
º
G1 E G
Pp q
E pG, G1 q
for the variety of all such classes of data for G. When there is no danger of confusion, we will continue to let G1 stand interchangeably for an endoscopic datum (which we will now usually identify with its strong isomorphism class) or a corresponding isomorphism class. For example, the
9.3. ON THE NORMALIZATION OF TRANSFER FACTORS
535
finite group OutG pG1 q makes sense whether we treat G1 as an element in E pGq or a representative in E pGq of a class in E pGq. We do, however, need to exercise some care. As we have noted, the transfer factors defined in [LS1] and [KS] do not come with a natural normalization if G is not quasisplit. Let us write T pG q
º
G1 E G
Pp q
T pG, G1 q
for the set of transfer factors for G. (Again, we apologize for an overlap with earlier notation, namely the set T pGq introduced in §3.5 in which T was meant to serve as an upper case τ .) An element in T pGq is then a pair pG1, ∆q, where G1 belongs to E pGq, and ∆ is a transfer factor for pG, G1q. For a given G1 , ∆ is determined only up to a scalar multiple in U p1q. In other words, the natural projection T pG q
ÝÑ E pGq
is a principal U p1q-bundle. The point is that this bundle does not generally have a continuous section, let alone a canonical section. To describe the obstruction, we recall some remarks from [A19, §3]. If α1 is an isomorphism between two data G11 and G12 in E pGq, and ∆ is a transfer factor for pG, G11 q, the function
pα1∆qpδ21 , γ q ∆ pα1q1δ21 , γ , δ21 P ∆G-reg pG12q, γ P Γreg pGq, is a transfer factor for pG, G12 q. In particular, we obtain an action of the finite group OutG pG1 q on the fibre in T pGq of a point G1 P E pGq. Given p sc of the G1 P E pGq and α1 P OutG pG1 q, let ssc and gsc be preimages in G p ad of the points s1 and g 1 in G p attached to G1 and α1 . We projections onto G can then write
1 gsc ssc gsc
where
ssczpα1q,
ÝÑ zpα1q Γ. is a homomorphism from OutG pG1 q into Zpsc
that
α1
P OutGpG1q, G1 P E pGq, where ∆ is any element in the fibre of G1 in T pGq, and ζpG is the character
(9.3.1)
pα1∆q ζpG zpα1q ∆,
It is then not hard to show
α1
Γ attached in §9.1 to the inner twist G of G . (See [A24].) on Zpsc The action of OutG pG1 q on the fibre of G1 in T pGq is therefore not trivial. (This point was overlooked in several of the papers leading up to the stable trace formula [A16], as we noted in [A19, §3]. The necessary modifications are minor, and will appear elsewhere.) It follows from (9.3.1) that we cannot assign transfer factors to pairs pG, G1 q in a consistent way as
536
9. INNER FORMS
G1 ranges over elements in E pGq in a given isomorphism class. To describe a formal resolution, it is suggestive to write S1
SG1 CentpLG1, Gpq Z pGp1qΓ
p of the L-subgroup L G1 of L G, in analogy with the for the centralizer in G 1 for the preimage group Sψ attached to a parameter ψ. We then define Ssc of the group p qΓ . S 1 S 1 { Z pG
This gives us an exact sequence 1
ÝÑ
Zpsc
ÝÑ
1 Ssc
S1
ÝÑ
ÝÑ
1
analogous to (9.2.1). It follows from the definitions that the point s1 attached to G1 lies in S 1 , and therefore projects to a point s¯1 in S 1 . With this notation, we define Esc pGq tpG1 , ssc q : G1
P Gp
P E pGqu,
1 of the point s¯1 P S 1 . Then Esc pGq where ssc ranges over the preimages in Ssc
is a covering space over E pGq, or more precisely, a principle Zpsc -bundle over E pGq. The fibre product Tsc pGq of T pGq and Esc pGq over E pGq is the set of triplets
tpG1, ∆, sscq : pG1, ∆q P T pGq, pG1, sscq P EscpGqu. It is clear that Tsc pGq is a principal U p1q-bundle over Esc pGq, which we can display over the original U p1q-bundle in a commutative diagram > U p1q > Tsc pGq > Esc pGq >1 1 } (9.3.2) ∨ ∨ > U p1q > T p Gq > E p Gq > 1. 1 This second principal bundle does have a section. Its existence amounts to the following elementary lemma. Lemma 9.3.1. There is a continuous mapping ρ : Tsc pGq
ÝÑ U p1q
such that (9.3.3)
ρpu∆, ssc z q uρp∆, ssc qζpG pz q1 ,
and (9.3.4) for any point in Tsc pGq.
ρ α1 ∆, Intpg 1 qssc
ρp∆, sscq,
p∆, sscq pG1, ∆, sscq
u P U p1q, z g1
g p α 1 q,
P Zpsc,
9.3. ON THE NORMALIZATION OF TRANSFER FACTORS
537
Proof. Suppose that pG1 , ssc q is the image in Esc pGq of a point p∆, ssc q in Tsc pGq, and that g 1 equals the point g pα1 q attached to an outer automorphism α1 P OutG pG1 q of G1 . It follows from (9.3.1) that ρ Intpg 1 q∆, Intpg 1 qssc
ρ ζpG zpα1q ∆, ssczpα1q , for any U p1q-valued function ρ on Tsc pGq. The condition (9.3.4) is then a
consequence of (9.3.3) in this case. Since the two conditions are otherwise independent, they are compatible with each other. We can therefore define the required section ρp∆, ssc q by (9.3.3). Indeed, we have only to specify its values at any set of points tp∆, ssc qu that projects bijectively onto the set of isomorphism classes G1 P E pGq. Remark. The isomorphism classes of endoscopic data G1 P E pGq parametrize the connected components Tsc pG, G1 q of Tsc pGq. It is clear that the section ρp∆, ssc q of the lemma is unique up to a U p1q-multiple on each component Tsc pG, G1 q. Consider the Langlands-Shelstad transfer correspondence f
f∆1 , f P HpGq, pG1, ∆q P T pGq. fixed pG1 , ∆q, it is the mapping from HpGq
ÝÑ
f1
We recall that for defined by the geometric transform
1 pδ 1 q f∆
¸
∆pδ 1 , γ qfG pγ q,
δ1
to S pG1 q
P ∆G-reg pG1q, γ P Γreg pGq,
γ
1 pδ 1 q can be regarded as a section introduced in §2.1. For fixed f and δ 1 , f∆ of a bundle, specifically the line bundle attached to T pG, G1 q by the action of its structure group U p1q on C. Rather than regard endoscopic transfer as a mapping that varies with a choice of ∆, we could normalize it with a section ρp∆, ssc q from Lemma 9.3.1. Given ρ and f , we define fρ1 to be the function in S pG1 q whose value at any δ 1 equals (9.3.5)
1 pδ 1 q. fρ1 pδ 1 q ρp∆, ssc q1 f∆
Observe that the function fρ1 remains unchanged if ∆ is replaced either by a U p1q-multiple u∆ or an OutG pG1 q-image α1 ∆, by the properties of ρp∆, ssc q in Lemma 9.3.1. It is therefore independent of ∆. It does, however, depend 1. on the implicit choice of a preimage ssc of s¯1 in Ssc 1 The transfer fρ does of course depend on the normalizing section ρ. However, Kaletha has introduced a finer normalization, following ideas of Kottwitz. Specifically, he has shown how to normalize the transfer factor ∆ for pG, G1 q (or equivalently, how to choose a section ρ) explicitly in terms of a transfer factor ∆ for the quasisplit pair pG , G1 q [Kal1, §2.2], [Kal2, §2.2]. I shall attempt to describe how Kaletha’s general constructions apply to the case at hand. Since some of his work is still in progress (and my own understanding of it leaves something to be desired), parts of the following discussion might for the moment be best taken conditionally. I am also indebted to Kottwitz for describing his unpublished ideas to me.
538
9. INNER FORMS
The field F remains local. Kaletha’s constructions depend on an inner twist with some extra structure. A pure inner twist over F is a triplet pG, ψ, zq, where pG, ψq is an inner twist of a connected, quasisplit group G over F , which we temporarily take to be arbitrary, and z P Z 1 pF, G q is a Galois 1-cocycle in G such that ψσ pψ q1
Int zpσq ,
σ
P ΓF .
