Invariant Differential Operators. Noncompact Semisimple Lie Algebras and Groups PDF

Invariant differential operators play a very important role in the description of physical symmetries – recall, e.g., the examples of Dirac, Maxwell, Klein–Gordon, d’Almbert, and Schrödinger equations. Invariant differential operators played and continue to play important role in applications to conformal field theory. Invariant superdifferential operators were crucial in the derivation of the classification of positive energy unitary irreducible representations of extended conformal supersymmetry first in four dimensions, then in various dimensions. Last, but not least, among our motivations are the mathematical developments in the last 50 years and counting. Obviously, it is important for the applications in physics to study these operators systematically. A few years ago we have given a canonical procedure for the construction of invariant differential operators. Lately, we have given an explicit description of the building blocks, namely, the parabolic subgroups and subalgebras from which the necessary representations are induced. Altogether, over the years we have amassed considerable material which was suitable to be exposed systematically in book form. To achieve portable formats, we decided to split the book in two volumes. In the present first volume, our aim is to introduce and explain our canonical procedure for the construction of invariant differential operators and to explain how they are used on many series of examples. Our objects are noncompact semisimple Lie algebras, and we study in detail a family of those that we call “conformal Lie algebras” since they have properties similar to the classical conformal algebras of Minkowski space-time. Furthermore, we extend our considerations to simple Lie algebras that are called “parabolically related” to the initial family.

Autor Vladimir K. Dobrev | Hermann Bondi | M.J.SMEAD | Συλλογικό | Macedonia is Greece and always be Greece

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Vladimir K. Dobrev Invariant Differential Operators

De Gruyter Studies in Mathematical Physics

Edited by Michael Efroimsky, Bethesda, Maryland, USA Leonard Gamberg, Reading, Pennsylvania, USA Dmitry Gitman, São Paulo, Brazil Alexander Lazarian, Madison, Wisconsin, USA Boris Smirnov, Moscow, Russia

Volume 35

Vladimir K. Dobrev

Invariant Differential Operators Volume 1: Noncompact Semisimple Lie Algebras and Groups

Mathematics Subject Classification 2010 17BXX, 17B45, 17B35, 17B67, 17B81, 16S30, 22EXX, 22E47, 22E15, 22E60, 81R05, 81R10, 32M15, 47A15, 47A46, 53A55, 70H33 Author Prof. Vladimir K. Dobrev Bulgarian Academy of Sciences Institute for Nuclear Research and Nuclear Energy Tsarigradsko Chaussee 72 1784 SOFIA Bulgaria [email protected] http://theo.inrne.bas.bg/∼dobrev/

ISBN 978-3-11-043542-9 e-ISBN (PDF) 978-3-11-042764-6 e-ISBN (EPUB) 978-3-11-042780-6 Set-ISBN 978-3-11-042765-3 ISSN 2194-3532 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de.

8

© 2016 Walter de Gruyter GmbH, Berlin/Boston Typesetting: Integra Software Services Pvt. Ltd. Printing and binding: CPI books GmbH, Leck Printed on acid-free paper Printed in Germany www.degruyter.com

Preface Invariant differential operators play a very important role in the description of physical symmetries – recall, e.g., the examples of Dirac, Maxwell, Klein–Gordon, d’Almbert, and Schrödinger equations. Invariant differential operators played and continue to play important role in applications to conformal ﬁeld theory. Invariant superdifferential operators were crucial in the derivation of the classiﬁcation of positive energy unitary irreducible representations of extended conformal supersymmetry ﬁrst in four dimensions, then in various dimensions. Last, but not least, among our motivations are the mathematical developments in the last 50 years and counting. Obviously, it is important for the applications in physics to study these operators systematically. A few years ago we have given a canonical procedure for the construction of invariant differential operators. Lately, we have given an explicit description of the building blocks, namely, the parabolic subgroups and subalgebras from which the necessary representations are induced. Altogether, over the years we have amassed considerable material which was suitable to be exposed systematically in book form. To achieve portable formats, we decided to split the book in two volumes. In the present ﬁrst volume, our aim is to introduce and explain our canonical procedure for the construction of invariant differential operators and to explain how they are used on many series of examples. Our objects are noncompact semisimple Lie algebras, and we study in detail a family of those that we call “conformal Lie algebras” since they have properties similar to the classical conformal algebras of Minkowski space-time. Furthermore, we extend our considerations to simple Lie algebras that are called “parabolically related” to the initial family. The second volume will cover various generalizations of our objects, e.g., the AdS/CFT correspondence, quantum groups, superalgebras, inﬁnite-dimensional (super-)algebras including (super-)Virasoro algebras, and (q-)Schrödinger algebras.

Contents 1 1.1 1.2 1.3 1.4

Introduction 1 Symmetries 1 Invariant Differential Operators Sketch of Procedure 8 Organization of the Book 10

2 2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6 2.1.7 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.4 2.5 2.5.1 2.5.2 2.5.3 2.5.4 2.5.5 2.5.6 2.5.7 2.6 2.6.1 2.6.2

11 Lie Algebras and Groups Generalities on Lie Algebras 11 Lie Algebras 11 Subalgebras, Ideals, and Factor-Algebras 12 Representations 13 Solvable Lie Algebras 13 Nilpotent Lie Algebras 14 Semisimple Lie Algebras 14 Examples 15 Elements of Group Theory 18 Definition of a Group 18 Group Actions 18 Subgroups and Factor-Groups 19 Homomorphisms 19 Direct and Semidirect Products of Groups 20 Structure of Semisimple Lie Algebras 21 Cartan Subalgebra 21 Lemmas on Root Systems 22 Weyl Group 25 Cartan Matrix 26 Classification of Kac–Moody Algebras 27 Realization of Semisimple Lie Algebras 34 Special Linear Algebra 34 Odd Orthogonal Lie Algebra 36 Symplectic Lie Algebra 38 Even Orthogonal Lie Algebra 40 Exceptional Lie Algebra G2 41 Exceptional Lie Algebra F4 42 Exceptional Lie Algebras E 44 Realization of Affine Kac–Moody Algebras 47 Realization of Affine Type 1 Kac–Moody Algebras Realization of Affine Type 2 and 3 Kac–Moody Algebras 52

7

47

VIII

2.6.3 2.7 2.8 2.9 2.10 2.10.1 2.10.2 2.10.3 2.10.4 2.10.5 2.10.6 2.10.7 2.11 2.11.1 2.11.2 2.11.3 2.11.4 2.11.5 2.11.6

Contents

Root System for the Algebras AFF 2 & 3 55 Chevalley Generators, Serre Relations, and Cartan–Weyl Basis Highest Weight Representations of Kac–Moody Algebras 61 Verma Modules 63 Irreducible Representations 71 A 72 C 73 B 73 D 75 E 77 F4 78 G2 78 Characters of Highest Weight Modules 78 Irreducible Quotients of Reducible Verma Modules 78 Embedding Patterns and Mulltiplets 79 Characters of Generic Highest Weight Modules 86 Characters for Nondominant Weights 88 Characters in the Affine Case 88 Example of A(1) 90 1

93 3 Real Semisimple Lie Algebras 3.1 Structure of Noncompact Semisimple Lie Algebras 93 3.1.1 Preliminaries 93 3.1.2 The Structure in Detail 95 3.2 Classification of Noncompact Semisimple Lie Algebras 96 3.3 Parabolic Subalgebras 100 3.4 Complex Simple Lie Algebras Considered as Real Lie Algebras 3.5 AI : SL(n, R) 103 3.6 AII : SU ∗ (2n) 106 3.7 AIII : SU(p, r) 107 3.7.1 Case SU(n, n), n > 1 108 3.7.2 Case SU(p, r), p > r ≥ 1 109 3.8 BDI : SO(p, r) 110 3.9 CI : Sp(n, R), n > 1 114 3.10 CII : Sp( p, r) 115 3.11 DIII : SO∗ (2n) 117 3.12 Real Forms of the Exceptional Simple Lie Algebras 119 3.12.1 EI : E6′ 119 3.12.2 EII : E6′′ 120 ′′′ 3.12.3 EIII : E6 122 3.12.4 EIV : E6iv 122 3.12.5 EV : E7′ 123

58

102

Contents

3.12.6 3.12.7 3.12.8 3.12.9 3.12.10 3.12.11 3.12.12 4 4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.1.6 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.4 4.5 4.6 4.6.1 4.6.2 4.7 4.7.1 4.7.2 4.7.3 4.8 4.8.1 4.8.2 4.8.3 4.8.4 4.8.5 4.8.6

124 EVI : E7′′ EVII : E7′′′ 126 ′ EVIII : E8 127 EIX : E8′′ 129 FI : F4′ 131 FII : F4′′ 132 ′ GI : G2 132 133 Invariant Differential Operators Lie Groups 133 Preliminaries 133 Classical Groups 134 Types of Lie Groups 135 Cartan Subgroups 136 Cartan and Iwasawa Decompositions 136 Parabolic Subgroups 137 Preliminaries on Group Representation Theory Representations and Modules 138 Reducibility and Irreducibility 138 Operations on Representations 139 Induced Representations 140 Elementary Representations 140 Compact Lie Groups 140 Noncompact Lie Groups 141 Knapp–Stein Integral Operators 142 ERs of Complex Lie Groups 144 Unitary Irreducible Representations 145 Associated Verma Modules 146 Invariant Differential Operators 149 Canonical Construction 149 Multiplets of GVMs and ERs 152 Example of SL(2,R) 152 Elementary Representations 152 Discrete Series and Limits Thereof 153 Positive Energy Representations 154 Explicit Formulae for Singular Vectors 154 A 155 D 156 E 157 B 157 C 158 F4 158

138

IX

X

4.8.7 4.8.8

Contents

G2 159 Nonstraight Roots

160

162 Case of the Anti-de Sitter Group Preliminaries 162 Lie Algebra 162 Finite-Dimensional Realization 164 Structure Theory 165 Lie Groups 166 Representations and Invariant Operators 166 Elementary Representations 166 Elementary Representations Induced from P0 168 Singular Vectors 168 Invariant Differential Operators 171 Reducible ERs 172 Holomorphic Discrete Series and Positive Energy Representations 175 5.2.7 Invariant Differential Operators and Equations Related to Positive Energy UIRs 176 5.2.8 Rac 177 5.2.9 Di 177 5.2.10 Massless Representations 178 5.3 Classification of so(5, C) Verma Modules and P0 -Induced ERs 5.4 Character Formulae 183 5.4.1 Character Formulae of AdS Irreps 183 5.4.2 Character Formulae of Positive Energy UIRs 186

5 5.1 5.1.1 5.1.2 5.1.3 5.1.4 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 5.2.6

6 6.1 6.1.1 6.1.2 6.1.3 6.1.4 6.1.5 6.1.6 6.1.7 6.1.8 6.2 6.2.1 6.2.2

188 Conformal Case in 4D Preliminaries 188 Realizations of the Group SU(2, 2) 188 Lie Algebra of SU(2, 2) 189 Restricted Root System, Bruhat and Iwasawa Decompositions 191 Restricted Weyl Group W(G, A0 ) 193 Parabolic Subalgebras 194 Complexified Lie Algebra 195 Compact and Noncompact Roots 198 Important Subgroups of G 199 Elementary Representations of SU(2, 2) 202 ERs from the Minimal Parabolic Subgroup P0 203 ERs from the Maximal Cuspidal Parabolic Subgroup P1 204

179

Contents

6.2.3 6.2.4 6.2.5 6.2.6 6.3 6.3.1 6.3.2 6.3.3 6.3.4 6.3.5 7 7.1 7.1.1 7.1.2 7.1.3 7.1.4 7.2 7.3 7.3.1 7.3.2 7.3.3 7.4 7.5 7.5.1 7.5.2 7.6 7.6.1 7.6.2 8 8.1 8.2 8.2.1 8.2.2

ERs from the Maximal Noncuspidal Parabolic Subgroup P2 205 Noncompact Picture of the ERs 207 Properties of ERs 209 Integral Invariant Operators 211 Invariant Differential Operators and Multiplet Classification of the Reducible ERs 215 Explicit Expressions for the Invariant Differential Operators 215 Multiplet Classification: Case P0 217 Multiplet Classification: Case P2 225 Holomorphic Discrete Series and Lowest Weight Representations 229 Multiplet Classification: Case P1 231 Kazhdan–Lusztig Polynomials, Subsingular Vectors, and Conditionally Invariant Equations 238 Subsingular Vectors 239 Preliminaries 239 Definition 240 Bernstein–Gel’fand–Gel’fand Example 242 The Other Archetypal sl(4, C) Example 243 Kazhdan–Lusztig Polynomials 248 Characters of LWM and Nontrivial KL Polynomials 251 Preliminaries on Characters of Lowest Weight Modules 251 Case sl(4, C) 253 Related Character Formulae 259 KL Polynomials and Subsingular Vectors: A Conjecture 260 Conditionally Invariant Differential Equations 262 Preliminaries 262 Conditionally Invariant Operators 263 Application to sl(4, C) 264 Equations Arising from the BGG Example 265 Equations Arising from the Other Archetypal sl(4, C) Example 265 Invariant Differential Operators for Noncompact Lie Algebras Parabolically Related to Conformal Lie Algebras 271 Generalities 271 The Pseudo-Orthogonal Algebras so(p,q) 274 Choice of Parabolic Subalgebra 274 Main Multiplets 275

XI

XII

8.2.3 8.2.4 8.2.5 8.2.6 8.3 8.3.1 8.3.2 8.4 8.4.1 8.4.2 8.4.3 8.4.4 8.4.5 8.4.6 8.5 8.5.1 8.5.2 8.6 8.6.1 8.6.2 8.7 8.7.1 8.7.2 9 9.1 9.2 9.3 9.3.1 9.3.2 9.3.3 9.4 9.5 9.5.1 9.5.2 9.5.3 9.5.4 9.6

Contents

Reduced Multiplets and Their Representations 280 Conservation Laws for so(p,q) 288 Remarks on Shadow Fields and History 290 Case so(3, 3) ≅ sl(4, R) 291 The Lie Algebra su(n,n) and Parabolically Related 292 Multiplets of su(3, 3) and sl(6, R) 293 Multiplets of su(4, 4), sl(8, R), and su∗ (8) 298 Multiplets and Representations for sp(n, R) and sp(r, r) 306 Preliminaries 306 The Case sp(3, R) 307 The Case sp(4, R) and sp(2, 2) 309 The Case sp(5, R) 313 The Case sp(6, R) and sp(3, 3) 317 Summary for sp(n, R) 325 SO∗ (4n) Case 325 Main Multiplets 327 Reduced Multiplets and Minimal Irreps 329 The Lie Algebras E7(–25) and E7(7) 332 Main Type of Multiplets 333 Reduced Multiplets 336 The Lie Algebras E6(–14) , E6(6) , and E6(2) 341 Main Type of Multiplets 343 Reduced Multiplets 346 Multilinear Invariant Differential Operators from New Generalized Verma Modules 353 Preliminaries 353 k-Verma Modules 355 Singular Vectors of k-Verma Modules 357 Definition 357 k=2 357 k=3 360 Multilinear Invariant Differential Operators 362 Bilinear Operators for SL(n, R) and SL(n, C) 363 Setting 363 Minimal Parabolic 364 SL(2, R) 366 SL(3, R) 370 Examples with k ≥ 3 372

Bibliography

375

Author Index

403

Subject Index

405

1 Introduction 1.1 Symmetries The notion of symmetry is a very old one. This is not surprising since there are many natural objects and living beings

which possess symmetry. So since the beginning of civilization people were inﬂuenced by this, and by 1200 B.C. symmetry was used extensively in Greek art. These were usually geometric symmetries such as discrete translational symmetry (when some ﬁgures were repeated from left to right (or top to bottom)); reﬂection symmetry with respect to some axis

(combined with translational symmetry); and discrete rotational symmetry (when a ﬁgure is not changed upon rotation of a ﬁxed angle). From the arts the notion of symmetry passed to the sciences. For instance, some symmetrical geometrical ﬁgures such as the circle and sphere were considered perfect by the Pythagoreans. Of course, it was clear that the real world is not exactly symmetric – e.g., take the human body as a nonexact symmetry. The ﬁrst appearances of symmetry in physics were of geometric nature. It was natural to think that the fundamental constituents of nature should possess some of these symmetries. Indeed, this is the case for many crystals and molecules, which in many cases are symmetrically arranged with respect to reﬂections as well as discrete translations and rotations. To this day, the study of such discrete symmetries is an interesting ﬁeld of science. The use of symmetries in mathematics and physics was enhanced when it was fully realized that symmetries can be described mathematically by expressing a set of transformations that leave a particular structure unchanged. This was especially important for the use of continuous symmetries.

2

1 Introduction

Thus the set of transformations which leaves the sphere unchanged is the set of rotations of arbitrary angle around the three axes in a three-dimensional Euclidean space. Mathematically, this is expressed as follows. The sphere of radius r (≠ 0) with the center at the beginning of the coordinate system is described as the points with coordinates x1 , x2 , x3 so that x12 + x22 + x32 = r2 , which can be written in matrix form as 6(x1 , x2 , x3 ) ≐ (x1 , x2 , x3 ) (x1 , x2 , x3 )t ⎛ ⎞ x1 ⎜ ⎟ = (x1 , x2 , x3 ) ⎝x2 ⎠ x3 = x12 + x22 + x32 and the fact that the rotations are preserving the sphere may be expressed as ˆ M(>) ˆ t (x1 , x2 , x3 )t = 6(x1 , x2 , x3 ), 6(x1′ , x2′ , x3′ ) = (x1 , x2 , x3 ) M(>) ⎛ ⎞ 100 ⎜ ⎟ ˆ M(>) ˆ t = I3 ≐ ⎝0 1 0⎠ , M(>) 00 1 ˆ depend on the three angles of rotation in the three where the 3 × 3 matrices M(>) ˆ possible planes in three dimensions, which is symbolically denoted by >. Using so-called Euler angles >1 , >2 , ;, the explicit dependence on the rotation angles is shown as follows:

⎛

cos > ⎜ M1 (>) = ⎝ sin > 0

ˆ = M1 (>1 ) M2 (;) M1 (>2 ), M(>) ⎞ ⎛ – sin > 0 1 0 ⎟ ⎜ cos > 0⎠ , M2 (;) = ⎝0 cos ; 0 1 0 sin ;

⎞ 0 ⎟ – sin ;⎠ , cos ;

where M1 and M2 are rotations in the planes (x1 , x2 ) and (x2 , x3 ), respectively, while the rotations in the plane (x3 , x1 ) are given by ⎛

cos ; 0 sin ;

⎞

⎜ ⎟ ⎟ M3 (>) = M1 (0/2) M2 (>) M1 (30/2) = ⎜ ⎝ 0 1 0 ⎠. – sin ; 0 cos ;

1.1 Symmetries

3

We note now the properties Mk (>) Mk (>′ ) = Mk (> + >′ ), Mk (>)t = Mk (–>), Mk (>) Mk (>)t = Mk (>)t Mk (>) = Mk (0) = I3 . The above properties may be expressed by the mathematical statement that the rotations form a group of transformations: A group G is a set of objects a, b, . . ., (e.g., transformations), for which there exists a rule by which each pair (a, b) of objects in G corresponds again to an object, say c in G, which is called the product of a, b, and we simply write c = ab. In the example above the product is the product of matrices. Then there is a special object e, called unit element (above I3 ), such that for every a in G we have ae = ea = a. Finally, for each element a there exists another element, called the inverse of a and denoted by a–1 , such that aa–1 = a–1 a = e; above we have ˆ –1 = M(>) ˆ t . Note that in general ab ≠ ba. Such groups, for which ab = ba for any M(>) choice of a and b, are called Abelian groups, otherwise a group is called non-Abelian. The group of rotations is not Abelian, e.g., M1 (>) M2 (;) ≠ M2 (;) M1 (>), as easily seen from Euler’s parametrization. The rotation group in three-dimensional Euclidean space is denoted in the literature by SO(3). Analogously is deﬁned the group of rotations SO(n) in n-dimensional Euclidean space. From these only SO(2) is Abelian – cf. the matrices Mk for the rotations in a ﬁxed plane. The fact that the rotations in a ﬁxed plane form by itself a group is expressed by saying that SO(2) is a subgroup of SO(3). Clearly, if m < n then SO(m) is a subgroup of SO(n). The groups of rotations are special in another respect. They are Lie groups. In general, this means that the elements of the group may be parametrized (the angles above) so that it would become an analytic manifold (real analytic here); moreover, the inverse element correspondence is an analytic function. More importantly for the exposition here is the fact that intimately related to the notion of a Lie group is the notion of a Lie algebra. In general, the Lie algebra G of the Lie group G is ﬁrst of all a linear space (over some ﬁeld of numbers F, here F = R – the real numbers, and below we shall also use F = C – the complex numbers) of dimension equal to the dimension of G. It may be identiﬁed with the tangent space of G at the unit element of G. Thus, for the group SO(3) the basis of this linear space may be represented as Tk ≐

∂ Mk (>) |> = 0 ∂>

4

1 Introduction

or explicitly: ⎛

0 ⎜ T1 = ⎝ 1 0

–1 0 0

⎞ 0 ⎟ 0⎠ , 0

⎛

0 ⎜ T2 = ⎝0 0

0 0 1

⎞ 0 ⎟ –1⎠ , 0

⎛

0 ⎜ T3 = ⎝ 0 –1

0 0 0

⎞ 1 ⎟ 0⎠ . 0

The elements Tk are also called (inﬁnitesimal) generators of the group. As in every algebra, a Lie algebra has also a product between its elements. For Lie algebras it is a special one called Lie bracket. Let X, Y, . . . be elements of a Lie algebra G, then [X, Y] denotes the Lie bracket of X, Y. It has the following special properties which are characteristic of a Lie algebra (over F = R, C): [X, Y] = –[Y, X] [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0. Properties above are called anticommutativity and Jacobi identity, respectively. In our situation the basis elements Tk are matrices, i.e., we have the ordinary associative product of matrices (for which we do not write the ⋅), as well as the commutators. Let us calculate the commutator of, e.g., T1 and T2 . This is a simple calculation which gives [T1 , T2 ] = T1 T2 – T2 T1 = T3 . Analogously, one obtains [Tj , Tk ] = %jk T ,

j, k, = 1, 2, 3,

where %jk is totally antisymmetric and %123 = 1. The Lie algebra of the group SO(n) is denoted by so(n). One may consider the same basis elements as generators of a Lie algebra over the complex numbers C. Then the analogs of so(n) are denoted by so(n, C). The notion of a subalgebra is analogous to the subgroup notion, e.g., so(m), resp., so(m, C), is subalgebra of so(n), resp., so(n, C), if m < n. Now for the algebra so(3, C) one may introduce the following basis: X ± ≡ – iT1 ∓ T2 ,

H ≡ – 2iT3 .

These generators have the following commutators: [H, X ± ] = ± 2X ± ,

[X + , X – ] = H.

1.1 Symmetries

5

This algebra is known also as the Lie algebra sl(2, C), the notation meaning special linear, and it has a realization as 2 × 2 matrices:

H=

1 0

0 , –1

+

X =

0 0

1 , 0

–

X =

0 1

0 . 0

The fact that the Lie algebras so(3, C) and sl(2, C) have the same commutators is expressed also by saying that they are isomorphic as Lie algebras, one writes so(3, C) ≅ sl(2, C). The most important applications of Lie groups and algebras are their representations. Let us recall that the representation of a Lie algebra G in a vector space V is a linear map > : G → End V such that >([X, Y]) = [>(X), >(Y)] ≡ >(X)○>(Y)–>(Y)○>(X). If dim V < ∞, then dim V is called the dimension of the representation. Above we had already examples of representations, e.g., two- and threedimensional representation of su(2) ≅ so(3), and of sl(2, C) ≅ so(3, C). A representation > is called reducible if V contains a nontrivial subspace V ′ ≠ V which is invariant under the action of >; otherwise > is called irreducible. The representations discussed above are irreducible. A representation is called completely reducible if V can be represented as the direct sum of invariant subspaces. A representation is called indecomposable if it is reducible but not completely reducible. For the applications of the Lie algebras it is important that they can be classiﬁed into two large classes: semisimple and solvable. The Lie algebras (over R) used in physics usually have a semisimple subalgebra, and this has a further su(2) ≅ so(3) subalgebra. We note that semisimple Lie algebras over C have triangular decomposition: G = G+ ⊕ H ⊕ G–, where G ± , H subalgebras of G, H is Abelian with the property [K, Y] = +(K)Y,

+ ∈ H∗ , ∀K ∈ H,

Y ∈ G±,

where H = l.s.{H}, G ± = l.s.{X ± }. We introduce lowest weight vector v0 : Xv0 = 0, Hv0 = D(H) v0 ,

X ∈ G–

H ∈ H,

D(H) ∈ H∗ ,

where D(H) is called lowest weight. A very important object is the Verma module V D , which in our example is the representation with states vk = (X + )k v0 , k ∈ Z+

6

1 Introduction

with the following action on the states: H vk = (D(H) + 2k)vk , X + vk = vk+1 , X – vk = k(D(H) + 1 – k)vk–1 . The lowest weight vector is v0 (since X – v0 = 0). Note that if the lowest weight is a positive integer D(H) = n ∈ Z+ then the representation is reducible since the states vk , k = n + 1, . . ., form an invariant subspace I D due to the fact that X – vn+1 = 0. Furthermore, the factor-space V D /I D is ﬁnite-dimensional of dimension n + 1 and spanned by the vectors vk , k = 0, . . . , n. For our purposes, we introduce a vector-ﬁeld representation Td of the algebra: X + = x2 ∂x – xd,

X – = –∂x,

Hd = 2x∂x – d.

These act on functions of the variable x. For simplicity, we take a representation Cd with basis u = x , ∈ Z+ . The action is Hd u = (2 – d)u , X + u = ( – d)u+1 , X – u = – u–1 . This is a lowest weight representation with lowest weigt equal to –d and lowest weight vector u0 . In the applications important role is played by the Casimir operator C2 which is quadratic in the generators. In our case, it is given by 1 C2 = Hd2 + X + X – + X – X + . 2 The importance of the Casimir operator is that it commutes with all generators. Because of this it has a constant value in a ﬁxed representation, in the above we have 1 C2 = d(d + 2). 2 Now we see that the Casimir for d = n and for d = – n – 2 has the same value C2 = 21 n(n + 2). Thus, the representation Tn and T–n–2 are said to be partially equivalent. The equivalence is realized by the invariant differential operator Dn = ∂xn+1 . Note that we shall use the notions invariant differential operator and intertwining differential operator as synonyms. Now we can check that the intertwining property Dn : Cn → C–n–2 ,

Dn Tn = T–n–2 Dn

1.2 Invariant Differential Operators

7

is fulﬁlled, e.g., Dn Hn = H–n–2 Dn . Next we note that the representation Tn is reducible. Indeed, the states u of Cn for = 0, . . . , n form a ﬁnite-dimensional invariant subspace En of dimension n + 1. Being ﬁnite-dimensional, it has also highest weight vector: un , with highest weight +n. Next we notice that En is annihilated by the invariant differential operator Dn . The rest of the states of En are mapped to all states of C–n–2 : Dn un+ =

A(n + + 2) u , A( + 1)

∈ Z+ .

Thus, we have the equivalence Cn /En ≅ C–n–2 .

1.2 Invariant Differential Operators Consider a Lie group G, e.g., the Lorentz, Poincaré, conformal groups, and differential equations with differential operators I and f , j functions on a manifold (or symbolically written collections of such objects): I f =j

(1.1)

which are G-invariant. These play a very important role in the description of physical symmetries. In particular, invariant differential operators play very important roles in the description of physical symmetries – starting from the early occurrences in the Maxwell, d’Allembert, Dirac, equations (for more examples, cf., e.g., [29, 96]) to the latest applications of (super-)differential operators in conformal ﬁeld theory, supergravity and string theory (for reviews, cf., e.g., [424, 577]). Thus, it is important for the applications in physics to study systematically such operators. To recall the notions, consider a semisimple Lie group G and two representations ′ T, T acting in the representation spaces C, C′ , which may be Hilbert, Fréchet, etc. An invariant (or intertwining) operator I for these two representations is a continuous linear map I : C → C′

(1.2)

such that T ′ (g) ○ I = I ○ T(g),

∀g ∈ G.

(1.3)

8

1 Introduction

This is what precisely is meant when we say that the equation (1.1) is a G-invariant equation. Note that ker I and im I are invariant subspaces of C and C′ , respectively. Such equations also exist for more general classes of Lie groups. However, if G is semisimple then there exist canonical ways for the construction of all intertwining operators which give all G-invariant equations. There are also integral invariant operators [352, 353] which we shall also use.

1.3 Sketch of Procedure Here we sketch our procedure for the semisimple Lie groups, illustrating the general notions in the example of the conformal group SU(2, 2). Let G be a real semisimple Lie group and G be the Lie algebra of G. We shall use so-called Bruhat decompositions of G ⊕M⊕A⊕N G =N

(1.4)

(considered as direct sum of linear spaces), where A is a noncompact Abelian subalgebra, M (a reductive Lie algebra) is the centralizer of A in G (mod A), and , N are nilpotent subalgebras preserved by the action of A. For the conformal N , M, A, and N are the subalgebras of translations, Lorentz group the subalgebras N transformations, dilatations, and special conformal transformations, respectively. Note that P =M⊕A⊕N

(1.5)

is called parabolic subalgebra of A. forms a conjugate parabolic subalgebra. In general, a real (Note that M ⊕ A ⊕ N noncompact Lie algebra G has more than one nonconjugate parabolic subalgebras; e.g., the conformal algebra has two more nonconjugate subalgebras. Of course, each parabolic subalgebra has an associate Bruhat decomposition, as above.) Let us now introduce the corresponding subgroups of G. Let K denote the maximal compact subgroup of G, and let K denote the Lie algebra of K. Then we have the ), and N = exp(N ). Further, M is simply connected subgroups A = exp(A), N˜ = exp(N the centralizer of A in G (mod A). (M has the structure M = Md Mr , where Md is a ﬁnite group, Mr is reductive with the same Lie algebra M as M.) Then P = MAN and P˜ = MAN˜ are conjugate parabolic subgroups of G. Let , be a ﬁnite-dimensional representation D, of M on the vector space V, . Let - be a (nonunitary) character of A, - ∈ A∗ . We call the induced representation 7 = IndGP (, ⊗ - ⊗ 1) an elementary representation of G. We use the following space of functions: C7 = {F ∈ C∞ (G, V, ) | F (gman) = e-(H) ⋅ D, (m–1 )F (g)},

(1.6)

1.3 Sketch of Procedure

9

where g ∈ G, m ∈ M, a = exp(H) ∈ A, (H ∈ A), n ∈ N. Then the elementary representation (ER) T 7 acts in C7 , as the left regular representation (LRR), by (T 7 (g)F )(g ′ ) = F (g –1 g ′ ),

g, g ′ ∈ G.

(1.7)

The importance of the ERs arises from the fact that every irreducible admissible representation of a real connected semisimple Lie group G with ﬁnite centre is equivalent to a subrepresentation of an ER of G, cf. [357, 358, 389]. (More precise deﬁnitions and statements are given in Section 4.3.) Another feature of the ERs which is crucial for our construction is the highest (also possibly lowest) weight module structure associated with them [126]. For this we introduce the right action of G C (the complexiﬁcation of G) by the standard formula: ˆ )(g) = (XF

d F (g exp(tX))|t=0 , dt

X ∈ GC,

F ∈ C7 ,

g ∈ G,

(1.8)

which is deﬁned ﬁrst for X ∈ G and then is extended to G C by linearity. We introduce C-valued realization C 7 of the space C7 by the formula: >(g) ≡ v0 , F (g),

(1.9)

where , is the M-invariant scalar product in V, and v0 is the highest weight vector of V, . On these functions the right action of G C is deﬁned by ˆ ˆ )(g) . (X>)(g) ≡ v0 , (XF

(1.10)

Part of the main result of [126] is as follows: Proposition: The functions of the C-valued realization C 7 of the ER C7 satisfy: ˆ = D(X) ⋅ >, X> ˆ = 0, X>

X ∈ HC ,

D ∈ (HC )∗ ,

X ∈ G+C ,

(1.11)

where D = D(7) is built canonically from 7 . Furthermore, special properties of a class of highest weight modules, namely, Verma modules, are immediately related with the construction of invariant differential operators. To be more speciﬁc let us recall that a Verma module is a highest weight module V D with highest weight D, such that V D ≅ U(G–C )v0 , where v0 is the highest weight vector, U(G–C ) is the universal enveloping algebra of G–C . We are interested in reducible Verma modules since these are relevant for the construction of differential equations. We recall the Bernstein–Gel’fand–Gel’fand [42] criterion according to which the Verma module V D is reducible iff 2D + 1, " = m", "

(1.12)

10

1 Introduction

holds for some " ∈ B+ and m ∈ N, where B+ denotes the positive roots of the root system (G C , cˆ ) and 1 is half the sum of the positive roots B+ . Whenever (1.12) is fulﬁlled there exists [120] in V D a unique vector vs , called singular vector, such that vs ≠ Cv0 and it has the properties (1.11) of a highest weight vector with shifted weight D – m": ˆ s = (D – m")(X) ⋅ vs , Xv ˆ s = 0, Xv

X ∈ HC ,

X ∈ G+C .

(1.13)

The general structure of a singular vector is [126]: vs = Pm" (X1– , . . . , X– )v0 ,

(1.14)

where Pm" is a homogeneous polynomial of U(G–C ) of weight m". Now we are in a position to deﬁne the invariant (aka intertwining) differential operators for semisimple Lie groups, corresponding to the singular vectors. Let the signature 7 of an ER be such that the corresponding D = D(7) satisﬁes (1.12). Then there exists an intertwining differential operator [126]: Dm" : C 7 → C 7′ ,

(1.15)

where 7′ is such that D′ = D′ (7′ ) = D – m". The important fact is that Dm" is explicitly given by [126]: Dm" >(g) = Pm" (Xˆ 1– , . . . , Xˆ – )>(g),

(1.16)

where Pm" is the polynomial from (1.14) and Xˆ j– denotes the right action (1.8). The procedure sketched above has the advantages of being both canonical and algebraic. Thus, it has been generalized to supersymmetry setting, quantum groups, and inﬁnite-dimensional (super-)algebras. We should also mention that this setting is most appropriate for the classiﬁcation of unitary representations of (super-)conformal symmetry in various dimensions, and it is also an ingredient in the AdS/CFT (anti-de Sitter/conformal ﬁeld theory) correspondence. These generalizations will be be shown in Volume 2.

1.4 Organization of the Book The idea of this book is to introduce and explain a canonical procedure for the construction of invariant differential operators and to explain how they are used in many series of examples. The topics are clear from the detailed Contents. Besides this ﬁrst chapter, there are eight chapters, Bibliography, Author Index, Subject Index. Each of the eight chapters has a summary which explains brieﬂy the contents and the extent of original material.

2 Lie Algebras and Groups Summary This chapter contains mostly standard introductory material based on mathematical sources [59, 85, 120, 327, 547, 624], and physics sources, notably the book by Barut–Ra¸czka [29]. Original results are cited appropriately. The actual exposition mostly follows the lecture course [139]. We introduce the notions of Lie algebras, subalgebras, ideals, factor-algebras, representations of Lie algebras, solvable Lie algebras, nilpotent Lie algebras, and semisimple Lie algebras. Further, we introduce some elements of group theory, the notion group, group actions, subgroups, factor-groups, homomorphisms, and direct and semidirect products of groups. Then we take up the structure of semisimple Lie algebras, introducing the notions of Cartan subalgebra, Weyl group, Cartan matrix, and the necessary for the further exposition lemmas on root systems. We then give the classification of Kac–Moody algebras and the realization of semisimple Lie algebras with all the relevant cases (the classical algebras: special linear algebra, orthogonal Lie algebra, symplectic Lie algebra, and the exceptional Lie algebras G2 , F4 , E6 , E7 , E8 ). Then comes the realization of affine Kac–Moody algebras with explicit root systems. We also introduce the notions of Chevalley generators, Serre relations, and Cartan–Weyl basis. We consider the highest weight representations of Kac–Moody algebras, especially Verma modules, and the irreducible representations of all classical and exceptional Lie algebras. Finally, we consider the characters of highest weight modules, starting with the irreducible quotients of reducible Verma modules, the latter embedding patterns, multiplets, then we give in detail characters in the generic case, then characters for nondominant weights, characters in the affine case, the latter most explicitly for A(1) 1 .

2.1 Generalities on Lie Algebras 2.1.1 Lie Algebras Let F be a ﬁeld of characteristic 0. Let A be an algebra over F, i.e., A is a linear space over F with additional bilinear operation: A × A ∋ (X, Y) ↦ X ⋅ Y ∈ A is called multiplication; X ⋅ Y is called the product of X, Y ∈ A. An algebra A is called an associative algebra if X⋅(Y ⋅Z) = (X⋅Y)⋅Z for all X, Y, Z ∈ A. An algebra A is called an Abelian algebra (or commutative algebra) if X ⋅ Y = Y ⋅ X for all X, Y ∈ A. An algebra A is called a Lie algebra if the following properties hold: X ⋅ X = 0,

(2.1a)

X ⋅ (Y ⋅ Z) + Y ⋅ (Z ⋅ X) + Z ⋅ (X ⋅ Y) = 0.

(2.1b)

These properties are called anticommutativity and Jacobi identity, respectively. From (2.1a) follows 0 = (X + Y) ⋅ (X + Y) = X ⋅ X + X ⋅ Y + Y ⋅ X + Y ⋅ Y = X ⋅ Y + Y ⋅ X

12

2 Lie Algebras and Groups

or X ⋅ Y = –Y ⋅ X.

(2.2)

(Actually (2.2) is equivalent to (2.1a) if charF ≠ 2.) Note that (2.1b) shows that a Lie algebra is not associative w.r.t. its product. Every associative algebra is also a Lie algebra w.r.t. the product: [X, Y] ≡ X ⋅ Y – Y ⋅ X

(2.3)

which is called the commutator of X and Y. Thus, further, in the case of Lie algebras we shall denote the product of X and Y by [X, Y]. Then (2.1) is written as [X, X] = 0,

(2.4a)

[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0.

(2.4b)

If A is a ﬁnite-dimensional algebra then the multiplication in A is determined by the multiplication of the basis elements Xi of A as follows: Xi ⋅ Xj =

n

ckij Xk ,

ckij ∈ F, i, j = 1, . . . , n,

(2.5)

k=1

where n = dimA. The elements ckij are called the structure constants of A (w.r.t. the basis {Xi }). For inﬁnite-dimensional algebras A, the relation (2.5) will also hold; however, n = n(Xi , Xj ). If A is a Lie algebra then the structure constants satisfy ckij = – ckji and a system of quadratic equations follows immediately from the Jacobi identity (2.1b). 2.1.2 Subalgebras, Ideals, and Factor-Algebras Let B be a vector subspace of A; B is called a subalgebra of A if X ⋅Y ∈ B for all X, Y ∈ B. Let B ⊂ A and C ⊂ A, then we deﬁne B⋅C ≡

k

ai Yi ⋅ Zi : k ∈ N, ai ∈ F, Yi ∈ B, Zi ∈ C .

(2.6)

i=1

A subalgebra I of A is called an ideal of A if A ⋅ I ⊂ I, I ⋅ A ⊂ I.

(2.7)

If A is a Lie algebra, only one of the two conditions in (2.7) is sufﬁcient for I to be an ideal (because of (2.2)).

2.1 Generalities on Lie Algebras

13

Let A be a Lie algebra and let I, J be ideals of A. Then [I, J ], called the commutator of ideals I and J , is also an ideal of A. Let A be an algebra and I its ideal. Then the factor vector space A/I is also an algebra, called the factor-algebra, with the product (X + I) ⋅ (Y + I) = X ⋅ Y + I. Let A and B be two algebras. The vector space A ⊕ B = {(X, Y) : X ∈ A, Y ∈ B} is an algebra by the product (X1 , Y1 ) ⋅ (X2 , Y2 ) ≡ (X1 ⋅ X2 , Y1 ⋅ Y2 ). The algebra A ⊕ B is called the direct sum of algebras of A and B. 2.1.3 Representations An endomorphism D of A is called a derivation of A if D(X ⋅ Y) = (DX) ⋅ Y + X ⋅ (DY), ∀X, Y ∈ A. Note that if D1 , D2 are derivations of A then D1 ○ D2 – D2 ○ D1 is also a derivation. A representation of a Lie algebra G in a vector space V is a linear map > : G → End V such that >([X, Y]) = [>(X), >(Y)] ≡ >(X) ○ >(Y) – >(Y) ○ >(X). A representation > is called a reducible representation or simple if V contains a nontrivial subspace V ′ , 0 ≠ V ′ ≠ V, which is invariant under the action of >, i.e., >(Y)v ∈ V ′ for all Y ∈ G, v ∈ V ′ ; otherwise > is called an irreducible representation. A representation is called a completely reducible or semisimple if V can be represented as the direct sum of invariant subspaces. A representation is called an indecomposable representation if it is reducible but not completely reducible. An important representation of G is the so-called adjoint representation of G acting in G itself: ad X : G → G; ad X : Y ↦ [X, Y], X, Y ∈ G.

(2.8)

It is easy to see that > : X ↦ ad X is a representation, i.e., that ad [X, Y] = (ad X) ○ (ad Y) – (ad Y) ○ (ad X). It is also clear that ad X is a derivation of G, i.e., that ad X([Y, Z]) = [ad X(Y), Z] + [Y, ad X(Z)]; the last equality is actually the Jacobi identity. The derivations ad X are called inner derivations of G. For further use, we shall note the equality ad DY = [D, ad Y] valid for every derivation D and ∀Y ∈ G. 2.1.4 Solvable Lie Algebras Let G be a Lie algebra. Deﬁne inductively G (1) = [G, G], G (k+1) = [G (k) , G (k) ], for k = 1, 2, . . . . It is clear that G (k+1) ⊂ G (k) and that all G (k) are ideals of G. (For the latter ﬁrst note that G (1) is an ideal as the commutator of two ideals; then by induction.) If G (k) = 0 for some k then G is called solvable. (Then all G (i) are solvable.) In particular, if G (1) = 0 then G is an Abelian algebra. Note that a solvable Lie algebra contains an Abelian ideal.

14

2 Lie Algebras and Groups

It can also be shown that a ﬁnite-dimensional Lie algebra is solvable iff its adjoint representation is triangular (i.e., there exists a basis in which all operators ad X are given by upper (or equivalently lower) triangular matrices with zeros below (above) the main diagonal). 2.1.5 Nilpotent Lie Algebras Let G be a Lie algebra. Deﬁne inductively G 1 = G, G k+1 = [G k , G], k = 1, 2, . . . . It is clear that G k+1 ⊂ G k and that all G k are ideals of G. (First note that G 2 ⊂ G and that G 2 = [G, G] is an ideal of G; then by induction.) If G k = 0 for some k > 1 then G is called a nilpotent. (Then also all G i are nilpotent.) Note that a nilpotent Lie algebra is solvable. (For this it is enough to show that G (k) ⊂ G k+1 for k = 1, 2, . . . . By deﬁnition G (1) = G 2 ; then by using induction we assume G (k–1) ⊂ G k and obtain G (k) = [G (k–1) , G (k–1) ] ⊂ [G k , G] = G k+1 .) A subalgebra Z of G is called the center of G if it is a maximal Abelian subalgebra of G such that [X, Y] = 0∀X ∈ Z, ∀Y ∈ G. Note that if G is nilpotent then it has a nontrivial center Z ≠ 0. It can be shown that a ﬁnite-dimensional Lie algebra is nilpotent iff its adjoint representation is strictly triangular (i.e., there exists a basis in which all operators ad X are given by upper (or lower) triangular matrices with zeros below (or above) and on the main diagonal). 2.1.6 Semisimple Lie Algebras Let G be a Lie algebra. Then G is called a simple algebra if it is not Abelian and has no ideals except for 0 and itself. G is called a semisimple if it has no Abelian ideals except for 0. Note that a semisimple Lie algebra G cannot have a solvable ideal S, therefore all ideals of S including the Abelian one will also be ideals of G. It can easily be shown that for a semisimple algebra G one has G k = G (k) = G for all k. Then it follows that G can be written as the direct sum of simple ideals. A Lie algebra G is called a reductive algebra if it can be written as G = Z ⊕ [G, G], where Z is the center of G and [G, G] is a semisimple ideal of G. Thus, a reductive Lie algebra is semisimple iff Z = 0. It can be shown that a ﬁnite-dimensional Lie algebra is reductive iff its adjoint representation is completely reducible. A Lie algebra G is called a pseudoreductive algebra if the factor-algebra S = G/Z is semisimple, where Z is the center of G. Then G is called a central extension of S. Clearly, a reductive algebra is pseudoreductive. A pseudoreductive algebra is reductive if the central extension is trivial, namely, if G ≅ Z ⊕ S. In the ﬁnite-dimensional case one can show that any central extension of a semisimple Lie algebra is trivial. Thus, in the ﬁnite-dimensional case the notions of reductive and pseudoreductive Lie algebra coincide.

15

2.1 Generalities on Lie Algebras

Let G be a ﬁnite-dimensional Lie algebra. Then G has a maximal solvable ideal 4(G) which contains all solvable ideals of G. The ideal 4(G) is called the radical of G. The factor-algebra G/4(G) is semisimple. There exists the following semidirect subalgebra decomposition: ⊎

G = S 4(G),

[S, 4(G)] ⊂ 4(G),

(2.9)

where S is a semisimple subalgebra of G, called a Levi factor of G. The latter is ﬁxed up to isomorphism of G. The above decomposition is called a Levi–Malcev decomposition. One may say that G is semisimple (resp. reductive) iff 4(G) = 0 (resp. 4(G) = Z). Let G be a ﬁnite-dimensional Lie algebra. Deﬁne for all X, Y ∈ G B(X, Y) ≡ tr ad X ad Y. Then B is a symmetric bilinear form on G × G called the Killing form of G. We recall that a bilinear form K on G × G is called invariant form under an endomorphism D of G if K(DX, Y) + K(X, DY) = 0 for all X, Y ∈ G. It is easy to see that the Killing form is invariant under any derivation D of G. We recall the fundamental result. Theorem 1: A ﬁnite-dimensional Lie algebra G is semisimple if and only if B is nondegenerate. 2.1.7 Examples –

If a Lie algebra is one-dimensional then it is Abelian.

–

If a Lie algebra S is two-dimensional then it is either Abelian or it has a basis X, Y with nontrivial commutation relation [X, Y] = Y. The latter algebra is called the afﬁne line algebra. The generator Y corresponds to inﬁnitesimal translations of the line R: x ↦ x+ b, x, b ∈ R; the generator X corresponds to inﬁnitesimal dilatations of R: x ↦ ax, x ∈ R, a ∈ R>0 . This algebra is solvable with S (1) = CY, S (2) = 0, but is not nilpotent since k S = CY ∀k > 1.

–

A three-dimensional Lie algebra which is solvable but not nilpotent is given by the commutation relations: [J, P1 ] = P2 ,

[J, P2 ] = –P1 ,

[P1 , P2 ] = 0

This algebra is called the Euclidean Lie algebra of R2 ; J is the generator of rotations, P1 , P2 are the generators of the two translations of R2 . It has a central extension, namely the last commutator may be replaced by [P1 , P2 ] = n. –

A three-dimensional nilpotent algebra is given by the commutation relations: [Pt , G] = Px ,

[Px , G] = 0,

[Px , Pt ] = 0

16

2 Lie Algebras and Groups

This algebra is called the Galilei Lie algebra of two-dimensional space-time; Pt , resp., Px , is the generator of time, resp., space translation, G is the generator of Galilean boost. It has two-dimensional central extension, namely, the last two commutators may be written as [Px , G] = m, [Px , Pt ] = n. –

The n × n matrices over F comprise an associative algebra denoted as Matn F. The product is the usual product of matrices. This algebra is ﬁnite-dimensional and dimA = n2 .

The algebra Matn F with the commutator [S1 , S2 ] = S1 S2 – S2 S1 , (S1 , S2 ∈ Matn F), is a Lie algebra which is denoted gl(n, F) and sometimes gl(n) if F = C. The endomorphisms (linear transformations) End V of a linear space V comprise an associative algebra. The product of two endomorphisms s1 , s2 is their composition s1 ○s2 . The algebra End V with the commutator [s1 , s2 ] = s1 ○s2 –s2 ○s1 as a product comprise a Lie algebra which is denoted as gl(V). If V is ﬁnite-dimensional of dimension n then End V ≅ Matn F, and gl(V) ≅ gl(n, F) as we show below. Let V be a ﬁnite-dimensional vector space over F, and e1 , . . . en be its basis. Let s ∈ gl(V) and let s : ej ↦

n

sjk ek , sjk ∈ F or

(2.10a)

⎞⎛ ⎞ s1n e1 ⎜ . ⎟ .. ⎟ ⎟⎜ ⎟ . ⎠ ⎝ .. ⎠ . en snn

(2.10b)

k=1

⎞ ⎛ s11 s e1 ⎜ . ⎟ ⎜ . ⎜ . ⎟=⎜ . ⎝ . ⎠ ⎝ . sn1 s en ⎛

... .. . ...

Then the map ⎛

s11 ⎜ . ⎜ s ↦ S, s ∈ gl(V), S = ⎝ .. sn1

... .. . ...

⎞ s1n .. ⎟ ⎟ . ⎠ ∈ gl(n, F) snn

(2.11)

is a (Lie) algebra isomorphism. The Lie algebra gl(n, F) is called the general linear algebra. It is reductive with one-dimensional center spanned by the unit n × n matrix which we shall denote by In . The importance of gl(n, F) stems from the following fundamental result of Ado [1]: Theorem: Every ﬁnite-dimensional Lie algebra over C is isomorphic to a subalgebra of gl(n, C) for some n. ♢ The above theorem is valid also for F = R. Of course, sometimes it is not easy to identify a ﬁnite-dimensional Lie algebra as a subalgebra of gl(n, F). Below we give important examples of subalgebras of gl(n, F) for n > 1.

17

2.1 Generalities on Lie Algebras

–

First we consider sl(n, F): sl(n, F) ≐ {X ∈ gl(n, F) : tr X ≡ x11 + . . . + xnn = 0},

(2.12)

called the special linear algebra; it is semisimple and is an ideal of gl(n, F) = CIn ⊕ sl(n, F). Let G+ (respectively G– ) be the subalgebra of G = sl(n, F) of matrices with only zero entries below (respectively above) and on the main diagonal. Then G+ , G– are nilpotent. –

Next we consider so(n, F): so(n, F) ≐ {X ∈ gl(n, F) : X + t X = 0},

(2.13)

called the orthogonal Lie algebra (t X is the matrix transposed to X). Since tr t X = tr X it is clear that so(n, F) ⊂ sl(n, F). The algebra so(n, F) is Abelian for n = 2, semisimple for n = 4 and simple for n = 3, n > 4. We also note:

–

Let Jn ≡

so(4, F) = so(3, F) ⊕ so(3, F),

(2.14a)

so(3, C) ≅ sl(2, C),

(2.14b)

so(6, C) ≅ sl(4, C).

(2.14c)

0 I n . Next is the algebra sp(n, F) –In 0 sp(n, F) ≐ {X ∈ gl(2n, F) : t XJn + Jn X = 0}

(2.15)

is called the symplectic Lie algebra. It is a simple Lie algebra. We note that

–

sp(1, F) ≅ sl(2, F),

(2.16a)

sp(2, C) ≅ so(5, C).

(2.16b)

Let W be the simple Lie algebra with basis dj , j ∈ Z and commutation relations [dj , dk ] = ( j – k)dj+k .

(2.17)

W is called the Witt algebra. Let ; be the standard parameter on the circle. Then eik; d/d; is a basis for the Lie algebra of vector ﬁelds on the circle with ﬁnite Fourier series. If we set dk = –ieik; d/d; then dk satisfy (2.17). The unique central extension of the Witt algebra is the well-known Virasoro algebra which we shall consider in some detail later.