In contrast to ordinary inner twists, pure inner twists do not necessarily exist. That is, an inner form G of G need not be the first component of a pure inner twist. Suppose, however, that G satisfies the condition that its p der G p sc ), and the further condition center Z pGq is connected (so that G p qΓ of the center of G p is discrete. The inner that the Γ-invariant part Z pG form G of G can then be inflated to a pure inner twist. With this structure, p qΓ Z pG p qΓ . The two conditions we of course have Z pGq Z pG q and Z pG can therefore be stated in terms of G , and therefore apply uniformly to all G. Assume that G represents a pure inner twist pG, ψ, z q of G . Suppose also that γ P GpF q and γ P G pF q are strongly G-regular elements that are stably conjugate, in the sense that for some g
ψ p γ q g γ p g q1 ,
P GpF q. Then the 1-cocycle σ ÝÑ g z pσ qσ pg q1 , σ P ΓF , takes values in the maximal F -torus T Gγ of G . Its image invpγ, γ q in H 1 pF, T q is the invariant of γ and γ (with respect to pψ, z q), defined
by Kaletha in [Kal1, §2.1]. Suppose also that G1 represents an endoscopic datum pG1 , G 1 , s1 , ξ 1 q, and that δ 1 is an image of γ , in the sense of [LS1, p. 236]. There is then an admissible embedding [LS1, p. 236] from the centralizer T 1 G1δ1 into G that maps δ 1 to γ (and that therefore maps T 1 isomorphically onto T ). Let s sT be the preimage in pTp qΓ of the p 1 qΓ of pTp1 qΓ , under the dual isomorphism element ξ 1 ps1 q in the subgroup Z pG from Tp to Tp1 . Then s maps to a component in the group π0 pTp qΓ , which we denote also by s . The Tate-Nakayama pairing between H 1 pF, T q and π0 pTp qΓ [K2, §1.1] then provides a complex number (9.3.6)
xinvpγ, γ q, sy,
which depends on the point δ 1 (as well as the cocycle z). This last pairing is one ingredient in Kaletha’s normalization of the transfer factors for G. The other is the choice of a family of transfer factors ∆ for the quasisplit group G , which we can assume are normalized by a fixed Whittaker datum for G . Given ∆ , we write (9.3.7)
∆K pδ 1 , γ q ∆ pδ 1 , γ qxinvpγ, γ q, s y1 ,
9.3. ON THE NORMALIZATION OF TRANSFER FACTORS
539
for γ, g and δ 1 as above. This is Kaletha’s normalization. He shows that it is independent of γ in [Kal1, Lemma 2.2.1], and proves that it is actually a transfer factor for pG, G q in [Kal1, §2.3–§2.5]. The construction (9.3.7) requires the two restrictions above on G (or p qΓ be discrete G ). For p-adic F , Kaletha removes the condition that Z pG in [Kal2] by using Kottwitz’s theory of isocrystals with additional structure [K4], [K7]. Kottwitz has pointed out to me that his results can be stated equivalently in terms of algebraic cohomology of Weil groups. This alternate formulation has the advantage that it applies also to the case that F is archimedean. I thank Kottwitz for the following description, which I hope I have interpreted correctly. Suppose that T is a torus over the local field F . The Tate-Nakayama pairing that provides the definition of (9.3.6) is a canonical isomorphism
H 1 pF, T q H 1 pΓK {F , T q Ý Ñ π0pTpΓq
def X π0 pTpΓ { q , where K {F is a large finite Galois extension and X pq denotes the finite def
K F
abelian group of (algebraic) characters on the finite abelian (diagonalizable) group π0 pTpΓ q. Its analogue for the Weil group is a canonical isomorphism
1 pWK {F , T q ÝÑ pTpΓq H 1 pWF , T q Halg def
def X pTpΓ { q, K F
1 pq denotes the cohomology group of classes represented by 1where Halg cocycles that are algebraic as morphisms of varieties over F , and X pq now represents the finitely generated abelian group of (algebraic) characters on the complex diagonalizable group TpΓ . More generally, suppose that G is a connected reductive group over F . Then the Tate-Nakayama pairing is a canonical mapping
H 1 pF, Gq H 1 pΓK {F , Gq def
ÝÑ
p qΓ π0 Z pG
def X
p qΓK {F π0 Z pG
,
which is a bijection when F is p-adic. (See [K5, §1.2].) Its Weil group analogue is a canonical mapping 1 H 1 pWF , Gq Halg pWK {F , Gq def
ÝÑ
p qΓ Z pG
def p qΓ X Z pG
{
K F
,
which is again a bijection if F is p-adic. In this case, the definition of 1 pW Halg E {F , Gq includes the supplementary condition that the underlying 1-cocycles map the subgroup E of WE {F to the subgroup Z pGq of G. Kaletha defines an extended pure inner twist to be a triplet pG, ψ, z q as above, but with the 1-cocycle appropriately weakened. In our terms, z is an element in the set of 1-cocycles 1 Z 1 pWF , G q Zalg pWK {F , Gq def
whose quotient is the Weil group cohomology set H 1 pWF , G q we have just described. If G represents an extended pure inner twist pG, ψ, z q, the pairing (9.3.6) makes sense. In [Kal2, Propositon 2.1], Kaletha shows that the
540
9. INNER FORMS
corresponding product (9.3.7) is a transfer factor for pG, G1 q. This removes p qΓ be discrete. the condition that Z pG The other restriction is that Z pGq be connected. To remove it, Kaletha embeds a general group G into a group G1 with connected center, whose representation theory is essentially the same as that of G. Following an idea of Kottwitz, he defines a z-embedding of G to be an F -extension
ÝÑ 1 of an induced torus C over F by G such that Z pG1 q is connected, and such (9.3.8)
1
that the mapping
ÝÑ
G
ÝÑ
H 1 F, Z pGq
(9.3.9)
G1
ÝÑ
ÝÑ
C
H 1 F, Z pG1 q
is an isomorphism. The reader will note that this is reminiscent of the well known definition [K2, §1] of a z-extension, which would in fact be dual to that of a z-embedding were it not for the extra condition on (9.3.9). The properties of z-embeddings are part of the work in progress [Kal4] of Kaletha. We shall briefly describe some of them. One of the simplest properties is that the restriction of representations from G1 pF q to GpF q gives a surjective mapping π1 Ñ π from ΠpG1 q to ΠpGq. This follows from the fact that G 1 pF q G p F qZ pG 1 , F q ,
where Z pG1 , F q Z pG1 q pF q is the centre of GpF q, a consequence in turn of the injectivity of (9.3.9) and the long exact cohomology sequence attached to the short exact sequence
ÝÑ Z pGq ÝÑ Z pG1q ÝÑ C ÝÑ 1, which together imply that the mapping from Z pG1 , F q to C pF q is surjective. Since the group ΠpC q of characters on C pF q acts transitively on the fibres of the mapping π1 Ñ π, the representation theory of G1 pF q is indeed essentially the same as that of GpF q. Kaletha also establishes finer endoscopic relations (9.3.10)
1
among the representations of the two groups. They arise naturally from the short exact sequence (9.3.11)
1
ÝÑ
p C
ÝÑ
p1 G
ÝÑ
p G
ÝÑ
1,
of dual groups, equipped with their compatible L-actions of WF . In particp 1 onto G p gives a mapping of Langlands parameters ular, the projection of G φ1 Ñ φ that takes ΦpG1 q to ΦpGq. Kaletha observes that this mapping p q by mulis surjective, that its fibres are torsors for the action of H 1 pW, C tiplication by cocycles, and that the centralizer groups of corresponding parameters are related by a short exact sequence
ÝÑ 1. Similar relations apply to the parameter sets ΨpG1 q and ΨpGq.
(9.3.12)
1
ÝÑ
pΓ C
ÝÑ
S φ1
ÝÑ
Sφ
I thank Kaletha for making his notes available to me.
9.3. ON THE NORMALIZATION OF TRANSFER FACTORS
541
There are parallel relations among endoscopic data. Given an endoscopic datum pG1 , s1 , G 1 , ξ 1 q for G, Kaletha constructs an endoscopic datum pG11, s11, G1, ξ11 q for G1 by defining s11 P Gp1 to be any preimage of s1, and pG11 , ξ11 q to be the pullback (fibre product) in the larger diagram 1
ÝÝÝÝÑ
p C
ÝÝÝÝÑ
LG
1
ÝÝÝÝÑ
p C
ÝÝÝÝÑ
G11
1
1 ξ1
ÝÝÝÝÑ
LG
ÝÝÝÝÑ
1
ÝÝÝÝÑ
G1
ÝÝÝÝÑ
1.
1 ξ
The quasisplit group G11 is then the pushout (fibre sum) in the diagram Z p Gq
ÝÝÝÝÑ Z pG1q
G1
ÝÝÝÝÑ G11 Conversely, given an endoscopic datum pG11 , s11 , G11 , ξ11 q for G1 , Kaletha takes pÑG p 1 is the restriction of the F -surjection G11 Ñ C whose dual injection C 1 p of ξ 1 pG 1 q. This gives the quasisplit group ξ11 to the subgroup C 1 1 G1 kerpG11 Ñ C q over F . He then inflates G1 to an endoscopic datum pG1 , s1 , G 1 , ξ 1 q for G p 1 q Ñ Z pG p1 q by setting s1 equal to the image of s11 under the mapping Z pG 1 attached to and pG 1 , ξ 1 q
p 1 impG p1 Ñ G p q, G 1 equal to the composition of three maps ξ11 p 1 Ñ G, p G 1 ãÑ G11 ÝÑ G
with G 1 defined in the natural way. Finally, he shows that these two correspondences give mutually inverse bijections
pG1, s1, G 1, ξ1q Ø pG11, s11, G11 , ξ11 q
between the strong isomorphism classes (as defined above) of endoscopic data for G and G1 . In the course of comparing endoscopic data, Kaletha observes that the mapping G1 Ñ G11 satisfies all the conditions of a z-embedding except for the connectedness of Z pG11 q. He calls such a mapping a pseudo-z-embedding. The properties of G1 we have just described all remain valid if the original z-embedding is weakened to a pseudo-z-embedding. In particular, a pseudoz-embedding of G into G1 (with quotient C) still gives a bijective correspondence G1 Ø G11 between the associated endoscopic data, taken up to strong isomorphism, and equipped with pseudo-z-embeddings G1 Ñ G11 (with the same quotient C). The supplementary condition on (9.3.9) required for a z-embedding is used only for the normalization (9.3.7) of transfer factors. Kaletha notes that z-embeddings (or pseudo-z-embeddings) can be chosen uniformly for inner twists. Suppose for example that G is quasisplit
542
9. INNER FORMS
over F , and that G Ñ G1 is a z-embedding. Suppose also that ψ: G Ñ G is an inner twist, with
ψσ pψ q1
Int upσq , σ P ΓF , for a 1-cocycle u P Z 1 pF, Gad q. Then there is a z-embedding G Ñ G1 , and an inner twist ψ1 : G1 Ñ G1 , with ψ1 σ pψ1 q1 Int upσ q , σ P ΓF , that fits into the diagram 1
ÝÝÝÝÑ
ÝÝÝÝÑ
G1
ÝÝÝÝÑ
C
G
ÝÝÝÝÑ
G1
ÝÝÝÝÑ
C
ψ
ψ1
ÝÝÝÝÑ
1
ÝÝÝÝÑ 1. The goal is to normalize the transfer factors for pG, ψ q in terms of the normalized transfer factors for an extended inner twist pG1 , ψ1 , z1 q. 1
ÝÝÝÝÑ
G
Let us first describe how Kaletha constructs z-embeddings. To be concrete, we shall treat only orthogonal and symplectic groups. We return therefore to our original setting, in which G represents an inner twist of G P Ersim pN q. It suffices to construct a z-embedding of the quasisplit group G . If G is of type Bn , it is isomorphic to a split group SOp2n 1q. In this case the center Z pG q is trivial, and we can take G1 G . We suppose therefore that G is of type Cn or Dn , and is consequently isomorphic to a split group Spp2nq or a quasisplit group SOp2nq. In these cases, the center Z pG q has order 2. The main ingredient of Kaletha’s construction, as it applies to our group G of type Cn or Dn , is a short exact sequence 1
ÝÑ Z pGq ÝÑ
Z
ÝÑ
C
ÝÑ
1,
where C is the induced torus of the z-embedding, and Z is an F -torus that will become the centre Z pG1 q of G1 . Let K {F be the compositum of the finite set of quadratic extensions of F , an abelian extension of F with Galois group ΓK {F pF q2 zF . We take the induced torus to be the restriction of scalars C
ResK {F pGm,K q.