18

2 Lie Algebras and Groups

2.2 Elements of Group Theory 2.2.1 Definition of a Group Definition: An abstract set G is called a group if: (1) G has the operation of multiplication ,:G × G → G. We write ( g, g ′ ) ↦ g ⋅ g ′ ∈ G for g, g ′ ∈ G. The multiplication is associative: ( g ⋅ g ′ ) ⋅ g ′′ = g ⋅ ( g ′ ⋅ g ′′ ), but not commutative, i.e., in general g ⋅ g ′ ≠ g ′ ⋅ g. (2) G contains an identity element (or unit) e such that g ⋅ e = e ⋅ g = g. (3) for every element g of G there exists an inverse g –1 ∈ G, so that g ⋅ g –1 = g –1 ⋅ g = e. ♢ A group G is called ﬁnite group if the number of elements of G is ﬁnite. In that case the number of elements of G is denoted by |G| and is called the order of the ﬁnite group G. A group G is called commutative group or Abelian if g ⋅ g ′ = g ′ ⋅ g for all g, g ′ ∈ G. In this case instead of g ⋅ g ′ often is used g + g ′ , and instead of e is used the symbol 0. Examples: (1) The set Z of all integers is an Abelian group w.r.t. addition of numbers (the unit being the number 0). (2) The set Z\{0} of all nonzero integers is an Abelian group w.r.t. multiplication (the unit being the number 1). (3) The set Aut V of all automorphisms of a vector space is a group (w.r.t. composition of automorphisms). (4) The general linear group GL(n, F), i.e., the set of all nondegenerate matrices n × n over the ﬁeld F is a group (w.r.t. multiplication of matrices). ♢ Further, for simplicity instead of g ⋅ g ′ we shall write simply gg ′ . 2.2.2 Group Actions We say that the group G is acting (from the left) on a set X if for every g ∈ G and x ∈ X is deﬁned an element gx ∈ X, so that (1) ex = x ; (2) ( gg ′ )x = g( g ′ x) for all g, g ′ ∈ G, x ∈ X. We shall call X a (left) G-space. Analogously is deﬁned a right G-space. The action of G is called effective action if the equality gx = x for all x ∈ X is possible only for g = e. Let X be a (left) G-space. For x, y ∈ X we denote x ∼ y if y = gx for some g ∈ G. Obviously, ∼ is a relation of equivalence on X. Thus, X is split into classes of equivalence w.r.t. the action of G. An equivalence class containing the point x ∈ X is called the orbit of x (relative to the group G). The space X is called homogeneous space if it consists of only one orbit. In that case we also say that the group G has transitive action on X.

2.2 Elements of Group Theory

19

2.2.3 Subgroups and Factor-Groups Let G be a group. A nonempty subset H ⊂ G is a called subgroup of the group G, if it is a group under the same multiplication. Let H be a subgroup of G. If we consider G as right H-space, then G splits into orbits: gH = {gh | h ∈ H},

g ∈ G,

which are called right congruence classes of G w.r.t. the subgroup H. The set of such congruence classes is denoted G/H and is called left factor-space of G w.r.t. H. Furthermore, G/H is a homogeneous space w.r.t. the left action of G. Furthermore, every homogeneous space X w.r.t. the left action of G is isomorphic to some G/H with the suitable choice of the subgroup H. Analogously, exchanging everywhere left ←→ right we obtain right factor-spaces H\G, where H is a subgroup of G. A subgroup H of G is called invariant subgroup if gH = Hg for all g ∈ G. Then for H there is no difference between left and right congruence classes, i.e., G/H = H\G. Furthermore the notion of multiplication is deﬁned for G/H since: ( gH)( g ′ H) = gHg ′ H = gg ′ HH = ( gg ′ )H. It is easy to check that G/H is a group w.r.t. this multiplication, H playing the role of e. This group is called the factor-group of G w.r.t. H, while H is called also normal divisor of G.

2.2.4 Homomorphisms Let G, H be two groups. A group homomorphism of G in H is any map > : G → H such that >( gg ′ ) = >( g)>( g ′ )

for all

g, g ′ ∈ G.

In particular, from this follows: >(eG )2 = >(eG )

⇒

>(eG ) = eH .

where eG , eH are the units of G, H, respectively. From (∗) also follows: >( g –1 ) = >( g)–1

for all

g ∈ G.

(∗)

20

2 Lie Algebras and Groups

A map 8 : G → H is called antihomomorphism if >( gg ′ ) = >( g ′ )>( g)

for all

g, g ′ ∈ G.

(∗∗)

Every bijective homomorphism of a group G on a group H is called an group isomorphism of the groups G and H. From the abstract point of view isomorphic groups may be considered to be the same. Every isomorphism of a group G on itself is called automorphism. The set Aut G of all automorphisms of a group G is also a group (w.r.t. the compositions of morphisms). An important example of automorphism of a group G is the map: g ↦ g0 gg0–1 (g ∈ G) with ﬁxed g0 ∈ G. These are called inner automorphisms. The set Aut0 G of all inner automorphisms is a normal divisor of Aut G. Examples: (1) The map >(x) = ex is an isomorphism of the additive group of the real numbers R to the multiplicative group of the positive real numbers R+ = (0, +∞). (2) The group Aut Rn of all automorphisms of the vector space Rn is isomorphic to GL(n, R). (3) The transposition map g ↦ t g is an antiautomorphism of the group GL(n, F) (4) The map g ↦ (t g)–1 is an automorphism of the group GL(n, F). ♢

2.2.5 Direct and Semidirect Products of Groups Let G, H be two groups. The Descartes product G × H becomes a group with the following component multiplication: ( g, h)( g ′ , h′ ) = ( gg ′ , hh′ ),

g, g ′ ∈ G, h, h′ ∈ H.

This group is called the direct product of G and H. Furthermore, the groups G, H are naturally identiﬁed with the normal divisors G × {eH }, {eG } × H, respectively., of the group G × H. If the group G acts on H by an automorphism h ↦ hg then again one can introduce multiplication in the Descartes product G × H: ( g, h)( g ′ , h′ ) = ( gg ′ , hh′g ),

g, g ′ ∈ G,

h, h′ ∈ H.

The resulting group is called the semidirect product of G and H, and is denoted by G ⋉ H. As before G × {eH }, {eG } × H, may be identiﬁed with G, H, respectively. Furthermore, H is a normal divisor of G ⋉H, and the factor-group (G ⋉ H)/H is isomorphic to G.

2.3 Structure of Semisimple Lie Algebras

21

2.3 Structure of Semisimple Lie Algebras 2.3.1 Cartan Subalgebra Let G0 be a ﬁnite-dimensional reductive Lie algebra over C or R, G0 = G ⊕ Z, where G = [G0 , G0 ] is a semisimple ideal of G0 , Z is the center of G0 . The description of the structure of G0 is based on a special class of commutative subalgebras H0 ⊂ G0 . A subalgebra H0 ⊂ G0 is called a Cartan subalgebra of G0 if: (1) H0 is a maximal Abelian subalgebra of G0 ; (2) ad X are diagonal in G0 for all X ∈ H0 . It is clear that H0 ⊃ Z, H0 = H ⊕ Z, where H is a Cartan subalgebra of G. A theorem of Cartan shows that each semisimple Lie algebra has a nontrivial Cartan subalgebra. We give an idea of the proof in order to introduce the so-called regular elements. For X ∈ G, let GX0 ≡ {Y ∈ G : (ad X)s (Y) = 0 for some s ∈ N}. X is called a regular element of G if dimGX0 = min dimGZ0 Z∈G

then one shows that regular elements exist and that GX0 is a Cartan subalgebra for every regular element. In fact, if H is a Cartan subalgebra and X ∈ H, then H = GX0 . Note that in the literature there are used also other deﬁnitions of a Cartan subalgebra. Equivalently, one may substitute condition (2) by: (2) ad X is completely reducible; or (1) and (2) by: (1) H is nilpotent and (2) H = nG (H), where for any subalgebra K ⊂ G nG (K) ≡ {X ∈ G : [X, K] ⊂ K}

(2.18)

is called the normalizer of K in G. Note that K is an ideal of nG (K), and that nG (K) is a subalgebra of G. By the deﬁnition of a Cartan subalgebra the elements ad H, H ∈ H can be simultaneously diagonalized. Denote by H∗ the dual algebra of H. Let for ! ∈ H∗ , G! ≐ {X ∈ G : [H, X] = !(H)X for all H ∈ H}. Note that G0 = H.

(2.19)

22

2 Lie Algebras and Groups

Let B ≐ {! ∈ H∗ : ! ≠ 0, G! ≠ {0}}.

(2.20)

The set B is called the root system of G relative to H, an element of B is called a root of G relative to H. Clearly G = H ⊕ ⊕ G! . !∈B

(2.21)

We shall call G! the root space of the root !, and the above decomposition shall be called the root space decomposition of G relative to H. (The name “root” originates from the fact that solving 0 = [Xi , Xj ] – !Xj = k (ckij – !$jk ) Xk is equivalent to solving the secular equation det|ckij – !$jk | = 0.) Let us denote by AutG the group of automorphisms of G. (The notion of “group” is deﬁned at the start of Subsection 2.2.1.) Let F = R, C. If D is a nilpotent derivation of G, then ( = exp D ∈ AutG. Let us denote by Aut0 G the group of inner automorphisms of G which is generated by the set {( = exp ad X : X ∈ G}. Obviously Aut0 G is a subgroup of AutG. Note that if G is semisimple then every derivation D of G is inner, i.e., D = ad X for some X ∈ G. Example: Let G = sl(n, F), ( : X ↦ –t X is not an inner automorphism of G. Theorem 2: Let F = C, let H and K be Cartan subalgebras of the semisimple Lie algebra G. Then there exists an inner automorphism ( ∈ Aut0 G, such that ((H) = K. Due to the above Theorem we shall say that all Cartan subalgebras of a semisimple Lie algebra G over C are conjugate via inner automorphisms. Consequently they all have the same dimension. For a semisimple Lie algebra G over C the rank of G, denoted rank G, is deﬁned by rank G = dimH, where H is a Cartan subalgebra of G. The same deﬁnition is applied for a reductive Lie algebra G0 = G ⊕ Z, where Z is the center of G0 ; clearly rank G0 = dimH0 = rank G + dimZ. 2.3.2 Lemmas on Root Systems Further some results will be formulated as simple lemmas. Lemma 1: (a) B(G! , G" ) = 0 if ! ≠ – "; (b) B(H, G! ) = 0 ∀! ∈ B; (c) B restricted to H is nondegenerate; (d) if ! ∈ B, then –! ∈ B and B is a nondegenerate pairing of G! with G–! . For further use let us denote E! ∈ G! , E–! ∈ G–! as elements for which B(E! , E–! ) = 1.

23

2.3 Structure of Semisimple Lie Algebras

Lemma 2: B spans H∗ . We can deﬁne for + ∈ H∗ , unique H+ : B(H+ , H) = +(H) ∀H ∈ H. Then + → H+ is an isomorphism between H∗ and H. We deﬁne inner products in H and H∗ by X, Y ≡ B(X, Y),

X, Y ∈ H

+, , ≡ H+ , H, (= +(H, ) = ,(H+ )), +, , ∈ H

(2.22) ∗

(2.23)

Note that H! ≠ 0 for ∀! ∈ B. Lemma 3: Let X ∈ G! , Y ∈ G–! , then [X, Y] = B(X, Y)H! .

(2.24)

It is clear that [G! , G–! ] = CH! because [E! , E–! ] = H! . Then ad H! = [ad E! , ad E–! ]. Lemma 4: tr ad H! |V = 0 for any subspace V ⊂ G invariant under both ad E! , ad E–! (and consequently under adH! ). Lemma 5: Let !, " ∈ B. Then ∃ a rational number q such that ", ! = q!, !; moreover !, ! is a rational number greater than zero. Lemma 6: dimG! = 1, ! ∈ B. If ! ∈ B, k! ∈ B, with k ∈ Z, then k = ±1. Lemma 7: Let HR =

RH! .

(2.25)

!

(1) (2)

Then dimR HR = dimH (≡ dimC H). Moreover, , is a positive deﬁnite scalar product on HR × HR . Each root is real-valued on HR . H = HR + iHR

The positive number !, ! will be called the length of the root !, or root length. Lemma 8: (1) Let !, " ∈ B, then 2"(H! )/!(H! ) = –r – s where r, s are integers, r ≤ 0 ≤ s, so that " + n! ∈ B, (n ∈ Z), for r ≤ n ≤ s, but " + (r – 1)! ∈/ B, " + (s + 1)! ∈/ B. (2) Let X! ∈ G! , X–! ∈ G–! , X" ∈ G" , then [X–! , [X! , X" ]) =

s(1 – r) !(H! )B(X! , X–! )X" . 2

Corollary: Let !, " ∈ B, !, " < 0. Then ! + " ∈ B.

24

2 Lie Algebras and Groups

Lemma 9: Let !, " ∈ B, " ≠ !. If " – ! is not a root, then !, " ≤ 0. Lemma 10: Let !, ", ! + " ∈ B. Then [G! , G" ] = G!+" . The vectors E! , ! ∈ B,

Hi = H!i ,

i = 1, . . . , dimH

(2.26)

are said to form the Cartan–Weyl basis of G. Let S ⊂ B. The hull of S, denoted S, is by deﬁnition the set of all roots in B of the form ±!, ±(! + ") where !, " run through S. Let us deﬁne N!" for !, " ∈ S such that ! + " ≠ 0 and either ! + " ∈ S or ! + " ∈/ B: [E! , E" ] = N!" E!+" if ! + " ∈ S, N!" = 0 if ! + " ∈/ B.

(2.27)

Clearly, N!" = –N"! . Note that B¯ = B. Then the constants N!" , !(Hi ) for i = 1, . . . , dimH, !, " ∈ B, determine the structure constants of G in the Cartan–Weyl basis, noting also that [Hi , E! ] = !(Hi )E! , [E! , E–! ] = H! . Lemma 11: Let !, ", # ∈ S, ! + " + # = 0. Then N!," = N",# = N#,! (all are deﬁned). Lemma 12: Let !, " ∈ S, ! + " ∈ B. Let " + n! be such that " + n! is a root for r ≤ n ≤ s, " + (r – 1)! ∈/ B, " + (s + 1)! ∈/ B. Then N!" N–!,–" = –

s(1 – r) !(H! ). 2

(2.28)

Lemma 13: Suppose !, ", #, $ are roots in S (not necessarily distinct) no two of which have sum 0. If ! + " + # + $ = 0. Then N!" N#$ + N"# N!$ + N#! N"$ = 0

(2.29)

(all N are deﬁned). Let HR be as above, let H1 , . . . H be a basis in HR , and let us introduce the lexicographical ordering in HR∗ , i.e. for +1 , +2 ∈ HR∗ , +1 > +2 if ∃k : 0 ≤ k ≤ – 1 such that +1 (Hi ) = +2 (Hi ), i = 1, . . . , k, +1 (Hk+1 ) > +2 (Hk+1 ). If + > 0, we say that + is positive. Let B+ ≡ {! ∈ B : ! > 0}, B– ≡ {! ∈ B : ! < 0}; then B = B+ ∪ B– , B+ ∩ B– = ∅. Proposition 1: Let G, G ′ be two semisimple Lie algebras, H, H′ their respective Cartan subalgebras, B, B′ their root systems. Let > be a linear isomorphism of H onto H′ such that >∗ B′ = B. Then > extends to an isomorphism of G onto G ′ .

2.3 Structure of Semisimple Lie Algebras

25

Let ! ∈ B, let 3! = {H ∈ H : !(H) = 0}; 3! is a hyperplane in H; H! ∈/ 3! because !(H! ) > 0. Let H!∨ = 2H! /!(H! ). Consider the reﬂection s! in the hyperplane 3! deﬁned by s! (H) = H – !(H)H!∨

(2.30)

s2! = 1, s! (H! ) = –H! , s! (H) = H, H ∈ 3! .

(2.31)

Clearly:

Note that s! leaves HR invariant since H – !(H)H!∨ ∈ HR if H ∈ HR . Thus, s! |HR is also the reﬂection of the null space of ! in HR , that is in 3!R = {H ∈ HR : !(H) = 0}. 2.3.3 Weyl Group Let W = W(G, H) be the group generated by s! , ! ∈ B, W is called the Weyl group of (G, H). W acts also in H∗ by 2+, ! ! = + – +, !∨ !, !, ! 2 + ∈ H∗ , ! ∈ B, !∨ ≡ !. !, !

s! (+) = + –

(2.32)

which follows from (2.30). Note that s! (±!) = ∓!. Lemma 14: Each element of W induces a permutation of B. Corollary: W is a ﬁnite group (as a subgroup of the permutations of a ﬁnite number of objects). A root ! ∈ B+ is called simple root if it cannot be written as ! = " + #, ", # ∈ B+ . It is clear that the simple roots form a basis of B+ , that is for any ! ∈ B+ we have ! = i ni !i with ni ∈ Z+ . Clearly the simple roots form also a basis for B– so that ∀! ∈ B– we have ! = i ni !i , ni ∈ Z– . So the simple roots form a basis of B, and of H∗ ; accordingly {H!i : !i a simple root} form a basis of H. Thus, the number of simple roots is equal to = dimH. Let 0 = {!1 , . . . ! } denote the set of simple roots. Let ! ∈ B+ , ! = n1 !1 + ⋅ ⋅ ⋅ + n ! . Let us call O(!) = n1 + ⋅ ⋅ ⋅ + n the order of root !. (It is called also the height of the root !, denoted as ht !.) Clearly O(!) ≥ 1 and O(!) = 1 iff ! ∈ 0. We shall say that a root ! ∈ B+ is a highest root of B if !+" is not a root for any " ∈ B+ . It can be easily seen that the highest root is unique and is of maximal order. ˜ Further the highest root will be denoted by !.

26

2 Lie Algebras and Groups

Proposition 2: Fix B, B+ , 0 for {G, H}. Then (1) for 1 ≤ i, j ≤ , i ≠ j, !i – !j ∈/ B, !i , !j ≤ 0; (2) for 1 ≤ i ≤ , s!i (B+ \{!i }) = (B+ \{!i }); (3) W is generated by s!1 , . . . s! and B = ∪i=1 W(!i ). Since the Weyl group is generated by the simple reﬂections then every element w ∈ W may be written as the product of some simple reﬂections. Every such product which uses a minimal number of simple reﬂections is called a reduced reﬂection expression or reduced form for w [59]. The number of simple reﬂections in the reduced form is called the length of reﬂection w and denoted by (w). 2.3.4 Cartan Matrix Let !i , !j ∈ 0, let A = (aij ) be the following matrix aij =

2!i , !j = !∨i , !j . !i , !i

(2.33)

The above matrix is called the Cartan matrix of B(G, H). Obviously A obeys the following: (1) aii = 2 for all i; (2) aij is an integer for all i, j; aij ≤ 0 if i ≠ j; (3) aij = 0 iff aji = 0; (4) det A ≠ 0. Two Cartan matrices A = (aij ) and A′ = (a′pq ) are said to be equivalent if they have the same rank, e.g., r, and if there is a permutation i ↦ i′ of {1, . . . , r} such that aij = a′i′ j′ , 1 ≤ i, j, i′ , j′ ≤ r. A Cartan matrix A of rank r is said to be reducible if we can ﬁnd a partition of {1, . . . , r} into two nonempty sets "1 , "2 such that aij = 0 for i ∈ "1 , j ∈ "2 ; otherwise A is called irreducible. If two Cartan matrices are equivalent, and one of them is irreducible, so is the other. Theorem 3: Two semisimple Lie algebras over C are isomorphic iff the corresponding equivalence classes of Cartan matrices are identical. A Lie algebra over C is simple iff the associated equivalence class of Cartan matrices consists entirely of irreducible elements. Corollary: Let 0′ ⊂ 0, 0 ′ ≠ 0. Then 0 ′ is the simple root system of a semisimple Lie algebra. Thus, we can be sure that if we classify all irreducible Cartan matrices then we obtain a classiﬁcation of the simple Lie algebras over C.

2.4 Classification of Kac–Moody Algebras

27

At this point we shall generalize the notion of a Cartan matrix and shall obtain a larger class of Lie algebras, including the widely used in applications afﬁne Kac– Moody algebras.

2.4 Classification of Kac–Moody Algebras Let A = (aij ) be a n × n complex matrix. The matrix A is called a generalized Cartan matrix if it satisﬁes the following: (1) aii = 2 for all 1 ≤ i ≤ n; (2) aij are nonpositive integers for i ≠ j; (3) aij = 0 iff aji = 0. Thus, the ordinary Cartan matrices are generalized Cartan matrices; a more precise statement is contained in Proposition 3 below. Following Kac [327] we shall classify all irreducible generalized Cartan matrices. Let u ∈ Rn , we write u > 0 if ui > 0, 1 ≤ i ≤ n, and u ≥ 0 if ui ≥ 0, 1 ≤ i ≤ n. Let us recall the following fundamental fact from linear algebra which we shall use in the following lemmas. Lemma 15: A system of real homogeneous linear inequalities +i > 0, 1 ≤ i ≤ m, has a solution iff there is no nontrivial linear dependence with nonnegative coefﬁcients among the +i . Lemma 16: If (aij ) is an arbitrary real m × s matrix for which there is no u ≥ 0, u ≠ 0, such that t Au ≥ 0, then there exists v > 0 such that Av < 0. Lemma 17: If A is an irreducible generalized Cartan matrix then Au ≥ 0, u ≥ 0 imply that either u > 0 or u = 0. Note that in Lemma 17 the property aii = 2 is not used. Theorem 4: Kac [327] Let A be a real n × n irreducible matrix with aij ≤ 0 for i ≠ j, aij = 0 iff aji = 0. Then one and only one of the following three possibilities holds for both A and t A: (FIN): det A ≠ 0, there ∃ u > 0 such that Au > 0; Av ≥ 0 implies v > 0 or v = 0; (AFF): rankA = n – 1, there ∃ u > 0 such that Au = 0; Av ≥ 0 implies Av = 0; (Ind): there ∃ u > 0 such that Au < 0; Av ≥ 0, v ≥ 0 imply v = 0. Referring to cases (FIN), (AFF), or (Ind) we shall say that A is of ﬁnite type, afﬁne type, or indeﬁnite type. Corollary: Let A be as in the theorem above. Then A is of ﬁnite (resp. afﬁne, or indeﬁnite) type iff there ∃u > 0 such that Au > 0 (resp. Au = 0, or Au < 0). Recall that a matrix of the form A" = (aij )ij ∈ " , " ⊂ {1, . . . , n} is called a principal submatrix of A = (aij ). The determinant of a principal submatrix is called a principal minor.

28

2 Lie Algebras and Groups

Lemma 18: If A is of ﬁnite or afﬁne type, then any proper principal submatrix of A decomposes into a direct sum of matrices of ﬁnite type. Lemma 19: A symmetric matrix A is of ﬁnite (resp. afﬁne) type iff A is positive deﬁnite (resp. positive semideﬁnite of rank n – 1). Definition: A matrix A is called symmetrizable matrix, if there exists a diagonal matrix D with positive entries such that DA is symmetric matrix. Lemma 20: A n × n matrix A = (aij ) is symmetrizable iff aij = 0 ⇒ aji = 0

(∗)

ai1 i2 ai2 i3 . . . aik i1 = ai2 i1 ai3 i2 . . . ai1 ik

(∗∗)

and

for all i1 , . . . ik . Lemma 21: Let A = (aij ) be a generalized Cartan matrix of ﬁnite or afﬁne type. Then A is symmetrizable. Moreover, if ai1 i2 ai2 i3 . . . ais i1 ≠ 0 for some i1 , . . . is ,

s ≥ 3,

(∗)

then A is of the form ⎛

2 ⎜ –1 ⎜ ⎜ 0 ⎜ ⎜ ⎜... ⎜ ⎝ 0 –1

–1 2 –1 ... 0 0

0 –1 2 ... 0 0

... ... ... ... ... ...

0 0 0 ... 2 –1

⎞ –1 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟. ...⎟ ⎟ –1 ⎠ 2

Let A = (aij ) be a generalized Cartan matrix. We associate to A a graph S(A) with n vertices, called the Dynkin diagram of A as follows. If aij aji ≤ 4, and |aij | ≥ |aji | the vertices i and j are connected by |aij | lines with an arrow pointing toward i if |aij | > 1, (i.e., the arrow (if any) points to the vertex corresponding to the root with smaller or equal norm). If aij aji > 4, the vertices i and j are connected by a boldfaced line equipped with an ordered pair of integers |aij |, |aji |. It is clear that A is irreducible iff S(A) is a connected graph. We say that S(A) is of ﬁnite, afﬁne or indeﬁnite type of rank if A is of that type. Proposition 3: Kac [327] Let A be an irreducible generalized Cartan matrix. (a) (b)

A is of ﬁnite type iff all its principal minors are positive. A is of afﬁne type iff all its proper principal minors are positive and det A = 0.

2.4 Classification of Kac–Moody Algebras

(c) (d) (e)

29

If A is of ﬁnite or afﬁne type, then any proper subdiagram of S(A) is a union of (connected) Dynkin diagrams of ﬁnite type. If A is of ﬁnite type, then S(A) contains no cycles. If A is of afﬁne type and has a cycle then S(A) is the cycle A(1) in the Table AFF. A is of afﬁne type iff ∃$ > 0 such that A$ = 0; such a $ is unique up to a constant factor.

Proof. To prove (a) and (b) note that by Lemma 21, if A is of ﬁnite or afﬁne type, it is symmetrizable. Then (a) and (b) follow from Lemma 19, (c) follows from Lemma 18, (d) follows from Lemma 21, (e) follows from Theorem 4. ∎ Now we can list all generalized Cartan matrices of ﬁnite and afﬁne type. Theorem 5: Kac [327] (a) (b) (c)

The Dynkin diagrams of all generalized Cartan matrices of ﬁnite type are listed in Table FIN. The Dynkin diagrams of all generalized Cartan matrices of afﬁne type are listed in the tables AFF 1–3 (all of them have + 1 vertices). The labels in Tables AFF 1–3 are the coordinates of the unique vector $ = (a0 , a1 , . . . a ) such that A$ = 0 and the ai are positive relatively prime integers. ♢

We omit the Proof mentioning only two lemmas which are of independent interest. Lemma 22: Let S(A) be of ﬁnite type of rank . Then the number of links is ≤ – 1. Lemma 23: Let S(A) be of ﬁnite type of rank . Then through each vertex there pass at most 3 links counting a double (triple) link as two (three) links. Remark 1: The Dynkin diagrams of the algebras G (1) are sometimes called extended Dynkin diagrams, since they are obtained from the Dynkin diagrams of the simple Lie algebras G by adding an additional node. ♢ Let a0 , a1 , . . . , a be the labels of the Dynkin diagrams in the Tables AFF. Let a∨i , (i = 0, 1, . . . , ), be the labels of the Dynkin diagrams of the dual algebras corresponding to the transposed Cartan matrices of afﬁne type. (They are obtained by reversing the direction of the arrows (if any) but keeping the enumeration of vertices.) Thus, the nontrivial dual pairs are : (2) B(1) and A2–1 ;

C(1) and D(2) +1 ;

F4(1) and E6(2) ; G(1) 2

and

D(3) 4 ;

when a0 = 2 while all other algebras are selfdual. Note that a0 = 1 except for A(2) 2 ∨ ∨ while a0 = 1 in all cases. The numbers h = i = 0 ai , respectively, h = i = 0 a∨i are

30

2 Lie Algebras and Groups

called the Coxeter number, respectively, the dual Coxeter number for the Cartan matrix and Dynkin diagram in consideration. Explicitly one has h = h∨ = + 1, for A(1) ;

h = 2, h∨ = 2 – 1, for B(1) ;

h = 2, h∨ = + 1, for C(1) ; h = h∨ = 12, for E6(1) ;

h = h∨ = 2 – 2, for D(1) ;

h = h∨ = 18, for E7(1) ;

h = 12, h∨ = 9, for F4(1) ;

h = h∨ = 30, for E8(1) ;

h = 6, h∨ = 4, for G(1) 2 ;

h = h∨ = 2 + 1, for A(2) 2 ;

h = 2 – 1, h∨ = 2, for A(2) 2–1 ;

h = + 1, h∨ = 2, for D(2) + 1;

h = 9, h∨ = 12, for E6(2) ;

h = 4, h∨ = 6, for D(3) 4 . Table FIN

A , ( ≥ 1), B , ( ≥ 3), C , ( ≥ 2),

○ ––– ○ –––⋯ ––– ○ ––– ○

!1

!2

!–1

!

○ ––– ○ ––– . . . ––– ○ ⇒ ○

!1

!2

!–1

!

○ ––– ○ ––– . . . ––– ○ ⇐ ○

!1

!2

!–1

!

!

D , ( ≥ 4),

E6

E7

E8 F4 G2

| ○ ––– ○ ––– . . . ––– ○ ––– ○

!1

!2

!–2

!–1

○!2 | ○ ––– ○ ––– ○ ––– ○ ––– ○

!1

!3

!4

!5

!6

○!2 | ○ ––– ○ ––– ○ ––– ○ ––– ○ ––– ○

!1

!3

!4

!5

!6

!7

○!2 | ○ ––– ○ ––– ○ ––– ○ ––– ○ ––– ○ ––– ○

!1

!3

!4

!5

○ ––– ○ ⇒ ○ ––– ○

!1

!2

○ ≡≡≡> ○

!1

!2

!3

!4

!6

!7

!8

2.4 Classification of Kac–Moody Algebras

Table AFF 1

A(1) 1

○ ⇐⇒ ○ 1

1

◦ 1

A(1) , ( ≥ 2),

◦ −−− ◦ −−− · · · · · · −−− ◦ −−− ◦

B(1) , ( ≥ 3),

○1 | ○ ––– ○ ––– ○ ––– ○ –––⋯ ––– ○ ⇒ ○

C(1) , ( ≥ 2),

○ ⇒ ○ –––⋯ ––– ○ ⇐ ○

D(1) , ( ≥ 2),

○1 ○1 | | ○ ––– ○ ––– ○ ––– . . . ––– ○ ––– ○

E6(1)

○1 | ○2 | ○ ––– ○ ––– ○ ––– ○ ––– ○

E7(1)

○2 | ○ ––– ○ ––– ○ ––– ○ ––– ○ ––– ○ ––– ○

E8(1)

○3 | ○ ––– ○ ––– ○ ––– ○ ––– ○ ––– ○ ––– ○ ––– ○

F4(1)

○ ––– ○ ––– ○ ⇒ ○ ––– ○

G(1) 2

○ ––– ○≡≡≡≡≡> ○

1

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

2

3

4

3

4

3

1

1

4

3

2

1

2

3

1

3

5

2

2

6

1

4

2

31

32

2 Lie Algebras and Groups

Table AFF 2 A(2) 2

○ ○< ≡≡≡≡≡≡≡ 2 ————- 1

A(2) 2 , ( ≥ 2),

○ ⇐ ○ ––– . . . ––– ○ ⇐ ○

A(2) 2–1 , ( ≥ 3),

○1 | ○ ––– ○ ––– ○ ––– . . . ––– ○ ⇐ ○

D(2) +1 , ( ≥ 2),

○ ⇐ ○ ––– . . . ––– ○ ⇒ ○

E6(2)

○ ––– ○ ––– ○ ⇐ ○ ––– ○

2

2

1

2

2

1

2

2

1

1

1

1

2

3

2

1

1

1

Table AFF 3

D(3) 4

○ ––– ○ 4. Thus, there are inﬁnitely many strictly hyperbolic type matrices of rank 2. Kac [327] has conjectured that there are only a ﬁnite number of hyperbolic matrices of order n ≥ 3 and that the order of a strictly hyperbolic (resp. hyperbolic) matrix is ≤ 5 (resp. ≤ 10). He has also given the 18 hyperbolic matrices of orders 7,8,9,10 (all are symmetrizable). Later, in [535] was given the complete classiﬁcation of such matrices. In the strictly hyperbolic case there are 31 matrices of order 3 (11 of them

2.4 Classification of Kac–Moody Algebras

33

○⇒===○ are symmetrizable), 3 of order 4, 1 of them is symmetrizable: | |, 1 of order 5 (not ○⇒===○ symmetrizable) and none of order 6 and higher. The question of symmetrizability was treated in [535] (and others) with some omissions, and was later settled in [77]. Putting together the results of [535] and [77] altogether in the hyperbolic case (including the strictly hyperbolic) there are 238 matrices, of which 142 are symmetrizable: 123 matrices of order 3 (44 symmetrizable), 53 matrices of order 4 (40 symmetrizable), 22 of order 5 (20 symmetrizable), 22 of order 6 (20 symmetrizable), and ﬁnally those that were given by Kac [327]: 4 of order 7, 5 of order 8, 5 of order 9, 4 of order 10 (all symmetrizable). Lemma 24: Let A be of type Tp,q,r , p ≥ q ≥ r ≥ 2, i.e., with Dynkin diagram : γ1 γr–1 δ α1

α2

αp–1

βq–1

β2

β1

which has p + q + r – 2 nodes. Set c = p1 + q1 + 1r . Then A is of ﬁnite, resp, afﬁne, resp. indeﬁnite type iff c > 1, resp. c = 1, resp c < 1. If A is indeﬁnite Tpqr type then its signature is (+, +, . . . , +, –). Most interesting are the Lie algebras En ≐ Tn–3,3,2 , n ≥ 6. Of course, for n = 6, 7, 8 these are the ﬁnite-dimensional En , E9 = E8(1) . In string physics of particular interest is the algebra E10 (cf., e.g., [343]) with diagram: ○ | ○ ––– ○ ––– ○ ––– ○ ––– ○ ––– ○ ––– ○––– ○ ––– ○ which gives the only symmetric hyperbolic matrix with determinant –1. It is interesting to consider also generalized Cartan matrices of inﬁnite order, i.e. A = (aij )ij ∈ Z . One has: Proposition 5: Kac [327] A complete list of generalized Cartan matrices of inﬁnite order such that any principal minor of ﬁnite order is positive is: A∞ ⋯ ––– ○ ––– ○ ––– ○ ––– ⋯ A+∞

○ ––– ○ ––– ○ –––⋯

B∞

○ ⇐ ○ ––– ○ ––– ○⋯

C∞ ○ ⇒ ○ ––– ○ ––– ○⋯ ○ | D∞ ○ ––– ○––– ○ ––– ○⋯

34

2 Lie Algebras and Groups

2.5 Realization of Semisimple Lie Algebras In this section, we give the realization of semisimple Lie algebras, i.e., Kac–Moody algebras of type FIN, and in the next section of the afﬁne Kac–Moody algebras of type AFF. Actually, we had already considered the example of sl(n, C), so(n, C), sp(n, C), all of which are subalgebras of gl(m, C). Let Eij ∈ gl(m, C) be a matrix with only nonzero entry, equal to 1, at the ith row and jth column, 1 ≤ i, j ≤ m, (Eij )st = $is $jt .

(2.34)

From this follow the commutation relations: [Eij , Ek ] = $jk Ei – $i Ekj .

(2.35)

2.5.1 Special Linear Algebra Here G = sl( + 1, C), = 1, 2 . . . , tr X = 0, X ∈ G. It is useful to show that G is semisimple. For this we show that B(X, Y) = 2( + 1)tr XY, X, Y ∈ G.

(2.36)

Indeed, let H denote the subalgebra of G consisting of all diagonal matrices. +1 Let H ∈ H, H = +1 = 0, then [H, Eij ] = (ki – kj )Eij . Hence i = 1 ki Eii , i = 1 ki = tr H +1 2 +1 2 2 = 2( + 1) 2 B(H, H) = Tr(ad H)2 = +1 (k – k ) i j i,j = 1 i = 1 ki – 2 ( i = 1 ki ) = 2( + 1)tr H . The latter is zero iff H = 0. Each X ∈ G with all eigenvalues different can be diagonalized so there ∃ a nonsingular matrix g such that gXg –1 ∈ H. By the invariance of B and Tr, B(X, X) = 2( + 1)Tr(X 2 ) and then by continuity this holds for all X ∈ G. By polarization, we have (2.36). This implies that G is indeed semisimple. Now we choose the following basis for G Hi = Eii – Ei+1,i+1 ∈ H, i = 1, . . . , , Ejk , j, k = 1, 2, . . . , + 1; j ≠ k.

(2.37)

Obviously H (spanned by {Hi }) is a Cartan subalgebra of G. Indeed, H is Abelian, it diagonalizes G: [Hi , Ejk ] = ($ij – $ik – $i+1,j + $i+1,k )Ejk

(2.38)

and no linear combination of Ejk , j ≠ k, can commute with H, so it is maximal. We know that the dual basis in H∗ will give us the simple roots and the Cartan matrix. Since all properties essentially depend on the Cartan matrix aij = 2!i , !j /!i , !i

35

2.5 Realization of Semisimple Lie Algebras

and !i , !j = B(Hi , Hj ) we notice that we can rescale the Killing form by a suitable factor so that the formulas look simpler. In this case we shall use ˜ B(X, Y) =

1 B(X, Y) = tr XY. 2( + 1)

(2.39)

Thus, for Hi , i = 1, . . . , , 0 = {!i , . . . ! }, introducing a rescaled scalar product (⋅, ⋅) also in H∗ , we have ⎧ ⎪ i=j ⎪ ⎨2 ˜ i , Hj ) = !i (Hj ) = !j (Hi ) = –1 |i – j| = 1 (!i , !j ) = B(H ⎪ ⎪ ⎩0 |i – j| ≥ 2

(2.40)

Let us denote by !jk , j ≠ k, the root corresponding to the subspace CEjk . Using (2.38) we see that !jk (Hi ) = $ij – $ik – $i+1,j + $i+1,k , !j,j+1 = !j = –!j+1,j ,

j = 1, . . . ,

!jk = !j + !j+1 + ⋅ ⋅ ⋅ + !k–1 = – !kj ,

(2.41a) (2.41b) (2.41c)

1 ≤ j < k + 1 ≤ + 2. Further, for simplicity we shall introduce an orthonormal basis in R+1 , %1 , . . . , %+1 such that (%i , %j ) = $ij ,

%i (Ekk ) = $ik .

(2.42)

Then we have %i (Hk ) = $ik – $ik+1 ,

!i = %i – %i+1 .

(2.43)

In terms of this basis the root system B is given by B+ = {!ij = %i – %j |i < j},

B– = {!ij = %i – %j |i > j}.

(2.44)

Note that all roots have the same length: (!ij , !ij ) = (%i – %j , %i – %j ) = 2, ∀i, j, (i ≠ j). One has the decomposition G = H ⊕ G+ ⊕ G– , G± = ⊕ G! ,

(2.45a)

dimG+ = |B+ | = dimG– = |B– | = ( + 1)/2,

(2.45b)

! ∈ B±

36

2 Lie Algebras and Groups

where |B| is the number of elements of the ﬁnite set B. Note that each G+ , G– is nilpotent. We note also that for i < j – 1 we have Eij = [Ei,i+1 , Ei+1,j ] = (ad Ei,i+1 ) Ei+1,j = [Ei,j–1 , Ej–1,j ] = (ad Ei,j–1 ) Ej–1,j ,

(2.46a)

Eji = [Ej,i+1 , Ei+1,i ] = (ad Ej,i+1 ) Ei+1,i = [Ej,j–1 , Ej–1,i ] = (ad Ej,j–1 ) Ej–1,i , [Eij , Eji ] = Hij ≡ Hi + Hi+1 + ⋅ ⋅ ⋅ + Hj–1 ,

(2.46b)

i < j – 1,

(2.46c)

[Eii+1 , Ei+1i ] = Hi .

(2.46d)

˜ ij , Eji ) = tr Eij Eji = 1, i ≠ j. B(E

(2.47)

The Cartan matrix aij = 2(!i , !j )/(!i , !i ) = (!i , !j ) is given by (2.40) and is of type A . Due to (2.46d) and (2.47) the elements Eij , i ≠ j, Hi , form the Cartan–Weyl basis of sl( + 1, C). Remark 2: Note that the n2 elements Eij , form the Cartan–Weyl basis of gl( + 1, C). The Weyl group of sl( + 1, C) is permuting the vectors %i , i.e., it is isomorphic to the symmetric group S+1 , i.e., |W| = ( + 1)!. 2.5.2 Odd Orthogonal Lie Algebra ˜ dimG = (2 + 1). Let G = so(2 + 1, C), = 1, 2, . . . , X˜ ∈ G, t X˜ = –X, First we make a unitary transformation in order to bring the Cartan subalgebra in diagonal form. Let U=

1 0

0 , U′

1 I U′ = √ 2 I

iI . –iI

(2.48)

˜ –1 to obtain: Obviously UU + = U + U = I2+1 , (U + )ij = (t U)ij . Then we set X = U XU ⎛

1 ⎜ t X = –>X>, > = U t U = ⎝0 0

0 0 I

⎞ 0 ⎟ I ⎠ = >–1 . 0

(2.49)

Thus ⎛

0 ⎜ X = ⎝–b –a

ta

A C

tb

⎞

⎟ B ⎠, a, b ∈ C , A, B, C ∈ gl(, C), – tA

t

B = –B,

is the general expression for an element of G in the new basis.

t

C = –C

(2.50)

2.5 Realization of Semisimple Lie Algebras

37

Obviously the Cartan subalgebra is spanned by ⎛ 0 ⎜ Hi = ⎝0 0

⎞ 0 ⎟ 0 ⎠ , i = 1, . . . , ; –Ai

0 Ai 0

Ai = diag(0, . . . 0, 1, –1, 0, . . . , 0), i = 1, . . . , – 1;

(2.51a)

(2.51b)

A = diag(0, . . . , 0, 1). Analogously to the sl case, we obtain B(X, Y) = (2 – 1)tr XY

(2.52)

and we set ˜ B(X, Y) =

1 1 B(X, Y) = tr XY. 2(2 – 1) 2

(2.53)

Then we have ⎧ ⎪ 2 i = j < , ⎪ ⎪ ⎪ ⎪ ⎨1 i = j = , ˜ i , Hj ) = !i (Hj ) = !j (Hi ) = (!i , !j ) = B(H ⎪ ⎪–1 |i – j| = 1, ⎪ ⎪ ⎪ ⎩0 |i – j| ≥ 2.

(2.54)

In terms of the orthonormal basis %i , such that %i (Hk ) = $ik – $ik+1 , the simple roots are 0 = {!i = %i – %i+1 , 1 ≤ i ≤ – 1, ! = % }

(2.55)

while the inverse formula is: %j = !j + ⋅ ⋅ ⋅ + ! .

(2.56)

Let E%i denote a matrix of type (2.50) with only nonzero entry for bi = 1, i = 1, . . . , ; let E–%i , E%i +%j , (i < j), E–(%i +%j ) , (i < j), E%i –%j , (i ≠ j), denote the matrices type (2.50) with only nonzero entries ai = 1, B = Eij – Eji , C = Eij – Eji , A = Eij , respectively. The described matrices together with Hi , i = 1 . . . , form a basis of G. The notation is suggestive, namely, one can check that [Hj , E±%i ] = ±%i (Hj )E±%i , [Hk , E±(%i +%j ) ] = ±(%i + %j )(Hk )E±(%i +%j ) , [Hk , E%i –%j ] = (%i – %j )(Hk )E%i –%j .

(2.57a) (2.57b) (2.57c)

38

2 Lie Algebras and Groups

Thus, the root system is determined by B+ = {%i ± %j , 1 ≤ i < j ≤ ; %k , 1 ≤ k ≤ }.

(2.58)

Note that one has roots of two different lengths, the length ratio is 2 : 1. With our choice of normalization of the bilinear form B˜ (2.53) the long roots %i ± %j have length 2, while the short roots %k have length 1. Finally, we check that ⎧ ⎪ 2 i = j, ⎪ ⎪ ⎪ ⎪ ⎨ –1 |i – j| = 1, i ≠ , 2(!i , !j ) (2.59) aij = = ⎪ (!i , !i ) ⎪–2 i = , j = – 1, ⎪ ⎪ ⎪ ⎩0 otherwise, which is the Cartan matrix of type B . Note dimG = (2 + 1), dimG+ = dimG– = 2 . For = 1 it is evident from (2.57) which reduces to [H1 , E±%1 ] = ± %1 (H1 )E±%1 = ± E±%1 , that H1 , E±%1 span an algebra isomorphic to sl(2, C) as we noted in Section 1.1. The Weyl group of so(2 + 1, C) is generated by permutations of the vectors %i and by the reﬂections %i → –%i , (i = 1, . . . , ), i.e., |W| = 2 !. 2.5.3 Symplectic Lie Algebra Let G = sp(, C), X ∈ gl(2, C), t XJ + J X = 0, dimG = (2 + 1). The general expression for X ∈ G is

B , A, B, C ∈ gl(, C), t B = B, t C = C. – tA

A X= C

(2.60)

A basis of the Cartan subalgebra is Ai 0

A′ H = 0 Hi =

0 , i = 1, . . . , – 1, –Ai 0 , –A′

(2.61a)

(2.61b)

A′ = 2A = (0, . . . , 0, 2),

(2.61c)

B(X, Y) = 2( + 1)tr XY

(2.62)

where Ai are given in (2.51). One has

2.5 Realization of Semisimple Lie Algebras

39

and we set ˜ B(X, Y) =

1 1 B(X, Y) = tr XY. 8( + 1) 4

(2.63)

Thus

˜ i , Hj ) = !i (Hj ) = !j (Hi ) = (!i , !j ) = B(H

⎧ ⎪ –1 |i – j| = 1, i, j < , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨–2 ij = ( – 1), 2 ⎪ ⎪ ⎪ ⎪ 4 ⎪ ⎪ ⎪ ⎩ 0

i = j < ,

(2.64)

i = j = , otherwise.

In terms of the orthonormal basis %i , such that %i (Ha ) = $ik – $i,k+1 , the simple roots are 0 = {!i = %i – %i+1 ,

1 ≤ i ≤ – 1;

! = 2% }.

(2.65)

Let E%i –%j , i ≠ j, E%i +%j , E–(%i +%j ) , denote the matrices of type (2.60) with the only nonzero entries A = Eij , B = Eij + Eji , C = Eij + Eji , respectively. These vectors span the root vector spaces of the roots %i – %j , (i ≠ j), %i + %j , –(%i + %j ), respectively. The positive roots are given by B+ = {%i ± %j ,

1 ≤ i < j ≤ ;

2%i , 1 ≤ i ≤ },

(2.66)

As in the previous case there are roots of two lengths with length ratio is 2 : 1. With our choice of normalization of the bilinear form B˜ (2.63) the long roots 2%k have length 4, while the short roots %i ± %j have length 2. We note also: dimG+ = |B+ | = dimG– = |B– | = 2 . Finally, we obtain

aij =

⎧ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎨–1

2(!i , !j ) = (!i , !i ) ⎪ –2 ⎪ ⎪ ⎪ ⎪ ⎩0

i = j, |i – j| = 1, j ≠ 1, j ≠ , i = – 1, j = ,

(2.67)

otherwise,

which is the Cartan matrix of type C . For = 1 it is clear that sp(1, C) ≅ sl(2, C). For = 2 the isomorphism sp(2, C) ≡ so(5, C) becomes obvious if we map (!1 , !2 )B2 → (!2 , !1 )C2 . The Weyl group of sp(, C) (as the one of so(2+1, C)) is generated by permutations of the vectors %i and by the reﬂections %i → –%i , (i = 1, . . . , ), i.e., |W| = 2 !.

40

2 Lie Algebras and Groups

2.5.4 Even Orthogonal Lie Algebra ˜ dimG = (2 – 1). Let G = so(2, C), = 2, 3, . . . , t X˜ = –X, (We consider the case > 1 since for = 1, dimG = 1, thus, so(2, C) is Abelian.) ˜ ′–1 , where U ′ is given in (2.48). Then Let X = U ′ XU t

X = –>′ X>′ , >′ = U ′ t U ′ =

0 I I 0

= >′–1 .

(2.68)

Thus, the general expression of an element of G is X=

A B , C – tA

t

B = –B, tC = –C.

(2.69)

The Cartan subalgebra is spanned by

A′i 0 Hi = , A′i = Ai , i = 1, . . . – 1, A′ = diag(0, . . . , 0, 1, 1); 0 –A′i

(2.70)

Ai are given in (2.51). We obtain B(X, Y) = 2( – 1)tr XY

(2.71)

1 1 B(X, Y) = tr XY, ( > 1). 4( – 1) 2

(2.72)

and we set ˜ B(X, Y) = Thus ⎧ ⎪ –1 |i – j| = 1, i, j ≠ , ⎪ ⎪ ⎪ ⎪ ⎨ ij = ( – 2), ˜ i , Hj ) = (!i , !j ) = B(H ⎪ 2 i = j, ⎪ ⎪ ⎪ ⎪ ⎩0 otherwise.

(2.73)

In terms of the orthonormal basis {%i } the simple root system is 0 = {!i = %i – %i+1 , 1 ≤ i ≤ – 1; ! = %–1 + % }

(2.74)

2.5 Realization of Semisimple Lie Algebras

41

while the inverse map to (2.74) is 1 %j = !j + ⋅ ⋅ ⋅ + !–2 + (!–1 + ! ), 2 1 %–1 = (!–1 + ! ), 2

1 ≤ j ≤ – 2,

1 % = (! – !–1 ). 2

(2.75)

1 ≤ i < j ≤ }.

(2.76)

The positive roots are B+ = {%i ± %j ,

Note that all roots have the same length: (%i ± %j , %i ± %j ) = 2, ∀i, j, (i ≠ j). The positive root space vectors E%i –%j , (i ≠ j), E%i +%j , E–(%i +%j ) are given by matrices of type (2.60) with the only nonzero entries A = Eij , (i ≠ j), B = Eij – Eji , C = Eij – Eji , respectively. The Cartan matrix aij =

2(!i , !j ) = (!i , !j ) (!i , !i )

(2.77)

is given by (2.73) and is of type D . Note dimG = (2 – 1), dimG+ = dimG– = ( – 1). It is clear from (2.73) that for = 2, (!1 , !2 ) = 0, so D2 ≅ A1 × A1 as we mentioned in Section 1.1, (each !i being identiﬁed with the single positive root of a copy of A1 ). For = 3, (!i , !j ) ≠ 0 for i ≠ j for (!1 , !2 ) = – 1 = (!1 , !3 ), thus, D3 ≅ A3 , (so(6, C) ≅ s(4, C)), by the mapping (!1 , !2 , !3 )D3 → (!2 , !1 , !3 )A3 . The Weyl group of so(2, C) is generated by permutations of the vectors %i , (i = 1, . . . , ), and by the pairs of reﬂections: %i → –%i , %j → –%j , i ≠ j, (i, j = 1, . . . , ), i.e., |W| = 2–1 !.

2.5.5 Exceptional Lie Algebra G2 2 –1 , (!1 , !1 ) = 3(!2 , !2 ) = –2(!1 , !2 ). –3 2 We choose (!2 , !2 ) = 2, then (!1 , !1 ) = 6, (!1 , !2 ) = –3 and by Lemma 8 !1 + !2 , !1 + 2!2 , !1 + 3!2 are positive roots (but not !1 + k!2 , k ≥ 4). Checking for negative products among these roots we ﬁnd only one more possibility (from Lemma 8): 2!1 + 3!2 . Thus, Let G = G2 , (aij ) =

B+ = {!1 , !2 , !1 + !2 , !1 + 2!2 , !1 + 3!2 , 2!1 + 3!2 }

(2.78)

42

2 Lie Algebras and Groups

or in terms of the orthonormal basis %1 , %2 , %3 B+ = {%1 – %2 , %2 – %3 , %1 – %3 , 2%1 – %2 – %3 , %1 + %3 – 2%2 , %1 + %2 – 2%3 }.

(2.79)

Thus, G2 is 14-dimensional (14 = |B|+ rank G2 ). For the simple roots we may choose !1 = %1 + %3 – 2%2 , !2 = %2 – %3 .