The norm NK {F then represents an F -homomorphism from C to Gm Gm,F . We take Z to be the fibre product Z
ÝÝÝÝÑ
C
Gm
p ÝÝÝÝ Ñ
Gm ,
N K {F
9.3. ON THE NORMALIZATION OF TRANSFER FACTORS
where p p2 is the homomorphism x diagram of short exact sequences
Ñ x2 .
It fits into a commutative
1
ÝÝÝÝÑ Z pGq ÝÝÝÝÑ
Z
ÝÝÝÝÑ
C
ÝÝÝÝÑ
1
1
ÝÝÝÝÑ Z pGq ÝÝÝÝÑ
Gm
p ÝÝÝÝ Ñ
Gm
ÝÝÝÝÑ
1.
543
N K {F
This in turn gives rise to the commutative diagram 1
ÝÝÝÝÑ Z pGq ÝÝÝÝÑ Z pF q ÝÝÝÝÑ C pF q ÝÝÝÝÑ
H 1 F, Z pG q
ÝÝÝÝÑ
1
ÝÝÝÝÑ Z pGq ÝÝÝÝÑ
F
H 1 F, Z pG q
ÝÝÝÝÑ
N K {F
F
p ÝÝÝÝ Ñ
ÝÝÝÝÑ
of long exact sequences of cohomology. The group Z is connected, since it is a torus over F . We have already noted that C is an induced torus, so in particular, H 1 pF, C q t1u. It follows from the continuation of the upper long exact sequence that the mapping H 1 F, Z pG q
ÝÑ
H 1 pF, Z q
is surjective. To prove that the mapping is injective, it suffices to show that the image of C pF q in H 1 F, Z pG q in the upper sequence is trivial. But the image of C pF q in upper sequence equals the image of NK {F C pF q in the lower sequence. By local class field theory, NK {F C pF q equals pF q2 , which is just the image of p in F . The image of NK {F C pF q in H 1 F, Z pG q is therefore trivial. It follows that the mapping above is an injection and hence an isomorphism, as will be required of (9.3.9). We have been following the argument of Kaletha, which is valid for any group over F . In our case that G is of type Cn or Dn , one sees explicitly that (9.3.13)
Z
pz, c1, . . . , ck q P Gkm 1 :
and that
z 2 t1 tk
(
(
1
,
pzp, pc1, . . . , pck q P pCqk 1 { pz2, z, . . . , zq : z P C C{t1u pCqk {C ,
Zp
(
where the subscripts index the Galois group ΓK {F
tσ1 1,
σ2 , . . . , σk u,
and the twisted Galois action on Z and Zp is provided by right translation of ΓK {F on these indices. Using the relations H 1 pF, Z q π0 pZpΓ q
π0 pCqk {C
ΓK {F
ΠpΓ K { F q,
one can then see directly that the mapping above is an isomorphism.
544
9. INNER FORMS
Once we have the torus Z, we can define G1 as the fibre sum of Z and G over Z pG q. This is easily seen to be an F -extension of C by G , which fits into the commutative diagram 1
ÝÝÝÝÑ Z pGq ÝÝÝÝÑ
Z
ÝÝÝÝÑ
C
ÝÝÝÝÑ
1
1
ÝÝÝÝÑ
G
G1
ÝÝÝÝÑ
C
ÝÝÝÝÑ
1.
ÝÝÝÝÑ
The mapping from Z to G1 is an injection that identifies Z with the centre Z pG1 q of G1 . The mapping of G into G1 then satisfies the required conditions of a z-embedding. This is what we set out to describe. As we have seen, it in turn gives a canonical z-embedding of our inner twist G of G , as well as a canonical pseudo-z-embedding of any endoscopic group G1 of G, both as above. The z-embedding for G we have just described gives us a group G1 with connected center. This was the condition for Kaletha’s construction (9.3.7) of normalized transfer factors. Suppose that pG1 , ψ1 , z1 q is an inflation of the inner twist pG1 , ψ1 q to an extended pure inner twist, and that t∆1 u is the family of transfer factors for the quasisplit group G1 normalized by a fixed Whittaker datum. The construction then gives a family of normalized transfer factors ∆1,K pδ11 , γ1 q for G1 . These functions are of course defined on subsets of products G11 pF q G1 pF q, parametrized by endoscopic data G11 for G1 . Suppose that G1 is an endoscopic datum for G that corresponds to G11 in the manner described above. We can then consider the restriction ∆1,K pδ 1 , γ q
of ∆1,K to the intersection of its domain with the subset G1 pF q GpF q of G11 pF q G1 pF q. Is it a transfer factor for pG, G1 q? Kaletha answers the question affirmatively in [Kal4] with a comparison of the relative transfer factors for pG1 , G11 q and pG, G1 q. He proves that these objects satisfy the identity ∆1 pδ 1 , γ; δ 1 , γ q ∆pδ 1 , γ; δ 1 , γ q,
where pδ 1 , γ q is any second pair in the intersection of the domain of ∆1,K with G1 pF q GpF q. Since ∆1,K pδ 1 , γ q is a transfer factor for pG1 , G11 q, it equals a product of the left hand side of the idenity with a factor in U p1q that is independent of pδ 1 , γ q. It therefore equals the product of the right hand side by the same factor, and is consequently a transfer factor for pG, G1 q. Our description of the normalized transfer factor ∆1,K , which Kaletha constructs for any connected reductive group, is the culmination of this section. We can now go back to the discussion leading up to Lemma 9.3.1. The z-embedding for G is given by the short exact sequence (9.3.8). The
9.4. STATEMENT OF THE LOCAL CLASSIFICATION
545
center Z of G1 defines a second short exact sequence (9.3.14)
ÝÑ
1
Z
ÝÑ
G1
ÝÑ
Gad
ÝÑ
1.
p 1 is simply connected. It equals Since Z is connected, the derived group of G p sc of G. p This becomes the first term in the the simply connected cover G dual short exact sequence
(9.3.15)
1
ÝÑ
p sc G
ÝÑ
p1 G
ÝÑ
Zp
ÝÑ
1,
which obviously bears the same relation to (9.3.11) as (9.3.14) bears to (9.3.8). Suppose that we start with G1 , which represents an endoscopic datum 1 pG , s1, G 1, ξ1q for G1. The corresponding endoscopic datum pG11, s11, G11 , ξ11 q for p 1 . In particular, G1 was defined by taking s11 to be any preimage of s1 in G 1 p p we can take s1 ssc to lie in the subgroup Gsc of G1 . We then define (9.3.16)
ρK p∆, ssc q ∆pδ 1 , γ q∆1,K pδ 1 , γ q1 ,
for any transfer factor ∆ for pG, G1 q. Since ∆1,K pδ, γ q is a transfer factor for pG, G1q, ρK p∆, sscq is independent of pδ1, γ q. Lemma 9.3.2. The function ρp∆, ssc q ρK p∆, ssc q satisfies the two conditions of Lemma 9.3.1. We shall leave the proof for [A28], even though it is not difficult. The main point is the Zpsc -equivariance condition in (9.3.3). This depends on our Γ to Z psc , in the choice of the extension (9.2.3) of the character ζpG from Zpsc case that G is not split. We note here only that in verifying (9.3.3) for ρK p∆, ssc q, one sees that the extension (9.2.3) is actually imposed on us by the chosen 1-cocycle z1 P Z 1 pWK {F , G1 q in pG1 , ψ1 , z1 q. The normalizing section ρK p∆, ssc q allows us to formulate theorems directly for G. It would be interesting to see how unique it is, beyond say, its dependence on the extension (9.2.3) and the transfer factors t∆ u for G . 9.4. Statement of the local classification We shall now formulate our local assertions for inner twists. Taken together, they represent a collective extension of Theorems 1.5.1 and 2.2.1 that encompasses the local refinements from Section 8.4. The first applies to generic parameters φ. It amounts to the local classification of representations conjectured by Langlands. But as we know, the local classification must be supplemented by further assertions for the general parameters ψ in order to establish the local framework for the global classification. The field F will be local throughout this section. As before, G stands for a fixed inner twist pG, ψ q of a quasisplit group G P Ersim pN q over F . We can think of the local classification for G as the specialization of general
546
9. INNER FORMS
r bdd pGq of Langlands parameters in Ψ r pGq. We local theorems to the subset Φ will need to formulate it from the perspective reached in Theorem 8.4.1. p is of the form SOp2n, Cq. Suppose for the moment that the group G As we explained in §8.3, there remains a “Z{2Z symmetry” between the Langlands parameters φ P Φbdd pG q and the associated stable characters on G pF q. In order to sidestep this ambiguity, we shall state the classification for G directly in terms of stable distributions. The set
Φ1dis pG q tφt : φ
P Φr 1bddpGq, t P T pφqu,
defined in the notation of Theorem 8.4.1, indexes certain stable tempered distributions on G pF q. We shall still denote these objects by φ , since they are bijective with the subset Φ1bdd pG q of Langlands parameters in Φbdd pG q r pGq Out r N pGq Z{2Z are trivial. It is this whose stabilizers in the group O r pGq on Φ1 pG q. bijection that is determined only up to the free action of O bdd r q of Φ1 pG q in Φbdd pG q is canonically bijective The complement Φbdd pG bdd with the associated family of stable distributions, a set we will denote by r q. We then write Φdis pG r q > Φ1 pG q. Φdis pG q Φdis pG dis
p is not of the form SOp2n, Cq, Φbdd pG q equals the set Φ r bdd pG q. If G By Theorem 2.2.1(a), it is again canonically bijective with a family Φdis pG q of stable distributions on G pF q. In all cases then, Φdis pG q represents a set tφ u of stable tempered distributions on G pF q. We denote them in the usual way by
f
ÝÑ
f p φ q,
f
P S p G q,
φ
P ΦdispGq.
This set is in fact a basis of the subspace of stable, tempered distributions on G pF q that are admissible, in the natural sense inherited from representation theory. Our interest is in the group G. We write Φdis pGq for the subset of distributions in Φdis pG q that correspond to parameters for G, that is, parameters for G that are G-relevant. We shall sometimes regard Φdis pGq as a separate set, equipped with an injection φ Ñ φ into Φdis pG q. This leaves us free to identify any φ P Φdis pGq with the stable distribution (9.4.1) on GpF q, where f G to G , and
f G p φ q e pG qf p φ q,
f
P H pG q,
Ñ f is the Langlands-Shelstad transfer mapping from
epGq
#
p1qqpGqqpGq, p1qrpGqrpGq,
if F is archimedean, if F is p-adic,
is the Kottwitz sign [K2]. We recall that q pGq equals one-half the dimension of the symmetric space attached to G over F , and rpGq equals the F -rank of the derived group of G (which is G itself in the case at hand).
9.4. STATEMENT OF THE LOCAL CLASSIFICATION
547
We have used the correspondence (1.4.11) (in both its local and global forms) repeatedly throughout the earlier chapters. It carries over to the present setting. If φ belongs to Φdis pG q, we write Sφ for the usual cenp of the image of a corresponding parameter. Then (1.4.11) can tralizer in G be interpreted as a correspondence
pG1, φ1q ÝÑ pφ, sq,
G1
P E pGq, s P Sφ,ss, in which φ1 and φ are now distributions in Φdis pG1 q and Φdis pG q respec-
tively. This version follows from (1.4.11) itself, its variant at the beginning of §8.4, and the definitions above. Similarly, we have a correspondence
pG1, φ1q ÝÑ pφ, sq, G1 P E pGq, s P Sφ,ss, for any φ P Φdis pGq, and its centralizer Sφ Sφ . Suppose that φ belongs to Φdis pGq. Its centralizer group Sφ has a quo-
(9.4.2)
tient S φ , which comes with the extension Sφ,sc by Zpsc defined as at the end of §9.2. The component group Sφ,sc π0 pSφ,sc q of Sφ,sc is in turn an extension of Sφ π0 pS φ q by the quotient Zpφ,sc of Zpsc , again as in §9.2. The local classification for G requires a packet Πφ Πtemp pGq of tempered representations for each φ P Φdis pGq, and an irreducible character x, π y for each π P Πφ . When G G is quasisplit, x, π y is a linear character on the abelian group Sφ , or equivalently, a Zpφ,sc -invariant linear character on Sφ,sc . For general G, however, x, π y will have to be an irreducible character on Sφ,sc that is equivariant under the pullback of ζpG to Zpφ,sc . For example, if φ belongs to the subset
tΦdisc,2pGq tφ P ΦdispGq : |Sφ| 8u of “square integrable” elements, Sφ,sc equals Sφ,sc , and Zpφ,sc equals Zpsc . Since the finite group Sφ,sc is often nonabelian, x, π y will generally be an irreducible character of higher degree, of the kind we described in §9.2. We discussed the Langlands-Shelstad transfer mapping for G in the last 1 is a function in S pG1 q section. For a given transfer factor ∆ P T pG, G1 q, f∆ that can be paired with any of the stable distributions φ1 P Φdis pG1 q. A 1 pφ1 q in f P HpGq. We have implicitly given φ1 thus provides a linear form f∆ agreed to normalize the transfer factors according to the ideas of Kaletha and Kottwitz. We therefore fix the section ρ ρK of Lemma 9.3.1 by the formula (9.3.16) based on the constructions in §9.3. This gives us the normalization f1
(9.4.3)
fK1 ρK p∆, sscq1f∆1
of the Langlands-Shelstad transfer mapping. We are interested in the corresponding normalization
ÝÑ f 1pφ1q ρp∆, sscq1f∆1 pφ1q, of the linear form attached to any φ1 P Φdis pGq. (9.4.4)
f
f
P H p G q,
548
9. INNER FORMS
Recall that the pair p∆, ssc q in (9.4.4) belongs to the covering space Tsc pGq in (9.3.2), or more precisely, to the fibre Tsc pG, G1 q of G1 P E pGq in Tsc pGq. The right hand side of (9.4.4) remains unchanged if ∆ is replaced either by a U p1q-multiple of t∆ or an OutG pG1 q-image of α1 ∆, by the properties of ρp∆, ssc q in Lemma 9.3.1. It is therefore independent of ∆. It does depend on the preimage ssc of the point s¯1 attached to G1 , as in the discussion preceding (9.3.2). For if ssc is replaced by the translate zssc by an element z P Zpψ,sc , the right hand side of (9.4.4) is replaced by its product with ζpG pz q, again by Lemma 9.3.1. The local theorems describe the decomposition of the ζpG -equivariant linear form (9.4.4) into irreducible characters. The local classification for G is formulated as the following theorem. It represents an analogue for G of the assertions of Theorems 1.5.1 and 2.2.1(b), specialized to generic parameters φ but with the local refinements of §8.4. Theorem 9.4.1. Assume that F is local, and that G is an inner twist of the quasisplit group G P Ersim pN q over F . (a) For each φ P Φdis pGq, there is a finite packet Πφ Πtemp pGq of irreducible tempered representations of GpF q, together with a mapping
ÝÑ x, πy, π P Πφ, from Πφ to the set ΠpSφ,sc , ζpG q of irreducible, ζpG -equivariant characters on π
Sφ,sc , with the following property. If ssc is a semisimple element in the group Sφ,sc with images s in Sφ , and xsc in Sφ,sc , and pG1 , φ1 q is the preimage of the pφ, sq under the correspondence (9.4.2), then (9.4.5)
f 1 pφ 1 q
¸
P
π Πφ
xxsc, πyfGpπq,
f
P H pG q .
(b) For any φ P Φdis pGq, the mapping from Πφ to ΠpSpφ,sc , ζpG q is injective, and bijective if F is nonarchimedean. Moreover, any representation in Πtemp pGq occurs in exactly one packet Πφ .
Remarks. 1. As we have explained, the left hand side f 1 pφ1 q of the spectral identity (9.4.5) depends on the underlying quasisplit inner twist of the group G, and the normalizing section ρK p∆, ssc q for the corresponding transfer factors. Once these data have been fixed, it is clear that (9.4.5) characterizes the packet Πφ and the pairing xxsc , π y. We can think of the coefficients xxsc, πy in this formula as “spectral transfer factors”, normalized by the functions ρK p∆, ssc q. 2. The theorem asserts that Πtemp pGq is a disjoint union over φ P Φdis pGq of finite subsets Πφ , which are in turn parametrized by irreducible characters in the sets ΠpSφ,sc , ζpG q. It does therefore represent a classification of the set Πtemp pGq of irreducible tempered representations of GpF q.
9.4. STATEMENT OF THE LOCAL CLASSIFICATION
549
3. If F is archimedean, the assertions of the theorem are no doubt implicit in the results [S7] of Shelstad. She actually establishes the Ranalogue of Theorem 9.4.1 (for general groups) with an abstract normalizing section ρp∆, ssc q, rather than the explicit factor ρK p∆, ssc q of Kaletha. We will need to assume the more precise version of her results (for our group G) in our global proof [A28] of the theorem. 4. As is well known, the endoscopic classification of tempered representations leads directly to the classification of the set ΠpGq of all irreducible representations. (See for example the remarks for quasisplit groups G P Ersim pN q at the beginning of §7.4 or in the discussion following (1.5.1).) For the group G here, we would extend the definition of the set Φdis pGq to a larger set Φdis pGq that is (noncanonically) bijective with the set ΦpGq all Langlands parameters. Theorem 9.4.1 implies that ΠpGq is a disjoint union over φ P Φ pGq of finite packets of Langlands quotients Πφ that are in canonical bijection with associated packets Πφ in Πtemp pGq given by the theorem and a suitable correspondence φ Ñ φ from Φdis pGq to Φdis pGq. However, the identity (9.4.5) is no longer valid in this case. (We recall that, alternatively, the identity (9.4.5) would remain valid if we defined Πφ as a packet of reducible standard representations, rather than irreducible Langlands quotients.) This circumstance can be seen as a local reflection of the need for a set Ψdis pGq that lies between Φdis pGq and Ψdis pGq. The results of §8.4 apply only to generic parameters φ. We have therer pGq of (O r pGq-orbits of) general fore to be content to work here with set Ψ parameters for G, which we introduced in §9.2. However, there is a problem that arises with this set. It concerns the transfer of automorphisms discussed in §9.1, and occurs specifically for indices in the local classification that contain subdiagrams (9.1.5) and (9.1.6). For in those cases, we cannot r unit pG q that we would use to formulate the define a G-analogue of the set Π general local theorem. Accordingly, we shall say that the inner twist G is symmetric if any outer automorphism θ of G transfers to an F -automorphism θ of G. This is automatic if G is of type Bn or Cn . For the case of type Dn , it means that G satisfies the equivalent conditions of Lemma 9.1.1. In general, then, G is symmetric if and only if its index from the tables of §9.1 does not contain a subdiagram (9.1.5) or (9.1.6). Assume that G is symmetric. If it is of type Dn , we write θr for a fixed F -automorphism of G of order 2 obtained by transfer of the automorphism θr of G . If it is of type Bn or Cn , we simply set θr 1. In all cases, we then r r pGq for the subalgebra of θ-invariant write H functions in the Hecke algebra r r pG q, Π r unit pGq and Π r temp pGq for the families of θHpGq. We can also write Π orbits in the sets ΠpGq, Πunit pGq and Πtemp pGq of irreducible representations r p G q, Ψ r pG q of GpF q. We are of course already familiar with the quotients Φ r and Φbdd pGq of the parameter sets ΦpGq, ΨpGq and Φbdd pGq. They are the
550
9. INNER FORMS
pG p , which are defined for orbits for the dual automorphism θr θr of G all of our G. We note that the symmetric, stable Hecke algebra SrpG q for p
p
the quasisplit group G can be defined in terms of either θr (and Langlands parameters φ ) or the original automorphism θr of G (and stable conjugacy classes δ ). Similarly, the symmetric algebras SrpG1 q for endoscopic groups G1 , which played such a key role in the first eight chapters, are defined in 1 terms of either the automorphism θr1 θrG of G1 or its dual. The F -automorphism θr of G is not uniquely determined by the original outer automorphism θr of G . Any choice of θr can always be modified by composition with an inner automorphism in Int Gad pF q . The normalizing section ρK is also not uniquely determined, and can be modified in a similar fashion. We will assume that they are both chosen so that for any G1 P E pGq, the transfer mapping f Ñ f 1 satisfies the identity p
pf θrq f 1 θr1,
P H p G q, r pGq into SrpG1 q. The linear and therefore takes the symmetric subalgebra H (9.4.6) form
f 1 pψ 1 q,
f
P HrpGq, r pGq is then defined for any θr1 -orbit ψ 1 P Ψ r pG 1 q. on H f
I am labelling the following assertion as a conjecture, since I have not written down the details of its proof. However, the methods of proof are parallel to those of the quasisplit case treated in Chapter 7, and should pose no new problems.