(2.80)

With the chosen normalization the roots !1 , !1 + 3!2 , 2!1 + 3!2 have length 6, while !2 , !1 + !2 , !1 + 2!2 have length 2. The dual roots are: !∨1 = a1 /3, !∨2 = a2 , (!1 + !2 )∨ = !1 + !2 = 3!∨1 + !∨2 , (!1 + 2!2 )∨ = !1 + 2!2 = 3!∨1 + 2!∨2 , (!1 + 3!2 )∨ = (!1 + 3!2 )/3 = !∨1 + !∨2 , (2!1 + 3!2 )∨ = (2!1 + 3!2 )/3 = 2!∨1 + !∨2 . The Weyl group of G2 is the dihedral group of order 12. This follows from the fact that (s1 s2 )6 = 1, where s1 , s2 are the two simple reﬂections (cf. next Section, (2.256)). 2.5.6 Exceptional Lie Algebra F4 Let G = F4 : ⎛

2 ⎜–1 ⎜ (aij ) = ⎜ ⎝0 0

–1 2 –2 0

0 –1 2 –1

⎞ 0 0⎟ ⎟ ⎟ –1⎠ 2

(2.81)

and (!1 , !1 ) = (!2 , !2 ) = 2(!3 , !3 ) = 2(!4 , !4 ). Next we construct the root system. We choose (!3 , !3 ) = 1, then !∨1 = 2!1 /(!1 , !1 ) = !1 , !∨2 = !2 , !∨3 = 2!3 , !∨4 = 2!4 ; (!1 , !2 ) = (!2 , !3 ) = –1, (!3 , !4 ) = –1/2. Thus !1 + !2 , !2 + !3 , !2 + 2!3 , !3 + !4 ∈ B+ .

(F4a)

(We note that for ", # ∈ B+ , (" + #)∨ = "∨ + #∨ iff (", ") = (#, #) = (" + #, " + #).) Further one has : (!1 , (!2 + !3 )∨ ) = –2, (!1 , !2 + 2!3 ) = –1 ⇒ !1 + !2 + !3 , !1 + 2!2 + 2!3 , !1 + !2 + 2!3 ∈ B+ ;

(F4b)

(!2 , (!3 + !4 )∨ ) = –2 ⇒ !2 + !3 + !4 , !2 + 2!3 + 2!4 ∈ B+ ;

(F4c)

(!1 + !2 , (!3 + !4 )∨ ) = –2, !2 + !3 , (!3 + !4 )∨ ) = –1 ⇒ !1 + !2 + !3 + !4 , !1 + !2 + 2!3 + 2!4 , !2 + 2!3 + !4 ∈ B+ ;

(F4d)

2.5 Realization of Semisimple Lie Algebras

43

(!1 + !2 + 2!3 , !∨4 ) = –2, (!∨4 , !1 + 2!2 + 2!3 ) = –2 ⇒ !1 + !2 + 2!3 + !4 , !1 + !2 + 2!3 + 2!4 , !1 + 2!2 + 2!3 + !4 , !1 + 2!2 + 2!3 + 2!4 ∈ B+ ;

(F4e)

(!∨3 , !1 + 2!2 + 2!3 + !4 ) = –1, (!∨3 , !1 + 2!2 + 2!3 + 2!4 ) = –2 ⇒ !1 + 2!2 + 3!3 + !4 , !1 + 2!2 + 3!3 + 2!4 , !1 + 2!2 + 4!3 + 2!4 ∈ B+ ;

(F4f )

(!2 , !1 + 2!2 + 4!3 + 2!4 ) = –1 ⇒ !1 + 3!2 + 4!3 + 2!4 ∈ B+ ;

(F4g)

(!1 , !1 + 3!2 + 4!3 + 2!4 ) = –1 ⇒ !˜ = 2!1 + 3!2 + 4!3 + 2!4 ∈ B+ .

(F4h)

One can check that there are no other possibilities and the 24 roots (in (F4) and 0) exhaust B+ . There are roots of two lengths with length ratio 2 : 1. The long roots are: !1 , !2 , !1 + !2 , !2 + 2!3 , !1 + !2 + 2!3 , !1 + 2!2 + 2!3 , !2 + 2!3 + 2!4 , !1 + !2 + 2!3 + 2!4 , !1 + 2!2 + 2!3 + 2!4 , !1 + 2!2 + 4!3 + 2!4 , !1 + 3!2 + 4!3 + 2!4 , 2!1 + 3!2 + 4!3 + 2!4 . With the chosen normalization they have length 2. The shorts roots are: !3 , !4 , !2 + !3 , !3 + !4 , !1 + !2 + !3 , !2 + !3 + !4 , !1 + !2 + !3 + !4 , !2 + 2!3 + !4 , !1 + 2!2 + 2!3 + !4 , !1 + !2 + 2!3 + !4 , !1 + 2!2 + 3!3 + !4 , !1 + 2!2 + 3!3 + 2!4 , and they have length 1. (Note that the short roots are exactly those which contain !3 and/or !4 with coefﬁcient 1, while the long roots contain !3 and !4 with even coefﬁcients.) The dual roots are: (!1 + !2 )∨ = !∨1 + !∨2 , (!2 + 2!3 )∨ = !∨2 + !∨3 , (!1 + !2 + 2!3 )∨ = ∨ !1 + !∨2 + !∨3 , (!1 + 2!2 + 2!3 )∨ = !∨1 + 2!∨2 + !∨3 , (!1 + !2 + 2!3 + 2!4 )∨ = !∨1 + !∨2 + !∨3 + !∨4 , (!2 + 2!3 + 2!4 )∨ = !∨2 + !∨3 + !∨4 , (!1 + 2!2 + 2!3 + 2!4 )∨ = !∨1 + 2!∨2 + !∨3 + !∨4 , (!1 + 2!2 + 2!3 + 2!4 )∨ = !∨1 + 2!∨2 + 2!∨3 + !∨4 , (!1 + 3!2 + 4!3 + 2!4 )∨ = !∨1 + 3!∨2 + 2!∨3 + !∨4 , (2!1 + 3!2 + 4!3 + 2!4 )∨ = 2!∨1 + 3!∨2 + 2!∨3 + !∨4 ; (!2 + !3 )∨ = 2!∨2 + !∨3 , (!3 + !4 )∨ = !∨3 + !∨4 , (!1 + !2 + !3 )∨ = 2!∨1 + 2!∨2 + !∨3 , (!2 + !3 + !4 )∨ = 2!∨2 + !∨3 + !∨4 , (!1 + !2 + !3 + !4 )∨ = 2!∨1 +2!∨2 +!∨3 +!∨4 , (!2 +2!3 +!4 )∨ = 2!∨2 +2!∨3 +!∨4 , (!1 +2!2 +2!3 +!4 )∨ = 2!∨1 +4!∨2 +2!∨3 +!∨4 , (!1 + !2 + 2!3 + !4 )∨ = 2!∨1 + 2!∨2 + 2!∨3 + !∨4 , (!1 + 2!2 + 3!3 + !4 )∨ = 2!∨1 + 4!∨2 + 3!∨3 + !∨4 , (!1 + 2!2 + 3!3 + 2!4 )∨ = 2!∨1 + 4!∨2 + 3!∨3 + 2!∨4 . In terms of the normalized basis %1 , %2 , %3 , %4 we have B+ = {%i , 1 ≤ i ≤ 4; %j ± %k , 1 ≤ j < k ≤ 4; 1 (%1 ± %2 ± %3 ± %4 ), all signs}. 2

(2.82)

44

2 Lie Algebras and Groups

The simple roots are 1 0 = {!1 = %2 – %3 , !2 = %3 – %4 , !3 = %4 , !4 = (%1 – %2 – %3 – %4 )}. 2

(2.83)

Thus, F4 is 52-dimensional (52 = |B| + rank F4 ). Note that the 16 roots on the ﬁrst line of (2.82) form the positive root system of B4 with simple roots %i – %i+1 , i = 1, 2, 3, %4 . The Weyl group of F4 is the semidirect product of S3 with a group which itself is the semidirect product of S4 with (Z/2Z)3 , thus, |W| = 27 32 .

2.5.7 Exceptional Lie Algebras E We use the orthonormal vectors %i , i = 1, . . . , 8. Thus, following Bourbaki [59] the root systems of E6 , E7 , E8 are given ﬁrst for E6 , and then by adding roots for E7 and E8 . 2.5.7.1 E6 The simple roots !i , i = 1, . . . , 6, are given as follows : 1 1 0 = !1 = (%1 + %8 ) – (%2 + %3 + %4 + %5 + %6 + %7 ), 2 2 1 1 (%2 – %1 ), !4 = (%3 – %2 ), 2 2 1 !6 = (%5 – %4 ) 2

!3 =

1 !2 = (%1 + %2 ), 2

1 !5 = (%4 – %3 ), 2 (2.84)

Clearly, (!i , !i ) = 2, i = 1, . . . , 6, while the nonzero products between noncoinciding simple roots are (!1 , !3 ) = (!3 , !4 ) = (!4 , !2 ) = (!4 , !5 ) = (!5 , !6 ) = –1.

(2.85)

This is consistent with the E6 Dynkin diagram from the Table FIN. The positive roots are B+ =

± %i + %j , 1 ≤ i < j ≤ 5 ,

1 (–1)-(i) %i , %8 – %7 – %6 + 2 5

5

i=1

i=1

-(i) ∈ 2Z .

In terms of the simple roots the positive roots are given as follows:

(2.86)

2.5 Realization of Semisimple Lie Algebras

45

ﬁrst there are 15 roots forming the positive roots of sl(6) with simple roots !1 , !3 , !4 , !5 , !6 , then there are the following 21 roots: ! 2 , ! 2 + ! 4 , ! 2 + !3 + ! 4 , !2 + !4 + ! 5 , ! 2 + ! 3 + ! 4 + !5 , !1 + !2 + !3 + !4 , !2 + !4 + !5 + !6 , !1 + !2 + !3 + !4 + !5 , ! 2 + !3 + !4 + ! 5 + !6 , ! 1 + ! 2 + ! 3 + !4 + !5 + ! 6 , !2 + !3 + 2!4 + !5 , !1 + !2 + !3 + 2!4 + !5 , !2 + !3 + 2!4 + !5 + !6 , !1 + !2 + !3 + 2!4 + !5 + !6 , !1 + !2 + 2!3 + 2!4 + !5 , !2 + !3 + 2!4 + 2!5 + !6 , !1 + !2 + 2!3 + 2!4 + !5 + !6 , !1 + !2 + !3 + 2!4 + 2!5 + !6 , !1 + !2 + 2!3 + 2!4 + 2!5 + !6 , !1 + !2 + 2!3 + 3!4 + 2!5 + !6 , ˜ !1 + 2!2 + 2!3 + 3!4 + 2!5 + !6 = !,

(2.87)

where !˜ denotes the highest root. Note that |B+ | = 36. Then dimE6 = 2|B+ | + 6 = 78. The order of the Weyl group of E6 is |W| = 27 34 5. 2.5.7.2 E7 The simple roots !i , i = 1, . . . , 7, are given as for E6 adding one more root: 1 1 1 0 = !1 = (%1 + %8 ) – (%2 + %3 + %4 + %5 + %6 + %7 ), !2 = (%1 + %2 ), 2 2 2 1 1 1 !3 = (%2 – %1 ), !4 = (%3 – %2 ), !5 = (%4 – %3 ), 2 2 2 1 1 !6 = (%5 – %4 ), !7 = (%6 – %5 ) . 2 2

(2.88)

Clearly, (!i , !i ) = 2, i = 1, . . . , 8, while the nonzero products between noncoinciding simple roots are (!1 , !3 ) = (!3 , !4 ) = (!4 , !2 ) = (!4 , !5 ) = (!5 , !6 ) = (!6 , !7 ) = –1.

(2.89)

This is consistent with the E7 Dynkin diagram from the Table FIN. The positive roots are B+ =

± %i + %j , 1 ≤ i < j ≤ 6,

%8 – %7 ,

1 (–1)-(i) %i , %8 – %7 + 2 6

6

i=1

i=1

-(i) ∈ 1 + 2Z .

(2.90)

46

2 Lie Algebras and Groups

The positive roots in terms of simple roots are given as follows: ﬁrst there are 21 roots forming the positive roots of sl(7) with simple roots !1 , !3 , !4 , !5 , !6 , !7 , then the 21 roots from (2.87), ﬁnally there are the following 21 roots (including both !2 and a7 ): ! 2 + ! 4 + ! 5 + !6 + !7 , ! 2 + ! 4 + ! 3 + ! 5 + !6 + ! 7 , !2 + !4 + !3 + !5 + !1 + !6 + !7 , !2 + 2!4 + !3 + !5 + !6 + !7 , !2 + 2!4 + !3 + !5 + !1 + !6 + !7 , !2 + 2!4 + !3 + 2!5 + !6 + !7 , !2 + 2!4 + 2!3 + !5 + !1 + !6 + !7 , !2 + 2!4 + !3 + 2!5 + !1 + !6 + !7 , !2 + 2!4 + 2!3 + 2!5 + !1 + !6 + !7 , !2 + 3!4 + 2!3 + 2!5 + !1 + !6 + !7 , 2!2 + 3!4 + 2!3 + 2!5 + !1 + !6 + !7 , !2 + 2!4 + !3 + 2!5 + 2!6 + !7 , !2 + 2!4 + !3 + 2!5 + !1 + 2!6 + !7 , !2 + 2!4 + 2!3 + 2!5 + !1 + 2!6 + !7 , !2 + 3!4 + 2!3 + 2!5 + !1 + 2!6 + !7 , 2!2 + 3!4 + 2!3 + 2!5 + !1 + 2!6 + !7 , !2 + 3!4 + 2!3 + 3!5 + !1 + 2!6 + !7 , 2!2 + 3!4 + 2!3 + 3!5 + !1 + 2!6 + !7 , 2!2 + 4!4 + 2!3 + 3!5 + !1 + 2!6 + !7 , 2!2 + 4!4 + 3!3 + 3!5 + !1 + 2!6 + !7 , ˜ 2!2 + 4!4 + 3!3 + 3!5 + 2!1 + 2!6 + !7 = !.

(2.91)

Note that |B+ | = 63. Then dimE7 = 2|B+ | + 7 = 133. The order of the Weyl group of E7 is |W| = 210 34 5.7. 2.5.7.3 E8 The simple roots !i , i = 1, . . . , 8, are given as for E7 adding one more root: 1 1 1 0 = !1 = (%1 + %8 ) – (%2 + %3 + %4 + %5 + %6 + %7 ), !2 = (%1 + %2 ), 2 2 2 1 !3 = (%2 – %1 ), 2

1 !4 = (%3 – %2 ), 2

1 !6 = (%5 – %4 ), 2

1 !7 = (%6 – %5 ), 2

1 !5 = (%4 – %3 ), 2 1 !8 = (%7 – %6 ) 2

(2.92)

2.6 Realization of Affine Kac–Moody Algebras

47

Clearly, (!i , !i ) = 2, i = 1, . . . , 8, while the nonzero products between noncoinciding simple roots are (!1 , !3 ) = (!3 , !4 ) = (!4 , !2 ) = (!4 , !5 ) = (!5 , !6 ) = (!6 , !7 ) = (!7 , !8 ) = –1

(2.93)

This is consistent with the E8 Dynkin diagram from the Table FIN. The positive roots are B+ = ± %i + %j , 1 ≤ i < j ≤ 8,

1 (–1)-(i) %i , %8 + 2 7

-(i) = 0, 1,

i=1

7

-(i) ∈ 2Z .

(2.94)

i=1

The positive roots in terms of simple roots are given as follows: ﬁrst there are 28 roots forming the positive roots of sl(8) with simple roots !1 , !3 , !4 , !5 , !6 , !7 , !8 , then the 21 roots from (2.91), ﬁnally there are 71 roots involving both !2 and a8 . Note that |B+ | = 120. Then dimE8 = 2|B+ | + 8 = 248. The order of the Weyl group of E8 is |W| = 214 35 52 7. Remark 3: We note that for the algebras A , D , E , all roots have the same length. As is customary we have ﬁxed the normalization so that this length is 2. For the other algebras there are roots of two different lengths, for B , C , F4 , the length ratio is 2 : 1, while for G2 the length ratio is 3 : 1. The algebras A , D , E , are called simply laced algebras, the rest are called nonsimply laced algebras.

2.6 Realization of Affine Kac–Moody Algebras 2.6.1 Realization of Affine Type 1 Kac–Moody Algebras Let G be a simple ﬁnite-dimensional Lie algebra. Let L = C[t, t–1 ] be the algebra of Laurent polynomials in t. Let L(G) denote the loop algebra: L(G) ≡ L ⊗ G

(2.95)

with bracket [P ⊗ X, Q ⊗ Y]0 = PQ ⊗ [X, Y],

P, Q ∈ L,

X, Y ∈ G.

(2.96)

This is an inﬁnite-dimensional Lie algebra with basis: tn ⊗ X,

n ∈ Z,

X∈G

(2.97)

48

2 Lie Algebras and Groups

Fix a nondegenerate ad-invariant symmetric bilinear C-valued form (⋅, ⋅) on G, i.e., ([X, Y], Z) = (X, [Y, Z]). We extend this form to an L-valued bilinear form on L(G) by (P ⊗ X, Q ⊗ Y) = PQ(X, Y)

(2.98)

d of L to a derivation of L(G) by Next we extend every derivation ts dt

ts

d dP (P ⊗ X) = ts ⊗X dt dt

(2.99)

Now we can deﬁne a C-valued 2-cocycle on L(G) by 8(a, b) = Res

da ,b , dt

(2.100)

where Res P = c–1 if P = k∈Z ck tk (with only ﬁnite number of ck are nonzero). We recall that a C-valued 2-cocycle on a Lie algebra G is a C-valued function 8 satisfying two conditions 8(a, b) = –8(b, a), ∀a, b ∈ G,

(2.101a)

8([a, b], c) + 8([b, c], a) + 8([c, a], b) = 0, ∀a, b, c ∈ G

(2.101b)

It is sufﬁcient to check (2.101) for a = P ⊗ X, b = Q ⊗ Y, c = R ⊗ Z, P, Q, R ∈ L, X, Y, Z ∈ G. Then (2.101a) follows from 8(a, b) = Res

dP dQ Q (X, Y) = –Res P (X, Y) = –8(b, a) dt dt

which follows from Res

dT = 0, dt

∀T ∈ L .

For (2.101b) we use the ad-invariance of (⋅, ⋅): 8([a, b], c) + 8([b, c], a) + 8([c, a], b) = d(PQ) dQR dRP = Res R + Res P + Res Q ([X, Y], Z) = dt dt dt dP dQ dR = 2 Res QR + Res PR + Res PQ ([X, Y], Z) = dt dt dt = 2Res

d (PQR) ([X, Y], Z) = 0 dt

2.6 Realization of Affine Kac–Moody Algebras

49

Let G˜ be the extension of L(G) by a one-dimensional center cˆ associated to the cocycle 8, i.e., G˜ = L(G) ⊕ Cˆc

(2.102)

[a + +ˆc, b + ,ˆc] = [a, b]0 + 8(a, b)ˆc

(2.103)

with bracket

or in terms of the basis

m

n

[t ⊗ X, t ⊗ Y] = t

m+n

dtm n ⊗ [X, Y] + Res t dt

(X, Y) cˆ

= tm+n ⊗ [X, Y] + m $m+n,0 (X, Y) cˆ ; [ˆc, tn ⊗ Y] = 0,

[ˆc, cˆ ] = 0.

(2.104) (2.105)

Definition: In general, a Lie algebra is called M-graded, where M is an Abelian group, if each generator Z of the algebra is assigned a degree, denoted deg Z and taking values in M, so that the degree is preserved by the commutation relations; the latter are also called M-graded. For example, the Witt algebra (2.17) is Z-graded, the degree being deﬁned as follows: deg dj ≐ j. ♢ Note that the Lie algebras G˜ are Z-graded, the degree being deﬁned as follows: deg tm ⊗ X ≐ m,

deg cˆ ≐ 0.

(2.106)

Obviously, the commutation relations (2.104) are Z-graded, in particular, each term in (2.104a) has degree equal to m + n. Remark 4: The algebras G˜ may be viewed as Fourier transforms of the current algebras of quantum ﬁeld theory. ♢ Finally, let Gˆ be the Lie algebra obtained from G˜ by adjoining a derivation dˆ which acts d on Gˆ as t dt and annihilates cˆ : Gˆ = L(G) ⊕ Cˆc ⊕ Cdˆ

(2.107)

with bracket ˆ t ⊗ Y + ,ˆc + -d] ˆ = [tk ⊗ X + 3ˆc + +d, = tk+ ⊗ [X, Y] + +t ⊗ Y – -ktk ⊗ X + kˆc$k,– (X, Y)

(2.108)

50

2 Lie Algebras and Groups

We shall show that Gˆ actually is the afﬁne Lie algebra of type G (1) if G is of type A , B , C , D , E , F4 , G2 . It is called also the afﬁnization of G. First let us record some observations for later use. Denote by d˜ s the endomorphism ˜ of G such that d d˜ s |L(G) = ts+1 , d˜ s (ˆc) = 0. dt

(2.109)

˜ Lemma 25: d˜ s is a derivation of G. Proof. Since d˜ s is a derivation of L(G) we have d˜ s ([a + +ˆc, b + ,ˆc]) = d˜ s ([a, b]0 ) = [d˜ s (a), b]0 + [a, d˜ s (b)]0 , but ˜ b)ˆc [d˜ s (a), b] = [d˜ s (a), b]0 + 8(ds(a), Hence it remains to check 8(d˜ s (a), b) + 8(a, d˜ s (b)) = 0 for which it is enough to set a = P ⊗ X, b = Q ⊗ Y).

∎

Note that –d˜ s , s ∈ Z, are a realization of the Witt algebra [–d˜ i , –d˜ j ] = (i – j)(–d˜ i + j )

(2.110)

which we introduced in Subsection 2.1.7 (cf. formula (2.17) with ds → –d˜ s . Let B0 be the root system of G, 00 = {!1 , . . . ! }, Ei , Fi , H˜ i , i = 1 . . . , be the canonical Cartan–Weyl generators corresponding to the simple roots, i.e., Ei = E!i , Fi = E–!i [Ei , Fi ] = H˜ i , !i (H˜ j ) = (!i , !j ). Let !˜ be the highest root of B0 . We choose E!˜ ∈ G!˜ , F!˜ ∈ G–!˜ so that (E!˜ , F!˜ ) = 1, then we have (cf. Lemma 3) [E!˜ , F!˜ ] = (E!˜ , F!˜ )H!˜ = H!˜ ˆ The elements F!˜ , E1 , . . . E generate G, (F!˜ ∈ G–!˜ ). It is clear that Cˆc is the centre of G, ˆ ˆ ˆ ˆ the centralizer of d in G is 1 ⊗ G ⊕ Cˆc ⊕ Cd; 1 ⊗ G is a subalgebra of G, which we identify with G. Thus, the Cartan subalgebra of Gˆ Hˆ = H ⊕ Cˆc ⊕ Cdˆ is ( + 2)-dimensional.

(2.111)

51

2.6 Realization of Affine Kac–Moody Algebras

ˆ = 0. Let d ∈ Hˆ ∗ be Next we consider Hˆ ∗ . We extend + ∈ H∗ to Hˆ ∗ by +(ˆc) = +(d) ˆ = 1. Let c ∈ H∗ be such that c deﬁned by dH⊕Cˆc = 0, d(d) = 0, c(ˆc) = 1. We also H⊕Cdˆ set (c, d) = 1, (c, c) = 0, (d, d) = 0, (c, !i ) = (d, !i ) = 0, i = 1, . . . , . Now we set: e0 = t ⊗ F!˜ ,

f0 = t–1 ⊗ E!˜ ,

e i = 1 ⊗ Ei ,

fi = 1 ⊗ Fi ,

i = 1...

(2.112)

Then we see [e0 , f0 ] = cˆ – H!˜ ≡ h0

(2.113)

Now we describe the root system and the root space decomposition of Gˆ w.r.t H B = {kd + #, k ∈ Z, # ∈ B0 } ∪ {kd, k ∈ Z\{0}},

(2.114)

Gˆ = Hˆ ⊕ ⊕ Gˆ! ,

(2.115)

Gˆkd+# = tk ⊗ G# , Gˆkdˆ = tk ⊗ H,

(2.116a)

dimGkd+# = 1, dimGˆkdˆ = = dimH = rankG

(2.116b)

!∈B

where

We also give the products of the roots: (kd + !, k′ d + ") = (!, ")(kd + !, k′ d) = 0,

(kd, k′ d) = 0

(2.117)

We see that the roots of type kd have zero product with every other root, including themselves. These roots are called imaginary roots, the rest are called real roots. The system of simple roots is ˜ !1 , . . . ! }. 0 = {!0 ≡ d – !,

(2.118)

Now we can check that A = (aij ) = (!∨i , !j )i,j=0,... is of type AFF 1. First we note that ˜ d – !) ˜ = (!, ˜ !), ˜ or equivalently: (!0 , !0 ) = (d – !, ˜ c – H!˜ ) = !(H ˜ !˜ ) = (!, ˜ !) ˜ (!0 , !0 ) = !0 (h0 ) = (d – !)(ˆ

(2.119)

Further, we use the highest root !˜ of the ﬁnite-dimensional Lie algebra G. For A1 , !˜ = !1 , !0 = d – !1 , (!0 , !1 ) = –2, ⇒ Gˆ = A(1) 1 . For A , > 1, !˜ = !1 + ⋅ ⋅ ⋅ + ! , (!0 , !1 ) = (!0 , ! ) = –1, (!0 , !i ) = 0 1 < i < , ˜ !) ˜ = 2, !˜ ∨ = !˜ = !∨1 + ⋅ ⋅ ⋅ + !∨ , (!k , !k+1 ) = –1, k = 1, . . . , – 1, ⇒ Gˆ = A(1) . Note that (!, ˜ !2 ) = 1 = –(!0 , !2 ), (!, ˜ !i ) = 0 = (!0 , !i ), i ≠ 2, ⇒ For B , !˜ = !1 + 2!2 + ⋅ ⋅ ⋅ + 2! , (!, ∨ + 2!∨ + ⋅ ⋅ ⋅ + 2!∨ + !∨ . ∨ =! ˜ ˜ ˜ ˜ Gˆ = B(1) . Note that ( !, !) = 2, ! = ! 1 2 –1

52

2 Lie Algebras and Groups

˜ !1 ) = 1 = –(!0 , !1 ), (!, ˜ !i ) = 0 = (!0 , !i ), For C , !˜ = 2!1 + 2!2 + ⋅ ⋅ ⋅ + 2!–1 + ! , (!, ˜ !) ˜ = 4, !˜ ∨ = 21 !˜ = !∨1 + ⋅ ⋅ ⋅ + !∨ . i > 1, a01 = –1, a10 = –2 ⇒ Gˆ = C(1) . Note that (!, ˜ !2 ) = 1 = –(!0 , !2 ), (!, ˜ !i ) = 0, For D , !˜ = !1 + 2!2 + ⋅ ⋅ ⋅ + 2!–2 + !–1 + ! , (!, ∨ + 2!∨ + ⋅ ⋅ ⋅ + 2!∨ + !∨ + !∨ . ∨ =! ˜ ˜ ˜ ˜ i ≠ 2 ⇒ Gˆ = D(1) . Note that ( !, !) = 2, ! = ! 1 2 –2 –1 ˜ !6 ) = 1 = –(!0 , !6 ), (!, !i ) = 0, i ≠ 6 For E6 , !˜ = !1 + 2!2 + 3!3 + 2!4 + !5 + 2!6 , (!, ⇒ Gˆ = E6(1) . ˜ !1 ) = 1, (!˜ 0 , !i ) = 0, i ≠ 1 For E7 , !˜ = 2!1 + 3!2 + 4!3 + 3!4 + 2!5 + !6 + 2!7 , (!, ⇒ Gˆ = E7(1) . ˜ !1 ) = 1, (!, ˜ !i ) = 0, i > 1 For E8 , !˜ = 2!1 + 3!2 + 4!3 + 5!4 + 6!5 + 4!6 + 2!7 + 3!8 , (!, ⇒ Gˆ = E8(1) . ˜ !1 ) = 1, (!, ˜ !i ) = 0, i > 1 ⇒ G˜ = F4(1) . Note that For F4 , !˜ = 2!1 + 3!2 + 4!3 + 2!4 , (!, ˜ !) ˜ = 2, !˜ ∨ = !˜ = 2!∨1 + 3!∨2 + 2!∨3 + !∨4 . (!, ˜ !1 ) = 3, (!, ˜ !2 ) = 0, a01 = a10 = –1 ⇒ Gˆ = G(1) For G2 , !˜ = 2!1 +3!2 = 2%3 –%2 –%1 , (!, 2 . 1 2 ∨ ∨ ∨ ˜ !) ˜ = 6, !˜ = 3 !˜ = 3 !1 + !2 = 2!1 + !2 . Note that (!, Note that the coefﬁcients ki of !˜ = :i=1 ki !i are given by the numbers corresponding to the simple roots of G in the Table AFF 1. For the simply laced algebras (i.e., all lines are of type ○ – – – ○) of type AFF (actually AFF 1) the number at any vertex equals the half sum of the numbers at all vertices connected with the given one. For the nonsimply laced algebras this rule also holds for the long roots and for the shorts roots which are connected only to short roots. The rule is generalized like this. Let ! be a simple root connected to !1 , . . . , !k , k ≤ 3; let p, p1 , . . . , pk be the numbers at the corresponding vertices; then: 1 %i pi , 2 k

p=

(2.120)

i=1

where %i = # of links connecting ! and !i if there is an arrow pointing from !i to !, otherwise %i = 1. 2.6.2 Realization of Affine Type 2 and 3 Kac–Moody Algebras Let G be a simple ﬁnite-dimensional Lie algebra. Let B be the root system of G, Aut(B) the group of automorphism of B. One has: Proposition 6: The factor group Aut(B)/W is isomorphic to the group of automorphism of the Dynkin diagram. One has: ⎧ ⎪ ⎪ ⎨Z1 , Aut(B)/W = Z2 , ⎪ ⎪ ⎩S , 3

G = B , C , E7 , E8 , F4 , G2 ,

(a)

G = A , D , E6 ,

(b)

G = D4 ,

(c)

where Zm is the cyclic group of order m, and S3 is the permutation group of three objects.

2.6 Realization of Affine Kac–Moody Algebras

53

Proof. W permutes the bases of B transitively, then Aut(B)/W is isomorphic to a subgroup of Aut(B) leaving a ﬁxed basis of B invariant. But this amounts to a subgroup permuting the vertices of a Dynkin diagram preserving the norms (!i , !i ) of the simple roots and the multiplicity of links. The diagrams of B , C , E7 , E8 , G2 admit no such permutation; for A , we use !k ↔ !+1–k , k ≤ /2; for E6 we use !1 ↔ !6 , !3 ↔ !5 ; for D we use !–1 ↔ ! ; for D4 : we can permute !1 , !3 , !4 in an arbitrary way. (In all cases we use the enumeration of indices as in Table FIN.) ∎ The realization of AFF 2 and AFF 3 Kac–Moody algebras proceeds in a similar way as for AFF 1. However, starting from a simple ﬁnite-dimensional algebra G one ﬁrst decomposes it w.r.t. the action of the groups in Proposition 6, from which we need to consider only cases (b,c). Let us write G = G0 ⊕ G1 , k = 2 for A , D , E6 , or

(2.121a)

G = G0 ⊕ G1 ⊕ G2 , k = 3 for D4 ,

(2.121b)

where Gj is the eigenspace of the automorphism - for eigenvalue %j , where % = exp 20i/k, -k = 1. Obviously [Ga , Gb ] ⊂ G(a+b)modk

(2.122)

Thus, G0 = -(G0 ) is a subalgebra of G. We shall give an explicit description of G0 for all relevant cases. Notation: Below Ei , Fi , H˜ i , 1 ≤ i ≤ will denote the canonical simple root Cartan–Weyl generators of the ﬁnite-deimensional simple Lie algebra G. 2.6.2.1 A2 Let G = A2 , ≥ 1; -(i) = 2 + 1 – i, -2 = 1. Here G0 is generated by Hi′ = H˜ i + H˜ 2+1–i ; 1 ≤ i ≤ , Ei′ = Ei + E2+1–i , Fi′ = Fi + F2+1–i , while the invariant bilinear form is deﬁned for > 1 by ⎧ ⎪ 2 i = j ≠ , ⎪ ⎪ ⎪ ⎪ ⎨ 1 i = j = , 1˜ ′ ′ B′ (Hi′ , Hj′ ) ≡ B(H i , Hj ) = ⎪ 2 –1 |i – j| = 1, ⎪ ⎪ ⎪ ⎪ ⎩0 otherwise, ˜ ′ , H ′ ) = 2. while for = 1 we set: B′ (H1′ , H1′ ) ≡ B(H 1 1

(2.123)

(2.124)

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2 Lie Algebras and Groups

The simple root system is 0 = {!′i , |1 ≤ i ≤ } with scalar products: (!′i , !′j ) = B′ (Hi′ , Hj′ ) = !′i (Hj′ ) = !′j (Hi′ ).

(2.125)

This gives the Cartan matrix of B (cf. (2.54), (2.59)), for ≥ 2 and of A1 for = 1. 2.6.2.2 A2–1 Let G = A2–1 , ≥ 3; -(i) = 2 – i. Here G0 is generated by Hi′ = H˜ i + H˜ 2–i , 1 ≤ i ≤ – 1, H′ = 2H˜ , √ Ei′ = Ei + E2–i , 1 ≤ i ≤ – 1, E′ = 2E , √ Fi′ = Fi + F2–i , 1 ≤ i ≤ – 1, F′ = 2F , while

⎧ ⎪ –1 |i – j| = 1, i, j < , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨–2 ij = ( – 1), 1˜ ′ ′ ′ ′ ′ B (Hi , Hj ) ≡ B(Hi , Hj ) = 2 i = j < , ⎪ 2 ⎪ ⎪ ⎪4 i = j = , ⎪ ⎪ ⎪ ⎩ 0 otherwise,

(2.126)

(2.127)

gives the Cartan matrix for C (cf. (2.64) and (2.67)). 2.6.2.3 D+1 Let G = D+1 , ≥ 2; -(i) = i, i ≤ – 1; -(, + 1) = ( + 1, ). Here G0 is generated by Hi′ = H˜ i , Ei′ = Ei , Fi′ = Fi , H′ while

1 < i ≤ – 1,

= H˜ + H˜ +1 , E′ = E + E+1 , F′ = F + F+1 ,

⎧ ⎪ ⎪ ⎪2 ⎪ ⎪ ⎨1 1˜ ′ ′ ′ ′ ′ B (Hi , Hj ) ≡ B(Hi , Hj ) = ⎪ 2 –1 ⎪ ⎪ ⎪ ⎪ ⎩0

(2.128)

i = j ≠ , i = j = , |i – j| = 1,

(2.129)

otherwise,

which gives the Cartan matrix of B (as above). 2.6.2.4 E6 Let G = E6 , -(1, 2, 3, 4, 5, 6) = (6, 2, 5, 4, 3, 1), -2 = 1. Here G0 is generated by H1′ = H˜ 1 + H˜ 6 , H2′ = H˜ 3 + H˜ 5 , H3′ = H˜ 2 , H4′ = H˜ 2 ,

(2.130)

2.6 Realization of Affine Kac–Moody Algebras

55

and analogously for Ei′ , Fi′ , while ⎧ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨1

i = j = 1,2,

i = j = 3,4, 1˜ ′ ′ B′ (Hi′ , Hj′ ) ≡ B(H , H ) = –1 ij = 2,6, i j ⎪ 2 ⎪ ⎪ ⎪ –1/2 ij = 12, ⎪ ⎪ ⎪ ⎩ 0 otherwise,

(2.131)

which gives the Cartan matrix of F4 via (!′i , !′j ) = B′ (Hi′ , Hj′ ) and aij =

2(!′i , !′j ) (!′i , !′i )

=

2B′ (Hi′ , Hj′ ) B′ (Hi′ , Hi′ )

.

2.6.2.5 D4 Let G = D4 , -(1, 2, 3, 4) = (4, 2, 1, 3), -3 = 1, Here G0 is generated by H1′ = H˜ 1 + H˜ 3 + H˜ 4 , H2′ = H˜ 2 ,

(2.132)

and analogously for Ei′ , Hi′ , while

⎧ ⎪ 6 ⎪ ⎪ ⎪ ⎪ ⎨2 ˜ ′, H′) = B′ (Hi′ , Hj′ ) ≡ B(H i j ⎪–3 ⎪ ⎪ ⎪ ⎪ ⎩0

i = j = 1, i = j = 2,

(2.133)

ij = 2, otherwise,

which gives the Cartan matrix for G2 . 2.6.3 Root System for the Algebras AFF 2 & 3 Let H, H0 , B, B0 , 0, 00 be the Cartan subalgebras, root systems and simple root systems of G, G0 respectively. Let p : H∗ → H0∗ be the map ⎧ ⎪ !i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪!i ⎨ p : !i ⎪ ⎪ ⎪ ⎪!i ⎪ ⎪ ⎪ ⎩ !i

→ !′i , 1 ≤ i ≤ ; !i → !′2+1–i , + 1 ≤ i ≤ 2; → → → →

!′i , 1 ≤ i ≤ ; !i → !′2–i , + 1 ≤ i ≤ 2 – 1; !′i , 1 ≤ i ≤ ; !+1 → !′ ; !′i , 1 ≤ i ≤ 3; !4 → !′2 ; !5 → !′1 ; !6 → !′4 ; !′i , 1 ≤ i ≤ 2; !3 → !′1 ; !4 → !′1 ;

for A2 ; for A2–1 ; for D+1 ; for E6 ; for D4 .

Thus, |p–1 (!′i )| = 1 or k, where k = 2, 3 is the order of the automorphisms in cases (b,c) of Proposition 6, respectively.

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2 Lie Algebras and Groups

Note that p–1 (B0 ) does not necessarily cover B. (For instance, for A2 , the roots "k = !k + ⋅ ⋅ ⋅ + !2+1–k , k = 1, . . . , are projected to 2(!k + ⋅ ⋅ ⋅ + ! ) ∈/ B0 , while for ∀!′ ∈ B+0 , |p–1 (!′ )| = 2. For a cross check note that |B+ | – 2|B+0 | = (2 + 1) – 22 = .) Further, we shall consider elements in p(B) together with their multiplicities. (Thus, if ! ∈ B0 is contained in p(B) with multiplicity n > 1, then ! ∈ p(B)\B0 with multiplicity n – 1.) In the construction below an important role is played by an element (0 ∈ H0∗ , such that (0 ∈ p(B)\B0 and (0 + !′i ∈/ p(B)\B0 , ∀!′i ∈ 00 . One may show that (0 is unique with this properties and (0 is contained in p(B)\B0 with multiplicity k – 1 (independently of whether (0 ∈ B or (0 ∈/ B0 ). Explicitly we have Table 2.1: Elements Used for the Root Systems of AFF 2 & 3 G A2 , ≥ 2, A2 A2–1 , ≥ 1, D+1 , ≥ 1, E6 D4

G0

(0

B A1 C B F4 G2

2(!′1 + ⋅ ⋅ ⋅ + !′ ) ∈/ B0 2!′1 ∈/ B0 !′1 + 2!2 ⋅ ⋅ ⋅ + 2!′–1 + !′ ∈ !′1 + ⋅ ⋅ ⋅ + !′ ∈ B0 !′1 + 2!′ + 3!′3 + 2!′4 ∈ B0 !′1 + 2!′2 ∈ B0

((0 , (0 )

B0

4 8 2 1 1 2

˜ ∈/ B0 , where !˜ is the highest root of B, while for the other cases For A2 (0 = p(!) ˜ because p(!) ˜ occurs always with multiplicity 1 and thus p(!) ˜ ∈/ p(B)\B0 if (0 ≠ p(!) ˜ ∈ B0 . Note also that (0 ∈ B0 ⇒ (0 ≠ !˜ 0 , the highest root of B0 , since in these cases p(!) ˜ = !˜ 0 . p(!) According to (2.122) we may consider the representation of G0 on G1 (and on G2 for D4 , k = 3) by the adjoint action and the above considerations actually show that this representation is irreducible (and for D4 those on G1 and G2 are equivalent). Let us deﬁne Gs,! = {X ∈ Gs : [H, X] = !(H)X, ∀H ∈ H0 },

s = 0, 1, 2;

Bs = {! ∈ H0∗ : ! ≠ 0, Gs,! ≠ 0},

(2.134a) (2.134b)

where Bs are called the (nonzero) weights of H0 in Gs , Gs,! are the corresponding weight spaces; naturally B0 is the root system of (G0 , H0 ), G0,! the root vector spaces of G0 , G0,0 = H0 . Next we write the weight space decomposition: Gs =

⊕

!∈Bs ∪{0}

Gs,! .

(2.135)

We choose E˜ 0 ∈ G1,(0 , F˜ 0 ∈ G1,–(0 and set H˜ 0 = [E˜ 0 , F˜ 0 ] One may check that the elements F˜ 0 , E1′ , . . . , E′ generate the Lie algebra G.

(2.136)

2.6 Realization of Affine Kac–Moody Algebras

57

Next we consider the twisted loop algebra: L(G, -) = ⊕ tj ⊗ G¯j , j∈Z

¯j = j mod k

(2.137)

which for k = 2 has the basis: {t2j ⊗ X | j ∈ Z, X ∈ G0 } ∪ {t2j+1 ⊗ X | j ∈ Z, X ∈ G1 }

(2.138)

while for k = 3 the basis is U 0 ∪ U1 ∪ U 2 ,

Uk ≡ {t3j+k ⊗ X | j ∈ Z, X ∈ Gk }

(2.139)

Note that L(G, -) is the ﬁxed point of the automorphism -˜ of L(G) deﬁned by -˜ (tj ⊗ X) = (%–j tj ) ⊗ -(X), j ∈ Z, X ∈ G

(2.140)

Analogously to the untwisted case we set ˆ g, -) ≡ L( g, -) ⊕ Cˆc ⊕ Cdˆ , gˆ = L(

(2.141)

Hˆ = H0 ⊕ Cˆc ⊕ Cdˆ ,

(2.142)

then

e0 = t ⊗ F˜ 0 ,

f0 = t–1 ⊗ E˜ 0 , ei = 1 ⊗ Ei′ ,

fi = 1 ⊗ Fi′ , 1 ≤ i ≤ ,

(2.143)

so that [e0 , f0 ] = cˆ – H˜ 0 , [ei , fi ] = H˜ i .

(2.144)

ˆ The root system and root space decomposition of L(G, -) w.r.t. H are B = { jd + # : j ∈ Z, # ∈ Bs , j = s mod k, s = 0, . . . , k – 1} ∪ ∪ { jd : j ∈ Z\{0}},

(2.145)

ˆ gˆ = L(G, -) = H ⊕ ⊕ Gˆ!- , !∈B

Gˆ -

jd+#

= tj ⊗ Gj,# , Gˆ - = tj ⊗ Gj,0 , j ≡ j(modk), jd

dimGˆjd+# = 1, dimGˆ - = dimGj,0 = jd

⎧ ⎨ = dimH0 , j = 0, ⎩

(dimH – dimH0 )/(k – 1), j ≠ 0.

(2.146)

(2.147a)

(2.147b)

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2 Lie Algebras and Groups

We set 0 = {!0 = d – (0 , !′i ∈ 00 }

(2.148)

We illustrate the above by the following table: Table 2.2: Subalgebras and dimensions relevant for the construction of AFF 2 & 3 G G0 dimG dimG0 dimG1 dimH0 dimG1,0

A2

A2

A2–1

D+1

E6

D4

B 4( + 1) (2 + 1) (2 + 3)

A1 8 3 5 1 1

C 42 – 1 (2 + 1) 22 – – 1 –1

B ( + 1)(2 + 1) (2 + 1) 2 + 1 1

F4 78 52 26 4 2

G2 28 14 7 = dimG2 2 1 = dimG2,0

It remains to check whether the above construction would really give us the Cartan matrices of AFF 2, AFF 3. For that we only have to check the properties of the root !0 . We note that (!0 , !0 ) = ((0 , (0 ) and by Table 2.1 this gives the correct values according to the place of !0 in the Dynkin diagrams in AFF 2, AFF 3. Then we must check that (!0 , !′i ) = 0, except for i = 1 for B , A1 , i = 2 for C , G2 , i = 4 for F4 . (At this point it is clear why for G = A2–1 , G0 = C we have the restriction ≥ 3. Indeed, for = 2, G = A3 , G0 = C2 , !0 would be attached to the long root, since (!0 , !′1 ) = 0, (!0 , !′2 ) = – 2 which would (2) produce the Dynkin diagram D(2) 3 and not A3 .)

2.7 Chevalley Generators, Serre Relations, and Cartan–Weyl Basis Let G be a Lie algebra of type FIN, AFF, resp., 0 the system of simple roots, 0 = {!1 , . . . , ! }, 0 = {!0 , !1 , . . . , ! }, resp. Further, we shall use uniform notation for the two cases distinguishing them only by the range of indices as in 0, e.g., dropping the 0 value in the ﬁnite case or using a range (0), 1, . . . , when a formula is valid for both cases. Consider the generators corresponding to the simple roots: E˜ i , F˜ i , H˜ i , i = (0), 1, . . . , have the relations: [H˜ i , H˜ j ] = 0, [H˜ i , E˜ j ] = !j (H˜ i )E˜ j , [H˜ i , F˜ j ] = –!j (H˜ i )F˜ j [E˜ i , F˜ j ] = $ij H˜ i

(2.149)

2.7 Chevalley Generators, Serre Relations, and Cartan–Weyl Basis

59

(note, e.g., in the afﬁne case H˜ i = hi , etc.). We know that these elements generate the algebra G, [G, G], resp. A more precise formulation is in terms of the Chevalley generators and using the Serre relations. The Chevalley generators [85] are denoted by Xi± , Hi , i = (0), 1, . . . , and they are expressed in terms of those above as follows: Xi+ = Xi– = Hi =

2 E˜ i (!i , !i ) 2 F˜ i (!i , !i )

2 H˜ i = H˜ i∨ (!i , !i )

(2.150)

Of course, in the simply-laced case (choosing customarily (!i , !i ) = 2) the two sets of generators coincide. The Chevalley generators have the following commutation relations: [Hi , Hj ] = 0, [Hi , Xj± ] = ±aij Xj± = ±(!∨i , !j ) Xj± [Xi+ , Xj– ] = $ij Hi

(2.151)

In order to be able to restore the initial algebra from this set one imposes also the Serre relations [547] for all cases i ≠ j: n

(–1)k

k=0

n k

Xi±

k

n–k Xj± Xi± = 0,

i ≠ j,

n = 1 – aij

(2.152)

Note that these relations are automatically fulﬁlled if we use the matrix realizations in the case FIN. They guarantee that the process of construction of nonsimple root vectors (from the simple root vectors) will not produce extra generators. One may prove that relations (2.151) and (2.152) determine G, [G, G] = G˜0 , resp., in the case FIN, AFF, resp., up to isomorphism. For the proof one uses that (2.152) may be rewritten as n times the adjoint action of Xi± on Xj± : (adXi± )n (Xj± ) = 0,

i ≠ j,

n = 1 – aij

(2.153)

or otherwise stated that !j + n!i = !j + (1 – !ij )!i is not a root which is then related to Lemma 8. Naturally, we shall use also the full basis of G consisting of Cartan-Weyl generators H! , E±! , ∀! ∈ B+ .

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2 Lie Algebras and Groups

In the ﬁnite-dimensional case, these were introduced after Lemma 10. The gen erators H! are expressed in terms of the Chevalley Hi as H! = j nj Hj , where the coefﬁcients nj come from the decomposition of the co-root of ! in co-simple roots, namely, !∨ = 2!/(!, !) = j nj !∨j . The other generators were normalized so that [H! , E!′ ] = (!∨ , !′ ) E!′ , [E! , E–! ] = H! , (E! , E–! ) = 1.

(2.154)

In the afﬁne case for the real roots kd + ! we choose Ekd+! ≐ tk ⊗E! .

(2.155)

Ekd+! , E–(kd+!) = (E! , E–! ) = 1.

(2.156)

and then we have

Below, instead of 1⊗X we write X. Then we deﬁne: Hkd+! ≐ Ekd+! , E–(kd+!)

(2.157)

and we get Hkd+! = [E! , E–! ] + k(E! , E–! ) cˆ = H! + kˆc, " ! [Hkd+! , Ed+!′ ] = t ⊗ H! , E!′ = (!∨ , !′ )Ed+!′ .

(2.158a) (2.158b)

For the imaginary roots one ﬁrst introduces an orthonormal basis Hˆ i in the ﬁnitedimensional Cartan subalgebra H in the case AFF 1, (in G1,0 in the cases AFF 2,3), i.e., (Hˆ i , Hˆ j ) = $ij . Then the basis vectors in Gkd are i = tk ⊗Hˆ i Ekd

(2.159)

Then we have using (2.98) and (2.104a):

j d

i Ekd ,E

= tk+l (Hˆ i , Hˆ j ) = tk+l $ij

(2.160a)

" ! ! i j " = tk+l ⊗ Hˆ i , Hˆ j + k $k+,0 (Hˆ i , Hˆ j ) cˆ = Ekd , E d

= k $k+,0 $ij cˆ

(2.160b)

Thus, we can deﬁne and get: i i i ≐ [Ekd , E–kd ] = kˆc Hkd

(2.161)

The degeneracy of the afﬁne case is manifest here since we get the same element of Hˆ for all root vectors in the root space of kd.

61

2.8 Highest Weight Representations of Kac–Moody Algebras

2.8 Highest Weight Representations of Kac–Moody Algebras Let E be a vector space over F. The tensor algebra T(E) over E is deﬁned as the free algebra generated by the unit element. We have ∞

T(E) = ⊕ Tk (E), Tk (E) ≡ E ⋅ ⋅ ⊗ E&, T0 (E) = F.1 # ⊗ ⋅

%$(2.162)

k=0

k

Further we shall write v1 . . . vr ≡ v1 ⊗ ⋅ ⋅ ⋅ ⊗ vr . The elements t ∈ Tk (E) are called covariant tensors of rank k t=

ti1 ...ik i1 . . . ik

(2.163)

where i ∈ S, S is a basis of E, ti1 ...ik ∈ F. (The rank of a covariant tensor does not depend on the choice of basis of E.) The tensor t is called symmetric (antisymmetric) tensor if ti1 ...ik is symmetric (antisymmetric) in all indices. Let us denote ∞

∞

S(E) = ⊕ Sk (E),

A(E) = ⊕ Ak (E)

k=0

(2.164)

k=0

where Sk (E) (Ak (E)) is the subspace of all symmetric (antisymmetric) tensors of rank k. Note that if dimE = n < ∞, then

n+k–1 (n + k – 1)! = , k k!(n – 1)! n n! = . dimAk (E) = k k!(n – k)! dimSk (E) =

(2.165a)

(2.165b)

Obviously dimAk (E) = 0, for k > n, dimA(E) = 2n . Let I be the two–sided ideal of T(E) generated by all elements of the type xy – yx, x, y ∈ E. Then we have T(E) = I ⊕ S(E), S(E) ≅ T(E)/I.

(2.166)

Consider now a Lie algebra G. The universal enveloping algebra U(G) of G is deﬁned as the unital associative algebra with generators i , where i forms a basis of G and the relations i j – j i =

ckij k ≡ [i , j ]

k

hold, where ckij are the structure constants of G.

(2.167)

62

2 Lie Algebras and Groups

Equivalently U(G) can be described as U(G) ≅ T(G)/J , where T(G) is the tensor algebra over G, J is the ideal generated by the elements [x, y] – (xy – yx). Since T(G) = J ⊕ U(G) = I ⊕ S(G) and J , I are isomorphic, then also U(G) ≅ S(G) as vector spaces. This is the content of the Poincaré–Birkhoff–Witt (PBW) theorem. Consequently U(G) has the following PBW basis: 0 = 1, i1 ...ik = i1 i2 . . . ik , i1 ≤ ⋅ ⋅ ⋅ ≤ ik .

(2.168)

(The latter gives a hint how to prove the PBW theorem by induction in k. Any element j1 . . . jk for which j1 ≤ ⋅ ⋅ ⋅ ≤ jk does not hold can be brought to this form by using the relation (2.167); if i > j, i j = j i + [i , j ]; now the ﬁrst term has the right order or indices j < i, and the second element is a tensor of degree k – 1.) Every representation 0 of G in the vector space V is uniquely extended to a representation of U(G) so that the elements 0(u) (u ∈ U(G)) are noncommutative polynomials of 0(i ) (i = 1, . . . n). Vice versa if 0 is a representation of U(G), then 0|G is a representation of G. Let Z(G) denote the centre of U(G). Let G0 be a complex semisimple Lie algebra, H0 is its Cartan subalgebra, Hi , H i j are dual bases in H w.r.t. the Killing form, i.e., B(Hi , H j ) = $i , i, j = 1, . . . , = rank G0 , E! , (! ∈ B0 ), are the root vectors of G0 is w.r.t. H, then the element of U(G0 ) C2 =

Hj H j +

(E! E–! + E–! E! )

(2.169)

!∈B+

j=1

is the only central element of U(G0 ) which is a homogeneous polynomial of degree 2; C2 is called the second-order Casimir element or the Casimir operator. There are also higher-order central elements of which are algebraically independent, = rank G0 (Chevalley theorem). In order to extend (2.169) to the afﬁne Kac–Moody case we ﬁrst rewrite it using [E! , E–! ] = H! : C2 =

Hj H j + 2

E–! E! +

!∈B+

j=1

H! .

(2.170)

!∈B+

Let us consider the following important element of H0∗ 10 ≡

1 ! 2 +

(2.171a)

!∈B

⇒ (10 , !i ) = (!i , !i )/2,

∀ ! i ∈ 00

(2.171b)

Let us denote by 1ˆ 0 the image of 10 in H0 , then we can rewrite (2.170) as C2 =

i=1

Hi H i + 2

!∈B+

E–! E! + 2ˆ10 .

(2.172)

2.9 Verma Modules

63

In the afﬁne case we deﬁne 1 ∈ H∗ (the analogue of 10 ) by (2.171b) since (2.171a) is meaningless: (1, !i ) = (!i , !i )/2, ∀!i ∈ 0 = {!0 , !1 , . . . , ! }.

(2.173)

Since the Cartan matrix is degenerated (2.173) does not ﬁx 1 completely. Then as 1 ∈ H∗ we deﬁne 1 = 10 !0 + 11 !1 + ⋅ ⋅ ⋅ + 1 ! + 1c c = 1d d + 1′1 !1 + ⋅ ⋅ ⋅ + 1′ ! + 1c c,

(2.174a) ˜ (!0 = d – !).

(2.174b)

It is clear that (2.173) cannot ﬁx 1d , because (d, !i ) = 0, i = 0, 1, . . . , . So we use either (2.173b) setting one of 1i equal to zero, or (2.173c) setting 1d = 0. Note that 1c = (1, d) is equal to the dual Coxeter number h∨ (cf. Section 2.4). The Casimir operator for G (introduced by Kac) is deﬁned for a class of G–modules, so-called restricted G-modules. A G-module V is called restricted if for ∀v ∈ V, G! ⋅ v ≠ 0 only for a ﬁnite number of roots ! ∈ B+ . Now we can deﬁne the Casimir operator for an afﬁne algebra G by K=

j=0

Hj H j + 2

mult

! !∈B+

(i) (i) E–! E! + 2ˆ1,

(2.175)

i=1

(i) ), form a basis of G! , (G–! ), and 1ˆ ∈ H is such that where mult ! = dimG! , E!(i) , (E–! ∗ +(ˆ1) = (+, 1) for each + ∈ H . It is clear that K is well deﬁned on any restricted module V over G and that K commutes with the action of G on V.

2.9 Verma Modules Let V be a vector space, 0 a representation of G in V. Equivalently we shall say that V is a G-module. A representation is called H-diagonalizable if V = ⊕ V, ,

(2.176a)

V, ≡ v ∈ V|0(H)v = ,(H)v, ∀H ∈ H .

(2.176b)

,∈H∗

If V, ≠ {0}, , is called a weight of 0, V, is called a weight space with weight ,, v ∈ V, is called a weight vector with weight ,. Let v be a weight vector with weight ,, then v′ = 0(E! )v is a weight vector with weight , + !, ! ∈ B.

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2 Lie Algebras and Groups

Suppose that there is a highest weight + in V, i.e., ++! is not a weight for any ! ∈ B+ . Then 0(E! )v0 = 0, ! ∈ B+ , for v0 of weight +; v0 is called a highest weight vector, (0, V) a highest weight module (HWM). Analogously if , is a lowest weight, 0(E! )v = 0, ! ∈ B– , for a lowest weight vector v, and a lowest weight module (LWM) (0, V). We also assume (as in [327]) that a highest (lowest) weight module V is cyclic module (or extremal), i.e., V = 0(U(G)) v0 ,

(2.177)

where v0 is the highest (lowest) weight vector. Using the standard decompositions G = G+ ⊕ H ⊕ G– and U(G) = U(G– ) ⊗ U(H) ⊗ U(G+ ) = U(G+ ) ⊗ U(H) ⊗ U(G– ),

(2.178)

one has V = 0(U(G– )) v0 if V is an HWM, and V = 0(U(G+ )) v0 if V is an LWM. Let dimG < ∞; if dimV < ∞, there is always a highest and lowest weight, otherwise there exists either a highest or lowest weight. Since there is an obvious duality between HWMs and LWMs, we shall study further the HWMs. Exercise: Show that the roots are the weights of the adjoint representation. If dimG < ∞, then the highest (lowest) weight is equal to the highest root !˜ ˜ (resp. –!). ♢ Further, we shall denote the representation (module) action of G on V by Xv, X ∈ G, v ∈ V. Then we shall say that V is a highest (or lowest) weight module when (0, V) is such a module. Let us consider G and a subalgebra B0 of G. Let V0 be a left B0 -module. Let us set V = Ind(G, V0 ) = U(G) ⊗U(B0 ) V0

(2.179)

We consider V as a left U(G)-module (w.r.t. multiplication in U(G)). V is called induced G-module from V0 . Obviously, V0 can be identiﬁed with a submodule of V w.r.t. the embedding V0 → 1 ⊗ V0 ≅ U(B0 ) ⊗U(B0 ) V0 . The algebra B = G+ ⊕ H is called a Borel subalgebra of G. Every character D of B is trivial on G+ and is uniquely determined by its values on H. Thus, the set of values D(Hk ) where Hk , k = 1, . . . , = dim H, is a basis of H determine D.

65

2.9 Verma Modules

A G-module induced from a character of a Borel subalgebra is called a Verma module [120, 599]. Let 1D denote the B-module 1D = Cv0 such that G+ v0 = 0, Hv0 = D(H)v0 , (dim1D = 1). Let V D be the corresponding Verma module, then V D = U(G) ⊗U(B) 1D .

(2.180)

U(G) = U(G– ) ⊗ U(B)

(2.181)

V D = U(G– ) ⊗ 1D .

(2.182)

Using

one has

Obviously V D is an HWM with highest weight D, and highest weight vector v0 , V D ≅ U(G– ) as vector spaces. Equivalently, V D may be deﬁned as V D = U(G)/ID , where ID is the left ideal of U(G) generated by the elements of G+ and elements of the form H – D(H) ⋅ 1, H ∈ H. Then v0 is the image of 1 ∈ U(G) w.r.t. the canonical projection, i.e., v 0 = 1 + ID . Further we have the following result by Dixmier [120]: Proposition 7: Let D ∈ H∗ , then one has (a) (b)

D , where H∗ ⊃ A ∋ ,, (,, !∨ ) ∈ Z , !∨ = 2! /(! , ! ), ∀! ∈ 0. V D = ⊕ VD–, + + i i i i i i ,∈A+ The weights of V D are of the type D – i=0 ni !i , where ni ∈ Z+ . We have D = P(,), dimVD–,

(c)

(2.183)

where P(,) is a generalized partition function P(,) = # of ways , can be presented ˆ j = dimG"j , as a sum of positive roots "j , each root taken with its multiplicity m and P(0) ≡ 1. For every , ∈ A+ : D VD, =

ˆ m

j

"j ∈ B+ ij =1 G"j =, "1 ≤"2 ≤⋅⋅⋅≤"n

(i )

(in ) E–"1 . . . E–" ⊗ 1D n 1

= U(G– )–, ⊗ 1D ,

(2.184a)

(2.184b)

ˆ j = 1 in the ﬁnite-dimensional case and for real roots in the afﬁne (recall that m case; for the imaginary roots cf. (2.116b) and (2.147b)). (d) VDD = 1 ⊗ 1D , V D = U(G– )VDD ,

U(G+ )VDD = 0

(2.185)

66

2 Lie Algebras and Groups

Proof. Let H ∈ H, then one has ! (i ) (i ) (in ) (in ) " ⊗ 1D + ⊗ 1D ) = H, E–"1 . . . E–" H ⋅ (E–"1 . . . E–" n n 1

(2.186)

1

(i ) (i ) (in ) (in ) + E–"1 . . . E–" H ⊗ 1D = – "1 (H) ⋅ ⋅ ⋅ – "n (H) E–"1 . . . E–" ⊗ 1D + n n 1

+

(i ) E–"1 1

1

(in ) . . . E–" D(H) ⊗ 1D = (D – "1 ⋅ ⋅ ⋅ – n

(i ) "n )(H)E–"1 1

(in ) . . . E–" ⊗ 1D . n

∎

This proves b). The rest is obvious. Corollary: A Verma module is restricted.

Proof. The basis of the Verma module consists of PBW monomials (i.e., elements of (i ) (in ) ˆ i !i . It is easy to check that for ⊗ 1D , with G"j = , = i m the PBW basis) E–"1 . . . E–" n 1 any ! = i ni !i ∈ B+ one has (i ) E! E–"1 1

(in ) . . . E–" n

⊗ 1D

≠0

ˆi ni ≤ m

0

otherwise.

∀i,

Thus, for a ﬁxed basis monomial only a ﬁnite number of positive root vectors E! have nonzero action. Since each vector v is a ﬁnite sum of basis monomials then again only a ﬁnite number of positive root vectors E! have nonzero action on v. ∎ Let V be a G-module. A vector subspace V ′ of V is called G-invariant or simply invariant if XV ′ ⊂ V ′ , ∀X ∈ G. The G-module V ′ is called a submodule of the module V. A G-module V is called reducible module if ∃ a proper submodule V ′ , i.e., {0} ≠ V ′ ≠ V. It is clear that a proper submodule of a Verma module V D cannot contain VDD and thus it cannot be an HWM with the same highest weight. Furthermore, one has Proposition 8: V D contains a unique proper maximal submodule I D . Proof. A sum of proper submodules of V D is again a proper submodule because every D D ∩ M D and proper submodule M D of V D has a weight space decomposition MD–, = VD–, D D also M ⊃/ VD . ∎ Verma modules have the following universality property: every HWM is isomorphic to a factor-module of the Verma module with the same highest weight. Thus, in particular, highest (lowest) weight modules are restricted. In particular, it follows from the above proposition that among the HWM with highest weight D there is a unique irreducible one, denoted by LD , i.e., LD = V D /I D . It is clear that LD is a quotient of any HWM with highest weight D.

(2.187)

2.9 Verma Modules

67

We need the following Lemma 26: On V D the Casimir operator has the property K = (D + 21, D) idV D .

(2.188)

Proof. Using (2.175) one has i + 2ˆ i ) + 2(D, 1) v H H 1 v = D(H )D(H Kv0 = 0 0 i i i=0 i=0 = =

i i=0 D(D(H )Hi )

+ 2(D, 1) v0 =

i i=0 (D, D(H )!i )

+ 2(D, 1) v0 =

= D, i=0 D(H i )!i + 21 v0 = (D, D + 21)v0 . Then, since K ∈ Z(G), (2.189) holds for any v ∈ V D .

(2.189) ∎

It may happen that besides v0 there are also other vectors vs for which Xvs = 0, ∀X ∈ G+ .

(2.190)

D , , ≠ 0, then v is called a singular vector and M D = U(G )v is obviously a If vs ∈ VD–, s – s proper submodule of V D , thus V D is reducible. Using the above lemma, we have

Kvs = (D + 21, D)vs

(2.191)

Kvs = (D + 21 – ,, D – ,)vs .

(2.192)

2(D + 1, ,) = (,, ,), , ∈ A+ .

(2.193)

and on the other hand

Thus, one has

Thus, (2.193) is obviously a necessary condition for the Verma module V D to be reducible. Results of Bernstein–Gel’fand–Gel’fand [42] for dimG < ∞ and of Kac–Kazhdan [328] for the afﬁne case show (as we shall see below) that (2.193) is a sufﬁcient condition for reducibility only if , is a quasi-root, i.e. if , = m", where m ∈ N, " ∈ B+ . Next we shall need a bilinear C-valued form on V D . We use the unique involutive antiautomorphism 3 on G such that 3(ei ) = fi , 3(fi ) = ei , i = 0, 1, . . . , , where ei , fi are the Cartan–Weyl generators corresponding to the simple roots, 3|H = id. It is clear that 3 can be uniquely extended to U(G). Using U(G) = U(H) ⊕ (G– U(G) ⊕ U(G)G+ )

(2.194)

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2 Lie Algebras and Groups

let " be the projection of U(G) on U(H) parallel to the second summand in (2.194). Obviously "(3(X)) = "(X), X ∈ U(G). Following [42, 548], we deﬁne a bilinear form 6 on U(G) with values in U(H): 6(X, Y) = "(3(X)Y), X, Y ∈ U(G).

(2.195)

6(U(G),1 , U(G),2 ) = 0 if ,1 ≠ ,2

(2.196)

Obviously

where U(G), is deﬁned analogously to (2.184b). Let us denote by 6, the restriction of 6 to U(G)–, . Since dimU(G), < ∞ we can treat 6, as a matrix with entries from U(H). For D ∈ H∗ we denote by 6, (D) a matrix obtained from 6, with every element H(∈ H) replaced by D(H). Following Shapovalov [548] one can show that rank 6, (D) = dimC (LD )D–, . Using 6 one can deﬁne a bilinear C-valued form 60 on V D : 60 (X1 v0 , X2 v0 ) ≡ 6(X1 , X2 )(D), X1 , X2 ∈ U(G– ).

(2.197)

The following properties are easily veriﬁed 60 (Xv, v′ ) = 60 (v, 3(X)v′ ), X ∈ U(G), v, v′ ∈ V D ,

(2.198)

60 (V,D1 , V,D2 ) = 0 if ,1 ≠ ,2 ,

(2.199)

and that I D (cf. Proposition 8) is the kernel of 60 . We have Lemma 27: Up to a nonzero constant factor the leading term of det 6, is ∞ '' (H! )P(,–k!) , H! = [E! , E–! ],

(2.200)

!∈B+ k=1

where the roots ! are taken with their multiplicities (i.e., for mult ! = dimG! > 1, H! (i) represents all H!(i) = [E!(i) , E–! ], i = 1, . . . mult !). Proof. The leading term of det 6, is obtained when every member of the basis as written in (2.184) is paired with itself. This means that this term is obtained from the following term in det 6, : ' (i ) (i ) (in ) (in ) , (2.201) 6 E–"1 . . . E–" , E–"1 . . . E–" n 1

1

1

where the product is taken over the terms in the sum (2.184a) and actually has the form '

(i )

H" 1 . . . H"(inn ) . 1

(2.202)

69

2.9 Verma Modules

Now let us suppress the superscripts (ij ) and consider a ﬁxed root !. To calculate the degree with which H! appears in (2.200) we consider separately the terms in (2.184b) for which exactly k of "1 , . . . , "n (k ≤ n) are equal to !. It is clear that there are exactly k! P(, – k!) such terms. Thus, the degree of H! is ∞ k=1 P(, – k!) (= k=1 P(, – k!), , – k! ! ∈ A+ , , – (k! + 1)! ∈/ A+ ). ∎ Further we need Lemma 28: Up to a nonzero constant, det 6, is equal to a product of linear factors of the form: H" + 1(H" ) – (", ")/2,

(2.203)

where " ∈ A+ is a quasiroot. Proof. Let V D be irreducible, then 60 is nondegenerate and (det 6, )(D) ≠ 0 for any , ∈ A+ \0 if D does not satisfy (2.193) for any " ∈ A+ \0. Thus, the polynomial det 6, has its zeroes in the union of the hyperplanes in H∗ given by the equations T" (D) = 0, where T" (D) = (D + 1, ") – (", ")/2, " ∈ A+ \0. Hence det 6, is a product of linear factors of the form T" , " ∈ A+ \0. By the previous Lemma only those T" appear for which " is a quasi–root. ∎ Combining all these results one has (up to a nonzero multiple) : ∞ '' (H! – 1(H! ) – k(!, !)/2)P(,–k!) .

det 6, =

(2.204)

!∈B+ k=1

Then we can obtain the results of [42] and [328] we mentioned above: Proposition 9: [42], [328] The Verma module V D over G is irreducible iff 2(D + 1, !) ≠ m(!, !)

(2.205)

for any ! ∈ B+ and any m ∈ N.

♢

For arbitrary D ∈ H∗ and " ∈ B+ , such that (", ") ≠ 0 we can deﬁne the numbers: m" (D) ≐ (D + 1, "∨ ),

"∨ =

2" . (", ")

(2.206)

The numbers are called the Harish-Chandra parameters of D. Since roots can be decomposed into simple roots analogously the subset of Harish-Chandra parameters corresponding to the simple roots !k : mk (D) ≐ (D + 1, !∨k )

(2.207)

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2 Lie Algebras and Groups

determine all Harish-Chandra parameters. The numbers from (2.207) are called the Dynkin labels of D (as they correspond to the nodes of the Dynkin diagram). Thus, the above Proposition means that if (", ") ≠ 0 and m" (D) ∈ N then the Verma module V D is reducible. Returning to the general situation, as we saw if a Verma module V D is reducible, i.e., 2(D + 1, ") = m(", "), m ∈ N, " ∈ B+ ,

(2.208)

D . Moreover the submodule M D = U(G– )vs then there exists a singular vector vs ∈ VD–m"

of V D is obviously isomorphic to the Verma module V D–m" = U(G– ) ⊗ v0′ where v0′ is the highest weight vector of V D–m" ; the isomorphism being realized by vs ↦ 1 ⊗ v0′ . Moreover using the Weyl reﬂection (2.32) in the case (", ") ≠ 0 we have D – m" = s" (D + 1) – 1.

(2.209)

Prompted by (2.209) we introduce for further use the “dot”-action: s" ⋅ (D) ≐ s" (D + 1) – 1 .

(2.210)

It is clear that a singular vector is given by ˆ m

vs =

j

"j ∈B+ ij =1 G"j = m" "1 ≤⋅⋅⋅≤"n

i ...in (i ) E 1 1 ..."n } –"1

C{"1

(in ) D . . . E–" ⊗ v0 ∈ VD–m" , n

(2.211)

n are ﬁxed (uniquely if (", ") ≠ 0) from the requirement where the coefﬁcients C"in ...i 1 ..."n (2.190). An alternative approach is to use only the negative Chevalley generators Xi– . Then the singular vector is given by

vs = Pm" (X0– , X1– . . . X– ) ⊗ v0 ,

(2.212)

where Pm" is a homogeneous polynomial in its variables of degrees mki , where ki ∈ Z+ come from " = ki !i , !i ∈ 0, the system of simple roots. D It is obvious that (2.212) satisﬁes vs ∈ VD–m" , while conditions (2.190) ﬁx the coefﬁcients of Pm" up to an overall multiplicative nonzero constant. Example: Let (2.208) hold for " = !i . Then up to a constant vs = (Xi– )m ⊗ v0

(2.213)

2.10 Irreducible Representations

71

Indeed using the Chevalley generators Xi+ , Xi– , Hi , (cf. (2.151), in particular, Hi ≡ H˜ i∨ = 2H˜ i /(!i , !i ) we obtain: [Xi+ , (Xi– )m ] =

m–1 m–1

(Xi– )m–1–j Hi (Xi– )j = (Xi– )m–1 (Hi – 2j) = j=0

j=0

= (Xi– )m–1 m(Hi – m + 1),

(2.214a)

Xi+ (Xi– )m ⊗ v0 = (Xi– )m ⊗ Xi+ v0 + (Xi– )m–1 ⊗ m(Hi – m + 1)v0 = = (Xi– )m–1 m(D(Hi ) – m + 1) ⊗ v0 = 0.

(2.214b)

The last equation follows because from (2.208) we have D(Hi ) = (D, !∨i ) = m – (1, !∨i ) = m – 1 , which follows from D(H˜ i ) = (D, !).

♢

General formulae for the singular vectors in the cases (", ") ≠ 0 are given below in Section 4.8. The cases (", ") = 0 which are possible in the afﬁne case (and in the supersymmetry case, cf. Volume 2) have some peculiarities. It is clear that if (2.208) holds for an imaginary root " = kd, (", ") = 0, then it holds for every imaginary root. According to (2.212) P is a homogeneous polynomial of degrees mki , ki ∈ Z, k0 = 1, ki , i ≥ 1, are determined from !˜ = i=1 ki !i in the AFF type 1 case or from (0 = i=1 ki !′i in the AFF type 2 or 3 cases (cf. Sec. 2.3). Example: [124] Consider A(1) 1 and let (2.208) hold for " = d. Then vsd = [(m + 1)f0 f1 + (1 – m)f1 f0 ] ⊗ v0 ,

(2.215)

where m ≡ (D + 1, !∨1 ) = –(D + 1, !∨0 ), d = !0 + !1 . If m ∈/ Z\{0}, V D is not reducible w.r.t. any real root. If m ∈ N, (resp. m ∈ –N), then V D is reducible w.r.t. " = !1 , (resp. " = !0 ). ♢ The formula for the singular vectors in the cases (", ") = 0 was obtained in [425] for A(1) 1 . There is a degeneracy, namely for " = kd, k ∈ N, these are p(k) linearly independent singular vectors, (where p(⋅) is the partition function, i.e., p(k) = # of ways k can be represented as the sum of positive numbers, p(0) ≡ 1).

2.10 Irreducible Representations Let us consider the irreducible quotients LD (cf. (2.187)) of Verma modules V D . If V D is irreducible then LD = V D . Thus, further we discuss LD , for which V D is reducible.

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2 Lie Algebras and Groups

Consider V D reducible w.r.t. every simple root (and thus w.r.t. all positive roots): (D + 1, !∨i ) = mi ∈ N, i = (0), 1, . . . , .

(2.216)

In the ﬁnite-dimensional case, i = 1, . . . , , LD is a ﬁnite-dimensional HWM. It has also a lowest weight D∗ = w0 (D + 1) – 1, where w0 ∈ W0 is the only element of the Weyl group W = W(G, H), which transforms the positive Weyl chamber C+0 ≡ {H ∈ H0 : !i (H) > 0, ∀!i ∈ 00 }

(2.217)

into –C+0 . Alternatively w0 may be deﬁned as the longest element of W (cf. Section 1.2). All ﬁnite-dimensional HWM are of the form LD , (2.216) holding for ∀!i ∈ 00 . Then from (2.208) also m" ≐ (D + 1, "∨ ) ∈ N for ∀" ∈ B+ . The trivial one-dimensional representation is obtained for D = 0, then mi = (1, !∨i ) = 1 for ∀!i ∈ 00 . For later use we denote m0" ≐ ( 1, "∨ ), (m0" ∈ N for ∀" ∈ B+ ). An important class of the case when (2.216) holds are so-called fundamental representations L(Di ), i = (0), 1, . . . , , characterized by (Di , !∨j ) = $ij , i.e., (Di + 1, !∨j ) = 1 + $ij = mj (Di ), or in a compact notation: [m1 , . . . , m ]i = [1, . . . , 2, . . . , 1],

where 2 is at the ith place.

(2.218)

The weights Di are often used as a basis for the highest weights. Among the fundamental irreps (≡ irreducible representations) are those with smallest nontrivial ( ≠ 1) dimension, which are called deﬁning irreps or vector irreps. The latter correspond to the matrix realizations of A , B ( ≥ 2), C ( ≥ 2), and D ( ≥ 4), given in Section 2.5. In the afﬁne case L(D0 ) is called the basic representation, it is trivial on H0 , D0 |H0 = 0. The trivial one-dimensional representation in the same notation is [m(0)1 , . . . , m ]0 = [m0(0)1 , . . . , m0 ] = [1, . . . , 1].

(2.219)

The dimension of an arbitrary ﬁnite-dimensional HWM LD is given by dimLD =

' " ∈ B+

(D + 1, "∨ )/

'

'

(1, "∨ ) =

"∈B+

'

m" /

" ∈ B+

" ∈ B+

m0" .

(2.220)

2.10.1 A In the case of A the general dimension formula is dimL(m1 , . . . , m ) =

–i+1 ' ' i+j–1

i=1 j=1 s=i

ms /

' t=1

t!

(2.221)

2.10 Irreducible Representations

while the dimension formula of the fundamental representations is +1 dimL(Di ) = , i = 1, . . . , i

73

(2.222)

Thus, the fundamental irreps come in pairs of equal dimension (except L(D(+1)/2 ) for odd ), which are conjugated under the exchange of simple roots (as used in Subsection 2.6.2) !k ↔ !+1–k , k ≤ /2. Thus, there are two conjugated vector irreps: L(D1 ) and L(D ); their dimension is + 1. 2.10.2 C In the case of C , ≥ 2, the dimension formula for the fundamental irreps is + 1 – i 2 + 2 dimL(Di ) = , i = 1, . . . , . +1 i

(2.223)

The deﬁning irrep is L(D1 ); it has dimension 2 (cf. Subsection 2.1.3). Note that this formula is meaningful for = 1 and consistently with the isomorphism C1 ≅ A1 gives the correct dimension of the two-dimensional fundamental irrep. 2.10.3 B In the case of B , ≥ 2, the general dimension formula is ( m 1≤i≤–1 (2mi + 2mi+1 + ⋯ + 2m–1 + m ) dimL(m1 , . . . , m ) = (2 – 1)!! ⎛ ' (mi + mi+1 + ⋯ + mj–1 ) ×⎝ × j–i 1≤i n–1 > ⋅ ⋅ ⋅ > n1 > 0, ≥ –1 ≥ ⋅ ⋅ ⋅ ≥ 1 ≥ 0. The representations with i integer and half-integer, resp. (or ni half-integer, integer, resp.), are called tensor representations and spinor representations, respectively. The terminology comes from the fact the latter are not irreps of the corresponding Lie group SO(2 + 1) but only of its double covering group Spin(2 + 1). Thus, the spinor irreps are those for which m is even. The trivial irrep is [n1 , . . . , n ]0 = [1/2, 3/2, . . . , – 1/2], [1 , . . . , ]0 = [0, . . . , 0], i.e., the ni and i differ by the components of 1. The signatures of the fundamental representations L(Di ) in these notations are ⎧ ⎪ ⎪ ⎨[1/2, 3/2, . . . , – i – 1/2, – i + 3/2, . . . , + 1/2], (2.227) [n1 , . . . , n ]i = 1 ≤ i ≤ – 1, ⎪ ⎪ ⎩[1, 2, . . . , ], i = ⎧ ⎪ [0, . . . , 0, 1, . . . , 1], ⎪ ⎨ [1 , . . . , ]i = # of zeroes = i, 1 ≤ i ≤ – 1, ⎪ ⎪ ⎩ [1/2, 1/2, . . . , 1/2], i = .

(2.228)

Their dimension formulae are 2+1 dimL(Di ) =

i

2 ,

,

i = 1, . . . , – 1 i=

The deﬁning irrep is L(D1 ); it has dimension 2 + 1.

(2.229)

2.10 Irreducible Representations

75

Note that all formulae are meaningful for = 1, 2 and moreover are consistent with the isomorphisms B1 ≅ A1 ≅ C1 , B2 ≅ C2 . In particular, for = 1 the fundamental (spinor) irrep is given by n1 = 21 m1 = 1, 1 = 1/2, and has dimension 2, i.e., it has smaller dimension than the three-dimensional deﬁning (vector) irrep of so(3, C). For = 2 we have to use the root mappings given after (2.67), which in terms of the mi signatures give: (m1 , m2 )B2 → (m2 , m1 )C2 ,

=2

In particular, the fundamental irreps (in B2 notation) are [m1 , m2 ]1 = [2, 1],

dim = 5;

(2.230a)

[m1 , m2 ]2 = [1, 2],

dim = 4

(2.230b)

and they are tensor, spinor, resp. (since m2 = 1, 2 resp.). Note that the irrep with smallest dimension (4) is spinor, while the ﬁve-dimensional irrep is the deﬁning (vector) irrep of so(5, C). Thus, for = 1, 2 the deﬁning irreps of so(2 + 1, C) are not the ones with smallest dimension, which gives a (representation-theoretic) reason to consider B for ≥ 3. 2.10.4 D In the case of D , ≥ 4, the general dimension formula is '

dimL(m1 , . . . , m ) =

n2j – n2i

1≤i 0.

(2.269)

i=(0)1

Let J = {i1 , . . . ik } ⊂ {(0), 1, . . . , } be such that mi = 0, i ∈ J and mi ≠ 0, if i ∈/ J. In this case the multiplet M is obtained by the action w(D′ + 1) – 1 for all w for which in any of their reduced expressions sj1 . . . sjp we have jp ∈/ J. For G afﬁne the set of such w is still ′′ inﬁnite and so is the corresponding multiplet. The situation for V D such that (2.252) holds and (D′′ + 1, !∨i ) = mi ∈ Z– ,

'

mi = 0,

i=(0)1

mi < 0

(2.270)

i=(0)1

is analogous. For G afﬁne the situations described in (2.253), (2.254), and (2.269), (2.270) exhaust ′ ′′ the cases when V D or V D exists but not both. Thus, for all further multiplets we ′ discuss only V D (cf. (2.251) and (2.252)). ′ Let V D and I ⊂ {(0), 1 . . . } be such that the inclusion is proper and (D′ + 1, !∨i ) = mi ∈ N, ′

(D +

1, !∨i )

∈/ Z,

i ∈ I,

i ∈/ I.

(2.271a) (2.271b)

Let I1 , . . . , Ik be such that I = I1 ∪ ⋅ ⋅ ⋅ ∪ Ik and si sj = sj si , if i ∈ Im , j ∈ In , m ≠ n. Let Wn be the subgroup of W generated by sj , j ∈ In . Then the multiplet M containing ′ V D is in 1-to-1 correspondence with the subgroup W1 × W2 × ⋅ ⋅ ⋅ × Wk of W. 2.11.2.7 Finally, there is the situation derived from the previous one with (2.271a) replaced by (D′ + 1, !∨i ) = mi ∈ Z+ , i ∈ I,

' i∈I

mi = 0.

(2.272)

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2 Lie Algebras and Groups

n denote the set of these (This is analogous to the passage from (2.253) to (2.269).) Let W elements w of Wn such that any of their reduced expressions is of the form sj1 . . . sjp ′ with mjp ∈ N. Then the multiplet M containing V D is in 1-to-1 correspondence with 1 × W 2 × ⋅ ⋅ ⋅ × W k . W 2.11.3 Characters of Generic Highest Weight Modules For all + ∈ H∗ we set +! ≡ (+, !∨ ), !∨ ≡ 2!/(!, !), ! ∈ B; +i ≡ +!i , !i ∈ 0. The element + ∈ H∗ is called integral element if +i ∈ Z for all i = 1, . . . , . The element + ∈ H∗ is called dominant element if +i ≥ 0 for all i = 1, . . . , . Denote by A the set of all integral elements of H∗ , by A+ the set of all integral dominant elements (cf. Proposition 7). Following [120, 327], let E(H∗ ) be the associative Abelian algebra consisting of the series ,∈H∗ c, e(,), where c, ∈ C, c, = 0 for , outside the union of a ﬁnite number of sets of the form D(+) = {, ∈ H∗ |, ≥ +}. Using some ordering of H∗ , e.g., the lexicographic one, the formal exponents e(,) have the properties: e(0) = 1, e(,)e(-) = e(, + -). We recall that for each invariant subspace V ⊂ U(G– ) ⊗ v0 ≅ V D we have the following decomposition: V = ⊕ V, , V, = {u ∈ V | Hk u = (D – ,)(Hk )u, ∀ Hk }. ,∈A+

(2.273)

(Note that V0 = C v0 .) The character of V is deﬁned by chV =

(dimV, )e(D – ,) = e(D)

,∈A+

(dim V, )e(–,).

(2.274)

,∈A+

Thus, from Proposition 7 we see that for a Verma module V D chV D = e(D)

P(,)e(–,) = e(D)

,∈A+

'

(1 – e(–!))–mult ! .

(2.275)

!∈B+

Using the results of the previous subsection we can also easily write the characters of the ﬁnite-dimensional irreducible representations when dimG < ∞ and of the integrable HWMs in the afﬁne case, i.e., of LD when (D + 1, !∨i ) ∈ N, ∀!i ∈ 0 (cf. (2.253)). Indeed, it is clear that chLD = chV D – chI D

(2.276)

(cf. (2.187)), and we can use (2.245) in this case. Now it is clear that chI D contains the terms

chV D–mi !i = chV si (D+1)–1 (2.277) i=0

i=0

87

2.11 Characters of Highest Weight Modules

but certainly the sum in (2.277) is larger than chI D because of the overlap. Indeed, V si (D+1)–1 = V D–mi !i has as submodules Vij = V D–mi !i –(mj –mi aji )aj , j ≠ i. However, the latter is also a submodule of V D–mj !j . Indeed, let ! = s!j !i = !i – aji !j , (!, !) = (!i , !i ), then !∨ = 2!/(!, !) = 2!i /(!i , !i ) – 2aji !j /(!i , !i ) = !∨i – aij !∨j , (D + 1 – mj !j , !∨ ) = (D + 1 – mj !j , !∨i – aij !∨j ) = mi ,

(2.278a)

s! (D + 1 – mj !j ) – 1 = D – mj !j – mi ! = D – mi !i – (mj – mi aji )!j .

(2.278b)

Thus, ch Vij is contained twice in (2.277) and in order to correct for this overlap one should subtract ch Vij once. So the ﬁrst two terms of ch V D should be

ch V si (D+1)–1 –

i=0

ch V si sj (D+1)–1 .

(2.279)

i,j i≠j

Certainly, there is overlap in the different Vij . Thus, one can convince himself that chI D =

(–1)(w) chV w(D+1)–1 .

(2.280)

w∈W w≠1

Finally, for LD we have chLD =

(–1)(w) chV w(D+1)–1

w∈W

= chV D =

(2.281a)

(–1)(w) e(w(D + 1) – D – 1

w∈W

(–1)(w) e(w(D + 1) – 1)/

'

(1 – e(–!))mult ! .

(2.281b) (2.281c)

!∈B+

w∈W

This is the classical Weyl character formula for the ﬁnite-dimensional irreducible HWM for dimG < ∞ and the Kac character formula for the integrable HWMs for G afﬁne. The complete proof of (2.281) may be found in [624] and [327], respectively. Further, we use the fact that for D = 0, dimLD = 1, and chLD = 1; then from (2.281c) we have a denominator identity (the Weyl denominator identity for dimG < ∞): ' !∈B+

(1 – e(–!))mult ! =

w∈W

(–1)(w) e(w(1) – 1).

(2.282)

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2 Lie Algebras and Groups

Thus, (2.281) can be further rewritten as chLD =

(–1)(w) e(w(D + 1))/

w∈W

(–1)(w) e(w(1)).

(2.283)

w∈W

2.11.4 Characters for Nondominant Weights If D ∈ A, then (2.281a) is replaced by chLD =

–1

(–1)(wD w) Pw,wD (1)chV w⋅(wD ⋅D) ,

D ∈ A,

(2.284)

w∈W w≤wD

where w ≤ wD is meant in the Bruhat order on W, wD is a unique element of W with minimal length such that the signature of D˜ = wD–1 ⋅ D is anti-semidominant: ˜ ), ˜ (0)1 , . . . , m 7˜ = (m

˜ K ) – 1 ∈ Z– ˜ k = D(H m

(2.285)

and Pw,wD (1) ∈ N is the value at 1 of the Kazhdan-Lusztig polynomials Pw,w′ (u) [332]. These polynomials are considered in detail in Chapter 7.

2.11.5 Characters in the Affine Case We further restrict to the afﬁne case in order to rewrite (2.281) in terms of theta functions. For # ∈ i=0 C!i let us deﬁne the following endomorphism on H∗ : t# (+) = + + (+, d)# – ((+, #) + (#, #)(+, d)/2)d.

(2.286)

One can easily see that t# t#′ = t#+#′ ;

(2.287a) –1

tw(#) = wt# w , w ∈ W.

(2.287b)

∗ , which is positive deﬁnite and integral, i.e., (,, -) ∈ Z, Let M be a Z-lattice in H0R ∀,, - ∈ M. Let M ∗ be the dual lattice of M, i.e., (,, -) ∈ Z, ∀, ∈ M ∗ , ∀- ∈ M. Obviously M ⊂ M ∗ . Let for m ∈ N

Pm = {+ ∈ H∗ : (+, d) = m, +H∗ ∈ M∗ }.

(2.288)

0

Then the classical C-function of degree m(= (+, d)) with characteristic +0 (= +H∗ ), i.e., 0

with + ∈ Pm , is deﬁned by the series

C+ = e–(+,+)d/2m

#∈M

et# (+)

(2.289)

2.11 Characters of Highest Weight Modules

89

with the property C+ = C++20im!+ad , ! ∈ M, a ∈ C.

(2.290)

In (2.289), following [327], we have replaced e(,) by e, which is a function on H, deﬁned by e, (H) = e,(H) . In this sense C+ is a holomorphic function on the domain Y = {H ∈ H : Re d(H) > 0}.

(2.291)

For + ∈ Pm , + = mc + dd + +0 , and +0 ∈ M ∗ ⊂ H0∗ , we can rewrite (2.286) as t# (+) = mc + d d + +0 + m# – ((+0 , #) + (#, #)m/2)d = mc + +0 + m# + ((+, +) – (+0 + m#, +0 + m#))d/2m

(2.292)

(actually (2.292) is valid for any m = (+, d) ≠ 0). Substituting (2.292) in (2.289) we obtain C+ = emc

e+0 +m#–(+0 +m#,+0 +m#)d/2m

(2.293a)

#∈M

= emc

em#–m(#,#)d/2 ,

(2.293b)

# ∈ M++0 /m

where we have changed # → # – +0 /m. Next we use the fact [327]: W = W0 | × T ,

(2.294)

where W0 is the Weyl group of G0 , and T is generated by ! ∈ M. We have

w∈W

(–1)(w) ew(1) =

(–1)(t# w) et# w(1) .

(2.295)

w ∈ W0 #∈M

The only generating element of W which is not contained in W0 is s0 = s!0 , !0 = d – ˜ the highest root of B+0 in the AFF 1 case, and ( = (0 in the AFF 2,3 cases. One (, ( = !, can verify the following: t(∨ = s!0 s(∨ , (∨ = 2(/((, ().

(2.296)

Then one can see that T is generated by wt(∨ w–1 , w ∈ W0 , and (2.295) can be rewritten as

w∈W

(–1)(w) ew(1) = e(1,1)d/21c

w ∈ W0

(–1)(w) Cw(1) .

(2.297)

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2 Lie Algebras and Groups

Analogously, we rearrange the nominator in (2.283) and obtain

chLD = e–sD d

(–1)(w) Cw(D+1) /

(–1)(w) Cw(1) ,

(2.298)

w ∈ W0

w∈W0

where sD = (D + 1, D + 1)/2(m + 1c ) – (1, 1)/21c .

(2.299)

Next let us choose an orthonormal basis v1 , . . . , v on H0R and let us denote the coordinates on H by v = –20i(4dˆ + uˆc + z1 v1 + ⋅ ⋅ ⋅ + z v ).

(2.300)

Then the domain Y is given by Y = {(z, 4, u) : z = (z1 , . . . z ) ∈ C , 4, u ∈ C, Im 4 > 0}.

(2.301)

Then C+ can be rewritten in its classical form (#i = #(vi )):

C+ (z, 4, u) = e–20imu

e0im4(#,#)–20mi(#1 z1 +⋅⋅⋅+# z ) .

(2.302)

# ∈ M++0 /m

2.11.6 Example of A(1) 1 ∗ In the setting above we restrict to G = A(1) 1 ; then = 1, M = Z!1 , M = Z!1 /2 = M/2, ((!1 , !1 ) = 2), + = d d + mc + n!1 /2, +0 = n!1 /2, # ∈ M + D0 /m = Z!1 + n!1 /2m, # = j!1 , j = #(v1 ), j ∈ Z + n/2m. For + = d d + mc + n!1 /2 write C+ = Cn,m , then we have a linear independent set of C–functions:

Cn,m (z, 4, u) = e–20imu

e20im( j

2 4–jz)

, n ∈ Z mod 2mZ.

(2.303)

j∈Z+n/2m

Let us introduce (2.303) in (2.298); W0 = {1, s1 }, D = dd + kc + (m1 – 1)!1 /2, 1 = 2c + !1 /2, (D + 1, !∨1 ) = m1 ∈ N, s1 (D + 1) – 1 = D – m1 !1 = d d + kc – (m1 + 1)!1 /2; (1, !∨1 ) = 1, s1 (1) = 1 – !1 = 2c – !1 /2. Further we set d = 0 in D. For sD we have sD = m21 /4(k + 2) – 1/8

(2.304)

and then chLD (v) = esD 20i4

(Cm1 ,k+2 – C–m1 ,k+2 ) (z, 4, u). (C1,2 – C–1,2 )

(2.305)

2.11 Characters of Highest Weight Modules

91

Note that using (2.249) we have k + 2 = m0 + m1 (m0 = (D + 1, !∨0 )), 1 ≤ m1 ≤ k + 2 – m0 ≤ k + 1.

(2.306)

It is instructive to obtain (2.305) directly from (2.281) using the explicit formulae (2.263)–(2.267). We have (D′ = D)

chLD = chV D –

∞

′

′

(chV D0n + chV D1n )

n=0

+

∞

(chV D0n + chV D1n )

(2.307a)

n=1 ∞ e–(n+1)((n+1)m0 +nm1 )d+((n+1)m0 +nm1 )!1 = chV D 1 – n=0

–

∞

e–n(nm0 +(n+1)m1 )d–(nm0 +(n+1)m1 )!1

n=0

+

∞

e–n(nm0 +(n–1)m1 )d–n(m0 +m1 )!1

n=1

+

∞

e–n(nm0 +(n+1)m1 )d+n(m0 +m1 )!1

(2.307b)

n=1

= chV D

e–n(n(k+z)+m1 )d+n(k+2)!1

n∈Z

–

e–n(n(k+2)–m1 )d+(n(k+2)–m1 )!1

(2.307c)

n∈Z 2

= chV D e–m1 d/4(k+2)–m1 !1 /2–(k+2)c (Cm1 ,k+2 – C–m1 ,k+2 ).

(2.307d)

For the last equality, we changed n = j–m1 /2(k+2) in the ﬁrst term and n = j+m1 /2(k+2) in the second. Analogously, using (2.275) and (2.282), we obtain chV D = e(D)/

(–1)(w) ew(1)–1 =

w∈W

= e(D)e

d/8+2c+!1 /2

/(C1,2 – C–1,2 );

(2.308)

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2 Lie Algebras and Groups

(note the fact that D + 1 → 1 means m1 = 1, k = 2, (1 = 2c + !1 /2)). Then using e(D) = ekc+(m1 –1)!1 /2 (d = 0), we obtain the desired result (with sD from (2.304)): chLD = e–sD d (Cm1 ,k+2 – C–m1 ,k+2 )/(C1,2 – C–1,2 ).

(2.309)

Finally, we combine (2.282), (2.297), and (2.307) to obtain a combinatorial identity. We set q = e–d , z = e–!1 and recall that each root is either kd or kd + !1 , k ∈ Z+ , or kd – !1 , k ∈ N. Thus, we have (mult ! = 1) ∞ ' (1 – qn )(1 – qn–1 z)(1 – qn z–1 )

(2.310a)

n=1

= q1/8 z1/2 e–2c (C1,2 – C–1/2 )

= qn(2n+1) z–2n – qn(2n–1) z2n–1 n∈Z

=

(–1)n qn(n–1)/2 zn .

(2.310b)

n∈Z

The equality between the product in (2.310a) and the sum in (2.310b) is the classical triple product identity.

3 Real Semisimple Lie Algebras Summary This chapter contains material based mostly on [59, 141, 538, 620]. Original results are cited appropriately. Here we start with the standard material on detailed structure of noncompact semisimple Lie algebras. We also recall the classification of the noncompact semisimple Lie algebras. Then following [141] we give case by case the parabolic subalgebras of each noncompact semisimple Lie algebra. We have done an exhaustive investigation on both cuspidal and maximal parabolic subalgebras, and for some cases we give all parabolic subalgebras.

3.1 Structure of Noncompact Semisimple Lie Algebras 3.1.1 Preliminaries Let G be a Lie algebra over R and let G C be its complexiﬁcation. Then G is called a real form of G C . Note that G C = G ⊕ iG. Facts: If G is reductive, semisimple, resp., then G C is reductive, semisimple, resp. If G is simple, then either G C is simple or G C = G1 ⊕ G2 , where G1 ≅ G2 ≅ G over R. In the latter case the algebra G itself has a complex structure, i.e., G is a complex simple Lie algebra but considered over R. ♢ Thus, we have the following theorem: Theorem: The simple Lie algebras over R are divided into two classes: – complex simple Lie algebras considered over R; –

real forms of the complex simple Lie algebras.

Thus, we consider further the second situation in detail. We start with a complex ˜ A subalgebra G of G˜ is called a real form of G˜ if G˜ ≅ G C . If Z ∈ G, ˜ simple Lie algebra G. then one can write uniquely Z = X + iY, where X, Y ∈ G. The operation 3(Z) = X – iY ˜ Alternatively, if 3 is an antilinear inis an antilinear involutive automorphism of G. ˜ ˜ Thus, volutive automorphism of G, then the set of its ﬁxed points is a real form of G. ˜ in order to classify all real forms of G it is enough to classify the antilinear involutive ˜ automorphisms of G. Next we point out two universal real forms. For every complex simple Lie algebra ˜ G there exist two nonisomorphic real forms. The split real form Gr ≐ {X ∈ G˜ | X¯ = X},

(3.1)

where ¯ is complex conjugation w.r.t the Cartan–Weyl basis of G˜ considered with real structure constants over R, i.e., Gr is like G˜ restricted to the real numbers.

94

3 Real Semisimple Lie Algebras

The compact real form Gk ≐ { X ∈ G˜ | X † = –X },

(3.2)

˜ where the Hermitian conjugate † is also deﬁned w.r.t. the Cartan–Weyl basis of G: ¯ the transposition being deﬁned as follows: t H! = H! , t E! = E–! , H! , E! , are X † ≐ t X, the elements of the Cartan–Weyl basis. Facts: All compact simple Lie algebras are obtained as compact real forms of the complex simple Lie algebras and the correspondence is 1-to-1. On the compact real form Gk the Killing form is negative deﬁnite. All other real forms of G˜ may be obtained from Gk (or from Gr ) using auto˜ Especially useful are the Cartan automorphisms ( ∈ Aut G, ˜ which are morphisms of G. involutive and commute with the Hermitian conjugation. Cartan Theorem: Each real form of the complex simple Lie algebra G˜ is conjugate w.r.t. Aut0 G˜ to one of the following forms: G3 = {X ∈ G˜ | 3(X) = X},

3(X) ≐ –((X † ),

where ( is a Cartan automorphism.

(3.3) ♢

In particular, in the case of Gr one has ((X) = – t X, while in the case of Gk one has ((X) = X. Using the Cartan automorphism one can make some conclusions about the structure of the real forms. Since G3 is invariant w.r.t. ( and (2 = Id, then G3 is decomposed into ± subspaces w.r.t. (. G3 = K ⊕ Q , K ≐ { X ∈ G3 | ((X) = X }, Q ≐ { X ∈ G3 | ((X) = –X }.

(3.4)

G3 = K ⊕ Q

(3.5)

The decomposition

is called Cartan decomposition of G3 . Obviously, [K, K] ⊂ K, [Q, Q] ⊂ K, and [K, Q] ⊂ Q, i.e., K is a subalgebra, while the subspace Q is an orthogonal supplement of K w.r.t. the Killing form B. The subalgebra K is the maximal compact subalgebra of G3 . It is unique up to conjugation from ˜ Aut0 G. If G = K ⊕ Q, then Gc ≐ K ⊕ iQ which is a compact real form of G C . Furthermore, Gc is the maximal compact subalgebra of G C considered over R. Notation: The compact real forms shall be denoted as their complexiﬁcations but by lowercase letters, i.e., an = su(n+1), bn = so(2n+1), cn = sp(n), dn = so(2n), en , f4 and g2

3.1 Structure of Noncompact Semisimple Lie Algebras

95

will denote the compact forms of complex algebras An = sl(n + 1, C), Bn = so(2n + 1, C), Cn = sp(n, C), Dn = so(2n, C), En , F4 , G2 . ♢ Naturally, a real form that is not compact will be called a noncompact real form. Further, when we say “noncompact” we may omit the word “real.” We have the following corollary of the theorem above: The simple noncompact Lie algebras are divided into two classes: – the complex simple Lie algebras considered over R; –

the noncompact real forms of the complex simple Lie algebras.

♢

3.1.2 The Structure in Detail Let G be a simple noncompact Lie algebra. Let ( be the Cartan involution of G, G = K ⊕ Q be the Cartan decomposition of G, so that (X = X, X ∈ K, (X = –X, and X ∈ Q. Let A0 be the maximal subspace of Q which is an Abelian subalgebra of G; r = dimA0 is the split rank (or real rank) of G, 1 ≤ r ≤ = rank G. The subalgebra A0 is called a Cartan subspace of Q. Next let M0 be the centralizer of A0 in K, i.e., M0 ≐ {X ∈ K|[X, Y] = 0, ∀Y ∈ A0 }. In general M0 is a compact reductive Lie algebra, and we can write M0 = M0s ⊕ M0a , where M0s ≐ [M0 , M0 ] is the semisimple part of M0 , and M0a is the Abelian subalgebra central in M0 . We mention also that a Cartan subalgebra H0m of M0 is given by H0m = H0s ⊕ M0a , where H0s is a Cartan subalgebra of M0s . Then a Cartan subalgebra H0 of G is given by H0 = H0m ⊕ A0 . It is the most noncompact among the nonconjugate Cartan subalgebras of G. Let BA0 be the restricted root system of (G, A0 ) + BA0 ≐ {+ ∈ A∗0 |+ ≠ 0, GA ≠ 0}, 0 + GA ≐ {X ∈ G|[Y, X] = +(Y)X, ∀Y ∈ A0 }. 0

(3.6)

The elements of BA0 are called A0 -restricted roots. (The terminology comes from the fact that things may be arranged so that these roots are obtained as restriction to A0 of some roots of the root system B of the pair (G C , HC ).) + + For + ∈ BA0 , GA are called A0 -restricted root spaces, dimR GA ≥ 1. 0 0 Next we introduce some ordering (e.g., the lexicographic one) in BA0 . Accordingly, the latter is split into positive and negative restricted roots: BA0 = B+A0 ∪ B–A0 . Furthermore, we introduce the simple restricted root system BRA0 , which is the simple root system of the restricted roots. Next we introduce the restricted Weyl reﬂections: for each root + ∈ B+A0 we deﬁne a reﬂection s+ in A0 ∗ .