Conjecture 9.4.2. Assume that F is local, and that G is a symmetric inner twist of the quasisplit group G P Ersim pN q over F . Assume also that θr and ρK are fixed so that for any G1 P E pGq, the transfer mapping f Ñ f 1 r pGq, there is a finite set Π r ψ over satisfies (9.4.6). Then for each ψ P Ψ r unit pGq, together with a mapping Π
ÝÑ x, πy, π P Πr ψ , r ψ to the set ΠpSψ,sc , ζpG q, such that from Π ¸ (9.4.7) f 1 pψ 1 q xsψ xsc, πyfGpπq, f P HrpGq, π
P
rψ π Π
for ssc P Sψ,sc , s P Sψ , and xsc P Sψ,sc as in the statement of Theorem 9.4.1, and pG1 , ψ 1 q the preimage of pψ, sq. r ψ are Remarks. 1. If F is archimedean, we again expect that the packets Π special cases for G of the general packets constructed in [ABV]. However, this is not presently known. r ψ are among those 2. If F is p-adic, at least some of the packets Π constructed by Moeglin, by quite different methods. As in the the quasisplit
9.4. STATEMENT OF THE LOCAL CLASSIFICATION
551
case, she proves the fundamental theorem that these packets are multiplicity r pGq [M4]. free, and are therefore simply subsets of Π 3. Observe that, as in the earlier quasisplit case, the theorem includes r the assertion the irreducible representations that comprise the θ-orbits in the r packets Πψ are unitary. This is a point of difference between the assertion here and the results described in Remarks 1 and 2. Another is that the proof of Conjecture 9.4.2 includes an extension to G of the local intertwining relation, which will in turn be required for the interpretation of the global trace formula. 4. The method for proving Conjecture 9.4.2 is global (as is that of Theorem 9.4.1). For this reason, both proofs are conditional on a global hypothesis that has not yet been established for the local normalizations of Kaletha. We shall state the hypothesis formally in the next section. r bdd pGq of Ψ r pGq, we remind ourselves that 5. If φ ψ lies in the subset Φ the point sφ (regarded as an element in either Sφ or Sφ,sc ) is trivial. The spectral identity (9.4.7) then reduces formally to its analogue (9.4.5) from the previous theorem, although the terms in the earlier identity of course have slightly different meanings. The conjecture does not account for the groups G that are not symmetric. These are the inner forms of a split even orthogonal group G SOpN q that do not extend to inner forms of the full orthogonal group OpN q. The case is of considerable interest, for the theory of Shimura varieties, among other things. Theorem 9.4.1 does apply to these groups, and gives a construction of their tempered L-packets Πφ . It would obviously be desirable to modify Conjecture 9.4.2 so that it applies as well. We shall describe a strategy for doing so, with the hope of carrying it out in [A28]. The problem is that we cannot define the symmetric Hecke subalgebra r pGq if G itself is not symmetric, since θr is then not an automorphism of G. H If f is a general function in HpGq, its endoscopic transfer f 1 does not belong r pG1 q, and cannot be paired with to the stable symmetric image SrpG1 q of H 1 the stable linear form ψ given by Theorem 2.2.1(a). In other words, the left r bdd pGq hand side f 1 pψ 1 q of (9.4.7) is undefined. It φ belongs to the subset Φ 1 1 r pGq, we do have associated linear forms φ P Φdis pG q on the full space of Ψ S pG1 q. This was one of the main points of §8.4, and is the reason that the left hand side f 1 pφ1 q of (9.4.5) is defined. But we have not extended the methods r pGq. Let us consider instead how we of §8.4 to general parameters ψ P Ψ might extend Conjecture 9.4.2, as an assertion for symmetric objects. We can start by trying to reconcile the assertion of Conjecture 9.4.2 with that of Theorem 9.4.1(a), in the case of a parameter ψ φ that is generic. We should first say a word about the transform represented by the formula (9.4.5) of Theorem 9.4.1. It maps a given function f P HpGq to the function fGE pG1 , φ1 q fGE pφ, ssc q,
552
9. INNER FORMS
defined by either of the two sides of the identity ¸
f 1 pφ 1 q
P
xxsc, xyfGpπq.
π Πφ
Recall that φ belongs to Φdis pGq, that ssc is a semisimple element in Sφ,sc with images s P Sφ and xsc in Sφ,sc , and that pG1 , φ1 q is the preimage of pφ, sq. As the notation suggests, the image fGE of f depends only on the projection fG of f onto the invariant Hecke algebra I pGq. Let us write I E pGq for the image of I pGq under this mapping, regarded as a space of functions fGE of either the variable pG1 , φ1 q or the variable pφ, xsc q. Using the trace Paley-Wiener theorem for G, one can describe I E pGq explicitly in terms of the appropriate Paley-Wiener spaces. The mapping
P I p G q, then becomes a topological isomorphism from I pGq onto I E pGq.
(9.4.8)
E IG : fG
ÝÑ
fGE ,
fG
How does the mapping (9.4.8) relate to automorphisms? If G is symmetr It acts on GpF q, and allows us to define ric, we have the F -automorphism θ. r pGq above. It also descends to a linear (topological) the symmetric Hecke H automorphism θrG of I pGq such that
pf θr1qG θrGfG,
f
P H p G q.
This allows us to define the symmetric invariant Hecke algebra IrpGq, as for r pGq in I pGq. But what is pertinent to the discussion example the image of H
p G p , and here is that we also have the dual automorphism θr θr of G this exists in all cases. It acts on the set Φbdd pG q of bounded Langlands parameters for G , and therefore on the set of stable distributions Φdis pG q. p
p
Suppose that G is symmetric. Then θr stabilizes the image of Φdis pGq in Φdis pG q. It therefore acts both on the set of pairs pG1 , φ1 q and the set of pairs pφ, ssc q, either of which serves as the domain for the space of functions E of I E pGq. One sees from (9.4.6) I E pGq. This gives a linear automorphism θrG E is the transfer of the automorphism θ rG θr1 to I E pGq, which is to that θrG G say that E θrG pIGE qθrG1pIGE q1. p
We emphasize that on the right hand side of this relation, θrG is defined in E on the left hand terms of the automorphism θr of G, while its counterpart θrG p
p The relation, which side is obtained from the dual automorphism θr of G. is only for motivation, allows us to compare Conjecture 9.4.2 and Theorem 9.4.1 in the special cases that ψ φ is generic and G is symmetric. Suppose now that G is not symmetric, the case at which the discussion is aimed. We would still like to define a symmetric subspace IrpGq of I pGq. According to Remark 1 following Lemma 9.1.1, we have an F -isomorphism _ r θ : G Ñ G_ of groups, in which G_ represents a second inner twist pG_, ψ_q of G that is not isomorphic (as an inner twist) to pG, ψq. The
9.4. STATEMENT OF THE LOCAL CLASSIFICATION
553
remarks above can be extended to the isomorphism θr_ . They lead to topological isomorphisms _ : I pG_ q ÝÑ I pGq θrG and _,E : I E pGq ÝÑ I E pG_ q, θrG such that
_,E θrG
(9.4.9)
pIGE _ qpθrG_q1pIGE q1.
p with G p by means of the dual It has been our convention to identify G p p _ with G p by using the isomorphism ψ of ψ. Let us agree also to identify G
p to G p_ . dual θr_ of θr_ . What remains here is dual isomorphism ψp_ from G p
r of θr . It gives pG p G p _ , it equals the dual θp As an automorphism of G rise to a topological isomorphism
,E : I E pG_ q θrG
ÝÑ
I E p G q,
since the dual automorphism θr is all that is needed to apply the original construction above. The composition p
E θrG
(9.4.10)
θrG,E θrG_,E
is then a topological linear automorphism of I E pGq. It in turn gives a topological linear automorphism
pIGE q1pθrGE qpIGE q : I pGq ÝÑ I pGq of the original invariant Hecke algebra I pGq. (9.4.11)
θrG
_,E and It is not hard to see explicitly how the two isomorphisms θrG ,E θrG transform functions in I E pG_ q and I E pGq on their respective domains tpφ_, x_ qu and tpφ, xscqu. This exercise, which we leave to the reader, leads sc
E and θ rG . In particular, one to concrete descriptions of the automorphisms θrG sees that θrG acts through an involution
π
ÝÑ
r θπ,
π
P ΠtemppGq
on Πtemp pGq that is given explicitly in terms of the endoscopic classification of Theorem 9.4.1. We can therefore define the symmetric subalgebra IrpGq r temp pGq of orbits in Πtemp pGq, of I pGq, and the corresponding quotient Π even though we do not have an underlying F -automorphism θr of G. This is what we were looking for. It allows us to extend the assertion of Conjecture 9.4.2 to the nonsymmetric group G simply by using an invariant function r pG q . fG P IrpGq in place of f P H The extension of Conjecture 9.4.2 will also require a modification for r unit pGq, since the involution θr described above is defined only for the set Π tempered representations. The simplest solution would be to postulate a natural extension of θr to Πunit pGq as part of the conjecture. However, this could be quite hard to prove without further hypotheses. The methods for
554
9. INNER FORMS
proving Conjecture 9.4.2, as it is stated for symmetric groups, will no doubt lead some result for the nonsymmetric group G, once we have the symmetric r ψ of orbits of irreducible invariant Hecke algebra IrpGq. But the packet Π unitary representations might then have to be replaced by something weaker. Could we expect to be able to define an involution on Πunit pGq (or Πtemp pGq) directly? There is of course a natural involution on the full set ΠpGq, the contragredient
ÝÑ
π_,
P ΠpG q . Is it possible to relate it to the involution on Πunit pGq we seek, and in particular, the involution θr on Πtemp pGq we have just described? D. Prasad [Pr] π
π
has given a conjectural description of the contragredient involution for any group in terms of the Langlands parametrization, which of course is also still conjectural in general. Kaletha [Kal3] has recently proved Prasad’s conjecture for general real groups (following earlier work of Adams and Vogan [AV]), and for quasisplit orthogonal and symplectic groups (using the version of the Langlands parametrization from §8.4). We are still thinking of a nonsymmetric inner twist G, necessarily of type Dn , for which there is no F -automorphism θr we can use to define our involution. The answer to the questions then depends on the parity of n. Suppose that n is odd. The index of G then has a subdiagram (9.1.6). In this case, θr equals the opposition involution [Ti, 1.5.1] (also called the p G p . Chevalley or duality involution), as an outer automorphism of G Prasad’s conjecture describes the contragredient involution on ΠpGq in terms of the action of the opposition involution on I E pGq. In our notation, it amounts to an assertion p
r θπ
ηπ_,
π
P ΠtemppGq,
where η is an explicit element in the finite group IntF pGq{Int GpF q
that is independent of π, and
pηπ_qpxq π_
η 1 pxq ,
x P G p A q.
Kaletha’s proof of Prasad’s conjecture in the p-adic case applies only to the quasisplit group G . His methods could presumably be combined with Theorem 9.4.1 to establish the conjecture for G. This would give a direct formula for our involution in terms of the contragredient of π. Its extension from Πtemp pGq to ΠpGq, a set that of course contains Πunit pGq, ought then to follow from the familiar properties of Langlands quotients. It thus seems likely that Conjecture 9.4.2 and its proof will extend directly to the nonsymmetric groups with n odd. Suppose now that n is even. The index of G then has a subdiagam of type (9.1.5). This is the unique case among all inner twists of G in which p the subset ΦpGq of Langlands parameters in ΦpG q is not θr -stable. In this
9.4. STATEMENT OF THE LOCAL CLASSIFICATION
555
pG p , since the latter case, θr is not equal to the opposition involution on G is trivial (as an outer automorphism). It follows from Prasad’s conjecture that our involution θr on Πtemp pGq has nothing to do with the contragredient. Is there anything more that one can say? I have no answer, other than to recall the example with n 2 from the end of §9.1. Set p
G G1
Spp2q Q1{t1u
in the earlier notation, and suppose that π
σbτ
is a representation in Π2 pGq, with factors σ
P Π2
Spp2q and τ
P Π2 pQ 1 q
that both have trivial central character. The automorphism θr acts on p
pG p G
Spp2, Cq Spp2, Cq{t1u by interchanging the two Spp2, Cq factors. It is then not hard to see that r τ b σ , θπ where τ is the Jacquet-Langlands “lift” of τ from Q1 to Spp2q, and σ is the Jacquet-Langlands “descent” of σ from Spp2q to Q1 . This is a rather
subtle operation. It is hard to see how its generalization could lead to a r direct way of characterizing the involution θ. We have discussed at some length how we might extend Conjecture 9.4.2 to all inner twists G of G . We will now close the section with a few words on the possibility of generalizing Theorem 9.4.1(a) to all parameters for G. For this, we may as well just assume that G is of type Dn , since the answer is otherwise given by Conjecture 9.4.2. Theorem 9.4.1 is based on the results of §8.4 for generic parameters r pG q. To generalize the theorem, one would need to establish similar φ P Φ r pG q. For the moment, we assume results for nongeneric parameters ψ P Ψ r that G G is quasisplit. Then G P Esim p2nq is a quasisplit, even orthogonal group over the general local field F . r unit pGq is the set of orbits in Πunit pGq of the group Recall that Π r pGq Out r N pGq Z{2Z. O
r of Π r unit pGq has a preimage Π in Πunit pGq. More generally, Any subset Π r is a set over Π r unit pGq, we can write Π for the corresponding set over if Π r pGq, the packet Π r ψ of Theorem 1.5.1(a) (which Πunit pGq. For any ψ P Ψ r unit pGq. Its preimage Πψ over was constructed in §7.4) is a finite set over Π r ψ and Πunit pGq over Π r unit pGq. The problem Πunit pGq is the fibre product of Π is to construct a section
(9.4.12)
rψ Π
ÝÑ
Πψ
that is compatible with endoscopic transfer for G. This is essentially the content of the assertions of Theorem 8.4.1 of §8.4, for the case that ψ φ
556
9. INNER FORMS
r pGq of generic local parameters. In the generic case, it lies in the subset Φ r φ that the is an easy consequence of the structure of the original packet Π r pGq. More section is uniquely determined up to the action of the group O r p G q. precisely, the set of such sections is a single orbit under the group O r pGq, one can construct a section (9.4.12) by For a general parameter ψ P Ψ following the global proof of Theorem 8.4.1. However, its uniqueness requires r ψ . This is a serious matter, since more information about the packet Π without the uniqueness, the global applications of the refinements (9.4.12) will fail. It is not hard to impose conditions on ψ that would imply the uniqueness of (9.4.12). Among these is a requirement that elements in the packet r ψ occur with multiplicity 1, or in other words, that Π r ψ be a subset of Π r unit pGq. For p-adic F , Moeglin [M4] has established that any packet Π rψ Π has multiplicity 1. This is a deep theorem, which describes a fundamental structural property of the packets. But if F R, the property has so far been elusive, even though one would expect it to be true. In any case, we can formally introduce a canonical set
Ψdis pGq
º
Ψdis pψ q,
ψ
r pGq ranges over parameters that satisfy the conditions of uniquewhere ψ P Ψ ness, and Ψdis pψ q is the set (of order 1 or 2) of OpGq-orbits of sections (9.4.12). For any ψ in Ψdis pGq, we define Πψ to be the image of ψ in Πψ , regarded as a packet of invariant linear forms on HpGq. The main point is that Πψ satisfies the appropriate refinements of the endoscopic character r ψ . In particular, the packet comes with a canonical stable relations for Π distribution
f
ÝÑ
f G pψ q
¸
P
π Πψ
xsψ , πyfGpπq,
f
P H pG q ,
which we identity with ψ itself. We can now go back to the setting at the beginning of the section. We let G revert to a general inner twist of the quasisplit group G P Ersim pN q of type Dn . We can then write Ψdis pGq for the subset of distributions in Ψdis pG q, which we now denote by ψ, that correspond to G-relevant parameters for G . The various definitions that precede the statement of Theorem 9.4.1 extend to elements ψ P Ψdis pGq. So does the assertion of Theorem 9.4.1(a), but with two obvious changes. The packet Πφ Πtemp pGq is replaced by a packet Πψ Πunit pGq, and the identity (9.4.5) becomes f 1 pψ 1 q
¸
P
π Πψ
xsψ xsc, πyfGpπq,
f
P H pG q .
Let us call this new assertion a conjecture, though the techniques for proving it should again be available. Keep in mind, however, that the set Ψdis pGq is
9.5. STATEMENT OF A GLOBAL CLASSIFICATION
557
r p G q, defined by conditions that have not been proved for all parameters in Ψ so the result would still have significant limitations. This completes the discussion. We have described possible extensions of Conjecture 9.4.2 and Theorem 9.4.1(a). In principle, a generalization of the theorem would be stronger than the extension of the conjecture. However, there are reasons to consider both. Some extension of Conjecture 9.4.2 ought to be within reach, as I have suggested above, while a complete generalization of Theorem 9.4.1(a) will require information that is not presently accessible. Moreover, the characters treated in Conjecture 9.4.2 satisfy local reciprocity laws that are more elementary, to the extent that they are explicitly related to twisted characters on GLpN, F q. The characters of Theorem 9.4.1 are defined in terms of stable characters on G pF q, which one can see from §8.4 are in general only indirectly related to twisted characters on GLpN, F q. We reiterate that the discussion has all been directed at groups of type Dn . If G is of type Bn or Cn , there is nothing further to say. In these cases, Conjecture 9.4.2 already represents the generalization of r pGq to the larger set ΨpGq Ψ r pG q . Theorem 9.4.1 from the set ΦpGq Φ
9.5. Statement of a global classification We come finally to the global setting. We shall state global analogues of Theorem 9.4.1 and Conjecture 9.4.2. The first is an extension of Theorem 8.4.2. It represents a refined classification for the subspace of the discrete spectrum attached to the global Langlands parameters φ. The full classification should be governed by the general parameters ψ, objects whose origins are unambiguously global. We shall formulate it as a conjecture that, as in the local case, applies to most but not all groups. The field F will be global throughout this final section. We fix a quasisplit group G P Ersim pN q. Our general interest is then in global inner twists pG, ψq of G. We will have to formulate the full classification of the global conjecture in terms of the locally symmetric subalgebra of the global Hecke algebra. This of course is analogous to Theorem 1.5.2. But for the inner forms G here, we will consequently be forced to restrict ourselves to groups that are locally symmetric, in the sense suggested by the last section. The global assertions will be formulated under the assumption that the global hypothesis of [L10, p. 149] (which was established in [LS1, §6.4]) extends appropriately to the local normalized transfer factors of Kaletha. We have already used the global results of [LS1] in our study of quasisplit groups. The reader will recall that in §3.2, we normalized the relevant local transfer factors by Whittaker data obtained by localization of a fixed global Whittaker datum. What we need here is a product formula for the normalizing sections (9.5.1)
ρv p∆v , ssc,v q,
p∆v , ssc,v q P TscpGv q,
558
9. INNER FORMS
attached to the localizations of a global inner twist pG, ψ q. The points
p∆v , ssc,v q pG1v , ∆v , ssc,v q depend implicitly on local endoscopic data G1v P E pGv q.
They could, for
example, be the localizations of a fixed global point
P Ssc1 , in which ∆ is the canonical adelic transfer factor for pG, G1 q [LS1, (6.3)], (9.5.2)
p∆, sscq pG1, ∆, sscq,
G1
P E pG q,
ssc
[KS, (6.3)].
Hypothesis 9.5.1. Suppose that pG, ψ q is a global inner twist of G over F . Then one can choose the normalizing sections (9.5.1) for the completions pGv , ψv q of pG, ψq so that (9.5.3)
¹
ρv p∆v , ssc,v q 1,
v
if p∆v , ssc,v q is the localization of a global point (9.5.2). One would expect the sections to equal 1 at places v with Gv quasisplit, so in particular, the product (9.5.3) would be over a finite set. In general, there are four sets of canonical data that ought to be part of the solution: the global transfer factor ∆ for pG, G1 q, its localizations ∆v at the quasisplit places v, the global transfer factor ∆ for pG , G1 q, and its localizations ∆v at all places. I have not tried to analyze these objects with the methods of [LS1, (4.2), (6.3)–(6.4)], partly for a lack of understanding of how to use the results of Kaletha at the remaining places v at which G is not quasisplit. I do not know whether Kaletha or Kottwitz have considered the global situation, but it seems reasonable to suppose that the questions will be resolved by the time the article [A28] has been written. In any case, we assume from now on that Hypothesis 9.5.1 holds. More precisely, we assume implicitly that the normalizing sections for the localizations Gv of G have been chosen to satisfy (9.5.3). The local assertions of Theorem 9.4.1 and Conjecture 9.4.2 are also to be regarded as conditional on the hypothesis, since as we noted at the time, their proofs will be by global means. r pGq, with semisimple points s P Sψ , Consider a global parameter ψ P Ψ and ssc P Sψ,sc that project to the same element in Sψ . Let pG1 , φ1 q be the preimage of the pair pψ, sq. On one hand, the global transfer mapping
1 pψ 1 q, f 1 pψ 1 q f∆
f
P H pG q,
from G to G1 is defined by the canonical adelic transfer factor ∆. On the other, we have agreed to define the local transfer mapping
1 pψ 1 q, fv1 pψv1 q fv,ρ v v
fv
P H pG v q ,
in terms of a fixed transfer section ρv . We are assuming that as v varies, the local transfer sections tρv u satisfy the condition (9.5.3) of Hypothesis 9.5.1.