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3 Real Semisimple Lie Algebras

s+ (,) ≡ , – 2

(+, ,) +, (+, +)

, ∈ A0 ∗ .

(3.7)

Clearly, s+ (+) = –+, s2+ =idA0 ∗ . The above reﬂections generate the restricted Weyl group W(G, A0 ). Next we introduce the corresponding nilpotent subalgebras. N0± ≐

+ GA . 0

⊕

+∈B± A

(3.8)

0

Next we introduce the Iwasawa decomposition of G. G = K ⊕ A 0 ⊕ N0 ,

N0 = N0±

(3.9)

and the minimal Bruhat decomposition. G = N0+ ⊕ M ⊕ A0 ⊕ N0– .

(3.10)

The relation between the two decompositions is given by the map: J : M0 ⊕ N0 → K, ′

′

(3.11) ′

J(X + X ) = X + X + (X ,

′

X ∈ M 0 , X ∈ N0 .

For further use we introduce some more notions recalling that B denotes the root system of G C . – compact roots of B relative to the minimal Bruhat decomposition are those roots that are zero when restricted to A0 . They are also called M0 – compact roots. –

noncompact roots of B relative to the minimal Bruhat decomposition are those that are nonzero when restricted to A0 . These are divided into real and complex roots: real roots of B are those that are zero when restricted to H0m . complex roots of B are the noncompact roots which are not real.

Remark 1: To avoid misunderstandings we recall that in the theory of root systems of afﬁne Lie algebras the term “real root” was used with completely different meaning. ♢

3.2 Classification of Noncompact Semisimple Lie Algebras In this section, we shall give explicit description of the real forms of classical complex simple Lie algebras. For this we shall use some additional notation. Here the operations of complex conjugation and transposition (consequently, Hermitian conjugate) are deﬁned as the same operations on matrices from gl(n, C), i.e., as in Subsection 2.1.7. We deﬁne

3.2 Classification of Noncompact Semisimple Lie Algebras

¯ r(X) ≐ X,

0 1k ; s≐ –1k 0

1p 0 ; 9≐ 0 –1q

9 0 . 4≐ 0 9

97

† –1

s(X) ≐ s X s , 9pq ≐ –9 X † 9, 4pq ≐ –4 X † 4,

(3.12)

Further, we describe the real forms G3 of the classical complex simple Lie algebras G˜ by the corresponding automorphism 3 such that G3 ≐ {X ∈ G˜ | 3(X) = X }. – The algebra gl(n, C) has the following real forms: gl(n, R), u(p, q), u∗ (n), corresponding to the automorphisms r, 9pq , (p + q = n), s, (n even), resp. –

The algebra sl(n, C) has the following real forms: sl(n, R), su(p, q), su∗ (n), corresponding to the automorphisms r, 9pq , (p + q = n), s, (n even), resp.

–

The algebra so(n, C) has the following real forms: so(p, q), so∗ (n), corresponding to the automorphisms 9pq , (p + q = n), s, (n even), resp.

–

The algebra sp(n, C) has the following real forms: sp(n, R), sp(p, q), corresponding to the automorphisms r, 4pq , (p + q = n).

The compact real forms are u(n) = u(n, 0) ≅ u(0, n), su(n) = su(n, 0) ≅ su(0, n), so(n) = so(n, 0) ≅ so(0, n), sp(n) = sp(n, 0) ≅ sp(0, n), resp. In terms of the notation introduced above, we have u(n) = su(n) ⊕ u(1), an–1 = su(n), bn = so(2n + 1), cn = sp(n), dn = so(2n). The split real forms are gl(n, R), sl(n, R), so(p, p) for n even, or so(p + 1, p) ≅ so(p, p + 1) for n odd, sp(n, R), resp. It is important to bear in mind the following isomorphisms: so(2n) ∩ sp(n) ≅ u(n), sp(p, q) ∩ u(2p + 2q) ≅ sp(p) ⊕ sp(q), sp(n, R) ∩ u(2n) ≅ u(n), so∗ (2n) ∩ u(2n) ≅ u(n), su(p, q) ∩ u(p + q) ≅ s(u(p) ⊕ u(q)), su∗ (2n) ∩ u(2n) ≅ sp(n). In the case of low dimensions there are isomorphisms that originate from the isomorphisms between complex Lie algebras. The low-dimension compact cases are 1-to-1 with the following complex cases: su(2) ≅ so(3) ≅ sp(1), su(4) ≅ so(6), so(4) ≅ so(3) ⊕ so(3), so(5) ≅ sp(2). The low-dimension noncompact cases are more interesting. – sl(2, R) ≅ su(1, 1) ≅ so(2, 1) ≅ sp(1, R), –

sl(4, R) ≅ so(3, 3),

–

su(3, 1) ≅ so∗ (6),

98

3 Real Semisimple Lie Algebras

–

su(2, 2) ≅ so(4, 2),

–

so(2, 2) ≅ so(2, 1) ⊕ so(2, 1),

–

so(3, 1) ≅ sl(2, C),

–

so(3, 2) ≅ sp(2, R),

–

so(4, 1) ≅ sp(1, 1),

–

so(5, 1) ≅ su∗ (4),

–

so(6, 2) ≅ so∗ (8),

–

so∗ (4) ≅ so(3) ⊕ so(2, 1).

Table 3.1: Classical Noncompact Semisimple Lie Algebras G

K

dimR Q

dimR A0

A,BD,C ≅ G˜ AI ≅ sl(n, R) AII ≅ su∗ (2n) AIII ≅ su(p, q) p≥q≥1 BDI ≅ so(p, q) p≥q≥1 DIII ≅ so∗ (2n) CI ≅ sp(n, R) CII ≅ sp(p, q) p≥q≥1

Gk so(n) sp(n) s(u(p) ⊕ u(q))

dimC G˜ = dimR Gk 1 2 (n – 1)(n + 2) (n – 1)(2n + 1) 2pq

rankC G˜ n–1 n–1 q

so(p) ⊕ so(q)

pq

q

u(n) u(n) sp(p) ⊕ sp(q)

n(n – 1) n(n + 1) 4pq

[n/2] n q

where A ≅ sl(n, C)R , BD ≅ so(n, C)R , C ≅ sp(n, C)R , are the classical complex algebras considered as real Lie algebras.

Table 3.2: Exceptional Noncompact Simple Lie Algebras G

K

dimR Q

dimR A0

E,F,G ≅ G˜ EI ≅ E6′ ≅ E6(6) EII ≅ E6′′ ≅ E6(2) EIII ≅ E6′′′ ≅ E6(–14) EIV ≅ E6iv ≅ E6(–26) EV ≅ E7′ ≅ E7(7) EVI ≅ E7′′ ≅ E7(–5) EVII ≅ E7′′′ ≅ E7(–25) EVIII ≅ E8′ ≅ E8(8) EIX ≅ E8′′ ≅ E8(–25) FI ≅ F4′ ≅ F4(4) FII ≅ F4′′ ≅ F4(–20) GI ≅ G′2 ≅ G2(2)

Gk sp(4) su(6) ⊕ su(2) so(10) ⊕ so(2) f4 su(8) so(12) ⊕ su(2) e6 ⊕ so(2) so(16) e7 ⊕ su(2) sp(3) ⊕ su(2) so(9) su(2) ⊕ su(2)

dimC G˜ = dimR Gk 42 40 32 26 70 64 54 128 54 28 16 8

rankC G˜ 6 4 2 2 7 4 3 8 4 4 1 2

where E ≅ (E n )R , F ≅ (F 4 )R , G ≅ (G2 )R . Here we have used the three kinds of notation that appear in the literature. The split real forms are E ′n , F ′4 , G′2 .

3.2 Classification of Noncompact Semisimple Lie Algebras

99

Here we have used the three kinds of notation that appear in the literature. The split real forms are E′n , F ′4 , and G′2 . Example:

su(2, 1) ⎛

1 ⎜ "2 ≡ ⎝0 0

0 0 1

⎞ 0 ⎟ 1⎠ 0

(3.13)

su(2, 1) ≐ { X ∈ gl(3, C) | X † " + " X = 0 , tr X = 0 }.

(3.14)

((X) ≐ "X" ⎞ ia z z ⎟ ⎜ K : ((X) = X ⇒ X = ⎝–¯z ia ib ⎠ , a, b ∈ R, z ∈ C; –¯z ib –2ia ⎞ ⎛ 0 v v ⎜ ⎟ Q : ((X) = –X ⇒ X = ⎝ v¯ d ic ⎠ , c, d ∈ R, v ∈ C; –¯v –ic –d ⎛ ⎞ 0 0 0 ⎜ ⎟ A 0 : D = ⎝0 d 0 ⎠ , d ∈ R; 0 0 –d ⎞ ⎛ ia 0 0 ⎟ ⎜ M0 : H = ⎝ 0 ia 0 ⎠ , a ∈ R. 0 0 –2ia ⎛

⎛

0 ⎜ N0+ : [D, X] = +(D)X ⇒ X = ⎝–¯v 0

0 0 0

⎞ v ⎟ ic⎠ , 0

+(D) = 2 ←→ c ∈ R, (mult.1), +(D) = 1 ←→ v ∈ C, (mult.2). ⎛

0 ⎜ – N0 : [D, X] = +(D)X ⇒ X = ⎝ 0 –¯z

(3.15)

z 0 ib

⎞ 0 ⎟ 0⎠ , 0

+(D) = –2 ←→ b ∈ R, (mult.1), +(D) = –1 ←→ z ∈ C, (mult.2).

(3.16)

100

3 Real Semisimple Lie Algebras

3.3 Parabolic Subalgebras Next we recall the minimal Bruhat decomposition (3.10). The subalgebra P0 ≐ M0 ⊕ A0 ⊕ N0– is called a minimal parabolic subalgebra of G. (Note that we may take equivalently N0+ instead of N0– .) We mention what happens in the case of the split real forms. For these we can use the same basis as for the corresponding complex simple Lie algebra G C , but over R. Restricting C → R one obtains the Bruhat decomposition of G (with M0 = 0) from the triangular decomposition of G C = G + ⊕ HC ⊕ G – and obtains the minimal parabolic subalgebras P0 from the Borel subalgebra B = HC ⊕ G + (or G – instead of G + ). Furthermore, in this case dimR K = dimR N0± . A standard parabolic subalgebra is any subalgebra P of G containing P0 . The number of standard parabolic subalgebras, including P0 and G, is 2r . Remark 2: In the complex case a standard parabolic subalgebra is any subalgebra P of G C containing B. The number of standard parabolic subalgebras, including B and G C , is 2 , = rankC G. ♢ Thus, if r = 1 the only nontrivial parabolic subalgebra is P0 . Thus, further in this section r > 1. Any standard parabolic subalgebra is of the form P = M ⊕ A ⊕ N–

(3.17)

so that M ⊇ M0 , A ⊆ A0 , N – ⊆ N0– ; M is the centralizer of A in G (mod A); N – is comprised from the negative root spaces of the restricted root system BA of (G, A). The decomposition (3.17) is called the Langlands decomposition of P. One also has the corresponding Bruhat decomposition (3.10). G = N + ⊕ M ⊕ A ⊕ N –,

(3.18)

where N + = (N – . Remark 3: Using the analogy with the minimal Bruhat decomposition we introduce pseudo-Iwasawa decomposition of G. G = K′ ⊕ A ⊕ N ,

N = N ±,

(3.19)

which is related to the nonminimal Bruhat decomposition analogously (3.18 and 3.11). J : M ⊕ N → K′ , J(X + Y) = X + Y + (Y,

(3.20) X ∈ M, Y ∈ N

♢

3.3 Parabolic Subalgebras

101

The standard parabolic subalgebras may be described explicitly using the restricted simple root system 0A , such that if + ∈ B+A (resp. + ∈ B–A ), one has +=

r

ni +i , +i ∈ BSA ,

all ni ≥ 0 (resp. all ni ≤ 0).

(3.21)

i=1

We shall follow Warner [620], where one may ﬁnd all references to the original mathematical work on parabolic subalgebras. For a short formulation one may say that the parabolic subalgebras correspond to the various subsets of BSA , hence their number 2r . To formalize this let us denote Sr = {1, 2, . . . , r}, and let C denote any subset of Sr . Let B±C ∈ BA denote all positive/negative restricted roots which are linear combinations of the simple restricted roots +i , ∀ i ∈ C. Then a standard parabolic subalgebra corresponding to C will be denoted by PC and is given explicitly as + N + (C) ≐ ⊕ GA .

PC = P0 ⊕ N + (C),

+∈B+C

(3.22)

Clearly, P∅ = P0 and PSr = G since N + (∅) = 0 and N + (Sr ) = N + . Further, we need to bring (3.22) in the form (3.17). First, deﬁne G(C) as the algebra generated by N + (C) and N – (C) ≐ (N + (C). Next, deﬁne A(C) ≐ G(C)∩A, and AC as the orthogonal complement (relative to the Euclidean structure of A) of A(C) in A. Then A = A(C) ⊕ AC . Note that dim A(C) = |C|, dim AC = r – |C|. Next, deﬁne NC+ ≐

⊕

+∈B+A –B+C

+ GA ,

NC– ≐ (NC+ .

(3.23)

Then N ± = N ± (C) ⊕ NC± . Next, deﬁne MC ≐ M ⊕ A(C) ⊕ N + (C) ⊕ N – (C). Then MC is the centralizer of AC in G (mod AC ). Finally, we can derive P0 ⊕ N + (C) = M ⊕ A ⊕ N – ⊕ N + (C) PC = = M ⊕ A(C) ⊕ AC ⊕ N – (C) ⊕ N C– ⊕ N + (C) = M ⊕ A(C) ⊕ N – (C) ⊕ N + (C) ⊕ AC ⊕ NC– MC ⊕ AC ⊕ NC– .

=

(3.24)

Thus, we have rewritten explicitly the standard parabolic PC in the desired form (3.17). The associated (generalized) Bruhat decomposition (3.18) is given now explicitly as G = N + ⊕ P0 = NC+ ⊕ N + (C) ⊕ P0 = NC+ ⊕ PC = NC+ ⊕ MC ⊕ AC ⊕ NC– .

(3.25)

Another important class are the maximal parabolic subalgebras which correspond to C of the form = Sr \{j}, 1 ≤ j ≤ r. Cmax j ) = r – 1, dim ACmax = 1. dim A(Cmax j j

(3.26)

102

3 Real Semisimple Lie Algebras

Facts: For further use, we mention without explanation. – First, we recall the fundamental result of Harish-Chandra [283] that G has discrete series representations iff rank G = rank K. ♢ –

We also recall that by Helgason [289] (G, K) is a Hermitian symmetric pair when the maximal compact subalgebra K contains a u(1) factor. Then G has holomorphic discrete series representations with lowest weight (and also conjugate antiholomorphic discrete series representations with highest weight). ♢

(The discrete series will be deﬁned in Section 4.4.) In the case of nonminimal parabolic M is a noncompact reductive Lie algebra. The parabolic subalgebra P = M ⊕ A ⊕ N is called cuspidal if M has discrete series representations. Obviously, the minimal parabolic subalgebra is cuspidal since M0 is compact.

3.4 Complex Simple Lie Algebras Considered as Real Lie Algebras Let Gc be a complex simple Lie algebra of dimension d and (complex) rank . We need the standard triangular decomposition Gc = N0+ ⊕ H ⊕ N0– .

(3.27)

We have dimC Gc = d, rankC Gc = dimC H = , dimC N0± = (d – )/2. Considered as real Lie algebras we have dimR Gc = 2d, rankR Gc = dimR H = 2, dimR K = d, rankR K = , dimR N0± = d – . Note that the maximal compact subalgebra K of Gc is isomorphic to the compact real form Gk of Gc . Thus, the complex simple Lie algebras do not have discrete series representations (and highest/lowest weight representations over R). Let Hj , j = 1, . . . , , be a basis of H, i.e., H = c.l.s. {Hj , j = 1, . . . , }, (where c.l.s. stands for complex linear span), such that each ad(Hj ) has only real eigenvalues. Let A0 ≐ HR = r.l.s. {Hj , j = 1, . . . , }, where r.l.s. stands for real linear span. Then the Iwasawa decomposition of Gc is G c = K ⊕ A 0 ⊕ N0 ,

N0 = N0± .

(3.28)

The centralizer M0 of A0 in K is given by M0 = u(1) ⊕ ⋯ ⊕ u(1),

factors.

(3.29)

In fact, the basis of M0 consists of the vectors {i Hj , j = 1, . . . , }. The Bruhat decomposition of Gc is Gc = N0+ ⊕ M0 ⊕ A0 ⊕ N0– .

(3.30)

3.5 AI : SL(n, R)

103

Comparing (3.27) and (3.30) we see that H = M0 ⊕ A0 .

(3.31)

The restricted root system (Gc , A0 ) looks the same as the complex root system (Gc , H), but the restricted roots have multiplicity 2, since dimR N0± = 2 dimC N0± . Let C be a string subset of S of length s. The MC factor of the corresponding parabolic subalgebra is MC = Gs ⊕ u(1) ⊕ ⋯ ⊕ u(1),

– s factors,

(3.32)

where Gs is a complex simple Lie algebra of rank s isomorphic to a subalgebra of Gc . Thus, the complex simple Lie algebras, considered as real noncompact Lie algebras, do not have nonminimal cuspidal parabolic subalgebras. The maximal parabolic subalgebras have MC factors as follows: MC = Gi ⊕ u(1),

i = 1, . . . , ,

(3.33)

where Gi is a complex semisimple Lie algebra of rank –1 which may be obtained from Gc by deleting the ith node of the Dynkin diagram of Gc .

3.5 AI : SL(n, R) In this section G = SL(n, R), the group of invertible n × n matrices with real elements and determinant is 1. Then its Lie algebra is G = sl(n, R) and the Cartan involution is given explicitly by (X = – t X, where t X is the transpose of X ∈ G. Thus, the maximal compact subalgebra K ≅ so(n) and is spanned by matrices (r.l.s. stands for real linear span) K = r.l.s.{Xij ≡ eij – eji ,

1 ≤ i < j ≤ n},

(3.34)

where eij are the standard matrices with only nonzero entry (=1) on the ith row and jth column, (eij )k = $ik $j . Note that sl(n, R) does not have discrete series representations if n > 2. Indeed, the rank of sl(n, R) is n – 1, and the rank of its maximal compact subalgebra so(n) is [n/2] and the latter is smaller than n – 1 unless n = 2. Further, the complementary space Q is given by Q = r.l.s.{Yij ≡ eij + eji , 1 ≤ i < j ≤ n, Hj ≡ ejj – ej+1,j+1 ,

1 ≤ j ≤ n – 1}.

(3.35) (3.36)

104

3 Real Semisimple Lie Algebras

The split rank is r = n – 1, and from (3.35) it is obvious that in this setting one has A0 = r.l.s.{Hj ,

1 ≤ j ≤ n – 1 = r}.

(3.37)

Since G is a maximally split real form of G C = sl(n, C), then M0 = 0, and the minimal parabolic subalgebra and the Bruhat decomposition, resp., are given as a Borel subalgebra and triangular decomposition of G C , but over R. G = N0+ ⊕ A0 ⊕ N0– ,

P0 = A0 ⊕ N0– ,

(3.38)

where N0+ , N0– , resp., are upper, lower, triangular, resp. N0+ = r.l.s.{eij , 1 ≤ i < j ≤ n},

N0– = r.l.s.{eij , 1 ≤ j < i ≤ n}.

(3.39)

The simple root vectors are given explicitly by Xj+ ≐ ej,j+1 , Xj– ≐ ej+1,j , 1 ≤ j ≤ n – 1 = r.

(3.40)

Note that matters are arranged so that [Xj+ , Xj– ] = Hj ,

[Hj , Xj± ] = ±2Xj± ,

(3.41)

and further we shall denote by sl(2, R)j the sl(2, R) subalgebra of G spanned by Xj± , Hj . The parabolic subalgebras may be described by the unordered partitions of n. (The parabolic subalgebras may also be described by the various ﬂags of Rn , F., e.g., [620], but we shall not use this description.) Explicitly, let -¯ ≐ {-1 , . . . , -s }, s ≤ n, be a partition of n: sj=1 -j = n. Then the set C corresponding to the partition -¯ and denoted by C(¯-) consists of the numbers of entries -j that are bigger than 1. C(¯-) = { j| -j > 1 }.

(3.42)

Note that in the case s = n all -j are equal to 1. This is the partition -¯ 0 = {1, . . . , 1} corresponding to the empty set: C(¯-0 ) = ∅ (corresponding to the minimal parabolic). Then the factor MC(¯-) in (3.24) and (3.25) is MC(¯-) = ⊕ sl(-j , R) = ⊕ sl(-j , R), 1≤j≤s -j >1

1≤j≤s

sl(1, R) ≡ 0.

(3.43)

Certainly, some partitions give isomorphic (though nonconjugate!) MC(¯-) subalgebras. The parabolic subalgebras in these cases are called associated parabolic subalgebras, and this is an equivalence relation. The parabolic subalgebras up to this equivalence relation correspond to the ordered partitions of n.

3.5 AI : SL(n, R)

105

The most important for us cuspidal parabolic subalgebras correspond to those partitions -¯ = {-1 , . . . , -s } for which -j ≤ 2, ∀ j. Indeed, if some -j > 2 then MC(¯-) will not have discrete series representations since it contains the factor sl(-j , R). A more explicit description of the cuspidal cases is given as follows. It is clear that the cuspidal parabolic subalgebras are in 1-to-1 correspondence with the sequences of r numbers n¯ ≐ { n1 , . . . , nr },

(3.44)

such that nj = 0, 1, and if for ﬁxed j we have nj = 1, then nj+1 = 0 (clearly from the latter follows nj–1 = 0, but we shall use this notation in other contexts as well). In the language above to each nj = 1 there is an entry -j = 2 in -¯ bringing an sl(2, R) factor to MC , i.e.: ¯ = { j | nj = 1, nj+1 = 0 }. C(n)

(3.45)

More explicitly, the cuspidal parabolic subalgebras are given as follows: MC(n) ¯ =

⊕ sl(2, R)jt ,

1≤t≤k

njt = 1,

1 ≤ j1 < j2 < ⋯ < jk ≤ r,

jt < jt+1 – 1.

(3.46)

± have dimensions The corresponding AC(n) ¯ and NC(n) ¯

dim AC(n) ¯ = n – 1 – k,

± 1 dim NC( ¯ = 2 n(n – 1) – k, n)

(3.47)

¯ was introduced in (3.46). where k = |C(n)| Note that the minimal parabolic subalgebra is obtained when all nj = 0, n¯ 0 = {0, . . . , 0}, then C(n¯ 0 ) = ∅, MC(n¯ 0 ) = 0, k = 0. Interlude: The number of cuspidal parabolic subalgebras of sl(n, R), n ≥ 2, including the case P = M = sl(n, R) when n = 2 is equal to F(n + 1), where F(n) and n ∈ Z+ are the Fibonacci numbers. Proof. First we recall that the Fibonacci numbers are determined through the relations F(m) = F(m – 1) + F(m – 2), m ∈ 2 + Z+ , together with the boundary values F(0) = 0, F(1) = 1. We shall count the number of sequences of r numbers ni , introduced above (r = n – 1). Let us denote by N(r) the number of the above-described sequences. Let us divide these sequences into two groups: the ﬁrst with n1 = 1 and the others with n1 = 0. Obviously the number of sequences with n1 = 1 is equal to N(r – 2) since n2 = 0, and then we are left with the above-described sequences but of r–2 numbers. Analogously, the number of sequences with n1 = 0 is equal to N(r – 1) since we are left with all the above-described sequences of r – 1 numbers. Thus, we have proved that N(r) = N(r – 1) + N(r – 2). This is the Fibonacci recursion relation and we have only to adjust

106

3 Real Semisimple Lie Algebras

the boundary conditions. We have N(1) = 2, N(2) = 3, i.e., N(r) = F(r + 2), or in terms of n = r + 1: N(n – 1) = F(n + 1). ∎ For further use we recall that there is explicit formula for the Fibonacci numbers [(n–1)/2]

n xn – (1 – x)n 1–n =2 F(n) = 5s , √ 2s + 1 5 s=0

(3.48)

√ where x is the golden ratio. x2 = x + 1, i.e., x = (1 ± 5)/2. Finally, we mention that the maximal parabolic subalgebras corresponding to C from (3.26) have the following factors: MCj = sl(j, R) ⊕ sl(n – j, R) , 1 ≤ j ≤ n – 1, dim ACj = 1,

dim NC±j

(3.49)

= j(n – j).

(Note that the cases j and n–j are isomorphic, or coinciding when n is even and j = 21 n.) Only one of the maximal ones is cuspidal, namely, for G = sl(4, R), n = 4 and j = 2 we have MC2 = sl(2, R) ⊕ sl(2, R).

(3.50)

3.6 AII : SU∗ (2n) The group G = SU ∗ (2n), n ≥ 2, consists of all matrices in SL(2n, C) which commute with a real skew-symmetric matrix times the complex conjugation operator C.

0 SU (2n) ≐ { g ∈ SL(2n, C) | Jn Cg = gJn C, Jn ≡ –1n ∗

1n }. 0

(3.51)

The Lie algebra G = su∗ (2n) is given by su∗ (2n) ≐ { X ∈ sl(2n, C) | Jn CX = XJn C }

a b ¯ = 0 }. | a, b ∈ gl(n, C) , tr (a + a) = {X = –b¯ a¯

(3.52)

dimR G = 4n2 – 1. We consider n ≥ 2 since su∗ (2) ≅ su(2), and we note that the case n = 2 (of split rank 1) will also appear below: su∗ (4) ≅ so(5, 1) (cf. the corresponding section). The Cartan involution is given by (X = –X † . Thus, the maximal compact subalgebra K ≅ sp(n)

a K ={X = –b†

b | a, b ∈ gl(n, C) , a† = –a , t b = b }. – ta

(3.53)

3.7 AIII : SU(p, r)

107

Note that su∗ (2n) does not have discrete series representations (rank K = n < rank su∗ (2n) = 2n – 1). The complimentary space Q is given by b † t t a | a, b ∈ gl(n, C), a = a , b = –b, tr a = 0 }.

a Q={X = b†

(3.54)

The split rank is n – 1 and the Abelian subalgebra A0 is given explicitly by

a A0 = { X = 0

0 a

| a = diag (a1 , . . . , an ), aj ∈ R ,

tr a = 0 }.

(3.55)

The subalgebras N0± which form the root spaces of the root system (G, A0 ) are of real dimension 2n(n – 1). The subalgebra M0 is given by

a M0 = {X = –b¯

b | a = i diag (61 , . . . , 6n ), 6j ∈ R, –a

(3.56)

b = diag (b1 , . . . , bn ), bj ∈ C} ≅ su(2) ⊕ ⋯ ⊕ su(2), n factors. Claim: All nonminimal parabolic subalgebras of su∗ (2n) are not cuspidal. Proof. Necessarily n > 2. Let C enumerate a connected string of restricted simple roots C = Sij = {i, . . . , j}, where 1 ≤ i ≤ j < n. Then the corresponding subalgebra MC is given by Mij = su∗ (2(s + 1)) ⊕ su(2) ⊕ ⋯ ⊕ su(2) , n – s – 1 factors, s ≡ j – i + 1.

(3.57)

In general C consists of such strings; each string of length s produces a factor su∗ (2(s + 1)), and the rest of MC consists of su(2) factors. ∎ The maximal parabolic subalgebras (cf. (3.26)), 1 ≤ j ≤ n – 1, contain MC subalgebras of the form Mmax = su∗ (2j) ⊕ su∗ (2(n – j)), j

(3.58)

then dim Nj± = 4j(n – j). (For j = 1 or j = n – 1 (3.58) coincides with (3.57) for s = n – 2 (and using su∗ (2) ≅ su(2)).

3.7 AIII : SU(p, r) In this section G = SU(p, r), p ≥ r, which standardly is deﬁned as follows:

1p SU(p, r) ≐ { g ∈ GL(p + r, C) | g "0 g = "0 , "0 ≡ 0 †

0 , det g = 1 }, –1r

(3.59)

108

3 Real Semisimple Lie Algebras

where g † is the Hermitian conjugate of g. We shall also use another realization of G differing from (3.59) by unitary transformation ⎛ 1p–r ⎜ " 0 ↦ "2 ≡ ⎝ 0 0

⎞ 0 ⎟ 1r ⎠ = U"0 U –1 , 0

0 0 1r

⎛

1 1 ⎜ p–r U≡ √ ⎝ 0 2 0

0 1r –1r

⎞ 0 ⎟ 1r ⎠ . 1r

(3.60)

The Lie algebra G = su(p, r) is given by (" = "0 , "2 ) su(p, r) ≐ { X ∈ gl(p + r, C) | X † " + " X = 0, tr X = 0 }.

(3.61)

The Cartan involution is given explicitly by (X = "X". Thus, the maximal compact subgroup K = U(1) × SU(p) × SU(q). The maximal compact subalgebra K = u(1) ⊕ su(p) ⊕ su(r) is given explicitly as (" = "0 )

u1 K ={X = 0

0 u2

| u†j = –uj , j = 1, 2; tr u1 + tr u2 = 0 }.

(3.62)

Note that su(p, r) has discrete series representations since rank K = 1 + rank su(p) + rank su(r) = p + r – 1 = rank su(p, r) and also due to the u(1) factor of K holomorphic discrete series representations (and thus highest/lowest weight representations). For r = 2 it has also quaternionic discrete series due to the normal subgroup SU(2) of K (cf. [270] and Section 4.4). The split rank is equal to r and the Abelian subalgebra A0 may be given explicitly by (" = "2 ) A0 = r.l.s.{ Hju ≡ ep–r+j,p–r+j – ep+j,p+j ,

1 ≤ j ≤ r }.

(3.63)

At this moment we need to consider the cases p = r and p > r separately, since the minimal parabolic subalgebras are different.

3.7.1 Case SU(n, n), n > 1 In this subsection G = su(n, n). We consider n > 1 since su(1, 1) ≅ sl(2, R) was already treated. The subalgebra M0 ≅ u(1) ⊕ ⋯ ⊕ u(1), (n – 1 factors) and is explicitly given as (" = "2 )

u M0 = { X = 0

0 u

| u = i diag (61 , . . . , 6n ), 6j ∈ R ; tr u = 0 }.

(3.64)

3.7 AIII : SU(p, r)

109

The subalgebras N0± which form the root spaces of the root system (G, A0 ) are of real dimension n(2n – 1). The simple root system (G, A0 ) looks as that of the symplectic algebra Cn ; however, the root spaces of the short roots have multiplicity 2. Further, we choose the long root of the Cn simple root system to be !n . Claim: The nontrivial cuspidal parabolic subalgebras are given by C of the form Cj = { j + 1, . . . , n }, 1 ≤ j < n.

(3.65)

Proof. First note that we exclude j = 0 since C0 = Sn . Consider now any C which contains a subset Sij = { i, . . . , j }, where 1 ≤ i ≤ j < n. Then the simple roots corresponding to Sij form a string subset 0ij of the simple root system of An–1 , but each root has multiplicity 2. Because of this multiplicity this string of simple roots will produce a subalgebra sl(j – i + 2, C) of MC . Since the simple Lie algebras sl(n, C) do not have discrete series representations, then PC is not cuspidal. Now note that PCj is cuspidal for all j since MCj ≅ su(n – j, n – j) ⊕ u(1) ⊕ ⋯ ⊕ u(1), j factors,

(3.66) ∎

(cf. the Remark above).

, j = 1, . . . , n, (cf. The maximal parabolic subalgebras correspond to the sets Cmax j (3.26)). The corresponding MC subalgebras are of the form = sl(j, C) ⊕ su(n – j, n – j) ⊕ u(1)1–$nj , Mmax j

(3.67)

where we use the convention: sl(1, C) = 0. We also have dim Nj± = j(4n – 3j). Note that Cmax = C1 and that the only cuspidal maximal parabolic subalgebra is PC1 . 1 3.7.2 Case SU(p, r), p > r ≥ 1 In this subsection G = su(p, r). We also include the case r = 1 although we noted that the case of split rank 1 is clear in general. The subalgebra M0 ≅ su(p – r) ⊕ u(1) ⊕ ⋯ ⊕ u(1) (r factors) and is explicitly given as (" = "2 ) ⎛

up–r ⎜ M0 = { X = ⎝ 0 0

0 u 0

⎞ 0 ⎟ 0⎠ | u†p–r = –up–r , u = i diag (61 , . . . , 6r ), u

6j ∈ R , tr up–r + 2tr u = 0 }.

(3.68) (3.69)

The subalgebras N0± which form the root spaces of the root system (G, A0 ) are of real dimension r(2p – 1). The restricted simple root system (G, A0 ) looks as that of the orthogonal algebra Br ; however, the root spaces of the long roots have multiplicity 2, the

110

3 Real Semisimple Lie Algebras

short simple root, say !r , has multiplicity 2(p – r), and there is also a root 2!r with multiplicity 1. Similarly to the su(n, n) case one can prove that the nontrivial cuspidal parabolic subalgebras are given by C of the form Cj = { j + 1, . . . , r }, 1 ≤ j < r, r > 1.

(3.70)

The corresponding cuspidal parabolic subalgebras contain the subalgebras MCj ≅ su(p – j, r – j) ⊕ u(1) ⊕ ⋯ ⊕ u(1), j factors.

(3.71)

The maximal parabolic subalgebras (cf. (3.26)) contain the MC subalgebras and are of the form Mmax = sl(j, C) ⊕ su(p – j, r – j) ⊕ u(1). j

(3.72)

Thus, the only cuspidal maximal parabolic subalgebra is PC1 . Note also that dim Nj± = j(2p + 2r – 3j).

3.8 BDI : SO(p, r) In this section G = SO(p, r), p ≥ r, which standardly is deﬁned as follows: SO(p, r) ≐

1p g ∈ SO(p + r, C) | g "0 g = "0 , "0 ≡ 0

0 –1r

†

,

(3.73)

where g † is the Hermitian conjugate of g. We shall use also another realization of G differing from (3.73) by unitary transformation ⎛

1p–r ⎜ "0 ↦ " 2 ≡ ⎝ 0 0

0 0 1r

⎞ 0 ⎟ 1r ⎠ = U"0 U –1 , 0

⎛

1 1 ⎜ p–r U≡ √ ⎝ 0 2 0

0 1r –1r

⎞ 0 ⎟ 1r ⎠. 1r

(3.74)

The Lie algebra G = so(p, r) is given by (" = "0 , "2 ) so(p, r) ≐ { X ∈ so(p + r, C) | X † " + " X = 0 }.

(3.75)

The Cartan involution is given explicitly by (X = "X". Thus, K ≅ so(p) ⊕ so(r), and more explicitly is given as (" = "0 )

u1 K={X = 0

0 u2

| u1 ∈ so(p), u2 ∈ so(r) }.

(3.76)

3.8 BDI : SO(p, r)

111

Note that so(5, 1) ≅ su∗ (4), so(4, 2) ≅ su(2, 2), so(3, 3) ≅ sl(4, R), so(4, 1) ≅ sp(1, 1), so(3, 2) ≅ sp(2, R), so(3, 1) ≅ sl(2, C), so(2, 2) ≅ sl(2, R) ⊕ sl(2, R), so(2, 1) ≅ sl(2, R), and (so(1, 1) is Abelian. Thus, below we can restrict to p + r > 4, since the cases p + r = 5 are not treated yet. Note that so(p, r) has discrete series representations except when both p and r are odd numbers, then rank K = rank so(p) + rank so(r) = 21 (p + r – 2) < rank so(p, r) = 1 2 (p + r). It has holomorphic discrete series (and thus highest/lowest weight representations) when p ≥ r = 2 and p = 2, r = 1. It has quaternionic discrete series (cf. Section 4.4) when p ≥ r = 4 and p = 4, r = 1, 2, 3. The split rank is equal to r and the Abelian subalgebra A may be given explicitly by (" = "2 ) A0 = r.l.s.{ Hju ≡ ep–r+j,p–r+j – ep+j,p+j ,

1 ≤ j ≤ r }.

(3.77)

The subalgebra M0 ≅ so(p – r) and is explicitly given as (" = "2 )

u M0 = { X = 0

0 0

| u ∈ so(p – r)}.

(3.78)

The subalgebras N0± which form the root spaces of the root system (G, A0 ) are of real dimension r(p – 1). Except in the case p = r the restricted simple root system (G, A0 ) looks as that of the orthogonal algebra Br ; however, the short simple root, say !r , has multiplicity p – r. Thus, we consider ﬁrst the case p > r > 1. First we note that the parabolic subalgebras given by Cj = { j }, j < r contain a factor: MCj = sl(2, R) ⊕ so(p – r). More generally, if r ∉ C then all possible cuspidal parabolic subalgebras are like those of sl(r, R), adding the compact subalgebra so(p – r). Suppose now that r ∈ C. In that case, C will include a set Cj of the form Cj = { j + 1, . . . , r }, 1 ≤ j < r.

(3.79)

That would bring a MC factor of the form so(p – j, r – j). Thus, all possible cuspidal parabolic subalgebras are obtained for those j, for which the number (p – j)(r – j) is even and for ﬁxed such j they would be like those of sl(j, R), adding the noncompact subalgebra so(p – j, r – j). Clearly, if both p and r are even (odd), then j must also be even (odd), while if one of p or r is even and the other one, i.e., p + r is odd, then j takes all values from (3.79). To be more explicit we ﬁrst introduce the notation n¯ s ≐ {n1 , . . . , ns }, 1 ≤ s ≤ r

(3.80)

(note that n¯ r = n¯ from (3.44)). Then we shall use the notation introduced for the sl(n, R) case, namely, C(n¯ s ) from (3.45). Then the cuspidal parabolic subalgebras are

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3 Real Semisimple Lie Algebras

given by the noncompact factors MC from (3.46)

Ms = MC(n¯ s ) ⊕ so(p – s, r – s) ,

⎧ ⎪ ⎪ ⎨s = 1, 2, . . . , r – 1 : p + r odd; s = 2, 4, . . . , r – 2 : p, r even; ⎪ ⎪ ⎩s = 1, 3, . . . , r – 2 : p, r odd.

(3.81)

Next we note that we can include the case when the second factor in MC is compact by just extending the range of s to r. Thus, all cuspidal parabolic subalgebras of so(p, r) in the case p > r will be determined by the following MC subalgebras:

Ms = MC(n¯ s ) ⊕ so(p – s, r – s),

⎧ ⎪ ⎪ ⎨s = 1, 2, . . . , r s = 2, 4, . . . , r ⎪ ⎪ ⎩s = 1, 3, . . . , r

: p + r odd; : p, r even;

(3.82)

: p, r odd.

The algebras MC have highest/lowest weight representations only when s = r – 2 or s = r, and then the second factor is so(p – r + 2, 2), so(p – r), resp. Finally, we note that the maximal parabolic subalgebras corresponding to (3.26) have MC factors given by = sl(j, R) ⊕ so(p – j, r – j) , Mmax j

j = 1, 2, . . . , r.

(3.83)

Thus, the maximal parabolic subalgebras are cuspidal (and can be found in (3.82)) when j = 1, 2 and the number (p – j)(r – j) is even. In addition, Mmax have j highest/lowest weight representations only when r – j = 0, 2, (or p – j = 2). Note also that dim Nj± = j(2p + 2r – 3j – 1)/2. Now we consider the split cases p = r ≥ 4. (Note that the other split-real cases, i.e., when p = r+1, were considered above without any peculiarities. The split cases p = r < 4 are not representative of the situation and were treated already; so(3, 3) ≅ sl(4, R), so(2, 2) ≅ so(2, 1) ⊕ so(2, 1), so(1, 1) is not semisimple.) We accept the convention that the simple roots !r–1 and !r form the fork of the so(2r, C) simple root system, while !r–2 is the simple root connected to the simple roots !r–3 , !r–1 and !r . Special care is needed only when C includes these four special roots, i.e.: ˆ s ≐ { s, . . . , r }, 1 ≤ s ≤ r – 4. C⊃C

(3.84)

In these cases, we have M factor of the form so(r–s, r–s), i.e., there will be no cuspidal parabolic if r – s is odd. For all other C the parabolic subalgebras would be like those of sl(r, R), when r ∉ C or r – 1 ∉ C, or like those of sl(r – 2, R) with possible addition of one or two sl(2, R) factors, (when r – 2 ∉ C). To describe the latter cases we need a modiﬁcation of

3.8 BDI : SO(p, r)

113

the notation (3.80) ¯ = {j | nj = 1, nj+1 = 0 if j ≠ r – 1}. Co (n)

(3.85)

Thus, the cuspidal parabolic subalgebras are determined by the following MC factors:

MC =

⎧ ⎪ ˆ s, ⎪MC(n¯ s ) ⊕ so(r – s, r – s) : C ⊃ C ⎪ ⎪ ⎪ ⎨ s = 2, 4, . . . , r – 4 : r even, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩M

¯ Co (n)

s = 1, 3, . . . , r – 4 ˆ s. : C ⊃/ C

: r odd,

(3.86)

Only the second subcase, namely, MCo (n) ¯ , has highest/lowest weight representations. The maximal parabolic subalgebras corresponding to (3.26) have MC factors given by ⎧ ⎪ sl(r, R) j = r – 1, r; ⎪ ⎪ ⎪ ⎪ ⎨sl(r – 2, R) ⊕ sl(2, R) ⊕ sl(2, R) j = r – 2; Mmax = j ⎪ sl(r – 3, R) ⊕ sl(4, R) j = r – 3; ⎪ ⎪ ⎪ ⎪ ⎩sl(j, R) ⊕ so(r – j, r – j) j ≤ r – 4.

(3.87)

The corresponding dimensions of N factors are ⎧ ⎪ r(r – 1)/2 j = r – 1, r; ⎪ ⎪ ⎪ ⎪ ⎨(r2 + 3r – 10)/2 j = r – 2; dim Nj± = ⎪(r2 + 5r – 24)/2 j = r – 3; ⎪ ⎪ ⎪ ⎪ ⎩j(4n – 3j + 1)/2 j ≤ r – 4.

(3.88)

Thus, the maximal parabolic subalgebras which are cuspidal occur for j = 1 and odd r ≥ 5 (4th case) or j = 2 and either r = 4 (2nd case) or even r ≥ 6 (4th case) Mmax = so(r – 1, r – 1), r = 5, 7, . . . 1 sl(2, R) ⊕ sl(2, R) ⊕ sl(2, R) r = 4 = Mmax 2 sl(2, R) ⊕ so(r – 2, r – 2) r = 6, 8, . . .

(3.89)

Of these, only Mmax for r = 4 has highest/lowest weight representations (it belongs to 2 the second subcase of (3.86)).

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3 Real Semisimple Lie Algebras

3.9 CI : Sp(n, R), n > 1 In this section G = Sp(n, R), the split real form of Sp(n, C). Both are standardly deﬁned by Sp(n, F) ≐ { g ∈ GL(2n, F)| t gJn g = Jn , det g = 1 },

F = R, C.

(3.90)

Correspondingly, the Lie algebras are given by sp(n, F) = {X ∈ gl(2n, F)| t XJn + Jn X = 0}.

(3.91)

Note that dimF sp(n, F) = n(2n + 1). The general expression for X ∈ sp(n, F) is

A X= C

B , A, B, C ∈ gl(n, F), t B = B, t C = C . – tA

(3.92)

A basis of the Cartan subalgebra H of sp(n, C) is

Ai Hi = 0

A′n Hn = 0

0 , i = 1, . . . , n – 1, Ai = diag(0, . . . 0, 1, –1, 0, . . . , 0), –Ai 0 , A′n = (0, . . . , 0, 2). (3.93) –A′n

The same basis over R spans the subalgebra A0 of G = sp(n, R), since rankF sp(n, F) = n. Note that sp(2, R) ≅ so(3, 2), sp(1, R) ≅ sl(2, R). The maximal compact subalgebra of G = sp(n, R) is K ≅ u(n), thus sp(n, R) has holomorphic discrete series representations (and thus highest/lowest weight representations). Explicitly,

A K = {X = 0

0 – tA

| A ∈ u(n)}.

(3.94)

The subalgebras N0± which form the root spaces of the root system (G, A0 ) are of real dimension n2 . Further, we choose the long root of Cn simple root system to be !n . The parabolic subalgebras corresponding to C such that n ∉ C are the same as the parabolic subalgebras of sl(n, R). The parabolic subalgebras corresponding to C such that n ∈ C contain a string C′s = {s + 1, . . . , n}. This string brings in MC a factor sp(n – s, R), which has discrete series representations. Thus, cuspidality depends on the rest of the possible choices and are the same as the parabolic subalgebras of sl(j, R). Thus, we have Cs = C(n¯ s–1 ) ∪ C′s ,

s = 1, . . . , n,

(3.95)

3.10 CII : Sp( p, r)

115

with the convention that C(n¯ 0 ) = ∅, C′n = ∅. Then the MC factors of the cuspidal parabolic subalgebras of sp(n, R) are given as follows: MCs = MC(n¯ s–1 ) ⊕ sp(n – s, R),

s = 1, . . . , n.

(3.96)

The minimal parabolic subalgebra for which MC = 0 is obtained for s = n and then MC(n¯ n–1 ) enumerates all cuspidal parabolic subalgebras of sl(n, R), including the minimal case MC = 0. The maximal parabolic subalgebras (cf. (3.26)), 1 ≤ j ≤ n, contain MC subalgebras of the form = sl(j, R) ⊕ sp(n – j, R), Mmax j

(3.97)

i.e., the only maximal cuspidal are those for j = 1, 2. We also have dim Nj± = j(4n – 3j + 1)/2.

3.10 CII : Sp(p, r) In this section G = Sp(p, r), p ≥ r, which standardly is deﬁned as follows:

"0 Sp(p, r) ≐ {g ∈ Sp(p + r, C) | g #0 g = #0 }, #0 = 0 †

0 , "0

(3.98)

and correspondingly the Lie algebra G = sp(p, r) is given by sp(p, r) ≐ { X ∈ sp(p + r, C)| X † #0 + #0 X = 0 }.

(3.99)

Note that Sp(p, 0) deﬁnes the compact group Sp(p). The Cartan involution is given explicitly by (X = #0 X#0 . Thus, the maximal compact subgroup is K = Sp(p) × Sp(r), while the maximal compact subalgebra K ≅ sp(p) ⊕ sp(r). Thus, G has discrete series representations. For p ≥ r = 1 it has quaternionic discrete series due to the normal subgroup Sp(1) ≅ SU(2) of K. More explicitly ⎛

u1 ⎜ 0 ⎜ K = {X =⎜ † ⎝–v1 0

0 u2 0 –v2†

v1 0 – t u1 0

⎞ 0 v2 ⎟ ⎟ ⎟| 0 ⎠ – t u2

u1 ∈ u(p), u2 ∈ u(r),t v1 = v1 ,t v2 = v2 }.

(3.100)

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3 Real Semisimple Lie Algebras

The split rank is equal to r and the Abelian subalgebra A0 may be given explicitly by A0 = r.l.s.{Hjs ≡ ep–r+j,p–r+j – ep+j,p+j – e2p+j,2p+j + e2p+r+j,2p+r+j ,

(3.101)

1 ≤ j ≤ r}. The subalgebras N0± which form the root spaces of the root system (G, A0 ) are of real dimension r(4p – 1). The subalgebra M0 ≅ sp(p – r) ⊕ sp(1) ⊕ ⋯ ⊕ sp(1), r factors. Here, just for a moment we distinguish the cases p > r and p = r, since the restricted root systems are different. When p > r the restricted simple root system (G, A) looks as that of Br ; however, the short simple root, say !r , has multiplicity 4(p – r), and the long roots have multiplicity 4. When p = r > 1 the restricted simple root system (G, A) looks as that of Cr ; however, the long root, say !r , has multiplicity 3, and the short roots have multiplicity 4. (We consider r > 1 since sp(1, 1) ≅ so(4, 1).) In spite of these differences from now on we can consider the two subcases together, i.e., we take p ≥ r. There are two types of parabolic subalgebras depending on whether r ∉ C or r ∈ C. Let r ∉ C. Then the parabolic subalgebras are like those of su∗ (2n). Let C enumerate a connected string of restricted simple roots C = Sij = { i, . . . , j }, where 1 ≤ i ≤ j < r. Then the corresponding subalgebra MC is given by Mij = su∗ (2(s + 1)) ⊕ sp(1) ⊕ ⋯ ⊕ sp(1), r – s – 1 factors, s ≡ j – i + 1.

(3.102)

In general C consists of such strings, each string of length s produces a factor su∗ (2(s + 1)), and the rest of MC consists of sp(1) ≅ su(2) factors. All these parabolic subalgebras are not cuspidal. Let r ∈ C and consider the various strings containing r Cj = { j + 1, . . . , r }, 1 ≤ j < r.

(3.103)

The corresponding factor in MC is given by algebra sp(p – j, r – j) which has discrete series representations. If C contains in addition some other string then it would bring some su∗ (2(s + 1)) factor and the corresponding MC will not have discrete series representations. Thus, the nontrivial cuspidal parabolic subalgebras are given by Cj from (3.103) and the corresponding MC is Mj ≅ sp(p – j, r – j) ⊕ sp(1) ⊕ ⋯ ⊕ sp(1),

j factors.

(3.104)

All these Mj do not have highest/lowest weight representations. The other factors in the cuspidal parabolic subalgebras have dimensions dim Aj = j,

3.11 DIII : SO∗ (2n)

117

dim Nj± = j(4p + 4r - 4j - 1). Extending the range of j we include the minimal parabolic subalgebra for j = r and the case M = P = G for j = 0. The maximal parabolic subalgebras corresponding to (3.26) have MC factors given by Mmax = su∗ (2j) ⊕ sp(p – j, r – j), 1 ≤ j ≤ r. j

(3.105)

The N ± factors in the maximal parabolic subalgebras have dimensions dim(Nj± )max = j(4p+4r–6j+1). The only cuspidal maximal parabolic subalgebra is PC1 , using su∗ (2) ≅ su(2) ≅ sp(1) and noting that (3.104) and (3.105) coincide for j = 1.

3.11 DIII : SO∗ (2n) The group G = SO∗ (2n) consists of all matrices in SO(2n, C) which commute with a real skew-symmetric matrix times the complex conjugation operator C: SO∗ (2n) ≐ { g ∈ SO(2n, C) | Jn Cg = gJn C}.

(3.106)

The Lie algebra G = so∗ (2n) is given by so∗ (2n) ≐ { X ∈ so(2n, C) | Jn CX = XJn C}

a b | a, b ∈ gl(n, C), t a = –a, b† = b . = X= –b¯ a¯

(3.107)

dimR G = n(2n – 1), rank G = n. Note that so∗ (8) ≅ so(6, 2), so∗ (6) ≅ su(3, 1), so∗ (4) ≅ so(3) ⊕ so(2, 1), and so∗ (2) ≅ so(2). Further, we can restrict to n ≥ 4 since the other cases are not representative. The Cartan involution is given by (X = – X † . Thus, the maximal compact subalgebra K ≅ u(n)

a K= X= –b

b t † ¯ ¯ | a, b ∈ gl(n, C), a = –a = –a, b = b = b , a

(3.108)

and so∗ (2n) has holomorphic discrete series representations (and thus highest/lowest weight representations). The complimentary space Q is given by

a Q= X= b

b ¯ b† = b = –b¯ . | a, b ∈ gl(n, C) ,t a = –a = a, –a

dimR Q = n(n – 1). The split rank is r ≡ [n/2].

(3.109)

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3 Real Semisimple Lie Algebras

The subalgebras N0± which form the root spaces of the root system (G, A0 ) are of real dimension n(n – 1) – [n/2]. Here, just for a moment we distinguish the cases n even and n odd since the subalgebras M0 and the restricted root systems are different. For n = 2r the split rank is equal to r ≥ 2, and the restricted root system is as that of Cr , but the short roots have multiplicity 4, while the long simple root !r has multiplicity 1. The subalgebra M0 ≅ so(3) ⊕ ⋯ ⊕ so(3), r factors. For n = 2r + 1 the split rank is equal to r ≥ 2, and the restricted root system is as that of Br , but all simple roots have multiplicity 4, and there is a restricted root 2!r of multiplicity 1, where !r is the short simple root. The subalgebra M0 ≅ so(2) ⊕ so(3) ⊕ ⋯ ⊕ so(3), r factors. In spite of these differences from now on we can consider the two subcases together. There are two types of parabolic subalgebras depending on whether r ∉ C or r ∈ C. If r ∉ C then the parabolic subalgebras are like those of su∗ (2r), and they are not cuspidal. Let r ∈ C and consider the various strings containing r Cj = { j + 1, . . . , r } , 1 ≤ j < r = [n/2].

(3.110)

The corresponding factor in MC is given by the algebra so∗ (2n – 4j) which has discrete series representations. Thus, all cuspidal parabolic subalgebras are enumerated by (3.110) and are Mj = so∗ (2n – 4j) ⊕ so(3) ⊕ ⋯ ⊕ so(3),

j factors, j = 1, . . . , r – 1.

(3.111)

All these Mj have highest/lowest weight representations. The other factors in the cuspidal parabolic subalgebras have dimensions dim Aj = j, dim Nj± = j(4n – 4j – 3). Extending the range of j we include the minimal parabolic case for j = r = [n/2], and the case M = P = G for j = 0 which is also cuspidal. The maximal parabolic subalgebras enumerated by Cmax from (3.26) have MC j factors as follows: = so∗ (2n – 4j) ⊕ su∗ (2j), j = 1, . . . , r. Mmax j

(3.112)

The N ± factors in the maximal parabolic subalgebras have dimensions dim(Nj± )max = j(4n – 6j – 1). Only the case j = 1 is cuspidal, noting that Mmax coincides with M1 from 1 ∗ (3.111), (su (2) ≅ su(2) ≅ so(3)).

3.12 Real Forms of the Exceptional Simple Lie Algebras

119

3.12 Real Forms of the Exceptional Simple Lie Algebras We start with the real forms of the exceptional simple Lie algebras. Here we cannot be so explicit with the matrix realizations. To compensate this we use Satake diagrams [538, 620], which we have omitted until now. A Satake diagram has as a starting point the Dynkin diagram of the corresponding complex form (as given in Table FIN of Section 2.4 or [59]). For a split real form it remains the same [620]. In the other cases some dots are painted in black – these black dots considered by themselves are Dynkin diagrams of the compact semisimple factors M of the minimal parabolic subalgebras. Further, there are arrows connecting some nodes which use the Z2 symmetry of some Dynkin diagrams. Then the reduced root systems are described by Dynkin–Satake diagrams which are obtained from the Satake diagrams by dropping the black nodes, identifying the arrow-related nodes, and adjoining all nodes in a connected Dynkinlike diagram, but in addition noting the multiplicity of the reduced roots (which is in general different from 1). More details can be seen in [620], and we have tried to make the exposition transparent (by repeating things).

3.12.1 EI : E6′ The split real form of E6 is denoted as E6′ , sometimes as E6(6) . The maximal compact subgroup is K ≅ sp(4), dimR Q = 42, dimR N0± = 36. This real form does not have discrete series representations. Since this is a split real form the Dynkin–Satake diagram is as the E6 Dynkin diagram in Table FIN of Section 2.4. Taking into account the enumeration of simple roots there the cuspidal parabolic subalgebras have MC factors as follows:

MC =

⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ sl(2, R)j , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪sl(2, R)j ⊕ sl(2, R)k , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

C = ∅, minimal C = {j}, j = 1, . . . , 6, C = {j, k} : j + 1 < k, {j, k} ≠ {2, 4},

(j, k) = (1, 2), (2, 3), ⎪ ⎪ ⎪ ⎪ sl(2, R)j ⊕ sl(2, R)k ⊕ sl(2, R) , C = {j, k, }, (j, k, ) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (1, 2, 5), (1, 2, 6), (1, 4, 6), ⎪ ⎪ ⎪ ⎪ ⎪ (2, 3, 5), (2, 3, 6), m ⎪ ⎪ ⎪ ⎩ so(4, 4) , C = {2, 3, 4, 5}.

(3.113)

where sl(2, R)j denotes the sl(2, R) subalgebra of G spanned by Xj± , Hj (using the same notation as in the section on sl(n, R)). All these MC , except the last case (so(4, 4)), have highest/lowest weight representations.

120

3 Real Semisimple Lie Algebras

The dimensions of the other factors are, respectively,

dim AC =

⎧ ⎪ 6 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨5 ,

4 ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎩ 2

dim NC± =

⎧ ⎪ 36 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨35 34 ⎪ ⎪ ⎪ ⎪ 33 ⎪ ⎪ ⎪ ⎩ 24

(3.114)

Taking into account (3.26) the maximal parabolic subalgebras are determined by Mmax ≅ Mmax ≅ so(5, 5), 1 6 Mmax 2 Mmax 3 Mmax 4

dim(NC± )max = 16,

(3.115)

dim(NC± )max = 21,

≅ sl(6, R),

dim(NC± )max = 25,

≅ Mmax ≅ sl(5, R) ⊕ sl(2, R), 5

dim(NC± )max = 29.

≅ sl(3, R) ⊕ sl(3, R) ⊕ sl(2, R),

Clearly, no maximal parabolic subalgebra is cuspidal.

3.12.2 EII : E6′′ Another real form of E6 is denoted as E6′′ , sometimes as E6(2) . The maximal compact subgroup is K ≅ su(6) ⊕ su(2), dimR Q = 40, dimR N0± = 36. This real form has quaternionic discrete series representations. The split rank is equal to 4, while M0 ≅ u(1) ⊕ u(1). The Satake diagram is ○!2 | ○ ––– ○ ––– ○ ––– ○ ––– ○ !1 !3 !4 !5 !6 # $% & # $% &

(3.116)

Thus, the reduced root system is presented by a Dynkin–Satake diagram which looks similar to the F4 Dynkin diagram ○ ––– ○ ⇒ ○ ––– ○

+1

+2

+3

+4

(3.117)

but the short roots have multiplicity 2 (the long – multiplicity 1). It is obtained from (3.116) by identifying !1 and !6 and mapping them to +4 , identifying !3 and !5 and mapping them to +3 , while the roots !2 and !4 are mapped to the F4 -like long simple roots +1 , +2 , resp.

3.12 Real Forms of the Exceptional Simple Lie Algebras

121

Using the above enumeration of F4 simple roots we give MC factors of all parabolic subalgebras ⎧ u(1) ⊕ u(1), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪sl(2, R)j ⊕ u(1) ⊕ u(1), ⎪ ⎪ ⎪ ⎪ ⎪ sl(2, C)j ⊕ u(1), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪sl(3, R) ⊕ u(1) ⊕ u(1), ⎪ ⎪ ⎪ ⎪ ⎪ sl(2, R)1 ⊕ sl(2, C)j ⊕ u(1), ⎪ ⎪ ⎪ ⎪ ⎨su(2, 2) ⊕ u(1), MC = ⎪ ⎪sl(2, R)2 ⊕ sl(2, C)4 ⊕ u(1), ⎪ ⎪ ⎪ ⎪ ⎪ sl(3, C), ⎪ ⎪ ⎪ ⎪ ⎪ so(4, 4) ⊕ u(1), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ sl(3, R) ⊕ u(1) ⊕ sl(2, C)4 , ⎪ ⎪ ⎪ ⎪ ⎪ sl(2, R)1 ⊕ sl(3, C), ⎪ ⎪ ⎪ ⎩ su(3, 3),

C = ∅, minimal C = {j}, j = 1, 2, C = {j}, j = 3, 4, C = {1, 2}, C = {1, j}, j = 3, 4, C = {2, 3},

(3.118)

C = {2, 4}, C = {3, 4}, C = {1, 2, 3}, C = {1, 2, 4}, C = {1, 3, 4}, C = {2, 3, 4}.

The dimensions of the other factors are, respectively, ⎧ 4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎨2 dim AC = ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎩ 1

,

⎧ 36 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 35 ⎪ ⎪ ⎪ ⎪ ⎪ 34 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 33 ⎪ ⎪ ⎪ ⎪ ⎪ 33 ⎪ ⎪ ⎪ ⎪ ⎨30 dim NC± = ⎪ 33 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 30 ⎪ ⎪ ⎪ ⎪ ⎪ 24 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 31 ⎪ ⎪ ⎪ ⎪ ⎪ 29 ⎪ ⎪ ⎪ ⎩ 21

.

(3.119)

The maximal parabolic subalgebras are given the last four lines in the above lists corresponding to Cmax , j = 4, 3, 2, 1 (cf. (3.26)). j

122

3 Real Semisimple Lie Algebras

The cuspidal parabolic subalgebras are those containing

MC =

⎧ ⎪ u(1) ⊕ u(1), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨sl(2, R)j ⊕ u(1) ⊕ u(1), su(2, 2) ⊕ u(1), ⎪ ⎪ ⎪ ⎪ su(3, 3), ⎪ ⎪ ⎪ ⎩ so(4, 4) ⊕ u(1)

C = ∅, minimal C = {j}, j = 1, 2, C = {2, 3},

(3.120)

C = {2, 3, 4}, C = {1, 2, 3}.

All these MC , except the last case, have highest/lowest weight representations.

3.12.3 EIII : E6′′′ Another real form of E6 is denoted as E6′′′ , sometimes as E6(–14) . The maximal compact subgroup is K ≅ so(10) ⊕ so(2), dimR Q = 32, dimR N0± = 30. This real form has holomorphic discrete series representations (and thus highest/lowest weight representations). The split rank is equal to 2, while M0 ≅ so(6) ⊕ so(2). The Satake diagram is ○!2 | ○ ––– ● ––– ● ––– ● ––– ○ !1 !3 !4 !5 !6 # $% &

(3.121)

Thus, the reduced root system is presented by a Dynkin–Satake diagram which looks similar to the B2 Dynkin diagram but the long roots (incl. +1 ) have multiplicity 6, while the short roots (incl. +2 ) have multiplicity 8, and there are also the roots 2+ of multiplicity 1, where + is any short root. It is obtained from (3.121) by dropping the black nodes (they give rise to M), identifying !1 and !6 and mapping them to +2 , while the root !2 is mapped to the long simple root +1 . The nonminimal parabolic subalgebras are given by MC =

so(7, 1) ⊕ so(2), su(5, 1),

C = {1} C = {2}

,

dim NC± =

24 21

.

(3.122)

Both are maximal (dim AC = 1) and the 2nd is cuspidal.

3.12.4 EIV : E6iv Another real form of E6 is denoted as E6iv , sometimes as E6(–26) . The maximal compact subgroup is K ≅ f4 , dimR Q = 26, dimR N0± = 24. This real form does not have discrete series representations. The split rank is equal to 2, while M0 ≅ so(8).

3.12 Real Forms of the Exceptional Simple Lie Algebras

123

The Satake diagram is ●!2 | ○ ––– ● ––– ● ––– ● ––– ○

!1

!3

!4

!5

!6

(3.123)

Thus, the reduced root system is presented by a Dynkin–Satake diagram which looks similar to the A2 Dynkin diagram but all roots have multiplicity 8. It is obtained from (3.123) by dropping the black nodes, while !1 , !6 , resp., are mapped to the A2 -like simple roots +1 , +2 . The two nonminimal parabolic subalgebras are isomorphic and given by MC = so(9, 1),

C = { j}, j = 1, 2,

dim NC± = 16.

(3.124)

Both are maximal (dim AC = 1) and not cuspidal. 3.12.5 EV : E7′ The split real form of E7 is denoted as E7′ , sometimes as E7(7) . The maximal compact subgroup K ≅ su(8), dimR Q = 70, dimR N0± = 63. This real form has discrete series representations. The Dynkin–Satake diagram is similar to the E7 Dynkin diagram in Table FIN of Section 2.4. Taking into account the above enumeration of simple roots the cuspidal parabolic subalgebras have MC factors as follows: ⎧ ⎪ 0 C = ∅, minimal ⎪ ⎪ ⎪ ⎪ ⎪ C = {j}, j = 1, . . . , 7; sl(2, R)j , ⎪ ⎪ ⎪ ⎪ ⎪ sl(2, R)j ⊕ sl(2, R)k , C = {j, k} : j + 1 < k; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ {j, k} ≠ {2, 4}; ⎪ ⎪ ⎪ ⎪ ⎪ {j, k} = {1, 2}, {, 3}; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ sl(2, R)j ⊕ sl(2, R)k ⊕ sl(2, R) , C = {j, k, } = {1, 2, 5}; ⎪ ⎪ ⎪ ⎪ ⎪ {1, 2, 6}, {1, 2, 7}, {1, 4, 6}; ⎪ ⎪ ⎪ ⎪ ⎨ {1, 4, 7}, {1, 5, 7}, {2, 3, 5}; MC = ⎪ {2, 3, 6}, {2, 3, 7}; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ {2, 5, 7}, {3, 5, 7}; ⎪ ⎪ ⎪ ⎪ ⎪ sl(2, R)1 ⊕ sl(2, R)2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⊕sl(2, R)5 ⊕ sl(2, R)7 , ⎪ ⎪ ⎪ ⎪ ⎪ sl(2, R)2 ⊕ sl(2, R)3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⊕sl(2, R)5 ⊕ sl(2, R)7 , ⎪ ⎪ ⎪ ⎪ ⎪ so(4, 4), C = {2, 3, 4, 5}; ⎪ ⎪ ⎪ ⎩ so(6, 6), C = {2, 3, 4, 5, 6, 7}.

(3.125)

124

3 Real Semisimple Lie Algebras

All these MC , except the last two cases (so(4, 4), so(6, 6)), have highest/lowest weight representations. The dimensions of the other factors are, respectively,

dim AC =

⎧ ⎪ 7 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 6 ⎪ ⎪ ⎪ ⎪ ⎪ 5 ⎪ ⎪ ⎪ ⎪ ⎨4 ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎩1

,

dim NC± =

⎧ ⎪ 63 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 62 ⎪ ⎪ ⎪ ⎪ ⎪ 61 ⎪ ⎪ ⎪ ⎪ ⎨60 ⎪ 59 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 59 ⎪ ⎪ ⎪ ⎪ ⎪ 51 ⎪ ⎪ ⎪ ⎪ ⎩33

.

(3.126)

Taking into account (3.26) the maximal parabolic subalgebras are determined by Mmax ≅ so(6, 6), 1

dim(NC± )max = 33;

Mmax ≅ sl(7, R), 2

dim(NC± )max = 42;

Mmax ≅ sl(6, R) ⊕ sl(2, R), 3

dim(NC± )max = 47;

Mmax ≅ sl(4, R) ⊕ sl(3, R) ⊕ sl(2, R), 4

dim(NC± )max = 53;

Mmax ≅ sl(5, R) ⊕ sl(3, R), 5

dim(NC± )max = 60;

Mmax ≅ so(5, 5) ⊕ sl(2, R), 6

dim(NC± )max = 42;

Mmax ≅ E6′ , 7

dim(NC± )max = 27.

(3.127)

Clearly, the only maximal cuspidal parabolic subalgebra is the one containing Mmax . 1

3.12.6 EVI : E7′′ Another real form of E7 is denoted as E7′′ , sometimes as E7(–5) . The maximal compact subgroup is K ≅ so(12) ⊕ su(2), dimR Q = 64, dimR N0± = 60. This real form has quaternionic discrete series representations. The split rank is equal to 4, while M0 ≅ su(2) ⊕ su(2) ⊕ su(2). The Satake diagram is ●!2 | ○ ––– ○ ––– ○ ––– ● ––– ○ ––– ●

!1

!3

!4

!5

!6

!7

(3.128)

3.12 Real Forms of the Exceptional Simple Lie Algebras

125

Thus, the reduced root system is presented by a Dynkin–Satake diagram which looks similar to the F4 Dynkin diagram (cf. (3.117)), but the short roots have multiplicity 4 (the long – multiplicity 1). Going to this Dynkin–Satake diagram we drop the black nodes of (3.128) (they give rise to M0 ), while !1 , !3 , !4 , !6 are mapped to +1 , +2 , +3 , +4 , resp., of (3.117). Using the above enumeration of F4 simple roots we shall give MC factors of all parabolic subalgebras ⎧ M = su(2) ⊕ su(2) ⊕ su(2), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ sl(2, R)j ⊕ M, ⎪ ⎪ ⎪ ⎪ ⎪ su∗ (4) ⊕ su(2)j+3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪sl(3, R) ⊕ M, ⎪ ⎪ ⎪ ⎪ ⎪ sl(2, R)1 ⊕ su∗ (4) ⊕ su(2)j+3 , ⎪ ⎪ ⎪ ⎪ ⎨so(6, 2) ⊕ su(2), MC = ⎪ ⎪sl(2, R)2 ⊕ su∗ (4) ⊕ su(2)7 , ⎪ ⎪ ⎪ ⎪ ⎪ su∗ (6), ⎪ ⎪ ⎪ ⎪ ⎪ so(7, 3) ⊕ su(2)6 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ sl(3, R) ⊕ su∗ (4) ⊕ su(2)7 , ⎪ ⎪ ⎪ ⎪ ⎪ sl(2, R)1 ⊕ su∗ (6), ⎪ ⎪ ⎪ ⎩ so∗ (12),

C = ∅, minimal C = {j}, j = 1, 2; C = {j}, j = 3, 4; C = {1, 2} C = {1, j}, j = 3, 4; C = {2, 3}; C = {2, 4};

(3.129)

C = {3, 4}; C = {1, 2, 3}; C = {1, 2, 4}; C = {1, 3, 4}; C = {2, 3, 4}.

The dimensions of the other factors are, respectively, ⎧ 4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎨2 dim AC = ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪2 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎩ 1

,

⎧ 60 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 59 ⎪ ⎪ ⎪ ⎪ ⎪ 56 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 57 ⎪ ⎪ ⎪ ⎪ ⎪ 55 ⎪ ⎪ ⎪ ⎪ ⎨50 dim NC± = ⎪ 55 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪48 ⎪ ⎪ ⎪ ⎪ ⎪ 42 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 53 ⎪ ⎪ ⎪ ⎪ ⎪ 47 ⎪ ⎪ ⎪ ⎩ 33

.

(3.130)

126

3 Real Semisimple Lie Algebras

The maximal parabolic subalgebras are the last four in the above list corresponding to Cmax , j = 4, 3, 2, 1 (cf. (3.26)). j The cuspidal parabolic subalgebras are those containing

MC =

⎧ ⎪ M = su(2) ⊕ su(2) ⊕ su(2), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨sl(2, R)j ⊕ M,

C = ∅, minimal C = {j}, j = 1, 2

⎪ ⎪ so(6, 2) ⊕ su(2), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩so∗ (12),

(3.131)

C = {2, 3} C = {2, 3, 4}

the last one also being maximal. All these MC have highest/lowest weight representations.

3.12.7 EVII : E7′′′ Another real form of E7 is denoted as E7′′′ , sometimes as E7(–25) . The maximal compact subgroup is K ≅ e6 ⊕ so(2), dimR Q = 54, dimR N0± = 51. This real form has holomorphic discrete series representations (and thus highest/lowest weight representations). The split rank is equal to 3, while M0 ≅ so(8). The Satake diagram is ●!2 | ○ ––– ● ––– ● ––– ● ––– ○ ––– ○

!1

!3

!4

!5

!6

!7

(3.132)

Thus, the reduced root system is presented by a Dynkin–Satake diagram which looks similar to the C3 Dynkin diagram ○ ⇒ ○ ––– ○

+1

+2

+3

(3.133)

but the short roots have multiplicity 8 (the long – multiplicity 1). Going to the C3 diagram we drop the black nodes in (3.132) (they give rise to M0 ), while !1 , !6 , !7 are mapped to +1 , +2 , +3 , resp., of (3.133). Using the above enumeration of C3 simple roots we shall give MC factors of all parabolic subalgebras

3.12 Real Forms of the Exceptional Simple Lie Algebras

MC =

⎧ ⎪ so(8), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ so(9, 1), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨sl(2, R)3 ⊕ so(8),

127

C = ∅ , minimal; C = {j}, j = 1, 2; C = {3};

(3.134)

⎪ ⎪ C = {1, 2}; E6iv , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪sl(2, R)3 ⊕ so(9, 1), C = {1, 3}; ⎪ ⎪ ⎪ ⎩so(10, 2), C = {2, 3}.

The dimensions of the other factors are, respectively, ⎧ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎨2 dim AC = ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪1 ⎪ ⎪ ⎪ ⎩ 1

,

⎧ ⎪ 51 ⎪ ⎪ ⎪ ⎪ ⎪ 43 ⎪ ⎪ ⎪ ⎪ ⎨50 dim NC± = ⎪ 27 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪42 ⎪ ⎪ ⎪ ⎩ 33

.

(3.135)

The last three give rise to the maximal parabolic subalgebras. The cuspidal parabolic subalgebras are those containing

MC =

⎧ ⎪ ⎪ ⎪so(8), ⎨

C = ∅, minimal;

sl(2, R)3 ⊕ so(8), C = {3}; ⎪ ⎪ ⎪ ⎩so(10, 2), C = {2, 3}

(3.136)

the last one also being maximal. All these MC have highest/lowest weight representations.

3.12.8 EVIII : E8′ The split real form of E8 is denoted as E8′ , sometimes as E8(8) . The maximal compact subgroup K ≅ so(16), dimR Q = 128, dimR N0± = 120. This real form has discrete series representations. The Dynkin–Satake diagram is same as the E8 Dynkin diagram in Table FIN of Section 2.4. Taking into account the enumeration of simple roots the cuspidal parabolic subalgebras have MC factors as follows:

128

3 Real Semisimple Lie Algebras

MC =

⎧ ⎪ C = ∅, minimal; ⎪0 ⎪ ⎪ ⎪ ⎪ ⎪ C = {j}, j = 1, . . . , 8; sl(2, R)j , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ C = {j, k} : j + 1 < k; sl(2, R)j ⊕ sl(2, R)k , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ {j, k} ≠ {2, 4}; ⎪ ⎪ ⎪ ⎪ ⎪ {j, k} = {1, 2}, {2, 3}; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪sl(2, R)j ⊕ sl(2, R)k ⊕ sl(2, R) , C = {j, k, } = {1, 2, 5}; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ {1, 2, 6}, {1, 2, 7}, {1, 2, 8}; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ {1, 4, 6}, {1, 4, 7}, {1, 4, 8}; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ {1, 5, 7}, {1, 5, 8}, {1, 6, 8}; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ {2, 3, 5}, {2, 3, 6}, {2, 3, 7}; ⎪ ⎪ ⎪ ⎪ ⎪ {2, 3, 8}, {2, 5, 7}, {2, 5, 8}; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ {2, 6, 8}, {3, 5, 7}, {3, 5, 8}; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ {3, 6, 8}, {4, 6, 8}; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪sl(2, R)1 ⊕ sl(2, R)2 ⎪ ⎪ ⎪ ⎨ ⊕ sl(2, R) ⊕ sl(2, R) , 5

7

⎪ ⎪ sl(2, R)1 ⊕ sl(2, R)2 ⎪ ⎪ ⎪ ⎪ ⎪ ⊕ sl(2, R)5 ⊕ sl(2, R)8 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ sl(2, R)1 ⊕ sl(2, R)2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⊕ sl(2, R)6 ⊕ sl(2, R)8 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪sl(2, R)1 ⊕ sl(2, R)4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⊕ sl(2, R)6 ⊕ sl(2, R)8 , ⎪ ⎪ ⎪ ⎪ ⎪ sl(2, R)2 ⊕ sl(2, R)3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⊕ sl(2, R)5 ⊕ sl(2, R)7 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ sl(2, R)2 ⊕ sl(2, R)3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⊕ sl(2, R)5 ⊕ sl(2, R)8 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ sl(2, R)2 ⊕ sl(2, R)3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⊕ sl(2, R)6 ⊕ sl(2, R)8 . ⎪ ⎪ ⎪ ⎪ ⎪ so(4, 4), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ so(6, 6), ⎪ ⎪ ⎪ ⎪ ⎩E′ , 7

(3.137)

C = {2, 3, 4, 5}; C = {2, 3, 4, 5, 6, 7}; C = {1, 2, 3, 4, 5, 6, 7}.

All these MC , except the last three cases (so(4, 4), so(6, 6), E7′ ), have highest/lowest weight representations.

3.12 Real Forms of the Exceptional Simple Lie Algebras

129

The dimensions of the other factors are, respectively,

dim AC =

⎧ ⎪ 8 ⎪ ⎪ ⎪ ⎪ ⎪ 7 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪6 ⎪ ⎪ ⎪ ⎪ ⎪ 5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪4 ⎪ ⎪ ⎪ ⎨4 ⎪ 4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 4 ⎪ ⎪ ⎪ ⎪ ⎪ 4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 4 ⎪ ⎪ ⎪ ⎪ ⎪ 4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎩1

,

dim NC± =

⎧ ⎪ 120 ⎪ ⎪ ⎪ ⎪ ⎪ 119 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪118 ⎪ ⎪ ⎪ ⎪ ⎪ 117 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 116 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪116 ⎪ ⎪ ⎪ ⎨116 ⎪ 116 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 116 ⎪ ⎪ ⎪ ⎪ ⎪ 116 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 116 ⎪ ⎪ ⎪ ⎪ ⎪ 108 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 90 ⎪ ⎪ ⎪ ⎩57

.

(3.138)

Taking into account (3.26) the maximal parabolic subalgebras are determined by Mmax ≅ so(7, 7), 1

dim(NC± )max = 78;

Mmax ≅ sl(8, R), 2

dim(NC± )max = 92;

Mmax ≅ sl(7, R) ⊕ sl(2, R), 3

dim(NC± )max = 98;

Mmax ≅ sl(5, R) ⊕ sl(3, R) ⊕ sl(2, R), 4

dim(NC± )max = 106;

Mmax ≅ sl(5, R) ⊕ sl(4, R), 5

dim(NC± )max = 104;

Mmax ≅ so(5, 5) ⊕ sl(3, R), 6

dim(NC± )max = 97;

Mmax ≅ E6′ ⊕ sl(2, R), 7

dim (NC± )max = 83;

Mmax ≅ E7′ , 8

dim(NC± )max = 57.

(3.139)

Clearly, the only maximal cuspidal parabolic subalgebra is the one containing Mmax 8 .

3.12.9 EIX : E8′′ Another real form of E8 is denoted as E8′′ , sometimes as E8(–24) . The maximal compact subgroup is K ≅ e7 ⊕ su(2), dimR Q = 112, dimR N0± = 108. This real form has quaternionic discrete series representations. The split rank is equal to 4, while M0 ≅ so(8).

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3 Real Semisimple Lie Algebras

The Satake diagram is ●!2 | ○ ––– ● ––– ● ––– ● ––– ○ ––– ○ ––– ○

!1

!3

!4

!5

!6

!7

(3.140)

!8

Thus, the reduced root system is presented by a Dynkin–Satake diagram which is similar to the F4 Dynkin diagram (cf. (3.117)), but the short roots have multiplicity 8 (the long – multiplicity 1). Going to the F4 diagram we drop the black nodes, while !1 , !6 , !7 , !8 are mapped to +4 , +3 , +2 , +1 , resp., of (3.117). Using the above enumeration of F4 simple roots we shall give MC factors of all parabolic subalgebras:

MC =

⎧ ⎪ M = so(8), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ sl(2, R)j ⊕ M, ⎪ ⎪ ⎪ ⎪ ⎪ so(9, 1), ⎪ ⎪ ⎪ ⎪ ⎪ sl(3, R) ⊕ M, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪sl(2, R)1 ⊕ so(9, 1), ⎪ ⎪ ⎨so(10, 2), ⎪ sl(2, R)2 ⊕ so(9, 1), ⎪ ⎪ ⎪ ⎪ ⎪Eiv , ⎪ 6 ⎪ ⎪ ⎪ ⎪ so(11, 3), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ sl(3, R) ⊕ so(9, 1), ⎪ ⎪ ⎪ ⎪ ⎪ sl(2, R)1 ⊕ E6iv , ⎪ ⎪ ⎪ ⎩E′′′ , 7

C = ∅ , minimal C = {j}, j = 1, 2; C = {j}, j = 3, 4; C = {1, 2}; C = {1, j}, j = 3, 4; C = {2, 3};

(3.141)

C = {2, 4}; C = {3, 4}; C = {1, 2, 3}; C = {1, 2, 4}; C = {1, 3, 4}; C = {2, 3, 4}.

The dimensions of the other factors are, respectively, ⎧ 4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎨2 dim AC = ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪2 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎩ 1

,

⎧ 108 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 107 ⎪ ⎪ ⎪ ⎪ ⎪ 100 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 105 ⎪ ⎪ ⎪ ⎪ ⎪ 99 ⎪ ⎪ ⎪ ⎪ ⎨90 dim NC± = ⎪ 99 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪84 ⎪ ⎪ ⎪ ⎪ ⎪ 78 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 97 ⎪ ⎪ ⎪ ⎪ ⎪ 83 ⎪ ⎪ ⎪ ⎩ 57

.

(3.142)

3.12 Real Forms of the Exceptional Simple Lie Algebras

131

The maximal parabolic subalgebras are the last four in the above lists corresponding to Cmax , j = 4, 3, 2, 1 (cf. (3.26)). j The cuspidal ones arise from ⎧ ⎪ so(8), ⎪ ⎪ ⎪ ⎪ ⎨sl(2, R) ⊕ so(8), j MC = ⎪ so(10, 2), ⎪ ⎪ ⎪ ⎪ ⎩E′′′ , 7

C = ∅ , minimal C = {j}, j = 1, 2; C = {2, 3};

(3.143)

C = {2, 3, 4}

the last one also being maximal. All these MC representations.

have highest/lowest weight

3.12.10 FI : F4′ The split real form of F4 is denoted as F4′ , sometimes as F4(4) . The maximal compact subgroup K ≅ sp(3) ⊕ su(2), dimR Q = 28, and dimR N0± = 24. This real form has quaternionic discrete series representations. Taking into account the enumeration of simple roots as in (3.117), the parabolic subalgebras have MC factors as follows: ⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ sl(2, R)j , ⎪ ⎪ ⎪ ⎪ ⎪ sl(3, R)jk , ⎪ ⎪ ⎪ ⎪ ⎨sl(2, R) ⊕ sl(2, R) , j k MC = ⎪ sp(2, R), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ so(4, 3), ⎪ ⎪ ⎪ ⎪ ⎪ sl(3, R) ⊕ sl(2, R), ⎪ ⎪ ⎪ ⎪ ⎩sp(3, R),

C = ∅, minimal C = {j}, j = 1, 2, 3, 4; C = {j, k} = {1, 2}, {3, 4}; C = {j, k} = {1, 3}, {1, 4}, {2, 4}; C = {2, 3};

(3.144)

C = {1, 2, 3}; C = {1, 2, 4}, {1, 3, 4}; C = {2, 3, 4}.

The dimensions of the other factors are, respectively, ⎧ ⎪ 4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎨2 dim AC = ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎩1

,

⎧ ⎪ 24 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 23 ⎪ ⎪ ⎪ ⎪ ⎪ 21 ⎪ ⎪ ⎪ ⎪ ⎨22 dim NC± = ⎪ 20 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 15 ⎪ ⎪ ⎪ ⎪ ⎪ 20 ⎪ ⎪ ⎪ ⎪ ⎩15

.

(3.145)

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3 Real Semisimple Lie Algebras

The maximal parabolic subalgebras are in the last three lines in the above lists corresponding to Cmax , j = 4, 3, 2, 1 (cf. (3.26)). j The cuspidal parabolic subalgebras arise from ⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ sl(2, R)j , ⎪ ⎪ ⎪ ⎪ ⎨sl(2, R) ⊕ sl(2, R) , j k MC = ⎪ sp(2, R), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ sp(3, R), ⎪ ⎪ ⎪ ⎩ so(4, 3),

C = ∅, minimal; C = {j}, j = 1, 2, 3, 4; C = {j, k} = {1, 3}, {1, 4}, {2, 4}; C = {2, 3};

(3.146)

C = {2, 3, 4}; C = {1, 2, 3}

the last two also being maximal. All these MC except the last one have highest/lowest weight representations. 3.12.11 F II : F4′′ Another real form of F4 is denoted as F4′′ , sometimes as F4(–20) . The maximal compact subgroup K ≅ so(9), dimR Q = 16, and dimR N0± = 15. This real form has discrete series representations. The split rank is equal to 1; thus, the minimal and maximal parabolics coincide, while M0 ≅ so(7). The Satake diagram is ● ––– ● ⇒ ● ––– ○

!1

!2

!3

!4

(3.147)

Thus, the reduced root system is presented by a Dynkin–Satake diagram which looks similar to the A1 Dynkin diagram but the roots have multiplicity 8. Going to the A1 Dynkin diagram we drop the black nodes (they give rise to M), while !4 becomes the A1 diagram. 3.12.12 GI : G′2 The split real form of G2 is denoted as G′2 , sometimes as G2(2) . The maximal compact subgroup K ≅ su(2) ⊕ su(2), dimR Q = 8, and dimR N0± = 6. This real form has quaternionic discrete series representations. The nonminimal parabolic subalgebras are given by MC = sl(2, R)j , C = {j}, j = 1, 2,

dim(Nj± )max = 5

(3.148)

They are cuspidal and maximal and have highest/lowest weight representations.

4 Invariant Differential Operators Summary This chapter is the most important one in the book. We explain in detail our canonical procedure for the construction of invariant differential operators based on [122, 126, 127]. The chapter contains standard material based mostly on [59, 120, 310, 547, 624, 639, 644]. We present necessary elements of the theory of Lie groups. Especially important for our procedure are the Cartan subgroups, the Cartan and Iwasawa decompositions, and parabolic subgroups. The latter are explained in parallel to algebraic considerations. We present necessary material on group representation theory: representations, modules, reducibility, irreducibility and induced representations. From the latter the most crucial are the elementary representations (ERs) of G and its Lie algebra G since they are the playing ground for all invariant operators. Besides our main object of invariant differential operators which act on reducible ERs there are important Knapp–Stein integral operators which are defined on any ER. We present the types of unitary irreducible representations. The cornerstone of our construction is the fact that there is a Verma module over G C (the complexification of G) which is associated canonically with each ER. The canonical crucial fact is that to every singular vector of a reducible Verma module there corresponds an invariant differential operator acting from the associated ER. Further we explain another important ingredient of our procedure, namely, the fact that partially equivalent ERs are grouped by the invariant differential operators in multiplets of ERs. These are sets of ERs on which the Casimir operators have the same value. These multiplets are also multiplets of the sets of associated Verma modules. We dedicate one section to the example of SL(2,R) which was developed by Bargmann and independently by Gel’fand et al. in 1947 itself. Finally, we give formulae for our building blocks – the singular vectors. We are able to do this explicitly for all cases of sl(n, C) and for roughly half of the cases of the other complex semisimple Lie algebras.

4.1 Lie Groups 4.1.1 Preliminaries A group G is called a Lie group (over F) if G is an analytical manifold (over F) and the function (g, g ′ ) ↦ gg ′–1 is analytic. The law of multiplication in the neighborhood of e ∈ G is given in local coordinates by analytic vector function f : U × U → F n , where U is a neighborhood of 0 ∈ F n (n = dimG), 0 ∈ F n corresponds to e ∈ G. Furthermore, f (x, 0) = f (0, x) = x, f (f (x, y), z) = f (x, f (y, z)). The function f may be expanded in Taylor expansions, i.e., we write f (x, y) = x + y + :(x, y) + . . . , where :(x, y) is a bilinear map F n × F n → F n , and the dots denote higher-order terms. The above map is an associative multiplication in F n . Thus, F n is a Lie algebra w.r.t. the commutator [x, y] = :(x, y) – :(y, x).

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4 Invariant Differential Operators

Thus, to each Lie group G over F there corresponds a Lie algebra G over F so that dimG = dimG. The Lie algebra is naturally identiﬁed with the tangent space at the point e ∈ G. The algebra G is called the Lie algebra of the Lie group G. 2

Example: The group GL(n, F) is an open set in F n . The local coordinates of g ∈ G are Descartes coordinates of the matrix x = g – 1n . Furthermore f (x, y) = x + y + xy, from where [x, y] = xy – yx. Thus, the Lie algebra of GL(n, F) is gl(n, F).

4.1.2 Classical Groups The groups that we shall use in this book will be linear groups, i.e., groups that are subsets of Matn F, the set of n × n matrices over F. (In Chapter 1 we used the set Matn F as an algebra of matrices.) Below we consider g ∈ Matn F. Following Jacobson [310] we give the conditions on g which distinguish some of the linear groups. (1) general linear group GL(n, F): det g ≠ 0; (2) special linear group SL(n, F): det g = 1; (3) orthogonal group O(n, F): t gg = 1n ; (4) special orthogonal group SO(n, F): t gg = 1n , det g = 1; (5) symplectic group Sp(m, F): t g3m g = 3m , n = 2m; (6) unitary group U(n) over C: g† g = 1n ; (7) special unitary group SU(n): g † g = 1n , det g = 1; (8) compact symplectic group Sp(m): t g3m g = 3m , g ∈ U(2m); (9) pseudoorthogonal group O(p, r) over R: t gsg = s, n = p + r; (10) special pseudoorthogonal group SO(p, r) over R: t gsg = s, n = p + r det g = 1; (11) pseudounitary group U(p, r) over C: g † "0 g = "0 , n = p + r; † (12) special pseudounitary group SU(p, r) over

C: g "0 g= "0 , n = p + r, det g = 1; 0 1n ; (13) SU ∗ (2n): g ∈ SL(2n, C), Jn Cg = gJn C, Jn ≡ –1n 0 (14) SO∗ (2n) | g ∈ SO(2n, C), Jn Cg = gJn C;

"0 0 † , (15) Sp(p, r) : g ∈ Sp(p + r, C), g #0 g = #0 , #0 = 0 "0

0 1m

1m 1p , "0 ≡ 0 0

0 , C is the –1r

where 1k denotes the unit k × k matrix, 3m =

1p 0 . complex conjugation operator and s = sp,r = 0 1r Note that the groups O(n, R), SO(n, R), U(n), SU(n), and Sp(m) are compact. The group SU ∗ (2n) sometimes is called linear quaternionic group SL(n, IH) since it can be realized as n × n matrices with quaternionic entries.

4.1 Lie Groups

135

Further F = C or F = R. All the above classical groups are topological groups in the Euclidean topology (induced by the Euclidean space Matn F). More importantly all of them are Lie groups since they are closed subgroups of GL(n, R) or GL(n, C). The above groups 1–15 are connected except GL(n, R), O(n, F), O(p, r), and SO(p, r). The group GL(n, R) has two components distinguished by det g > 0, det g < 0, the ﬁrst component containing the unit matrix. The group O(n, F) has two components distinguished by det g = 1, (SO(n, F)), and det g = –1. The group O(p, r) has four components distinguished by det g = ±1 and Bp = ±1, where Bp is a principal minor of the matrix g comprised by its ﬁrst p rows and columns. The group SO(p, r) has two components distinguished by Bp = ±1, and the component with Bp = +1 contains the unit element and is denoted by SOe (p, r). The groups SL(n, C), SU(n), and Sp(m, C) are simply connected. The Lie algebras of the groups in the above list are denoted by the same notation with small letters, e.g., gl(n, F), sl(n, F), o(n, F), etc. If a group G′ is the connected component of G, then they have the same Lie algebra, e.g., o(n, F) = so(n, F), o(p, r) = so(p, r). The groups GL(n, C), SL(n, C), SO(n, C), and Sp(n, C) are called classical complex Lie groups. The groups U(n), SU(n), SO(n), and Sp(n) are classical compact Lie groups. The latter are also the maximal compact subgroups of the corresponding complex groups. The groups SL(n, C) (n > 1), SO(n, C) (3 ≤ n ≠ 4), and Sp(n, C) are the classical complex simple Lie groups. 4.1.3 Types of Lie Groups Here F = C or F = R. Let G be a connected Lie group, G its Lie algebra. The group G is called simple, semisimple, reductive, solvable, nilpotent, and Abelian if its Lie algebra is simple, semisimple, reductive, solvable, nilpotent, and Abelian, respectively. Thus, the groups SL(n, F) (n > 1), SO(n, F) (3 ≤ n ≠ 4), Sp(n, F), SU(n) (n > 1), Sp(n), SU ∗ (2n), SO∗ (2n), SU(p, r), SOe (p, r), and Sp(p, r) are simple connected Lie groups. If G is a reductive connected Lie group, G = RS, S is semisimple, the radical R is a connected component of the center of G and is given by R = AT = A × T, where A is simply connected Abelian group (a vector space), T is a compact Abelian group (a torus). The group G is compact iff A = {e} and S is compact. The group G is simply connected iff T = {e} and S is simply connected. The simply connected covering group of a reductive group is reductive. A connected semisimple Lie group G has a discrete center, and the simply connected covering group of G is semisimple. The adjoint group Ad G is locally isomorphic to G and has trivial center (= {e}). If the group G is linear, then its center is ﬁnite. Every complex semisimple Lie group is linear.

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4 Invariant Differential Operators

Each compact Lie group is reductive and has ﬁnite center. Each compact Lie group is linear. 4.1.4 Cartan Subgroups Let G be a semisimple connected Lie group with ﬁnite center and G its Lie algebra. A subgroup H ⊂ G is reductive in G, if its adjoint representation in G is completely reducible. The maximal commutative subgroups H ⊂ G, which are reductive in G, are called Cartan subgroups of G. Each Cartan subgroup is closely connected and is given by exp H where H is a Cartan subalgebra of G. Thus, the Cartan subgroups are determined and classiﬁed by Cartan subalgebras. Each Cartan subgroup is given by H = AT = A × T, where A is simply connected Abelian group (a vector space), T is a maximal compact connected Abelian subgroup of H (a torus). The subgroup A is called vector subgroup of H, the subgroup T is called torus subgroup of H. A Cartan subgroup H = AT is called main Cartan subgroup if its vector part is of maximal dimension. Then A is called the main vector subgroup of G. All main Cartan subgroups are conjugated w.r.t. inner automorphisms. The dimension = dimH of H does not depend on the choice of H and is called the rank of G. The dimension r = dimA of the main vector subgroup A does not depend on the choice of A and is called the real rank of G (or split rank). If G is complex, then all Cartan subgroups H = AT are conjugate and dimA = dimT (equals the rank of G as a complex group). 4.1.5 Cartan and Iwasawa Decompositions The treatment in this subsection and the next parallels the algebraic theory developed earlier. Let G be a noncompact semisimple Lie group with Lie algebra G. Let K denote a maximal compact subgroup of G with Lie algebra K. We recall the Cartan decomposition of G = K ⊕ Q. Let Q = exp Q, then we have the Cartan decomposition of G = KQ = QK. This decomposition is global (exists for every g ∈ G) and is unique up to internal automorphisms. Next we have the Iwasawa decomposition of G = KA0 N0 , where A0 is Abelian simply connected subgroup of G, N0 is a nilpotent simply connected subgroup of G preserved by the action of A0 . The subgroup A0 is isomorphic to the main vector subgroup of G and its Lie algebra is isomorphic to A0 . The Iwasawa decomposition is global (exists for every g ∈ G) and is unique up to internal automorphisms. The subgroup A0 is ﬁxed up to automorphism a ↦ k–1 ak (a ∈ A0 , k ∈ K). The subgroup N0 is determined up to the exchange of B+R ←→ B–R , i.e., changing N0 to the conjugate group N˜ 0 . Finally, one may choose different order of the three factors K, A0 , and N0 (this gives six possibilities).

4.1 Lie Groups

137

The groups K, A0 , and N0 are unimodular. The decomposition g = kan g ∈ G, k ∈ K, a ∈ A0 , and n ∈ N0 induces the following Haar measure: dg = |a|21 dk da dn,

(4.1)

where dk, da, and dn are the normalized Haar measures on K, A0 , N0 , and |a|+ is a character on A0 and is deﬁned as follows: |a|+ = | exp x|+ = e +(x) and 21 is the sum of positive restricted roots B+R . Such decompositions remain valid for real reductive groups as well.

4.1.6 Parabolic Subgroups In representation theory very important role is played by closed subgroups P ⊂ G with compact factor-space G/P. Such subgroups are called parabolic subgroups (see the deﬁnition below). The exposition below parallels to some extent the exposition for parabolic subalgebras in Section 3.3. (1) If G is complex, then the role of minimal parabolic subgroup is played by a Borel subgroup B = exp B, where B is a Borel subalgebra of G. Thus, B = HN = NH, where H is the Cartan subgroup of G, N is a maximal nilpotent subgroup of G. Note that B is maximal solvable group of G. The set G0 = NH N˜ is dense in G and is called the Gauss decomposition of G0 . Over C it parallels the triangular decomposition of the complex Lie algebra G. (2) In the general case let A0 be a main vector subgroup of G. Further, let M0 be the centralizer of A0 in K. Then the subgroup P0 = M0 A0 N0 is a minimal parabolic subgroup of G. A parabolic subgroup P = MAN is any closed subgroup of G (including G itself) which contains a minimal parabolic subgroup. Furthermore, we have N ⊂ N0 , A ⊂ A0 , and M ⊃ M0 . The number of nonconjugate parabolic subgroups is 2r , where r = rank A (cf., e.g., [620]). Note that in general M is a reductive Lie group with structure M = Md Ms Ma , where Md is a ﬁnite group, Ms is a semisimple Lie group, and Ma is an Abelian Lie group central in M. Note that if M ≠ M0 , then Ms is a noncompact semisimple Lie group, i.e., the same class of groups that we study.

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4 Invariant Differential Operators

Finally, for further use we mention that the restricted Weyl group W(G, A0 ) (cf. Section 3.1) is isomorphic to the factor-group M0′ /M0 , where M0′ is the normalizer of A0 in K.

4.2 Preliminaries on Group Representation Theory 4.2.1 Representations and Modules A representation of the group G in the vector space V is any homomorphism of G in the group End V. In more detail, let > : G → End V; then > is a representation if >(gg ′ ) = >(g)>(g ′ )

for all g, g ′ ∈ G,

>(e) = idV .

The space V is called representation space. Sometimes instead of >(g) v one writes g v, (g ∈ G, v ∈ V). Then the space V is called G-module. The dimension of V is called the dimension of >. If V is ﬁnite-dimensional one denotes dim> = dimV. Every one-dimensional representation is called a character of G. If the character : is such that :(g) = 1 for all g ∈ G, then such character is called the trivial representation. A vector v ∈ V is called a weight vector if >(g)v = +(g)v for all g ∈ G, where + is a character of G. Example: Let X be a right G space. Let F X be the set of all functions f : X → F. Then F X is a G-module w.r.t. the representation (F (g)f )(x) ≐ f (xg),

x ∈ X, g ∈ G.

Analogously, let X be a left G space. Then F X is a G-module w.r.t. the representation (F (g)f )(x) ≐ f (g –1 x),

x ∈ X, g ∈ G.

4.2.2 Reducibility and Irreducibility Let V be a vector space. A subspace V ′ ⊂ is called invariant subspace w.r.t. the operator A ∈ End V if AV ′ ⊂ V ′ , i.e., Ax ∈ V ′ for all x ∈ V ′ . Let V be a vector space and > a representation of G on V. The representation > is called reducible representation if there exists a nontrivial subspace V ′ of V (V ′ ≠ {0}, V) which is invariant w.r.t. >(g) for all g ∈ G. In that case, the restriction >′ = >|′ V is a representation of G in the vector space V ′ which is called a subrepresentation.

4.2 Preliminaries on Group Representation Theory

139

Furthermore, the representation >F deﬁned on the factor-space V/V ′ by >F (g)(x+V ′ ) ≐ >(f )x + V ′ is called a factor-representation. Then V ′ is called a submodule, and the factor-space V/V ′ is called a factor-module. If the representation > is not reducible, then it is called irreducible representation. The representation > is called completely reducible if each nontrivial invariant subspace V ′ has complementary invariant subspace V ′′ so that V is the direct sum of these invariant subspaces V = V ′ ⊕ V ′′ . If the representation > is reducible, but not completely reducible, the > is called indecomposable representation. Let V be a vector space and > a representation of G on V. For every x ∈ V the linear envelope Vx of the orbit >(G)x is a submodule of the module V. In that case Vx is called a cyclic submodule of V with cyclic vector x. The module V is called a cyclic module if V = Vx for some x ∈ V. The module V is irreducible iff V = Vx for all x ∈ V. The representation > is called exact representation if ker> = {e}. We say that the group G has exact linear representation if G has exact representation over R or C. In that case the group G is called linear group. In particular, every subgroup of GL(n, R) or GL(n, C) is called a linear (or matrix) group. 4.2.3 Operations on Representations Let U,V be vector spaces and 8, > be representations of G on U,V, respectively. Then an operator A ∈Hom(U, V) is called intertwining operator for the representations 8, > if 6(g) A = A 8(g)

for all g ∈ G.

(4.2)

Remark 1: Note that we shall use the notions of intertwining operator and invariant operator as synonyms. ♢ In the setting above the representations 8, > are called partially equivalent. If the operator A in (4.2) is bijective, then 8, > are called equivalent representations. If two representations are partially equivalent, but not equivalent, then at least one of them is reducible. If ker A ≠ {0} then ker A is an invariant subspace of U. Analogously, if A is not onto, then the image Im A is an invariant subspace of V. The representations 8, > are said to be in duality w.r.t. the bilinear (or sesquilinear) form ⋅, ⋅ on U × V, if U, V are in duality w.r.t. this form and 8(g)y, >(g)x = y, x

for all g ∈ G, y ∈ U, x ∈ V.

(4.3)

In particular, every representation > is in duality with the representation ˆ >(g) ≐ >(g)∗ –1

(4.4)

ˆ is called the contragredient represon the conjugated space V ∗ . The representation > entation (or conjugate) of >.

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4 Invariant Differential Operators

A representation > in the Hilbert space V is called unitary representation if all operators >(g) are unitary, i.e., >(g)∗ = >(g)–1 . If U, V are Hilbert spaces, then the representations 8, > are called unitarily equivalent if the intertwining operator A is bijective and isometric. Then if 8 is unitary, so is >. If > is an irreducible representation in a ﬁnite-dimensional space V, then the scalar product in V w.r.t. which > is unitary is determined (if it exists) uniquely up to a multiplicative constant. A unitary irreducible representation 0(g) of the unimodular group G on the Hilbert space V is said to belong to the discrete series if its matrix coefﬁcients 0(g) ⋅ v, w, v, w ∈ V are square-integrable on G w.r.t. the Haar measure.

4.2.4 Induced Representations Let H be a subgroup of G, let 4 be a representation of H in the vector space V. Let V4 (G) be the space if V-valued functions on G fulﬁl >(gh) = 4(h) >(g)

for all h ∈ H, g ∈ G.

(4.5)

Take the operators of the left regular representation (LRR): (L(g)>)(g ′ ) = >(g –1 g ′ ).

(4.6)

This representation is called induced representation of G from the subgroup H and is denoted by IndGH (4). The property (4.5) means that in fact the functions > do not depend on their values on H except on the unit (note that >(h) = 4(h)>(e)). Thus, there is an equivalent realization on the homogeneous space X = G/H.

4.3 Elementary Representations 4.3.1 Compact Lie Groups Let Gc be a compact semisimple Lie group with Lie algebra Gc . Compact Lie groups will play an auxiliary role in our considerations, most often as subgroups of inducing parabolic subgroups. Furthermore their interesting representations are ﬁnite-dimensional unitary representations. The latter may be identiﬁed with the class of holomorphic ﬁnite-dimensional representations of the complex Lie algebras which we studied in detail in Section 2.8.