9.5. STATEMENT OF A GLOBAL CLASSIFICATION
It then follows from (9.3.5) that f1
(9.5.4)
¹
559
1 , fv,ρ v
v
±
for a product f fv . In particular, although the factors on the right depend on the point ssc , the product does not. Before describing the finer structure of the discrete spectrum for G, we need a coarse decomposition analogous to that of the quasisplit case in Chapter 3. Extending the definition of §3.4 prior to Corollary 3.4.3, we set L2disc,ψ GpF qzGpAq
L2disc,tpψq,cpψq GpF qzGpAq , r pN q. for our inner twist G and a general global parameter ψ P Ψ
The next proposition is the generalization to G of the assertion (3.4.4) of Corollary 3.4.3. It is proved by making modest changes in the methods of §3.4–§3.5. Proposition 9.5.2. The discrete spectrum for our inner twist pG, ψ q of G P Ersim pN q over F has the following decomposition L2disc GpF qzGpAq
à
P p q
r N ψ Ψ
L2disc,ψ GpF qzGpAq .
What remains to describe is an explicit decomposition of any of the 2 invariant subspaces Ldisc,ψ GpF qzGpAq into irreducible representations. As
r pGq of in the last section, our first main assertion applies to the subset Φ generic parameters. We need to formulate it from the perspective of the global Theorem 8.4.2. r pG q for the quasisplit group G , we For any global parameter φ P Φ have the set T pφ q defined in §8.3. This leads to the mapping
t
ÝÑ
φt
â v
φv,tv ,
t P T pφ q ,
from T pφ q to the space of linear forms on the global Hecke algebra S pG q, defined prior to the statement of Theorem 8.4.2. Following our local convention from the last section, we now write ΦpG q φt : φ
P Φr pGq, t P T pφq
(
.
Then ΦpG q represents a family of stable linear forms on the global Hecke algebra HpG q, equipped with mappings φt Ñ φv,tv to families Φdis pGv q of stable linear forms on the local Hecke algebras HpGv q. It is to be regarded as a refined substitute for our set of global Langlands parameters for G , r pG q that have served us until now represent since elements in the set Φ r pGq-orbits of parameters. There is no need to append the subscript only O “dis” to the new set, as there is no set of true Langlands parameters to distinguish it from, the global Langlands group LF not having been defined. These remarks are of course only of interest in the case that G is of type r pG q are otherwise the same. Dn , since the global sets ΦpG q and Φ
560
9. INNER FORMS
r p Gq For our global inner twist G, we define ΦpGq to be the preimage of Φ in ΦpG q. It is therefore the subset of global parameters for G that are locally G-relevant. We then have localization mappings
φ
ÝÑ
φ P Φ pG q ,
φv ,
from ΦpGq to the local sets Φdis pGv q. Recall from Remark 4 following the statement of Theorem 9.4.1 that Φdis pGv q is a set of stable distributions on GpFv q that is in (noncanonical) bijection with the set ΦpGv q of all local Langlands parameters. Since φv comes from a parameter φ in the global subset ΦpGq of ΦpG q, it actually lies in the local subset Φdis,unit pGv q Φdis pGv q X Φdis,unit pGv q
of Φdis pGv q attached to representations of GLpN, Fv q that are unitary. (See the definitions following the statement of Theorem 1.5.1.) The point of this distinction is that elements in Φdis,unit pGv q should satisfy a version of Conjecture 8.3.1, which would imply that representations in the packet Πφv are irreducible (and unitary). A parameter φ P ΦpGq has its global centralizer quotient Sφ , with its extension Sφ,sc by the quotient Zpφ,sc of Zpsc . It too has localization mappings xsc
ÝÑ
xsc,v ,
xsc
P Sφ,sc,
from Sφ,sc to the local centralizer extensions Sφv ,sc . Given Theorem 9.4.1, we can define the global packet Πφ
!
π
â
πv : πv
v
P Πφ , x, πv y 1 for almost all v v
of representations of GpAq attached to φ. Any representation π this packet then determines a character (9.5.5)
xx, πy
¹
)
xxsc,v , πv y,
xsc
v
 πv in
P Sφ,sc,
on Sφ,sc . It follows easily from Hypothesis 9.5.1 that, as the notation suggests, xx, π y depends only on the image x of xsc in Sφ . Theorem 9.5.3. Assume that F is global, and that G is an inner twist of the quasisplit group G P Ersim pN q over F . Suppose also that ψ φr lies in the subset r 2 p Gq Φ r p Gq X Φ r ell pN q Φ
r pN q. Then there is an of generic, square-integrable parameters for G in Ψ HpGq-module isomorphism
(9.5.6)
L2disc,ψ GpF qzGpAq
à
à
P p q P
mφ p1, π qπ,
φ Φ ψ π Πφ
where Φpψ q is the preimage of ψ in ΦpGq (of order 1 or 2), and mφ p1, π q |Sφ |1
¸
P
x Sφ
xx, πy
9.5. STATEMENT OF A GLOBAL CLASSIFICATION
561
is the multiplicity of the trivial representation of Sφ in the irreducible decomposition of the character x, π y. The theorem applies to the generic part of the discrete spectrum. This is the subspace of the automorphic discrete spectrum of G whose irreducible constituents are believed to satisfy the analogue of Ramanujan’s conjecture, in the sense that they are locally tempered. The theorem extends the generic case of Theorem 1.5.2 in two ways. It generalizes the earlier assertion to inner twists G of the quasisplit group G , and it refines it to the full global Hecke algebra HpGq. This is of course parallel to the relation that Theorem 9.4.1 bears to the generic part of the assertion of Theorem 2.2.1(b). For general parameters ψ, we need a global version of Conjecture 9.4.2. It applies to the case of a global inner twist G that is locally symmetric. In other words, each completion Gv of G is symmetric, in the sense of the last section. As in the local case, this is automatic if G is type Bn or Cn . If G is of type Dn , it means that each completion Gv of G satisfies the equivalent conditions (i)–(iv) of Lemma 9.1.1. From the condition (iii) and the injectivity of the mapping H 1 pF, Gad q
ÝÑ
à v
H 1 pF, Gad q,
we see that G itself satisfies the four equivalent conditions of the lemma. In particular, G is locally symmetric if and only if the automorphism θr of G transfers to an F -automorphism θr of G, in the sense of the condition (i) of Lemma 9.1.1. Assume that G is locally symmetric. For a given choice of F -automorphism θr of G (in which we take θr 1 if G is of type Bn or Cn ), we have the usual subspace r pG q H
â v
r p Gv q H
p q
of locally θrv -invariant functions in the global Hecke algebra H G . We also have the the quotient
r pG q Π
¹
r p Gv q Π
v
of (restricted tensor products of) local θrv -orbits of representations in ΠpGq, r unit pGq and Π r temp pGq of the subsets as well as corresponding quotients Π Πunit pGq and Πtemp pGq. As always, we obtain a well defined pairing
pf, πq ÝÑ
fG pπ q tr π pf q ,
P HrpGq, π P Πr pGq, in which the image is independent of the representative in ΠpGq of the orbit f
π.
The next assertion will again be stated as a conjecture, even though I do not expect any serious difficulties with the proof. However, there is one further point to mention. We are assuming that the local normalizing sections (9.5.1) are chosen to satisfy Hypothesis 9.5.1. We want to assume
562
9. INNER FORMS
in addition that they are compatible with the localizations θrv of the chosen global F -automorphism θrv , in the sense that the local transfer condition (9.4.6) holds for each v. I have not verified that this is possible, so it will have to be regarded as part of the conjecture. The assertion is formulated in terms of the objects that we now know well. For our locally symmetric inner twist G, we have the localization mappings r p G q, ψ ÝÑ ψv , ψPΨ
r pGq of general global parameters to the local sets Ψ r from the set Ψ unit pGv q, r pGq has its centralizer quotient Sψ , defined as in §1.5. A parameter ψ P Ψ with localization mappings
xsc
ÝÑ
xsc,v ,
xsc
P Sψ,sc,
from the extension Sψ,sc of Sψ to the corresponding local extensions Sψv ,sc . Assuming Conjecture 9.4.2, we form the global packet rψ Π
!