4.3 Elementary Representations

141

The reader who is interested in more details may look in the classic book by Zhelobenko [639]. 4.3.2 Noncompact Lie Groups Let G be a noncompact semisimple Lie group. Let P = MAN be a cuspidal parabolic subgroup of G. Let , ﬁx a discrete series representation D, of M on the Hilbert space V, , or the so-called nondegenerate limit of a discrete series representation (cf. [349]). Let - be a (nonunitary) character of A, - ∈ A∗ , where A is the Lie algebra of A. Note that in general, , is actually a triple (:, 3, $), where : is the signature of the character of Md , 3 gives a character of Ma , $ ﬁxes a discrete or ﬁnite-dimensional irrep of Ms on V, (the latter depends only on $). Definition: We call the induced representation 7 = IndGP (, ⊗ - ⊗ 1) an elementary representation of G [159]. (These are called generalized principal series representations (or limits thereof) in [349].) For practical purposes it is useful to extend the deﬁnition to induction from noncuspidal parabolics, and we shall do so. In that case, we use for the induction ﬁnite-dimensional irreps of M. ♢ The spaces of functions of the ERs are C7 = {F ∈ C∞ (G, V, )|F (gman) = e-(H) ⋅ D, (m) F (g)},

(4.7)

where a = exp(H) ∈ A, H ∈ A, m ∈ M, and n ∈ N. The special property of the functions of C7 is called right covariance [126, 159] (or equivariance). Because of this covariance the functions F actually do not depend on the elements of the parabolic subgroup P = MAN. Thus, we can use right covariance to pass from functions on the group G ˆ on the coset space G/P. Note that G/P is a completion to so-called reduced functions > ˜ of the group N (conjugate to N) and in practical calculations one is usually using the ˜ local coordinates of N. Remark 2: It is well known that when V, is ﬁnite-dimensional C7 can be thought of as the space of smooth sections of the homogeneous vector bundle (also called vector G-bundle) with base space G/P and ﬁber V, (which is an associated bundle to the principal P-bundle with total space G). We shall not need this description for our purposes. ♢ The ER T 7 acts in C7 as the (LRR) by (T 7 (g)F )(g ′ ) = F (g –1 g ′ ),

g, g ′ ∈ G.

(4.8)

One can introduce in C7 a Fréchet space topology or, if the representation is unitary, complete it to a Hilbert space (cf. [620]).

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4 Invariant Differential Operators

The ERs differ from the LRR (which is highly reducible) by the speciﬁc representation spaces C7 . In contrast, the ERs are generically irreducible. The reducible ERs form a measure zero set in the space of the representation parameters ,, -. (Reducibility here is topological in the sense that there exists nontrivial (closed) invariant subspace.) The irreducible components of the ERs (including the irreducible ERs) are called subrepresentations. Remark 3: Note that one may exchange the left and right actions in the above considerations, i.e., consider the representations acting as right regular representations with properties of left covariance. Independently, one may use lowest weight representa˜ where P˜ = MAN. ˜ tions, and then one uses the coset G/P, ♢ The importance of the parabolic subgroups and the ERs is due to the following result: Theorem: Every irreducible admissible representation of a real connected semisimple Lie group G with ﬁnite center is equivalent to a subrepresentation of an ER of G. ♢ The above result was obtained ﬁrst by Langlands [389] and then reﬁned by Knapp– Zuckerman [357, 358]. Remark 4: A representation of G is called admissible if it is K-ﬁnite (K is the maximal compact subgroup) and Z(G)-ﬁnite (Z(G) is the center of G). K-ﬁniteness means that the representation space C contains a dense subspace CK = ⊕#∈Kˆ E# (direct sum), where Kˆ is the set of equivalence classes of irreducible representations of K, and the representation spaces are ﬁnite-dimensional: dimE# < ∞. Z(G)-ﬁniteness means that there exists a polynomial p such that p(T (X)) = 0, ∀X ∈ G. The admissibility condition is usually fulﬁlled in the physically interesting examples. ♢ We shall also need inﬁnitesimal version of LRR (XL F )(g) ≐

d F (exp(–tX)g)|t=0 , dt

(4.9)

where F ∈ C7 , g ∈ G, and X ∈ G – the Lie algebra of G; then we use complex linear extension to extend the deﬁnition to a representation of the complexiﬁcation G C of G. 4.3.3 Knapp–Stein Integral Operators The main topic of this book is the invariant differential operators which we shall introduce below and which intertwine reducible ERs. On the other hand there are Knapp– Stein (KS) integral intertwining operators introduced by Knapp and Stein [352, 353] which are deﬁned for arbitrary ERs. We shall call them KS operators hereafter. Let G be a noncompact semisimple Lie group with Lie algebra G. Let P = MAN – be a parabolic subgroup of G. Let BA and W(G, A) be the restricted root system

4.3 Elementary Representations

143

and restricted Weyl group relative to the Lie algebra A of A. These are deﬁned in complete analogy as done in Subsection 3.1.2 for their counterparts w.r.t. the main vector subalgebra A0 . Let s ∈ W(G, A). Then there is an integral intertwining operator deﬁned as follows: A7 : Cs(7) → C7 , * A7 F (g) ≡ #s (7) F (g3(s)n+s ) dn+s ,

(4.10)

Ns+

where s(7) denotes the action of s on the signatures 7, F ∈ Cs(7) , 3(s) is a matrix representation of s, n+s ∈ Ns+ , the latter is the submanifold of N + that is not invariant under s, i.e., Ns+ ≡ 3(s)–1 N – 3(s) ∩ N + ,

(4.11)

dn+s is the Haar measure on Ns+ . The intertwining property is A7 ○ Ts(7) = T7 ○ A7 .

(4.12)

We shall write down explicit examples of such operators in Section 4.7 (though for the construction of SL(2, R) there was given 20 years before the general approach was formulated), in Section 5.2.5 and especially in Section 6.2.6 where W(G, A) had seven nontrivial elements. Now we only mention that these integral operators may be cast in the form * A7 f (x) ≡ K7 (x, x′ ) f (x′ ) dx′ , (4.13) Ns+

where x and x′ are suitable coordinates on Ns+ , dx′ is the Haar measure in these coordinates, and K7 (x, x′ ) is a generalized function (cf. [253]) depending meromorphically on the parameters of 7. In particular, for the conformal group K7 (x, x′ ) is given by the two-point function of conformal ﬁeld theory. The ERs C7 and Cs(7) are partially equivalent. If both C7 and Cs(7) are irreducible then they are equivalent and furthermore hold A7 ○ As(7) = IdC7 ,

As(7) ○ A7 = IdCs(7) .

(4.14)

Finally, note that there is a multiplet of ERs corresponding to W(G, A) M(G, A) ≐ { Cs(7) : s ∈ W(G, A) }.

(4.15)

This multiplet exists independently from the reducibility of the ERs in it, though it may be reduced for some degenerate signatures.

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4 Invariant Differential Operators

Similarly to the multiplets of Verma modules (cf. Subsection 2.11.2), it is convenient to depict multiplets of ERs by a connected graphs, the vertices of which correspond to the ERs and the lines between the vertices correspond to the KS operators between the ERs. Each line between two vertices may be depicted as a bidirectional arrow since there are two KS operators between two members of M(G, A) (cf. (4.14)). Furthermore, when the ERs of M(G, A) are reducible, then it may be identiﬁed with a submultiplet of the multiplet of Verma modules corresponding to W(G C , HC ). This will be given more content after we introduce the notion of associated Verma modules in Section 4.5.

4.3.4 ERs of Complex Lie Groups As we have seen in Section 3.4, the complex simple Lie algebras, considered as real noncompact Lie algebras, do not have nonminimal cuspidal parabolic subalgebras. Thus, it is enough to consider ERs induced from the minimal parabolic subgroup P0 = M0 A0 N0 , where M0 ≅ U(1) × . . . × U(1) ( factors), A0 ≅ SO(1, 1) × . . . × SO(1, 1) ( factors), and N0 ≅ exp N0± . Thus, the signature 7 = [,, -] consists of integer numbers ,i ∈ Z giving the unitary character , = (,1 , . . . , , ) of M0 , and of complex numbers -i ∈ C giving the character - = (-1 , . . . , - ) of A0 , -j = -(Hj ). Thus, if H = j 3j Hj , 3j ∈ R, is a generic element of A0 , then for the corresponding factor in (4.7) we have e-(H) = exp j 3j -j . Analogously, if m = exp i j 6j Hj ∈ M, 6j ∈ R, then we have D, (m–1 ) = exp i j 6j ,j . To relate with the general setting of the previous subsection we must introduce ˜ such that D(Hj ) = +j and D(H ˜ j ) = +˜ j . Let us use (3.31) and two weight functionals D, D, H = j (3j + i6j )Hj ∈ H. Thus, the ERs (in particular, the right covariance conditions) for a complex semisimple Lie group Gc are given by an analog (4.7). ¯ ⋅ F (g) ˜ H) CD,D˜ = {F ∈ C∞ (Gc )|F (gman) = exp D(H) + D(

= exp (3j + i6j )+j + (3j – i6j )+˜ j ⋅ F (g)}, j

-j = +j + +˜ j ,

,j = +j – +˜ j ∈ Z

(4.16)

and the last condition in (4.16) stresses that we have uniqueness on the compact subgroup M0 of the Cartan subgroup Hc = M0 A0 of Gc . ˜ The ER is reducible when (2.208) holds for either D or D. The ERs for which D˜ = 0 are called holomorphic, and those for which D = 0 are called antiholomorphic.

4.4 Unitary Irreducible Representations

145

More information can be found in [640, 643] from where we mention some important statements. – All irreducible representations of a complex semisimple Lie group are obtained as subrepresentations of the ERs induced from the minimal parabolic subgroup. –

All ﬁnite-dimensional irreps of a complex semisimple Lie group are obtained as subrepresentations when all +j , +˜ j ∈ Z+ .

4.4 Unitary Irreducible Representations There is no explicit general classiﬁcation of the unitary irreducible representations of all real semisimple Lie groups (cf. [354]). However, all known cases may be summarized as follows: – principal series of unitary representations These are ERs in the cases when the character - of the vector part A is imaginary (such characters being called unitary). These ERs are irreducible (or possibly completely reducible for - = 0). –

complementary series of unitary representations These are ERs in the cases when the character - ≠ 0 of the vector part A is real and restricted (in open intervals). These ERs are irreducible.

–

limiting points of the complementary series of unitary representations These are subrepresentations of reducible ER at the boundaries of the open intervals of the complementary series. They are listed separately, since their properties are very different.

–

discrete series of unitary representations and their limits These are subrepresentations of reducible ER. Among these the most studied are the holomorphic discrete series and their limits. We will also discuss about the quaternionic discrete series in the following sections.

–

the trivial representation.

Some details: – A holomorphic discrete series representation is a discrete series representation of a semisimple Lie group that can be represented in a natural way as a Hilbert space of holomorphic functions. The simple Lie groups with holomorphic discrete series are those whose symmetric space is Hermitian and the maximal compact subalgebra has the form K = K′ ⊕ u(1) [279]. Holomorphic (antiholomorphic) discrete series representations are the easiest discrete series representations to study because they have lowest (highest) weights, which makes their behavior similar to that of ﬁnite-dimensional representations of compact Lie groups. Bargmann [24] and Gel’fand et al. [248, 250] found the ﬁrst examples of holomorphic discrete series representations studying SL(2, R). Harish-Chandra [279] classiﬁed them

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4 Invariant Differential Operators

for all semisimple Lie groups. Furthermore he gave exact conditions expressed in terms of the representation data, as we shall point out in the relevant cases below. Martens [430] and Hecht [287] described the characters of some holomorphic discrete series representations. For the role of parabolic subgroups in the direct construction of unitary representations, we refer to [629]. –

The quaternionic discrete series representation is a discrete series representation of a semisimple Lie group G associated with a quaternionic structure on the symmetric space of G (cf. Gross–Wallach [269, 270]).

Quaternionic discrete series representations exist when the maximal compact subgroup K of the group G has a normal subgroup isomorphic to SU(2). Every complex simple Lie group has a real form with quaternionic discrete series representations. In Chapter 2, we have mentioned without explanation all real forms for which quaternionic discrete series exist. In particular, the classical groups SU(2, n), SO(4, n) and Sp(1, n) have quaternionic discrete series representations, and so do the exceptional cases E6(2) , E7(–5) , E8(–24) , F4(4) , and G2(2) . From the above eight cases, only SU(2, n) has also holomorphic discrete series.

4.5 Associated Verma Modules The main feature of the ERs which makes them important for our construction is the structure of highest (or lowest) weight module (over C) associated with them [126]. To display this structure, we shall use the right action of G C (the complexiﬁcation of G) by the standard formula (XR F )(g) ≐

d F (g exp(tX))|t=0 , dt

(4.17)

where F ∈ C7 , g ∈ G, X ∈ G; then we use complex linear extension to extend the deﬁnition to a representation of G C . Note that this action takes F out of C7 for some X but that is exactly why it is used for the construction of the invariant differential operators. We specialize (4.17) for X = H ∈ A, and obtain using the right covariance property in (4.7): (HR F )(g) ≐

d d -(tH) F (g) |t=0 F (g exp(tH))|t=0 = e dt dt

= -(H) F (g), HR F = -(H) F ,

or

(4.18a)

H ∈ A.

(4.18b)

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4.5 Associated Verma Modules

Next we consider (4.17) for X ∈ N : (XR F )(g) ≐

d d F (g) |t=0 = 0, F (g exp(tX))|t=0 = dt dt

(4.19a)

or XR F = 0,

X ∈ N.

(4.19b)

Next we consider (4.17) for X ∈ Ma , so that exp X ∈ Ma : d d 3(tX) F (g) |t=0 F (g exp(tX))|t=0 = e dt dt = 3(X) F (g)

(XR F )(g) ≐

(4.20a)

or XR F = 3(X) F ,

X ∈ Ma .

(4.20b)

Before we apply the above to Ms we recall the connection between the structure of a noncompact semisimple Lie algebra and its complexiﬁcation. We ﬁrst recall that C MC = MC s ⊕ Ma ,

(4.21)

and then write the standard triangular decomposition of the complexiﬁcation MC s of Ms : C C C MC s = Ms+ ⊕ Hs ⊕ Ms– ,

(4.22)

then insert this into the complexiﬁcation of the Bruhat decomposition C C G C = N+C ⊕ MC s+ ⊕ A ⊕ N–

=

N+C

⊕ MC +

⊕ HsC

⊕ MC –

(4.23a) ⊕ MC a

⊕A

C

⊕ N–C .

(4.23b)

Comparing the above to the standard triangular decomposition of G C , we obtain G±C = N ±C ⊕ MC ±,

C HC = HsC ⊕ MC a ⊕A .

(4.24)

Extending by complexiﬁcation (4.18), (4.19), and (4.20), we see that they hold for AC , C N C , and MC a , respectively. Our aim is to obtain an analogue (4.18) and (4.20) for Ma C and of (4.19) for M+ . At this moment we restrict temporarily to the case of minimal parabolic subgroup P0 = M0 A0 N0 . In this case, M0 is compact and V, = V$ is ﬁnite-dimensional. Then we can choose a vector v0 which is a highest weight vector w.r.t. the representation of MC 0s obtained as follows. We deﬁne the representation $ D˜ (m) ≡ D$ (m–1 )+

(= D$ (m) for m ∈ M0s ).

(4.25)

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4 Invariant Differential Operators

Then we consider the action of X ∈ M0s d ˜$ XR ⋅ v ≡ , D (exp tX) ⋅ v dt |t=0

v ∈ V$

(4.26)

and extend this action to MC 0s . The vector v0 fulﬁlls the following conditions: X ∈ HsC ,

XR ⋅ v0 = $(X) v0 , XR ⋅ v0 = 0,

X∈

(4.27a)

MC 0+ .

(4.27b)

Using the above we introduce C-valued realization C 7 of the space C7 by the formula >(g) ≡ v0 , F (g),

(4.28)

where , is the M0 -invariant scalar product in V, . (If M0 is Abelian or discrete then V, is one-dimensional and C 7 coincides with C7 ; so we set > = F .) On these functions the right action of G C is deﬁned by (XR >)(g) ≡ v0 , (XR F )(g).

(4.29)

Remark 5: In the geometric language we have replaced the homogeneous vector bundle with base space G/P0 and ﬁber V, with a line bundle again with base space G/P0 (also associated to the principal P0 -bundle with total space G). The functions > can be thought of as smooth sections of this line bundle. ♢ It is clear that the functions > inherit properties (4.27) from v0 , i.e., formulae (4.27) hold with v0 are replaced by >(g). In the same way formulae (4.18), (4.19), and (4.20) hold for >(g) (note that complex conjugation in , affects only the ﬁrst entry). Now we extend -, 3, $ to HC by setting -|HC = 0, 0m

3|AC ⊕HC = 0, 0

0s

$|AC ⊕MC = 0, 0

0a

(4.30)

C = HC ⊕ MC is the Cartan subalgebra of MC . where H0m 0s 0a 0 Then we deﬁne the element D ∈ (H0C )∗

D ≡ - + 3 + $.

(4.31)

Part of the main result of [126] is Proposition: The functions of the C-valued realization C 7 of the ER C7 satisfy XR > = D(X) ⋅ >, XR > = 0,

X∈

X ∈ HC , D ∈ (HC )∗ G+C ,

(4.32a) (4.32b)

4.6 Invariant Differential Operators

149

where D = D(7) is built canonically from 7. (It contains all the information from 7, except about the character : of the ﬁnite group Md . In the case of G being a complex Lie group we need two weights to encode 7 (cf. Section 4.3.4).) ♢ Remark 6: Note that here we are using highest weight modules instead of the lowest weight modules used in [126]; also the weights are shifted by 1 w.r.t. the notation of [126]. ♢ For simplicity we have restricted to the minimal parabolic subgroup. However, the construction is extended immediately to any cuspidal parabolic subgroup P = MAN when inducing from discrete series representations of M. The point is that instead of the inducing discrete series representation of M we shall consider the ﬁnitedimensional irrep of M which lies on the same orbit of the Weyl group (in other words, which has the same Casimirs). (Thus, we are using the so-called translation by action of the Weyl group [349].) The same mechanism is applicable when P is noncuspidal, again by inducing from ﬁnite-dimensional irreps of M. Thus, we obtain the necessary highest weight D as above in (4.31). Thus, since our ERs may be induced from ﬁnite-dimensional representations V, of M the associated Verma modules are always reducible. Thus, it is more convenient to use generalized Verma modules (GVMs) V D such that the role of the highest/lowest weight vector v0 is taken by the (ﬁnite-dimensional) space V, v0 . For the GVMs the reducibility and invariant differential operators are only w.r.t. M-noncompact roots. In the case when M itself has highest weight representations (and thus holomorphic discrete series) we can use the above construction directly using the scalar product of the discrete series and the corresponding highest weight vector. Note that we can also apply the above to the case of induction from limits of discrete series. This amounts to some analytic limit of the weight D. Finally, we note that the ERs of complex Lie groups (Subsection 4.3.4) are “richer” since they have two Verma modules associated with an ER, one “holomorphic” V D and ˜ ˜ one “antiholomorphic” V D . The ER is reducible when either V D or V D are reducible.

4.6 Invariant Differential Operators 4.6.1 Canonical Construction The basic fact for the construction of invariant differential operators is the association of Verma modules over C to the ERs described in the previous section. We start by reminding that conditions (4.32) are the deﬁning conditions for the highest weight vector of a highest weight module over G C with highest weight D (cf. Chapter 3). Of course, it is enough to impose (4.32b) for the simple root vectors Xj+ .

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4 Invariant Differential Operators

As we have seen in Chapter 1 the Verma module V D is reducible iff (cf. (2.208)) D + 1, "∨ = m holds for some " ∈ B+ , m ∈ N, and then in V D exists a unique singular vector vs , , which has the properties of a highest weight vector with shifted weight D – m". X vs = (D – m")(X) ⋅ vs , X vs = 0,

X∈

X ∈ HC ;

G+C .

(4.33a) (4.33b)

The general structure of a singular vector was given in [126] and above in (2.212). Explicit expressions are given in detail in Section 4.8. Here we write vs = Pm" (X1– , . . . , X– )v0 ,

(4.34)

where Pm" is a homogeneous polynomial in its variables of degrees mki , where ki ∈ Z+ come from " = ki !i , !i ∈ 0, the system of simple roots. Now we are in a position to deﬁne the invariant (aka intertwining) differential operators for semisimple Lie groups, corresponding to the singular vectors. Let the signature 7 of an ER be such that the corresponding D = D(7) satisﬁes (2.208) for some " ∈ B+ and some m ∈ N. Then there exists an intertwining differential operator [126] Dm" : C 7 → C 7′ ,

(4.35)

where 7′ is such that D′ = D′ (7′ ) = D – m". Remark 7: If " is a real root, (i.e., "|HC = 0, where Hm is the Cartan subalgebra of M), m then some conditions are imposed on the character : representing the ﬁnite group Md (cf. [564] and some explicit examples in next chapters). ♢ The most important fact is that (4.35) is explicitly given by [126] Dm" >(g) = Pm" ((X1– )R , . . . , (X– )R ) >(g),

(4.36)

where Pm" is the same polynomial as in (4.34) and (Xj– )R denotes the right action (4.17). One important simpliﬁcation in our procedure is that in order to check the intertwining properties of the operator in (4.36) it is enough to work with the inﬁnitesimal versions of (4.7) and (4.8), i.e., work with representations of the Lie algebra. This is important for using the same approach to superalgebras and quantum groups, and to any other (inﬁnite-dimensional) (super-)algebra with triangular decomposition.

4.6 Invariant Differential Operators

151

Thus, in each such situation we have an invariant differential equation of order m = m" D m " > = >′ ,

> ∈ C 7(D) ,

>′ ∈ C 7(D–m") .

(4.37)

In most of these situations, the invariant operator Dm " has a nontrivial invariant kernel in which a subrepresentation of G is realized. Thus, studying the equations with trivial right-hand side (RHS) > ∈ C 7(D) ,

Dm " > = 0,

(4.38)

is also very important. Note that when we induce from ﬁnite-dimensional irreps of Ms this means that (2.208) are fulﬁlled for all simple roots (and then for all positive roots) of the root sysC tem of MC s . We recall that the root system BM of Ms consists from the compact (w.r.t. C C Ms ) roots of the root system B of G , i.e., roots ! such that !|AC ⊕MC = 0. a

Let us denote by #k , k = 1, . . . , rM =rank MC s , the simple roots of the system BM . Then we have the following from (2.208): nk ≡ D + 1, #k∨ ∈ N.

(4.39)

Since these are also simple roots for B, then the singular vectors are vkM = (XkM )nk v0 ,

k = 1, . . . , rM

(4.40)

and the corresponding differential operators are DkM = ((XkM )R )nk .

(4.41)

Finally, we note a peculiarity here – since the above conditions are part of the induction process, they are intrinsic for the resulting representation, i.e., the functions of the representation fulﬁll [126]: DkM > = ((XkM )R )nk > = 0.

(4.42)

This is consistent with the fact that for such ERs the associated Verma modules are actually the GVMs deﬁned in Section 4.5. Remark 8: After our paper [126] was submitted for publication Prof. B. Kostant kindly informed the author about his paper [367] in which he has proved a bijection between the set of all differential operators intertwining multiplier representations of G on

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4 Invariant Differential Operators

C∞ (G/P) (called quasi-invariant differential operators in [367]) and the set of all singular vectors in the Verma modules over G C (called leading weight vectors in [367]). At that time we were also informed about similar construction in the paper [641]. ♢ 4.6.2 Multiplets of GVMs and ERs In the setting of the previous subsection we know that reducible GVMs and ERs are grouped in multiplets. The multiplets MW of GVMs correspond to the complex Weyl group W ≡ W(G C , HC ) or subsets of it. The multiplets MA of ERs correspond to the restricted Weyl group WA ≡ W(G, A) or subsets of it. It is important that WA may itself be identiﬁed as a subset of W or several subsets of W. Actually, since to every GVM there corresponds an ER, this means that the multiplets MW of GVMs are also multiplets of ERs. In particular, the latter means that some ′ arrows between two GVMs V D → V D correspond a bidirectional KS arrow between the ERs with the same weights: C7(D) ←→ C7(D′ ) . More than this, in such a case the arrow ′ V D → V D coincides with the arrow C7(D) → C7(D′ ) . Analytically, this means that the KS operator C7(D) → C7(D′ ) degenerates to an invariant differential operator, which happens when the kernel K7(D′ ) (x, x′ ) (from (4.13)) has to be regularized as a generalized function. These topics are illustrated and used for the classiﬁcation of ERs/GVMs in Section 4.7 and Chapters 5, 6, and 8.

4.7 Example of SL(2,R) 4.7.1 Elementary Representations We give as an important example the lowest conformal case G = SOe (2, 1) ≅ SL(2, R)/Z2 . We consider G = SL(2, R) following Gel’fand et al. [248] (the original results are in [24, 250]). The Lie algebra so(2, 1) ≅ sl(2, R) is maximally split and the subalgebra M is trivial. Since r = 1 there is only one nontrivial parabolic. The minimal/maximal parabolic subgroup is P = MAN, where M ≅ Z2 . The parabolic subalgebra P = A ⊕ N , dimR A = dimR N = 1. The ERs of SL(2, R) are parametrized by a complex number s representing A and a signature : = 0, 1, representing M. We shall denote the ERs by D7 , 7 = [s, :]. The space of functions is Vs,: = { 6 ∈ C∞ (R)|6(x)

→ |x| → ∞

C |x|s–1 sign: (x) }

(4.43)

on which the action of G is $x – " T s,: (g)6 (x) = |! – #x|s–1 sign: (! – #x) 6 , ! – #x

! " ∈ SL(2, R). g= # $

(4.44)

4.7 Example of SL(2,R)

153

The representations 7+ = [s, :] and 7– = [–s, :] are partially equivalent. The operators realizing this partial equivalence are given by A± : D7∓ → D7± , *+∞ dx′ (A± 6)(x) = |x′ – x|±s–1 sign: (x′ – x) 6(x′ ) . A(–s)

(4.45) (4.46)

–∞

We could call these operators KS operators, but we refrain from this since they were introduced in this setting more than 20 years before [352, 353]. Since for s ≠ 0 the doublet 7± form a partially equivalent pair they have coinciding Casimirs. These ERs are irreducible except when s ∈ Z and : = s(mod2). The ERs 7± = [±s, :] are equivalent when irreducible.

4.7.2 Discrete Series and Limits Thereof In the reducible case we introduce notation D±s ≡ D[±s,:=s(mod2)] ,

s ∈ Z+ .

(4.47)

For s ∈ N the reducible ER Ds contains a ﬁnite-dimensional (nonunitary) subspace Es of dimension s. In that case the integral operator A– : Ds → D–s degenerates to an invariant differential operator (A– 6)(x) =

ds 6(x) = ∂xs 6(x). dxs

(4.48)

The discrete unitary series are realized as subspaces of D–s when s ∈ N. More precisely, ± , which are invariant subspaces of D . there are two discrete series UIRs F–s –s + The UIRs realized in F–s nowadays are called holomorphic discrete series, they – nowadays are called antihohave a lowest weight vector, the UIRs realized in F–s lomorphic discrete series, and they have a highest weight vector. Their direct sum + ⊕ F – is an invariant subspace of D . F–s = F–s –s –s The kernel of A– is the space Es and its image is the space F–s . The kernel of A+ is the space F–s and its image is the space Es . Note also that the factor-space Fs = Ds /Es is the direct sum of two further ±. subspaces Fs = Fs+ ⊕ Fs– , and that the operator A– maps Fs± onto F–s + – For s = 0 = : one has D0 = F0 = F0 ⊕F0 (setting E0 = 0). (In that case formula (4.48) is valid trivializing to A– = IdD0 .) The UIRs F0± are not related to ﬁnite-dimensional representations; correspondingly, they do not fulﬁl Harish-Chandra’s criterion for discrete series [283], and thus, nowadays they are called limits of discrete series.

154

4 Invariant Differential Operators

4.7.3 Positive Energy Representations As we have mentioned, the positive energy representations are conveniently realized via the lowest weight Verma module V D over the complexiﬁcation G C = sl(2, C) of the Lie algebra G = sl(2, R), and the weight D = D(7) = D(s) is determined uniquely by 7. For s ∈ N the Verma module V D(s) is reducible; it has a ﬁnite dimensional factorrepresentation Es+ = V D(s) /I D(s) of real dimension s (which naturally can be identiﬁed with Es from above). By introducing the conjugation singling out G we can use the Shapovalov form on U(G + ), (G + are the raising generators of G C ), to deﬁne a scalar product in V D and then positivity produces the list of lowest weight modules. Using as parameter d = – s the condition for positivity is d ≥ –1. The point d = –1 is the ﬁrst reduction point (FRP), which happens in the Verma module V D(1) (cf. above). There the factor-UIR is trivial (being one-dimensional). For d > –1 the Verma modules V D(–d) are irreducible and unitarizable. They are called positive energy representations, the parameter d being the energy or conformal weight. They are also + . In particular, for called analytic continuation of the holomorphic discrete series F–s D(–d) d = –s ∈ N the Verma module V is inﬁnitesimally equivalent to the holomorphic + , while for d = 0 it is inﬁnitesimally equivalent to the limit of discrete series irrep F–s holomorphic discrete series irrep F0+ . All this is illustrated on the ﬁgure below: FRP

LDS

DS

d = −1

d=0

d=1

•

•

•

...

DS

•

d=k

A similar construction exists if we take the conjugate highest weight Verma modules, + , E+ being played by F – , E– . In that case we are speaking about negative the role of F–s s –s s energy representations, with parameter d = s, so that we have d ≤ 1.

4.8 Explicit Formulae for Singular Vectors The basic source for this section is [127]. We give explicit formulae for singular vectors of Verma modules over any complex simple Lie algebra G. (In the literature they are also called null or extremal vectors.) In [126] we gave the general formula for the singular vectors which however was not so explicit. Some explicit formulae for singular vectors for G = A and for some rank two algebras G ≠ A were presented in [122]. Here we give explicit formulae for the singular vectors for arbitrary G corresponding to a class of positive roots of G which we shall call straight roots. In some special cases we give singular vectors corresponding to arbitrary positive roots (cf. Subsection 4.8.8). As in [122, 126] we use a special basis of U(G – ). This basis seems more economical than the Poincaré–Birkhoff–Witt type of basis used by Malikov, Feigin and

155

4.8 Explicit Formulae for Singular Vectors

Fuchs [425] for the construction of singular vectors. Furthermore our basis turns out to be special of a general basis introduced for other reasons by Lusztig [401] for the quantum group deformation of U(B – ), where B – is a Borel subalgebra of G. It is well known [59] that every root may be expressed as the result of the action of an element of the Weyl group W on some simple root. More explicitly, for any " ∈ B+ we have " = w(!p ) = sp1 sp2 . . . spr (!p ),

(4.49)

s" = wsp w–1 = sp1 . . . spr sp spr . . . sp1 ,

(4.50)

and consequently

where !p is a simple root, and the element w ∈ W is written in a reduced form. The positive root " is called a straight root if all numbers p, p1 , p2 , . . . pr in (4.49) are different. Note that they may be different from (4.49) presentations of " involving other elements w′ and !p′ ; however this deﬁnition does not depend on the choice of these elements. Obviously, any simple root is a straight root. The case of simple roots was treated in (2.213). Other easy examples of straight roots are those which are sums of simple roots with coefﬁcients not exceeding 1, i.e., " = i ni !i , with ni = 1 or 0. All straight roots of the simply laced algebras A , D , and E are of this form. In what follows we shall use the following notion. A root #′ ∈ B+ is called a subroot ′′ of # ∈ B+ if #′′ – #′ ≠ 0 may be expressed as a linear combination of simple roots with nonnegative coefﬁcients.

4.8.1 A We start with the case G = A with roots system given in Section 2.5.1. We recall that for A the highest root is given by !˜ = !1 + !2 + ⋯ + ! and that every root " ∈ B+ is the highest root of a subalgebra of A . Thus, it is enough to consider the highest root. Every root of A is straight. In particular, for the highest root we have, e.g., !˜ = s1 s2 . . . s–1 (! ).

(4.51)

(For different presentations of !˜ we refer to [127].) ˜ but not for any other positive root Let us have condition (2.208) fulﬁlled for !, [(+ + 1, !˜ ∨ ) – m] = 0, m ∈ N,

(4.52a)

˜ ∀m′ ∈ Z+ . [(+ + 1, "∨ ) – m′ ] ≠ 0, " ≠ !,

(4.52b)

156

4 Invariant Differential Operators

(The reason for the appearance of Z+ in (4.52ab) instead of N will become clear in Subsection 4.8.8.) Now one can check that the singular vector is given by ˜ = v!,m

m

k1 =0

⋯

m

ak1 ...k–1 (X1– )m–k1

(4.53)

k–1 =0

– m–k–1 – k–1 × . . . (X–1 ) (X– )m (X–1 ) . . . (X1– )k1 v0 m m ak1 ...k–1 = (–1)k1 +⋯+k–1 a ⋯ k1 k–1

×

(D + 1)(H 1 ) (D + 1)(H –1 ) ... , 1 (D + 1)(H ) – k1 (D + 1)(H –1 ) – k–1

H s = H + H–1 + ⋯ + Hs , Hk is the Cartan generator corresponding to the simple root !k . For = 2, 3 the above formula was given in [122, 126], and for arbitrary in [127]. The above formulae for the singular vectors are actually valid for many other choices of G as we shall see below. Furthermore, it would become clear that the correspondence between the presentations (4.51) and the structure of the formula for the singular vector (4.53) is quite general. Thus, we have the following Recipe: the order of the factors (Xv– )u in (4.51) and of their analogues below corresponds to the order of the factors sv in (4.51); the degrees of homogeneity are according to (2.212); the factors involving Xp–j appear two times and the degree of homogeneity is distributed among them, hence the summation in kj . We have the Conjecture that the above recipe should be valid also for nonstraight positive roots which is supported by the special cases in Subsection 4.8.8.

4.8.2 D Let G = D , ≥ 4, as given in Section 2.5.4. The straight roots which are not positive roots of An subalgebras of D are "i = !i + !i+1 + ⋯ + ! . As in the previous case it is enough to consider a representative root; here, "˜ = "1 = !1 + !2 + ⋯ + ! for which we take the following presentation a la (4.50): "˜ = s1 s2 . . . s–3 s–1 s (!–2 ). Let us have condition (2.208) fulﬁlled for "˜ ∨ (+ + 1, "˜ ) = m

˜ Now one can check that following the recipe above the but not for any subroot # of ". singular vector is given by

4.8 Explicit Formulae for Singular Vectors

˜

v",m =

m

k1 =0

⋯

m

157

– m–k–3 dk1 ...k–1 (X1– )m–k1 ⋯ (X–3 )

k–1 =0

– m–k–1 – m × (X–1 ) (X– )m–k–2 (X–2 ) (X– )k–2

dk1 ...k–1

– k–1 – k–3 × (X–1 ) (X–3 ) ⋯ (X1– )k1 ⊗ v0 , m m = (–1)k1 +⋯+k–1 d ⋯ k1 k–1

×

(+ + 1)(H 1 ) (+ + 1)(H –3 ) ... 1 (+ + 1)(H ) – k1 (+ + 1)(H –3 ) – k–3

×

(+ + 1)(H ) (+ + 1)(H–1 ) , (+ + 1)(H ) – k–2 (+ + 1)(H–1 ) – k–1

(4.54)

where H s = H1 + H2 + . . . + Hs . 4.8.3 E Let G = E , = 6, 7, 8, with root system as given in Section 2.5.7. There is only one straight root which is not a positive root of an An or Dn subalgebra, namely, "˜ = !1 + !2 + ⋯ + ! = s1 s2 . . . s–1 (! ). ˜ Now one Let us have condition (2.208) fulﬁlled for "˜ but not for any subroot # of ". can check that following the recipe above the singular vector is given by ˜

v",m =

m

k1 =0

ek1 ...k–1

⋯

m

ek1 ...k–1 (X1– )m–k1 (X2– )m–k2 ⋯

(4.55)

k–1 =0

– m–k–1 – k–1 ) (X– )m (X–1 ) ⋯ (X2– )k2 (X1– )k1 ⊗ v0 , × (X–1 m m = (–1)k1 +⋯+k–1 e ⋯ k1 k–1

×

(+ + 1)(H 1 ) (+ + 1)(H –1 ) ⋯ , 1 (+ + 1)(H ) – k1 (+ + 1)(H –1 ) – k–1

where H s = H–1 + ⋯ + Hs . 4.8.4 B Let G = B , ≥ 2, with root system as given in Section 2.5.2. The straight roots are of two types: "in = !i +!i+1 +⋯+!i+n–1 , 1 ≤ i ≤ , 1 ≤ n ≤ –i+1, and "i = !i +⋯+!–1 +2! , 1 ≤ i < . It is enough to consider "1 = !1 +⋯+! = s1 . . . s–1 (! ), and "′ = "1 = !1 +. . .+!–1 +2! = s1 . . . s–2 s (!–1 ) = s . . . s2 (a1 ). Let us have condition (2.208) fulﬁlled for "1 or "˜ but not for any of their subroots. The formula for the singular vector corresponding to "1 is given by the sl(n) formula

158

4 Invariant Differential Operators

(4.53). The formula for the singular vector corresponding to "′ = !1 + ⋯ + !–1 + 2! is given by ′

v" ,m =

m

⋯

k1 =0

bk1 ...k–1

m

2m

– m–k–2 bk1 ...k–1 (X1– )m–k1 ⋯ (X–2 )

k–2 =0 k–1 =0

– m – k–2 × (X– )2m–k–1 (X–1 ) (X– )k–1 (X–2 ) ⋯ (X1– )k1 ⊗ v0 , m m 2m k1 +⋯+k–1 = (–1) b ⋯ k1 k–2 k–1

×

[(+ + 1)(H –2 )] [(+ + 1)(H 1 )] . . . [(+ + 1)(H 1 ) – k1 ] [(+ + 1)(H –2 ) – k–2 ]

×

[(+ + 1)(H )] . [(+ + 1)(H ) – k–1 ]

(4.56)

For = 2 formula (4.56) was given in [122]. 4.8.5 C Let G = C , ≥ 3, with root system as given in Section 2.5.3. The straight roots are of two types : "in = !i +!i+1 +⋯+!i+n–1 , 1 ≤ i ≤ , 1 ≤ n ≤ –i+1, and "i = 2!i +⋯+2!–1 +! , 1 ≤ i < . It is enough to consider "1 = !1 + ⋯ + ! = s1 . . . s–2 s (!–1 ) = s . . . s2 (a1 ), and "′′ = "1 = 2!1 + ⋯ + 2!–1 + ! = s1 . . . s–1 (! ). Let us have condition (2.208) fulﬁlled for "1 or "′′ but not for any of their subroots. The formula for the singular vector corresponding to "1 is given by the sl(n) formula (4.53). The formula for the singular vector corresponding to "′′ is given by v"

′′ ,m

=

2m

k1 =0

ck1 ...k–1

⋯

2m

– 2m–k–1 ck1 ...k–1 (X1– )2m–k1 ⋯ (X–1 )

k–1 =0

– k–1 × (X– )m (X–1 ) ⋯ (X1– )k1 ⊗ v0 , 2m 2m = (–1)k1 +⋯+k–1 c ⋯ k1 k–1

×

(4.57)

[(+ + 1)(H 1 )] [(+ + 1)(H –1 )] ... . 1 [(+ + 1)(H ) – k1 ] [(+ + 1)(H –1 ) – k–1 ]

4.8.6 F4 Let G = F4 , with root system as given in Section 2.5.6. The nonsimple straight roots which are not positive roots of subalgebras of F4 are the two roots "˜ = !1 + !2 + !3 + ′ !4 = s1 s2 s4 (!3 ), and "˜ = !1 + !2 + 2!3 + 2!4 = s1 s3 s4 (!2 ).

4.8 Explicit Formulae for Singular Vectors

159

′

Let us have condition (2.208) fulﬁlled for "˜ or "˜ but not for any of their subroots. Then one can check that the singular vectors are given by ˜

v",m =

m m m

fk1 k2 k3 (X1– )m–k1 (X2– )m–k2 (X4– )m–k3

k1 =0 k2 =0 k3 =0

fk1 k2 k3

× (X3– )m (X4– )k3 (X2– )k2 (X1– )k1 ⊗ v0 , m m m = (–1)k1 +k2 +k3 f 4 k1 k2 k3 ×

˜′

v" ,m =

(4.58)

[(+ + 1)(H 1 )] [(+ + 1)(H 2 )] [(+ + 1)(H4 )] , 1 [(+ + 1)(H ) – k1 ] [(+ + 1)(H 2 ) – k2 ] [(+ + 1)(H4 ) – k3 ]

2m 2m m

fk′1 k2 k3 (X1– )m–k1 (X4– )2m–k2 (X3– )2m–k3

k1 =0 k2 =0 k3 =0

fk′1 k2 k3

× (X2– )m (X3– )k3 (X4– )k2 (X1– )k1 ⊗ v0 , m 2m 2m k1 +k2 +k3 ′4 = (–1) f k1 k2 k3 ×

(4.59)

[(+ + 1)(H1 )] [(+ + 1)(H ′4 )] [(+ + 1)(H ′3 )] . [(+ + 1)(H1 ) – k1 ] [(+ + 1)(H ′4 ) – k2 ] [(+ + 1)(H ′3 ) – k3 ]

4.8.7 G2 Let G = G2 with root system as given in Section 2.5.5. The nonsimple straight roots are ′′′ the two roots "˜ = !1 + !2 = s1 (!2 ), and "˜ = !1 + 3!2 = s2 (!1 ). ′′′ Let us have condition (2.208) fulﬁlled for "˜ or "˜ but not for any of their subroots. The formula for the singular vector corresponding to "˜ is given by formula (4.53) for ′′′ = 2. The formula for the singular vector corresponding to "˜ is given by [122] ˜ ′′′ ,m

v"

=

3m

gk (X2– )3m–k (X1– )m (X2– )k ⊗ v0 ,

k=0

k

gk = (–1) g ′′

2

3m k

(4.60)

[+(H2 ) + 1] . [+(H2 ) + 1 – k]

Note that for the nonstraight root "˜ = !1 + 2!2 = s2 s1 (!2 ), and with condition (2.208) fulﬁlled for m = 1 the formula for the singular vector is given by (4.54) for = 2 and m = 1 (cf. [122]).

160

4 Invariant Differential Operators

4.8.8 Nonstraight Roots Let us say that condition (2.208) is almost fulﬁlled if it is satisﬁed for m = 0. The singular vectors given in the previous subsections are in the generic situation, i.e., when condition (2.208) is fulﬁlled for ", but not for any subroot of ". In this subsection, we shall discuss the situation when condition (2.208) is fulﬁlled for " and is fulﬁlled or almost fulﬁlled for any subroot of ". Consider the explicit expansion of " ∈ B+ into simple roots " = j=1 nj !j , with nj ∈ Z+ . Deﬁne J" ≡ {j | nj ≠ 0} and assume that in addition to (2.208) we have (+ + 1, !∨j ) = mj , j ∈ J" , mj ∈ Z+ ,

mj ∈ N,

j∈J"

i.e., condition (2.208) is fulﬁlled in addition for at least one of the roots !j , j ∈ J" , since j∈J" mj = 0 would contradict fulﬁllment of (7) for ". In this situation, we can give the formula for the singular vector for arbitrary positive roots, i.e., not only for straight roots. The formula that takes up notation from the presentation (4.49) is v",m = c",m (Xp–1 )mnp1 –m˜ 1 ⋯(Xp–r )mnpr –m˜ r (Xp– )mnp (Xp–r )m˜ r ⋯(Xp–1 )m˜ 1 ⊗ v0 ,

(4.61)

˜ s is deﬁned consequently for s = 1, . . . r + 1: where m ˜ s ≡ (+ + 1 – m

s–1

˜ t !pt , !∨ps ) ( = (sps–1 . . . sp1 (+ + 1), !∨ps ) for s > 2 ). m

(4.62)

t=1

˜ 1 = mp1 , m ˜ 2 = mp2 – mp1 ap2 p1 . For = 2 all possible ˜ s ∈ Z+ , and, e.g., m Note that m cases of formula (4.61) were given explicitly in [122]. This formula is almost obvious. First, in the cases described in the previous subsections, it follows directly from the explicit expressions there. In the general case, for ﬁxed s formula (4.62) means that condition (2.208) is fulﬁlled or almost fulﬁlled w.r.t. the root !ps in a Verma mod ˜ t ! pt ule V +s with highest weight shifted by the Weyl dot reﬂection +s ≡ + – s–1 t=1 m ˜s > 0 = sps–1 . . . sp1 ⋅ +, s = 1, . . . r + 1, (+1 = +); w ⋅ + ≡ w(+ + 1) – 1. Thus if m then (Xp–s )m˜ s ⊗ v0s is a singular vector of V +s (v0s denotes the highest weight vector p of V +s ). Analogously, (Xp– )mnp ⊗ v0 is a singular vector of V +r+1 . Here one should ∨ ˜ r+1 ≡ (+r+1 + 1, !p ) = mnp . Next one deﬁnes consecutively for s = r, . . . 1, show that m ˜ ′t !pt , !∨ps ) = (sps+1 . . . spr sp spr . . . sp1 (+ + 1), !∨ps ), and ˜ ′s ≡ (+r+1 – mnp !p + 1 – rt=s+1 m m ′ ˜ s . Finally, one considers Verma modules V +s , with ˜ ′s = mnps – m one should show that m ˜ ′t !pt = sps+1 . . . spr sp spr . . . sp1 ⋅ +, s = r, . . . 1, highest weight +′s ≡ +r+1 – mnp !p – rt=s+1 m ˜ ′s – mn – m ˜ s > 0. Recalling also that for which (Xps ) ps s ⊗ v0 is a singular vector if mnps – m + – m" = s" ⋅ + = sp1 . . . spr sp spr . . . sp1 ⋅ + we note that these singular vectors describe ′ ′ the nesting of Verma modules V +–m" ⊂ V +1 ⊂ . . . ⊂ V +r ⊂ V +r+1 ⊂ V +r ⊂ . . . ⊂ V +1 = V + .

4.8 Explicit Formulae for Singular Vectors

161

Such nestings of Verma modules were extensively described and used in many papers of ours and again in the following chapters. In the above situation, there are also available mixed forms of the singular vectors. We shall illustrate this in the case A2 for " = !1 + !2 , mj = (+ + 1, !∨j ) ∈ Z+ , m = m1 + m2 ∈ N. The singular vector is given by v",m = c1 (X1– )m2 (X2– )m (X1– )m1 ⊗ v0 = =

=

(4.63)

c2 (X2– )m1 (X1– )m (X2– )m2 ⊗ v0 m2

– m2 –k (X2– )m1 a′1 (X2– )m2 (X1– )k+m1 ⊗ v0 k (X1 ) k=0 m1

– m1 –k (X1– )m2 a′0 (X1– )m1 (X2– )k+m2 ⊗ v0 , k (X2 ) k=0

0 where the ﬁrst two lines give the two forms of (4.61) in this case, and a′1 k and ak are given by (4.53). The four expressions in (4.63) are used to prove commutativity of certain embedding diagrams, in particular, the hexagon diagram (2.261) for sl(3, C).

Remark 9: Note that the formulae for singular vectors were given in [127] in the setting of Verma modules over quantum algebras, i.e., involving polynomials of the q-deformation Uq (G–C ) of U(G–C ). Thus, all formulae for singular vectors in this section may be obtained from those in [127] by setting q = 1. ♢

5 Case of the Anti-de Sitter Group Summary The anti-de Sitter group SO(3, 2) and algebra so(3, 2) = AdS4 have attracted attention very early since they have truly remarkable unitary representations known as singletons, which were first discovered by Dirac in 1963 [118]. These representations have been extensively studied by Fronsdal, Flato, and Evans [197, 216–218, 240–243]. There are two singleton representations, called Di and Rac. In terms of the lowest energy value E0 and the spin s0 , the Di has E0 = 1 and s0 = 1/2 while the Rac has E0 = 1/2 and s0 = 0. These representations have remarkably reduced spectrum (weight spaces). Consequently, the singleton field theory has a very large gauge symmetry which enables one to gauge away the singleton fields everywhere except on the boundary of the anti-de Sitter space [218]. Moreover, the direct product of two singletons decomposes into infinitely many massless states of the anti-de Sitter group [216]. However, the methods applied to the above results are not suitable for our purposes. Thus, the aim of this chapter is (following mostly [140] and [142]) to apply systematically to AdS4 the general modern tools that were exposed in the previous chapters. Thus, we introduce the necessary representations of the AdS4 algebra and group. We give explicitly all singular (null) vectors of the reducible AdS4 Verma modules. These are used to obtain the AdS4 invariant differential operators. Using this we display the main multiplets of elementary representations – a diagram involving four partially equivalent reducible representations one of which contains all finite-dimensional irreps of the AdS4 algebra. We study in more detail the cases involving the positive energy UIRs, in particular, the Di and the Rac singletons, and the massless UIRs. In the massless case, we discover the structure of sets of 2s0 – 1 conserved currents for each spin s0 UIR, s0 = 1, 32 , . . . All massless cases are contained in a one-parameter subfamily of the quartet diagrams mentioned above, the parameter being the spin s0 . Further, we give the classification of the so(5, C) irreps presented in a diagrammatic way which makes easy the derivation of all character formulae.

5.1 Preliminaries 5.1.1 Lie Algebra We start with the complexiﬁcation G C = so(5, C) = B2 of the algebra G = so(3, 2). We use the standard deﬁnition of G C given in terms of the Chevalley generators Xi± and Hi (i = 1, 2) by the relations [Hj , Hk ] = 0, [Hj , Xk± ] = ±ajk Xk± , [Xj+ , Xk– ] = $jk Hj , n

m m=0 (–1)

n m

Xj±

m

(5.1)

n–m Xk± Xj± = 0, j ≠ k, n = 1 – ajk ,

where

(ajk ) =

(!∨j , !k )

2 –2 = –1 2

(5.2)

5.1 Preliminaries

is the Cartan matrix of G C , !∨j ≡

2!j (!j ,!j )

163

is the co-root of !j and (⋅, ⋅) is the scalar

product of the roots, so that the nonzero products between the simple roots are (!1 , !1 ) = 2, (!2 , !2 ) = 4, and (!1 , !2 ) = –2. The elements Hi span the Cartan subalgebra H of G C , while the elements Xi± generate the subalgebras G ± . We shall use the standard triangular decomposition G C = G+ ⊗ H ⊕ G– , G± ≡ ⊕ G! , !∈B±

(5.3)

where B+ and B– are the sets of positive and negative roots, resp. Explicitly, we have B± = {±!1 , ±!2 , ±!3 , ±!4 },

!3 = !1 + !2 , !4 = 2!1 + !2 .

(5.4)

Let us denote the root space vector of G! by X! , or more explicitly: Xk± ≡ X±!k , where k = 1, 2, 3, 4. To give the full Cartan–Weyl basis, we need to deﬁne also Xk± and (k = 3, 4) for which we follow [127]: 1 X4± = ± [X1± , X3± ]. 2

X3± = ±[X1± , X2± ],

(5.5)

Then we have H3 ≡ [X3+ , X3– ] = H1 + 2H2 , H4 ≡ [X4+ , X4– ] = H1 + H2 .

(5.6)

Note that for Hk also holds +(Hk ) = (+, !∨k ),

∀ + ∈ H∗ ,

k = 1, 2, 3, 4.