π
â
πv : πv
v
P Πr ψ , x, πv y 1 for almost all v
)
v
r of (adelic  orbits of) representations in Πunit pGq attached to ψ. Any element πv in this packet then determines a character π v
(9.5.7)
xx, πy
¹
xxsc,v , πv y,
xsc
v
P Sψ,sc,
on Sψ,sc , which depends only on the image x of xsc in Sψ . Conjecture 9.5.4. Assume that F is global, and that G is a locally symmetric inner twist of the quasisplit group G P Ersim pN q over F . Assume also that the local transfer sections of Hypothesis 9.5.1 are chosen so that they are compatible with the localizations θrv of the fixed F -automorphism θr r pGq-module isomorphism of G. Then there is an H (9.5.8)
L2disc GpF qzGpAq
where mψ t1, 2u and εψ : Sψ Theorem 1.5.2, and (9.5.9)
à
à
P p q P
r2 G π Π rψ ψ Ψ
mψ mψ pεψ , π q π,
Ñ t1u are as defined after the statement of
mψ pεψ , π q |Sψ |1
¸
P
εψ pxqxx, π y
x Sψ
is the multiplicity of the one-dimensional representation εψ of Sψ in the irreducible decomposition of the character x, π y. The proof of the conjecture will be considerably simpler that that of Theorem 1.5.2. This is because we already have the stable multiplicity formula of Theorem 4.1.2 for G . In particular, we can apply Corollary 4.1.3 to the quasisplit endoscopic groups G1 for G . We are also free to apply Theorem 1.5.3, as well as the sign Lemmas 4.3.1 and 4.4.1, since they
9.5. STATEMENT OF A GLOBAL CLASSIFICATION
563
are the same for G and G . The proof should consequently amount to an adaptation of the earlier parts of the discussion of §4.3 and §4.4, or if one prefers, the specialization to G of the general arguments of §4.8. We remind ourselves that the assertion of Conjecture 9.5.4 applies to all global inner twists G of type Bn or Cn . It also applies to any G of type Dn whose local indices at places v (which we can of course restrict to the finite set of places at which G is quasisplit) do not contain subdiagrams (9.1.5) or (9.1.6). The tables in §9.1, together with the global reciprocity law (9.1.4), allow us to quantify the remaining exceptional set of global inner twists. Conjecture 9.5.4 places any G in its complement on the same footing as the quasisplit group G following the proof of Theorem 1.5.2. Theorem 9.5.3 and Conjecture 9.5.4 of course require their local analogues, Theorem 9.4.1 and Conjecture 9.4.2, even to state. At the end of the last section, we discussed how these local assertions might possibly be strengthened to include all parameters for all inner twists. This would govern how the global assertions could also be strengthened. The supplementary discussion for Conjecture 9.4.2 concerned how to extend the assertion (and its expected proof) to inner twists Gv that are not symmetric. The corresponding extension of Conjecture 9.5.4 would then apply to all global inner twists G whose localizations Gv either are locally symmetric, or satisfy the extended local assertion. The closing remarks on Theorem 9.4.1 concerned the possibility of generalizing part (a) of the theorem (and its proof) to local r pGv q. The corresponding generalization of Theoparameters ψv in the set Ψ r pGq whose localizations rem 9.5.3 would apply to all global parameters ψ P Ψ ψv each satisfy the generalized local assertion. In particular, any ψv would r v q of the subset need either to lie in the complement ΨpG r 1 pGv q ψv Ψ
P Ψr pGv q :
mψv
2
(
r pGv q, or to satisfy conditions that ensure an extension of the local results of Ψ of §8.4. An interesting aspect of global inner twists G is the presence of higher multiplicities in the discrete spectrum. Consider the generic case represented by Theorem 9.5.3. It consists of an explicit multiplicity formula (9.5.6) for the contribution to the automorphic discrete spectrum of a generic global parameter
P EsimpNiq, r 2 pGq, or if one prefers, of the automorphic family c cpψ q of semisimple in Φ ψ
ψ1 `
`
ψr ,
ψi
P Φr simpGiq,
Gi
conjugacy classes in L G attached to any such ψ. The right hand side of the formula is a double (direct) sum over φ P Φpψ q and π P Πφ . Let us review the terms in these sums explicitly. The sum over φ in (9.5.6) is quite similar to its quasisplit analogue discussed in §8.3. The indexing set Φpψ q is of order 1 unless G is of type Dn and the degrees Ni of the irreducible constituents ψi of ψ are all even, in
564
9. INNER FORMS
which case Φpψ q has order 2. In this latter case, the sum over φ contributes a factor of 2 to the multiplicities of the interior terms unless one of the associated local sets Φpψv q has order 2. If some Φpψv q does have order 2, the set Φpψ q indexes two distinct global L-packets Πφ , and therefore does not increase the inner multiplicities. The conditions for a local set Φpψv q to have order 2 are that the simple factors ψv,i of ψv all have even degree, and the two parameters φv for Gv in Φpψv q both be relevant to Gv . The second of these conditions here is automatic unless the local index of Gv contains a subdiagram of the form (9.1.5). The inner sum over π P Πφ in (9.5.6) is more interesting here than in the quasisplit case. This is because the function x, π y in the summand is the ± restriction to the abelian 2-group Sφ of an irreducible character x, πv y on v
a group that could be nonabelian, namely the product of the local extensions Sφv ,sc . The multiplicity mφ p1, π q of the trivial representation in this restricted character could then be greater than 1. We discussed the groups Sφv ,sc in §9.2. If G SOp2n 1q is of type p Spp2n, Cq SppN, Cq is simply connected. The Bn , the dual group G group Sφv Sφv ,sc is then abelian, and there are no higher multiplicities. p equals If G Spp2nq is of type Cn , or G SOp2nq is of type Dn , G an orthogonal group SOpN, Cq, where N is respectively equal to p2n 1q or 2n. In these cases, Sφv ,sc is a nontrivial extension of Sφv that is often nonabelian. Suppose for simplicity that
φv,1 ` ` φv,r , φv,i P Φdis,simpGiq, Gv,i P ErsimpNv,iq, belongs to the subset Φdis,2 pGv q of Φdis pGv q. As we then recall from §9.2, the group Sφ ,sc Sφ ,sc is nonabelian if and only if the set Iv,o ti : Nv,i is oddu φv
v
v
v
has order greater than 2, a condition that precludes the case that Fv is archimedean. Assume that Sφv ,sc is nonabelian. Assume also that the character ζpGv on Zpsc , which then determines Gv as a p-adic inner twist, is nontrivial on the derived group
pSφ ,scqder t1u. v
It then follows from Lemma 9.2.2 that the degree of any irreducible representation in the set ΠpSφv ,sc , ζpGv q equals the positive integer dφv
#
2p|Iv,o |1q{2 , if N is odd, 2p|Iv,o |2q{2q , if N is even.
This is the contribution of v to the degree of the character x, π y. A modest extension of the formula, which we did not include in §9.2, applies to any localization φv of φ. Our conclusion is that the multiplicity mφ p1, π q on the
9.5. STATEMENT OF A GLOBAL CLASSIFICATION
565
right hand side of (9.5.6) is equal to a product mφ p1, π q
(9.5.10)
¹
dφv εpπ q,
v
where v is taken over the finite set of places v such that |Iov | ¥ 3 and ζpGv p1q 1, dφv is given by an explicit formula like that above, and εpπ q P t0, 1u is defined by a global reciprocity law that is entirely parallel to that of the quasisplit case. r 2 pGq will The analysis of multiplicities for arbitrary parameters ψ P Ψ be similar, if slightly more complicated. We do not presently have general results that are as sharp as Theorem 9.5.3, but suppose that the multiplicity formula (9.5.8) of Conjecture 9.5.4 has been established. In principle, any r unit pGv q in the local packet Π r ψ of a completion of ψ could element πv P Π v occur in the packet with higher multiplicity µψv pπv q. This would break into a sum ¸ µψv pξ, πv q µψv pπv q ξ
r ψ over the set ΠpSψ , ζpG q of irreducible of multiplicities in the fibres of Π v v characters, parametrized as in §9.2. These summands would in turn contribute appropriately to the sum over π in the global multiplicity formula (9.5.8). Suppose, however, that none of the local multiplicities µψv pπv q is greater than one, as seems likely. Then (9.5.8) will simplify to something close to the formula we have described for the right hand side of (9.5.6). In particular, the coefficient mψ in (9.5.8) could no doubt be replaced by a sum over the set Ψpψ q, like the outer sum in (9.5.6). The only remaining difference would then be the sign character εψ in (9.5.8), which we recall was defined in §1.5 in terms of symplectic ε-factors, in place of the trivial character from (9.5.6). The analysis of its multiplicity mψ pεψ , π q would be identical to the discussion of mφ p1, π q above that led to (9.5.10). Higher multiplicities for global inner forms of the group Spp2q appeared in the seminal paper [LL] by Labesse and Langlands on the stabilization of the trace formula for this and related groups. (See also [S1, §12, §15].) Higher multiplicities for more general orthogonal and symplectic groups were later discovered by Jian-Shu Li [Li1], [Li2]. He found many such examples by using the theta correspondence. They arise in Li’s constructions from a phenomenon we described in §9.1, the existence of nonisomorphic groups over F that are locally isomorphic. Suppose that G and G1 are inner forms of the split groups Spp2nq and SOp2nq. We then have the usual embeddings p1 G
SOp2n, Cq SOp2n
p 1, Cq G
of their dual groups. If G1 were itself split, it would of course be an endoscopic group for G. But pG1 , Gq is in general a dual reductive pair, which
I thank Steve Kudla for these references, and for conversations on their contents.
566
9. INNER FORMS
comes with a theta correspondence of automorphic representations. As we saw in §9.1, we can sometimes choose G1 so that it belongs to a larger family tG1 u of nonisomorphic groups over F that are all locally isomorphic. Li chooses G, and certain families π1 P Π2 pG1 q of locally isomorphic, automorphic representations of the groups G1 , such that the theta correspondence for the various pairs pG1 , Gq take the representations π1 to distinct, locally isomorphic, automorphic representations π P Π2 pGq of the group G. This, if I have understood it, is the source of Li’s constructions of automorphic representations π P Π2 pGq that occur with higher multiplicities in the discrete spectrum of G. For us, the inner forms G1 simply represent different groups over F . For any G1 , an adelic isomorphism from G1 to G1 gives a bijection between the associated sets Π GpAq and Π G1 pAq of irreducible adelic representations. This does not a priori have anything to say about the associated automorphic representations. The implicit question here amounts to a comparison of automorphic spectra for the locally compact group G1 pAq (identified with G1 pAq by the adelic isomorphism) relative to a pair G1 pF q and G1 pF q of different discrete subgroups. The group G1 is not locally symmetric, according to the remarks at the end of §9.1, so it is not included in the classification of automorphic representations of Conjecture 9.5.4. However, one can hope that the extensions of the assertion proposed above and in the last section would lead to an isomorphism between the automorphic discrete spectra of G1 and G1 . In any case, it seems quite surprising that the two methods for establishing higher multiplicities are so different. In Theorem 9.5.3 and Conjecture 9.5.4, they arise from a local phenomenon, the fact that the local centralizer groups Sψv ,sc can be nonabelian. In [Li1] and [Li2], they are a consequence of the existence of locally isomorphic groups that are not globally isomorphic, a global property. The two general methods of course have quite different sources, the trace formula on one hand, and the theta correspondence on the other. It would be very interesting to relate the classification of local and global representations of this volume with the explicit constructions provided by the theta correspondence and its generalizations. A full answer is probably still far away. For some preliminary results, a reader can consult the papers [Ji] of Jiang.
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