(5.7)

The algebra G = so(3, 2) is the split real form of G C so we can use the same basis (but over R) and the same root system. Thus, we can use the second-order Casimir of G C : C2 =

1 + – 1 (X X + X1– X1+ ) + X2+ X2– + X2– X2+ + (X3+ X3– + X3– X3+ ) 2 1 1 2 1 + X4+ X4– + X4– X4+ + H12 + H22 + H1 H2 . 2

(5.8)

It useful to relate the Cartan–Weyl basis given above with the so(3, 2) generators XAB : H1 = 2iX12 , X1± = X10 ± iX20 , H2 = X34 – iX12 , X2± = ∓

X4± =

1 (X23 ± X24 + iX14 ± iX13 ), 2

(5.9a) (5.9b)

X3± = ∓ i (X03 ± X04 ) ,

(5.9c)

1 (X24 ± X23 ∓ iX14 – iX13 ) . 2

(5.9d)

164

5 Case of the Anti-de Sitter Group

It is easy to see that the ten generators XAB = – XBA (A, B = 0, 1, 2, 3, 4) satisfy the standard so(3, 2) commutation relations [XAB , XCD ] = 'AC XBD + 'BD XAC – 'AD XBC – 'BC XAD ,

(5.10)

where '11 = '22 = '33 = –'00 = –'44 = 1, 'jk = 0 if j ≠ k. (Note that often are used also the generators MAB = iXAB ) 5.1.2 Finite-Dimensional Realization It is useful to have a ﬁnite-dimensional realization of G X12 = –

i 33 0 , 2 0 33

Xa3 =

X04 =

i 0 3a , a = 1, 2, 2 3a 0

i 0 12 ; 2 12 0

X34 =

1 12 0 , 2 0 –12

Xa0 =

1 3a 0 , a = 1, 2; 2 0 –3a

X03 =

i 0 12 , 2 –12 0

Xa4 =

i 0 3a , a = 1, 2. 2 –3a 0

(5.11a)

(5.11b)

The hermiticity properties of this deﬁning realization are † = –X XAB AB † =X XAB AB

for for

(A, B) = (0, 4), ( j, k), (A, B) = (0, j), ( j, 4),

(5.12a) (5.12b)

where j, k = 1, 2, 3. Note that the four generators in (5.12a) (or (5.11a)) are compact. They span the maximal compact subalgebra K = so(3) ⊕ so(2), the so(3) being spanned by the generators Xjk (j, k = 1, 2, 3) and the so(2) being spanned by the generator X04 . The six generators in (5.12b) (or (5.11b)) are noncompact. Thus, in this basis, we can identify – up to sign – the Hermitian conjugation with the Cartan involution ( which in general is deﬁned by (:X↦X ( : X ↦ –X

if X if X

is compact, is non compact.

(5.13)

From the above, we have explicitly a ﬁnite-dimensional representation for the Cartan– Weyl basis: 33 0 3+ 0 3– 0 H1 = , X1+ = , X1– = ; (5.14a) 0 33 0 –3+ 0 –3–

5.1 Preliminaries

e2 0 H2 = , 0 –e1 X3+ = X4+ =

X2+

0 3– = , 0 0

X2–

0 0 = ; 3+ 0

0 12 , 0 0

X3– =

0 3+ , 0 0

X4– =

0 0 , 12 0

1 10 , e1 ≡ (1 + 33 ) = 00 2 1 01 , 3+ ≡ (31 + i32 ) = 00 2

(5.14b)

12 0 ; 0 –12

(5.14c)

e1 0 ; 0 –e2

(5.14d)

H3 = H1 + 2H2 =

0 0 , 3– 0

165

H4 = H1 + H2 =

1 00 e2 ≡ (1 – 33 ) = ; 01 2 1 00 3– ≡ (31 – i32 ) = . 10 2

5.1.3 Structure Theory The Iwasawa decomposition of G = so(3, 2) is G = K ⊕ A 0 ⊕ N0 ,

(5.15)

so that dim A0 = 2, dim N0 = 4. Since G is maximally split, then the centralizer M0 of A0 in K is zero; thus, the minimal parabolic P0 and the corresponding Bruhat decomposition are P = A0 ⊕ N0+ ,

G = A0 ⊕ N0+ ⊕ N0– ,

(5.16)

where N0– ≅ N0+ . Further, the algebra so(3, 2) has two maximal parabolic subalgebras, but it turns out that they coincide (cf. (3.83) for j = 1, 2 = r). Thus, we have the maximal cuspidal parabolic and the corresponding Bruhat decomposition P = M ⊕ A ⊕ N– ,

G = N+ ⊕ M ⊕ A ⊕ N–

(5.17)

in which all subalgebras have physical meaning, namely, the subalgebra M is the Lorentz subalgebra of three-dimensional Minkowski space-time M 3 , i.e., M = so(2, 1), the subalgebras N + and N – are (three-dimensional) Abelian and represent the translations of M 3 and special conformal transformations of M 3 , and the algebra A represents the dilatations of M 3 (A commutes with M). Explicitly, we choose the following basis: M : {X12 , Xa3 , a = 1, 2},

A : {H3 },

N ± : {Xk± , k = 2, 3, 4}.

(5.18)

Note that the generators H1 and X1± are complex linear combinations of those of M (cf. (5.9aa)), i.e., span the complexiﬁcation MC = so(3, C) of M, and actually

166

5 Case of the Anti-de Sitter Group

± C ± C represent a triangular decomposition of MC = MC + ⊕Mh ⊕M– : X1 spanning (M ) , H1 spanning the Cartan subalgebra Mh . Matters are arranged so that the factors from (5.3) are related to the complexiﬁcation of (5.17) in the following obvious manner:

G ± = N ± ⊕ (M± )C ,

H = A ⊕ Mh .

(5.19)

5.1.4 Lie Groups Finally, we introduce the corresponding connected Lie groups: G = SOe (3, 2) with Lie algebra G = so(3, 2), K = SO(3) × SO(2) is the maximal compact subgroup of G, A0 = exp(A0 ) = SOe (1, 1) × SOe (1, 1) is Abelian simply connected, A = exp(A) = SOe (1, 1) is also Abelian simply connected, N0± = exp(N0± ) are Abelian simply connected subgroups of G preserved by the action of A0 , and N ± = exp(N ± ) are Abelian simply connected subgroups of G preserved by the action of A. The group M ≅ SOe (2, 1) (with Lie algebra M) commutes with A. The subgroup P0 = A0 N0 , where N0 = N0+ or N = N0– , is a minimal parabolic subgroup of G. The subgroup P = MAN, where N = N + or N = N – , is a maximal cuspidal parabolic subgroup of G.

5.2 Representations and Invariant Operators 5.2.1 Elementary Representations Following the general scheme of the previous chapter, we work with elementary representations (ERs) induced from representations of P = MAN + . We take p = 0, 1, 2, . . . to ﬁx a ( p + 1)-dimensional representation of M, and c ∈ C to ﬁx a (nonunitary) character of A. This data is enough to determine a weight D ∈ H∗ as follows: D(H1 ) = p, D(H3 ) = c. Thus, we shall denote the ERs by CD . Using the fact that the group G is maximally split, we can work with GC instead. Thus, the ERs may be considered also holomorphic ERs of GC , and their functions can be taken to be complex-valued C∞ functions on G or GC . We use the right covariance to pass from functions on the group G to the so-called ˆ on the coset space GC /B, where B = exp(H) exp(G+ ) is a Borel reduced functions > C subgroup of G . Note that GC /B is a completion of G– = exp(G– ), and as usual, we shall use the local coordinates of G– , ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

⎛

1

⎜ ⎜ z G– = g = ⎜ ⎜ ⎪ ⎪ ⎝ . + zv/2 ⎪ ⎪ ⎩ ' – vz2 /6

0

0

1

0

v

1

. – zv/2

–z

⎞⎫ 0 ⎪ ⎪ ⎟⎪ ⎪ 0⎟⎬ ⎟ , ⎟ 0⎠⎪ ⎪ ⎪ ⎪ ⎭ 1

(5.20)

obtained from exponentiation of the general term: zX1– + vX2– + . X3– + 'X4– of G– . ˆ v, . , ') are polynomials in the variable z of degree p and smooth The functions >(z,

5.2 Representations and Invariant Operators

167

functions in the other three variables. Consistently with the above, we have for the right action of the Cartan algebra: ˆ = p >, ˆ p = 0, 1, . . . , Hˆ 1 >

(5.21a)

ˆ = c >. ˆ Hˆ 3 >

(5.21b)

ˆ is also calculated easily and after some changes of variables The right action of G– on > (z, v, . , ') ↦ (z, v, x, u) we obtain Xˆ 1– = ∂z , Xˆ 2– = ∂v – z∂x + z2 ∂u , Xˆ 3– = ∂x – 2z∂u , Xˆ 4– = ∂u .

(5.22)

ˆ is derived in a straightforward way, and after the The left (representation) action on > same changes of variables we have 0D (X1+ ) = z2 ∂z + 2x∂v + u∂x – zp,

(5.23a)

1 0D (X2+ ) = –(x + zv)∂z + v2 ∂v + vx∂x + x2 ∂u – (c – p)v, 2

(5.23b)

0D (H1 ) = 2z∂z – 2v∂v + 2u∂u – p,

(5.23c)

1 0D (H2 ) = –z∂z + 2v∂v + x∂x – (c – p), 2

(5.23d)

0D (X1– ) = –∂z + v∂x + 2x∂u ,

(5.23e)

0D (X2– ) = –∂v .

(5.23f)

In addition, we have 0D (X3+ ) = [0D (X1+ ), 0D (X2+ )] = (vz2 – u)∂z + 2xv∂v + (x2 + uv)∂x + 2xu∂u – (x + zv)p – (c – p)x 0D (X3– ) = [0D (X2– ), 0D (X1– )] = – ∂x

(5.24a) (5.24b)

0D (H3 ) = [0D (X3+ ), 0D (X3– )] = 0D (H1 ) + 20D (H2 ) = 2x∂x + 2v∂v + 2u∂u – c 0D (X4+ ) =

1 [0D (X1+ ), 0D (X3+ )] = (xz2 + uz)∂z + x2 ∂v + xu∂x 2 1 + u2 ∂u – (xz + u)p – (c – p)u 2

(5.24c)

(5.24d)

0D (X4– ) = [0D (X3– ), 0D (X1– )] = – ∂u

(5.24e)

0D (H4 ) = [0D (X4+ ), 0D (X4– )] = 0D (H1 ) + 0D (H2 )

(5.24f)

1 = z∂z + x∂x + 2u∂u – (c + p). 2

(5.24g)

168

5 Case of the Anti-de Sitter Group

Since the left and right actions commute, we can calculate the value of the secondorder Casimir in either of them and obtain 1 0D (C2 ) = Cˆ2 = ( p + 1)2 + (c + 3)2 . 4

(5.25)

5.2.2 Elementary Representations Induced from P0 We would like consider also ERs induced from the minimal parabolic subgroup P0 = A0 N0+ . In that case we consider C∞ functions over G/P0 which is locally isomorphic to N0– . In fact, as in the case P-induction, we work on G– which is comˆ 0 (z, v, . , ') or > ˆ 0 (z, v, x, u), plexiﬁcation of N0– . Thus, we shall use also functions > ˆ 0 functions are not necessarily polynomials in z. Thus, the difference being that the > the left and right vector ﬁelds above are valid also for P0 -induced case, the difference being that p may be arbitrary complex number. Thus, the P-induced ERs are subrepresentations of P0 -induced ERs when p ∈ N. Next, the associated Verma modules of the P0 -induced ERs will be the Verma modules over G C = so(5, C). In fact, since G = so(3, 2) is a maximally split subalgebra of G C then the P0 -induced representations form the same set as the Verma modules over G C . Thus, we postpone the study of reducible P0 -induced ERs to Section 5.3 where we classify the Verma modules.

5.2.3 Singular Vectors We now consider our main tool – the singular vectors of Verma modules V D of highest weight D. The general reducibility conditions (2.208) for V D spelled out for the four positive roots in our situation are m1 = m1 (D) ≐ D(H1 ) + 1 ∈ N,

(5.26a)

m2 = m2 (D) ≐ D(H2 ) + 1 ∈ N,

(5.26b)

m3 = m3 (D) ≐ D(H3 ) + 3 = m1 + 2m2 ∈ N,

(5.26c)

m4 = m4 (D) ≐ D(H4 ) + 2 = m1 + m2 ∈ N.

(5.26d)

The singular vectors corresponding to these cases are v!1 ,m1 = (X1– )m1 v0 ,

m1 ∈ N,

(5.27a)

v!2 ,m2 = (X2– )m2 v0 ,

m2 ∈ N,

(5.27b)

v!3 ,m3 =

m3

k=0

ak (X2– )m3 –k (X1– )m3 (X2– )k v0 ,

m3 ∈ N,

5.2 Representations and Invariant Operators

ak =

v!4 ,m4 =

⎧ 2 ⎨a0 (–1)k m3 mm–k ,

m2 ∉ { 0, . . . , m3 },

⎩ a0 $k,m2 ,

m2 ∈ { 0, . . . , m3 },

k

2m4

2

bk (X1– )2m4 –k (X2– )m4 (X1– )k v0 ,

169

(5.27c)

m4 ∈ N,

k=0

bk =

⎧ 1 ⎨b0 (–1)k 2m4 mm–k , k

⎩

b0 $k,m1 ,

1

m1 ∉ { 0, . . . , 2m4 },

(5.27d)

m1 ∈ { 0, . . . , 2m4 },

(Note that in each of the four cases (5.27a) only the relevant mj must be a natural number (as displayed).) Formulae (5.27a,b) are the general expressions valid for any simple root [126], while (5.27c,d) are given in [127]. Certainly, (2.208) may be fulﬁlled for several positive roots, even for all of them if (2.208) is fulﬁlled for all simple roots. For future use, we note one important special case of (5.27c) when m3 is even, and m1 = 1, then m2 = 21 (m3 – 1) ∉ Z. In this case, (5.27c) can be written as follows: 1m v!3 ,m3 = (X3– )2 – 4X4– X2– 2 3 v0 + P(G– ) X1– v0 ,

m3 ∈ 2N, m1 = 1,

(5.28)

where in order to achieve this simple form we have used also the nonsimple root vectors X3– , X4– , and P(G– ) is a polynomial with exact form not important for our purposes. It is useful for future reference to write down the numbers corresponding to the four cases of D′ , i.e., to write down m1 (D′ ), m2 (D′ ) for the four cases of (5.26a). In fact, though it is redundant, the picture becomes more instructive, if we write down also m3 (D′ ), m4 (D′ ). Explicitly, we have for mi ≡ mi (D)

m1 (D′ ), m2 (D′ ), m3 (D′ ), m4 (D′ ) = (–m1 , m4 , m3 , m2 ) ,

D′ = D – m1 !1 , m1 ∈ N, m1 (D′ ), m2 (D′ ), m3 (D′ ), m4 (D′ ) = (m3 , –m2 , m1 , m4 ) ,

(5.29a)

D′ = D – m2 !2 , m2 ∈ N, m1 (D′ ), m2 (D′ ), m3 (D′ ), m4 (D′ ) = (m1 , –m4 , –m3 , –m2 ) ,

(5.29b)

D′ = D – m3 !3 , m3 ∈ N, m1 (D′ ), m2 (D′ ), m3 (D′ ), m4 (D′ ) = (–m3 , m2 , –m1 , –m4 ) ,

(5.29c)

D′ = D – m4 !4 , m4 ∈ N.

(5.29d)

170

5 Case of the Anti-de Sitter Group

Now we shall express conditions (2.208) for the four positive roots taking into account the signatures of our representations (cf. [170]): 1 1 s0 = p ∈ Z+ , 2 2 1 m2 = D(H2 ) + 1 = 1 – E0 – s0 , E0 = – c, 2 m3 = D(H3 ) + 3 = m1 + 2m2 = 3 – 2E0 , m1 = D(H1 ) + 1 = 2s0 + 1 ,

m4 = D(H4 ) + 2 = m1 + m2 = 2 – E0 + s0 ,

(5.30a) (5.30b) (5.30c) (5.30d)

where we have introduced also the traditionally used energy E0 and spin s0 . For future reference, we record an explicit expression for D: 1 1 D = (m1 – 1) !1 + (m3 – 3) !3 = s0 !1 – E0 !3 . 2 2

(5.31)

The eigenvalue of H3 is called the energy since upon contraction of so(3, 2) to the Poincaré algebra, the generator H3 goes to the translation operator P0 . Analogously, H1 is the third component of the angular momentum. In terms of E0 , s0 (and taking into account their relation to p, c in (5.30a)) the Casimir from (5.25) becomes C2 = E0 (E0 – 3) + s0 (s0 + 1) + 5/2

(5.32)

which was the preferred usage in [216–218]. In the latter papers, the irreducible representations of so(3, 2) are denoted by D(E0 , s0 ) which we shall also use. Next we note that if m1 , m2 ∈ N, then the irreducible representations with nonpositive energy E0 = (3 – m1 – 2m2 )/2, and spin s0 = (m1 – 1)/2, are the nonunitary ﬁnite-dimensional representations of G. Their dimension is m1 m2 m3 m4 /6. (For example, m1 = m2 = 1 gives the trivial one-dimensional representation; the fundamental representations are obtained for m1 = 1, m2 = 2 and m1 = 2, m2 = 1) We are interested mostly in the positive energy UIRs of G given as follows [118, 197, 216–218, 240–242] (with s0 ∈ 21 Z+ ): Rac : D(E0 , s0 ) = D(1/2, 0), D(E0 ≥ 1, s0 = 0),

Di : D(E0 , s0 ) = D(1, 1/2),

D(E0 ≥ 3/2, s0 = 1/2),

D(E0 ≥ s0 + 1, s0 ≥ 1).

(5.33a) (5.33b) (5.33c)

The ﬁrst two are the singleton representations discovered by Dirac [118] and the last ones for E0 = s0 + 1 correspond to the spin-s0 massless representations. Comparing the list (5.30a) with (5.33a) we note that (5.30a) holds in all cases of (5.33a), while (5.30b) never holds because m2 ≤ 1/2. Next, we note that m3 is a positive integer only for E0 = 1/2, 1, in which case m3 = 2, 1, respectively. Similarly, m4 is a positive integer only for E0 – s0 = 1, and that integer is m4 = 1.

5.2 Representations and Invariant Operators

171

The singular vectors corresponding to these cases are (choosing the normalization constants appropriately) v!1 ,m1 = (X1– )m1 v0 , m1 = 2s0 + 1 ∈ N,

(5.34a)

v!3 ,1 = (s0 X3– + X2– X1– ) v0 , m3 = 1,

(5.34b)

v!3 ,2 = {(2s0 – 1)(2s0 + 3) X2– (X1– )2 X2– – 1 – (2s0 + 1)(2s0 + 3) (X2– )2 (X1– )2 2 1 – (2s0 – 1)(2s0 + 1) (X1– )2 (X2– )2 }v0 2 = { (1 – 4s20 ) (X3– )2 – 4(1 – 2s0 ) X4– X2– + 4(1 – 2s0 ) X3– X2– X1– – 4(X2– )2 (X1– )2 } v0 , m3 = 2;

(5.34c)

va4 ,1 = { 2s0 (2s0 – 1) X4– + (2s0 – 1) X3– X1– + X2– (X1– )2 } v0 ,

(5.34d)

m4 = 1, where the nonsimple root vectors are used in the appropriate ordering. Note that (5.34b) for s0 = 0, (5.34c) for s0 = 1/2, and (5.34d) for s0 = 0, 1/2 are composite singular vectors being descendants of (5.34a). (The latter follow also from the general formulae (5.27a).) 5.2.4 Invariant Differential Operators The main ingredient of our general procedure is that to every singular vector there corresponds an invariant differential operator. Thus, following (4.35) and (4.36) we know that to the singular vector vs = v!,m! corresponds an invariant differential operator D!,m! : CD → CD–m! ! . These operators give rise to the G-invariant equations: ˆ => ˆ ′, D!,m! >

ˆ ∈ CD , > ˆ ′ ∈ CD–m! ! , >

ˆ = 0, D!,m! >

ˆ ∈ CD , >

! noncompact;

! compact.

(5.35a) (5.35b)

We recall that a compact root is deﬁned by the property that it has zero value on the dilatation subalgebra A, i.e., in our case, on the generator H3 , which means that only the root !1 is compact. In fact, equation (5.35ab) just expresses the fact that we have induction from ﬁnite-dimensional irreps of the Lorentz group M. In our case, the ˆ = 0, which is trivially satisﬁed since all our functions counterpart of (5.35ab) is ∂zm > are polynomials of degree 2s0 = m – 1 = p.

172

5 Case of the Anti-de Sitter Group

For future use, we mention one important case in which arises the d’Alembert operator. Namely, we consider m1 = 1, m3 ∈ 2N, the singular vectors are (5.27a) and (5.28) (the latter equivalent to (5.27c)). Thus, the relevant equations are ˆ = 0, ∂z >

m1 = 1,

(5.36a)

ˆ = {(∂x – 2z∂u )2 – 4 ∂u (∂v – z∂x + D!3 ,m3 > 1m ˆ => ˆ ′, = ∂x2 – 4∂u ∂v 2 3 >

1 z2 ∂u )} 2 m3

ˆ + P(Gˆ– ) ∂z > ˆ >

m1 = 1, m3 ∈ 2N.

(5.36b)

Note that the operator ∂x2 – 4∂v ∂u is the d’Alembert operator in M 3 if we identify the coordinates as follows: x = y0 ,

u = y1 – iy2 ,

v = y1 + iy2 .

(5.37)

Then indeed we have ∂y2 – 4∂v ∂u = ∂02 – ∂12 – ∂22 ≡ ◻.

(5.38)

Analogously, the Minkowski length is given by x2 – uv = y02 – y12 – y22 ≡ y2 .

(5.39)

5.2.5 Reducible ERs The reducible P-induced ERs are grouped in quartets, doublets, and singlets. The main multiplet of reducible representations here is a quartet which is represented in the following ﬁgure: + (q, – q − k) χqk

− (q, k) χqk

kα 4

kα 2

ʹ− (q + 2k,−k) χqk

qα 3

+

ʹ (q + 2k, –q − k) χqk

5.2 Representations and Invariant Operators

173

The quartets are parametrized by the positive integer Dynkin labels m1 , m2 (2.207) which we denote here by q, k ∈ N, resp. Thus, the E0 , s0 signatures of the ERs of the quartet are ± 7q,k =

′± = 7q,k

.

.

/ 1 1 (3 ± (q + 2k)), (q – 1) , 2 2 / 1 1 (3 ± q), (q + 2k – 1) , 2 2

+, (q, –q – k) , –, (q, k)

(5.40)

+, (q + 2k, –q – k) , –, (q + 2k, –k)

where we have given also the Dynkin labels for all members of quartet. The corresponding ﬁnite-dimensional irrep is denoted as Eq,k , it has dimension q k (q + k) (q + – . They are obtained in the following way: 2k)/6, and it is contained in the ER 7q,k Eq,k = {f ∈ C

– 7q,k

: q, k ∈ N,

q

D!1 ,q f = ∂z f = 0,

D!2 ,k f = 0}.

(5.41)

Relatedly, the quartets also contain the holomorphic discrete series representations in + . This follows the results of Harish-Chandra [279] the holomorphic discrete series 7q,k happen when the Harish-Chandra parameters m! (2.206) are negative integers for all noncompact roots, i.e., when m2 , m3 , m4 < 0. The ﬁgure contains also the Knapp–Stein (KS) integral operators depicted on the ﬁgure by dashed arrows. They are related to the restricted root system of G w.r.t. A, where the root spaces are N ± . In our case (cf. (5.18)), this restricted root system has just one root which coincides with !3 restricted to A due to the facts [H3 , X1± ] = 0 and [H3 , Xk± ] = ± 2 Xk± , k = 2, 3, 4. The action of this root on the signatures naturally coincides with the action of !3 , but the action itself is valid for arbitrary signatures. The KS integral operator involves the conformal two-point function GD , D ≅ (m1 , m2 , m3 , m4 ). For scalar functions (m1 = 1), the latter is given by GD (y) =

N(D) 1

(y2 ) 2 (3+m3 )

,

y2 = y02 – y12 – y22 ,

(5.42)

where N(D) is a normalization constant related to the Plancherel measure, and the KS integral operator itself is given by ˜ AD : CD → CL , L˜ = D – m3 !3 , * ˆ ′ ) d3 y ′ , ˆ ∈ CD . ˆ (AD >)(y) > = GD (y – y′ ) >(y

(5.43)

174

5 Case of the Anti-de Sitter Group

If m3 ∉ Z the operators AD and AL˜ are inverse to each other: AL˜ ○ AD = idCD ,

AD ○ AL˜ = idCL˜ .

(5.44)

′– to 7′+ deFor m3 = q ∈ N as happens in the quartet, the operator AD acting from 7q,k q,k generates to the differential operator Dm3 !3 in the same situation (cf. (5.36a), (5.37), and (5.38)). In particular, for m3 ∈ 2N we use

N(D(m3 + :))

lim GD(m3 +:) (y) = lim

: ↦+0

: ↦+0

1 (y2 ) 2 (3+m3 +:)

∼ lim

: ↦+0

: 1 (y2 ) 2 (3+m3 +:)

1

∼ ◻ 2 m3 $3 (y) (cf. [159, 164, 253]) and then we have * 1 1 ˆ ′ ) d3 y′ = ◻ 2 m3 >(y) ˆ ˆ = D!3 ,m3 >(y). ˆ (AD >)(y) ∼ ◻ 2 m3 $3 (y – y′ ) >(y

(5.45)

(5.46)

For the same values m3 ∈ N the operator AL˜ remains integral and acts in the opposite direction w.r.t. Dm3 !3 as is clear on the quartet ﬁgure. (Note that (5.45) remains valid also for m3 = 0 when D′ = D and the operator AD reduces to the identity operator idCD .) Next we consider multiplets – doublets, of which there are two types. The doublets of the ﬁrst type are denoted by 7q± , and the expression for their ± or from 7′± by setting k = 0, i.e., signatures can be obtained from the signatures of 7q,k q,k 7q± =

.

/ 1 1 (3 ± q), (q – 1) , q ∈ N. 2 2

(5.47)

Here in 7q+ are contained the limits of holomorphic discrete series. The limits of the holomorphic discrete series happen when some of the noncompact Harish-Chandra parameters m! become zero, while the rest of the noncompact numbers m! remain negative. In our case: m2 , m3 = –q < 0, m4 = 0. ± The doublets of the second type are denoted by 7q,,/2 , and the expression for their ± ′± by setting 2k ↦ , ∈ signatures can be obtained from the signatures of 7q,k or from 7q,k 2N – 1, i.e., . / 1 1 ± 7q,,/2 = (3 ± (q + ,)), (q – 1) , (5.48a) 2 2 . / 1 1 ′± (3 ± q), (q + , – 1) , q ∈ N, , ∈ 2N – 1. (5.48b) 7q,,/2 = 2 2 ± ′± is not related to the doublet 7q,,/2 by any operator. Note that the doublet 7q,,/2 In all doublets there is a differential operator (degenerated KS operator as above) from 7– to 7+ and a KS operator from 7+ to 7– .

5.2 Representations and Invariant Operators

The singlets have the signature . / 3 7ns = , (n – 1)/2 , 2

n ∈ N,

175

(5.49)

± The expression for their signatures can be obtained either from the signatures of 7q,k by setting q = 0, k = n/2, when n is even, or from (5.48ab) for q = 0, , = n, when n is odd.

5.2.6 Holomorphic Discrete Series and Positive Energy Representations We will now discuss how the positive energy representations displayed in (5.33a) ﬁt in the multiplets, and when they are inﬁnitesimally equivalent to holomorphic discrete series. – since E = 3 – 1 q – k ≤ 0, and 7′– First, we note that no such irreps can ﬁt 7q,k 0 q,k 2 2 since E0 = 32 – 21 q ≤ 1 but s0 = 21 (q – 1) + k ≥ 1. We notice that all positive energy irreps from (5.33ab) that would ﬁt some multiplet can be parametrized with one parameter k ∈ Z, k ≥ – 1. The parametrization is as follows: m1 = q = 2s0 + 1,

m2 = –q – k

⇒

E0 = 2 + s0 + k.

+ , q = 2s + 1 (cf. (5.40)), which In fact, for k ≥ 1 we obtain tautologically all ERs 7q,k 0 contain all holomorphic discrete series. For k = 0 we obtain all ERs 7q+ , q = 2s0 + 1 (cf. the table) which contain all limits of holomorphic discrete series. Finally, for k = –1 we obtain the positive energy irreps D(1, 0), D( 32 , 21 ), and D(s0 + 1, s0 ) (s0 ≥ 1), which are ′+ contained in the ERs 71– , 71s , and 72s . 0 –1,1 From this we see that the above would ﬁt the Enright–Howe–Wallach [194] picture with A(+0 ) = 1, (the parameter z from [194] corresponds to our –k). Accordingly the irreps D(1, 0), D( 32 , 21 ), and D(s0 + 1, s0 ) (s0 ≥ 1) are in GVMs which are FRPs. In fact, exceptionally, the UIRs D(1, 0), D( 32 , 21 ), are isomorphic to the corresponding GVMs since they happen to be irreducible. (See the details in [140].) Finally, from the list of positive energy irreps, it remains to discuss the Rac and the Di from (5.33aa). They are found in doublets (5.48a). With respect to the positive energy spectrum they are isolated points below – by 21 -spacing – the FRPs D(1, 0), D( 32 , 21 ), resp. The Rac is in the reducible ER 7– 1 . This ER is partially equivalent to the ER 7+ 1 1, 2

1, 2

(denoted as Rac*). The latter’s GVM is irreducible and also of positive energy: s0 = 0, E0 = 52 . The invariant operator acting from the Rac ER to Rac* is the d’Alembert operator (obtained by reduction of a KS operator [140]). The Di is in the reducible ER 7′–1 . This ER is partially equivalent to the ER 7′+1 1, 2

1, 2

(denoted as Di*). The latter’s GVM is irreducible and also of positive energy: s0 = 21 , E0 = 2. All this is illustrated in the following ﬁgure

176

5 Case of the Anti-de Sitter Group

. . . . . . . . .

E0 = s0 + k HDS

. . . . . . . . .

E0 = s0 + k HDS

E0 = s0 + 3

HDS

E0 = s0 + 3 HDS

E0 = s0 + 2

LHDS

E0 = s0 + 2 LHDS

E0 = s0 + 1

FRP

E0 = s0 + 1 FRP

E0 = s0 + s0 = 0,

1 2

Rac, Di

1 2

s0 = 1,

3 , ... 2

5.2.7 Invariant Differential Operators and Equations Related to Positive Energy UIRs The differential operators related to the unitary cases correspond to the singular vectors in (5.34a) and are given as follows: D!1 ,m = ∂zm ,

m = m1 = p + 1 = 2s0 + 1 ∈ N,

D!3 ,1 = s0 (∂x – 2z∂u ) + (∂v – z∂x + z2 ∂u ) ∂z ,

(5.50a) m3 = 1,

(5.50b)

D!3 ,2 = (1 – 4s20 ) (∂x – 2z∂u )2 – 4(1 – 2s0 ) ∂u (∂v – z∂x + z2 ∂u ) + 4(1 – 2s0 ) (∂x – 2z∂u ) (∂v – z∂x + z2 ∂u ) ∂z – 4(∂v – z∂x + z2 ∂u )2 ∂z2 ,

m3 = 2,

(5.50c)

Da4 ,1 = 2s0 (2s0 – 1) ∂u + (2s0 – 1) (∂x – 2z∂u ) ∂z + (∂v – z∂x + z2 ∂u ) ∂z2 , Next we analyze the important cases separately.

m4 = 1.

(5.50d)

5.2 Representations and Invariant Operators

177

5.2.8 Rac In this case the energy and spin are: E0 = 21 , s0 = 0. The equations are (5.50a,c) (cf. also (5.36)): ˆ = 0, ∂z >

(5.51a)

ˆ = {(∂x – 2z∂u )2 – 4 ∂u (∂v – z∂x + z2 ∂u ) D!3 ,2 > ˆ + 4 (∂x – 2z∂u ) (∂v – z∂x + z2 ∂u ) ∂z – 4(∂v – z∂x + z2 ∂u )2 ∂z2 } > 2 ˆ=◻> ˆ => ˆ ′. = ∂x – 4∂u ∂v >

(5.51b)

′

Note that the target space CD , D′ = D – 2!3 , is again of scalar functions (s′0 = 0) and ′ unitary (E0′ = 52 ). Thus, CD is reducible, since the image ∂x2 – 4∂u ∂v CD can not be ′ ′ ˆ ′ on the RHS belong to a proper subspace of CD . Note that onto CD , the functions > ′ the Verma module V D is reducible only w.r.t. (5.30a) so that m′1 = m1 = 1, thus, the ′ ˆ ′ = 0. functions of CD obey only one differential equation: ∂z > The Rac case is the lowest case of the subfamily in which the invariant operator is directly related to the d’Alembertian, cf. (5.36).

5.2.9 Di In this case the energy and spin are: E0 = 1, s0 = 21 . The equations are (5.50a,b)

ˆ= D!3 ,1 >

01 2

ˆ = 0, ∂z2 >

(5.52a) 1

ˆ => ˆ ′. (∂x – 2z∂u ) + (∂v – z∂x + z2 ∂u ) ∂z >

(5.52b)

The invariant equation (5.52b) has the advantage over the usual writing of the equation for the “Di” ﬁeld (cf. [242]) since it is a scalar equation encompassing all information. It is easy to decompose the equation in components by setting ˆ Di = > ˆ0 + z> ˆ 1, >

ˆ i = 0, ∂z >

(5.53)

ˆ ′ in (5.52b). Substituting the above in (5.52b) and extracting the and analogously for > coefﬁcients of the resulting polynomial in z we have 1 ˆ ′0 , ˆ 0 + ∂v > ˆ1 => ∂x > 2

(5.54a)

1 ˆ ′1 . ˆ1 => ∂x > 2

(5.54b)

ˆ0 + ∂u >

The ﬁrst equation is the coefﬁcient at z0 of (5.52b), the second is at z1 , while the coefﬁcient at z2 is identically zero.

178

5 Case of the Anti-de Sitter Group

′

Note that the target space CD in (5.52b), D′ = D – !3 , is again of two-component ′ functions (s′0 = 21 ) and unitary (E0′ = 2). Thus, CD is reducible, as in the Rac case, and ′ again the Verma module V D is reducible only w.r.t. (5.30a) so that m′1 = m1 = 2, thus, ′ ˆ ′ = 0. the functions of CD obey only one differential equation: ∂z2 > Now we rewrite (5.54) in matrix form: ⎞ ⎛ 1

∂ ∂ ˆ0 ˆ0 > x v ˆ ′0 > ⎟ > ⎜2 =⎝ = . (5.55) A ⎠ 1 ˆ ′1 > ˆ1 ˆ1 > > ∂u ∂x 2 Further we multiply both sides of (5.55) by the operator ⎛

⎞

1 ∂ ⎜2 x B=⎝

–∂v

⎟ , 1 ⎠ ∂x 2

–∂u

A B = det A 1

(5.56)

and we obtain: det A

ˆ0 > ˆ1 >

=

=B

ˆ ′0 >

1 2 ∂ – ∂u ∂v 4 x

ˆ ′1 >

ˆ0 > ˆ1 >

⎞

⎛

1 ∂ ⎜2 x =⎝ –∂u

–∂v 1 ∂x 2

⎟ ⎠

1 = ◻ 4

ˆ ′0 >

ˆ0 >

ˆ1 >

ˆ ′1 >

.

(5.57)

We now consider the kernel of the differential operator D!3 ,1 , i.e., when the RHS of (5.52) and (5.54) are zero. Clearly, from (5.57) we see that on the kernel the “Di” fulﬁlls the d’Alembert equation: 2 ˆ Di = 0. ∂x – 4∂v ∂u >

(5.58)

Of course, the full equations (5.52) (or in components (5.54)) contain more information than (5.58). 5.2.10 Massless Representations In this case the energy and spin are: E0 = s0 + 1, s0 = 1, 32 , . . . . The equations are (5.50a,d) p+1

∂z

ˆ = 0, >

p = 2s0 = 2, 3, . . . ,

(5.59a)

ˆ = { p( p – 1) ∂u + ( p – 1)(∂x – 2z∂u ) ∂z D!4 ,1 > ˆ => ˆ ′. + (∂v – z∂x + z2 ∂u ) ∂z2 } > ′

(5.59b)

The target space CD is unitary but not massless: it has E0′ = s′0 + 3, s′0 = s0 – 1, (p′ = p – 2).

5.3 Classification of so(5, C) Verma Modules and P0 -Induced ERs

179

Thus, from the point of view of applications to physics these invariant equations relate a ﬁeld of spin s0 to a ﬁeld of spin s0 – 1. In the simplest case a vector ﬁeld s0 = 1 is coupled to a scalar ﬁeld. ˆ = pj=0 zj > ˆ j, It is useful to write out (5.59b) in components using: > p–2 j ′ ′ ˆ = j=0 z > ˆ j . The result is p – 1 equations: > ˆ j + ( j + 1)( p – j – 1) ∂x > ˆ j+1 ( p – j) ( p – j – 1) ∂u > ˆ ′j , ˆ j+2 = > + ( j + 1) ( j + 2) ∂v >

(5.60)

j = 0, 1, . . . , p – 2.

Further we restrict to the kernel of D!4 ,1 . In the simplest case p = 2 there is only one equation ˆ 0 + ∂x > ˆ 1 + 2∂v > ˆ 2 = 0, 2∂u >

(5.61)

which can be rewritten as equation for conserved current with the substitution ˆ 0 = J1 – iJ2 , >

ˆ 1 = – J0 , >

ˆ 2 = J1 + iJ2 , >

(5.62)

using also (5.37) ∂0 J0 – ∂1 J1 – ∂2 J2 = 0.

(5.63)

Similarly, for arbitrary p = 2, 3, . . . , there are p – 1 independent conserved currents J p,j , j = 0, . . . , p – 2, with components as follows: p,j

ˆ j+1 , J0 = –( j + 1)( p – j – 1) > p,j

J1 = p,j

J2 =

1 ˆ j+2 + ( p – j)( p – j – 1) >ˆ j }, {( j + 1)( j + 2) > 2 i ˆ j }. ˆ j+2 – ( p – j)( p – j – 1) > {( j + 1)( j + 2) > 2

(5.64)

Finally, we would like to mention that all massless representations appear on the quartet diagram 5.2.5 in the case k = 1. We would like to parametrize these diagrams by the spin of the massless irrep, i.e., by s0 = 1, 32 , . . .. Thus, q = 2s0 – 1 and the signature – of the ER 7q,k=1 on the top-left corner is (2s0 – 1, 1, 2s0 + 1, 2s0 ) and this ER contains a ﬁnite-dimensional irrep of dimension: s0 (4s20 – 1)/3. The massless UIR is contained in ′+ in the bottom-right corner of the quartet diagram, while the massive UIR of the ER 7q,1 + in the top-right corner. spin s0 – 1 is in the ER 7q,1

5.3 Classification of so(5, C) Verma Modules and P0 -Induced ERs Here we classify the Verma modules over G C = so(5, C). This also provides the classiﬁcation of the P0 -induced ERs since the restricted Weyl group W(G, A0 ) related to the

180

5 Case of the Anti-de Sitter Group

minimal parabolic subalgebra P0 = A0 N0 (cf. (5.16)) is isomorphic to the Weyl group W(G C , AC 0 ) (since G = so(3, 2) is maximally split). The classiﬁcation can be summarized as follows. There are four types of multiplets of reducible Verma modules: type Fm1 ,m2 (m1 , m2 ∈ N); type Sm1 ,m3 (m1 , m3 ∈ N, 1 2 (m1 + m3 ) ∉ Z); type R with four subtypes: Rk , k = 1, 2, 3, 4; and type L with two subtypes: Lk , k = 1, 2. We now give all of them below. Multiplets of type Fm1 ,m2 are parametrized by two natural numbers m1 and m2 . They are given as follows: Λm1,m2 s2

s1 1

2

Λm1,m2 s4

s3

s2

Λm1,m2

(–m1, m4, m3, m2) m4α2 m2α4

s1

12

Λm1,m2

(m1, m2, m3, m4) m1α1

21

s4

s3

s1

Λm1,m2

(m3 , –m4 , –m1 , m2) m3α1 m2α4

s2

121

212

Λm1,m2 s2

s1

Λm1,m2

(–m3 , m2 , –m1 , –m4) m2α2

Λ+m1,m2

m2α2 (m3, – m2, m1, m4) m1α3 m α 3 1 (–m3 , m4 , m1 , –m2) m1α3 m α 4 2 (m1 , –m4 , –m3 , –m2) m1α1

(–m1, –m2, –m3 , –m4)

where we have given the multiplets in two ways: on the left the Verma modules are depicted by their highest weights, while on the right they are given by the HarishChandra parameters. The numbers at the embedding arrows on the left indicate w.r.t. which reﬂection is the embedding; on the right we have given the weight m" of the + embedding. All Verma modules of these multiplets, except V D , are reducible and their weights are given explicitly as follows: Dm1 ,m2 ,

mk = mk (Dm1 ,m2 ) ∈ N, ∀k,

(5.65)

Dim1 ,m2 = Dm1 ,m2 – mi !i , i = 1, 2, ij

Dm1 ,m2 = Dm1 ,m2 – mi !i – mj+2 !j , (i, j) = (1, 2), (2, 1), iji

Dm1 ,m2 = Dm1 ,m2 – (mi + mi+2 )!i – mj+2 !j , (i, j) = (1, 2), (2, 1). +

The weights of the irreducible modules V D are D+m1 ,m2 = Dm1 ,m2 – (m1 + m3 )!1 – (m2 + m4 )!2 = Dm1 ,m2 – 2m4 !1 – m3 !2 . Note that only embeddings which are not compositions of other embeddings are given on the ﬁgure.

5.3 Classification of so(5, C) Verma Modules and P0 -Induced ERs

181

The same ﬁgure depicts also an octet of P0 -induced ERs with the same weights as the above Verma modules. Of course, in the ER picture instead of the embedding ′ ′ V D → V D we have the invariant differential operator D(D′ – D) : CD → CD . In fact, this invariant differential operator is a degeneration of the KS operator corresponding to the reﬂection s as given on the left-hand side (LHS) of the ﬁgure. More than this, there is a nondegenerate KS operator corresponding to the same reﬂection but acting ′ from CD to CD . This will be true for all invariant differential operators shown below in this section: every invariant differential operator is also an invariant differential operator between two P0 -induced ERs, and there is a nondegenerate KS operator acting in the opposite direction. Remark 1: The ﬁgure shown above contains also the quartet of P-induced ERs. To restore the quartet, we need to exclude the four ERs with P-incompatible Dynkin label m′1 ∉ N, as well as to restore the KS operator from Dm1 ,m2 to D212 ♢ m1 ,m2 . Multiplets of type Sm1 ,m3 are parametrized by two natural numbers m1 , m3 of different parity so that 21 (m1 + m3 ) ∉ Z. They are given in the ﬁgure below, where as above we have given the multiplet in two ways, and again the parametrizing numbers m1 , m3 are s related to the Verma module V D on the top: mk = mk (Ds ), where k = 1, 3. Note that the supplementary condition on these numbers ensure that mk = mk (Ds ) ∉ N (k = 2, 4) since m2 (Ds ) = 21 (m3 – m1 ) and m4 (Ds ) = 21 (m3 + m1 ). The Verma modules of these ′′ multiplets, except V D , are reducible and their weights are given explicitly as follows: Ds ,

mk = mk (Ds ) ∈ N, k = 1, 3, mk = mk (Ds ) ∉ Z, k = 2, 4,

Dsk = Ds – mk !k , k = 1, 3.

(5.66) ′′

The weights of the irreducible modules V D are D′′ = Ds – m1 !1 – m3 !3 .

s1

Λs

(m1, m2, m3, m4)

s3

Λs1

m1α1 Λs3

s1

s3 Λ+s

(−m1, m4, m3, m2) m3α3

m3α3 (m1, −m4, −m3, −m2) m1α1

(–m1, –m2, –m3, –m4)

Remark 2: To obtain the P-induced counterpart of the ﬁgure, we need to exclude the two spaces with P-incompatible Dynkin label m′1 ∉ N. Thus, we reduce to one of the pairs given by (5.48) parametrized by q and ,. More precisely, we reduce to (5.48a) when m2 > 0, then (m1 , m2 ) ↦ (q, ,/2), and to (5.48b) when m2 < 0, then (m1 , m2 ) ↦ (q + ,, –,/2).

182

5 Case of the Anti-de Sitter Group

The second case contains also a P-induced ER singlet when m1 is odd, m3 = q = 0 and m2 = – 21 m1 , then we have (5.49) with n = m1 = ,. Multiplets of type R are given as follows. Fix k = 1, 2, 3, 4 to ﬁx a subtype Rk . Then the multiplets of this subtype are doublets parametrized by the natural number mk and are given as follows: V Dk → V Dk –mk !k , Dk

LDk = V /V

mk (Dk ) = mk ∈ N,

Dk –mk !k

mj (Dk ) ∉ N, j ≠ k,

.

(5.67)

The modules V Dk –mk !k are irreducible. From these multiplets only the module V D1 contains a P-induced ER. Multiplets of type L are given as follows. Fix k = 1, 2 to ﬁx a subtype Lk . Then the multiplets are parametrized by a natural number m and are given as follows. The multiplets of subtype L1 are shown in the following ﬁgure. The weights of the reducible Verma modules are given as follows: D1m ,

m1 (D1m ) = 0, m2 (D1m ) = m4 (D1m ) = m ∈ N, m3 (D1m ) = 2m, 1 D12 m = Dm – m!2 ,

1 D121 m = Dm – m!4 .

(5.68)

′

The weights of the irreducible modules V Dm are D′m = D1m – 2m!3 . Note that this embedding picture is actually a special case of the factorization of the singular vector v!3 ,m3 according to the last line of (5.27c) for m1 = 0, m2 = m. Note that D12 m contains a P-induced ER singlet (cf. (5.49)), for n = 2m. Λ1m

s2

(0, m, 2m, m)

mα2 Λ12 m

s1

(2m, –m, 0, m) 2mα1

Λ121 m

s2

(–2m, m, 0, –m)

mα2 Λʹm

(0, –m, –2m, –m)

The multiplets of subtype L2 are shown in the following ﬁgure. The weights of the reducible Verma modules are given as follows: D2m ,

m1 (D2m ) = m3 (D2m ) = m4 (D2m ) = m ∈ N, m2 (D2m ) = 0, 2 D21 m = Dm – m!1 ,

2 D212 m = Dm – m!3 .

(5.69)

5.4 Character Formulae

183

′′

The weights of the irreducible modules V Dm are D′′m = D2m – m!4 . Note that this embedding picture is a special case of the factorization of the singular vector v!4 ,m4 according to the last line of (5.27d) for m2 = 0, m1 = m. Note that here are contained the P-induced – , and D212 contains the doublets (5.47) for q = m, since D2m contains the P-induced ER 7m m + P-induced ER 7m . Λ2m

(m, 0, m, m)

s1

mα1 Λ21 m

(–m, m, m,0)

s2

mα2 Λ212 m

(m, –m, –m, 0)

s1

mα1 Λʺm

(–m, 0, –m, –m)

5.4 Character Formulae 5.4.1 Character Formulae of AdS Irreps The question of reducibility of Verma modules is closely related to the corresponding Weyl groups. In particular, whenever (2.208) is fulﬁlled, then the shifted weight is given by the corresponding Weyl reﬂection (cf. (2.210)). In our case, G C = so(5, C) = B2 , the Weyl group WB2 has eight elements explicitly given in reduced form by (sk ≡ s!k ) WB2 = {1, s1 , s2 , s1 s2 , s2 s1 , s3 = s2 s1 s2 , s4 = s1 s2 s1 , s1 s2 s1 s2 = s2 s1 s2 s1 }.

(5.70)

Then the classical Weyl character formula (2.281) becomes in our case: chLm1 ,m2 = chV Dm1 ,m2 (1 – e!1 m1 – e!2 m2 + e!1 m1 e!2 (m1 +m2 ) + e!1 m1 +2m2 e!2 m2 – e!4 m4 – e!3 m3 + e2!1 m4 e!2 m3 ).

(5.71)

The character formulae for the inﬁnite-dimensional irreducible highest weight representations involve less terms than in (5.71) since the maximal invariant submodules I D of V D are smaller. It is easy to derive these using the same considerations as above, so we just list the results.

184

–

5 Case of the Anti-de Sitter Group

The character formulae for the irreps with highest weights D1 from (5.67), D212 m1 ,m2 2 212 from (5.65), Ds3 from (5.66), D12 m from (5.68), Dm , and Dm from (5.69) are chLD =

(–1)(w) chV w⋅D = chV D (1 – em1 !1 ),

w∈W1

W1 = {1, s1 }

(5.72)

where D denotes all highest weights under consideration and m1 should be replaced by m for the cases from (5.68) and (5.69). –

The character formula for the irreps with highest weights D2 from (5.67), D121 m1 ,m2 21 from (5.65), D1m , D121 m from (5.68), and Dm from (5.69) is

chLD =

(–1)(w) chV w⋅D = chV D (1 – em2 !2 ),

w∈W2

W2 = {1, s2 },

(5.73)

where D denotes all highest weights under consideration and m2 should be replaced by m for the cases from (5.68) and (5.69). –

The character formula for the irreps with highest weights D3 from (5.67) and Ds1 from (5.66) is

chLD =

(–1)(w) chV w⋅D = chV D (1 – em3 !3 ),

w∈W3

W3 = {1, s3 },

(5.74)

where D denotes all highest weights under consideration. –

The character formula for the irreps with highest weights D4 from (5.67) is

chLD4 =

(–1)(w) chV w⋅D4 = chV D4 (1 – em4 !4 ),

w∈W4

W4 = {1, s4 }. –

(5.75)

The character formula for the irreps with highest weights Ds from (5.66) is chLDs =

s

(–1)(w) chV w⋅D

w∈W s s

= chV D ( 1 – e!1 m1 – e!3 m3 + e!1 m1 +!3 m3 ), W s = {1, s1 , s3 , s1 s3 } = W1 × W3 .

(5.76)

185

5.4 Character Formulae

–

The character formula for the irreps with highest weights D12 m1 ,m2 from (5.65) is

= chLD12 m ,m 1

2

12

(–1)(w) chV w⋅Dm1 ,m2

(5.77)

w∈W 12 12

= chV Dm1 ,m2 ( 1 – e!1 m3 – e!4 m2 + e!1 m3 +!2 m2 ), W 12 = {1, s1 , s2 s1 , s1 s2 s1 }. –

The character formula for the irreps with highest weights D21 m1 ,m2 from (5.65) is

= chLD21 m ,m 1

2

21

(–1)(w) chV w⋅Dm1 ,m2

(5.78)

w∈W 21 21

= chV Dm1 ,m2 ( 1 – e!2 m4 – e!3 m1 + e!1 m1 +!2 m4 ), W 21 = {1, s2 , s1 s2 , s2 s1 s2 }. –

The character formula for the irreps with highest weights D1m1 ,m2 from (5.65) is chLD1m

1 ,m2

=

1

(–1)(w) chV w⋅Dm1 ,m2

(5.79)

w∈W 1 1

= chV Dm1 ,m2 ( 1 – e!2 m4 – e!4 m2 + e!1 m3 +!2 m4 + e!2 m4 +!4 m2 – e!1 m3 +!2 m3 ), W 1 = {1, s2 , s1 s2 , s2 s1 s2 , s1 s2 s1 , s2 s1 s2 s1 }. –

The character formula for the irreps with highest weights D2m1 ,m2 from (5.65) is chLD2m

1 ,m2

=

2

(–1)(w) chV w⋅Dm1 ,m2

(5.80)

w∈W 2 2

= chV Dm1 ,m2 ( 1 – e!1 m3 – e!3 m1 + e!1 m3 +!2 m4 + e!1 m3 +!3 m1 – e2!1 m4 +!2 m4 ), W 2 = {1, s1 , s2 s1 , s2 s1 s2 , s1 s2 s1 , s2 s1 s2 s1 }. Note that each of the Weyl groups Wk (k = 1, 2, 3, 4) is isomorphic to the A1 = sl(2) Weyl group, while the Weyl group W s is the direct product of two such A1 Weyl groups. In contrast, the subsets of W over which is carried the summation in the last four cases, namely, W 1 , W 2 , W 12 , and W 21 are not considered subgroups of W, since then the elements of these subsets will generate the whole W.

186

5 Case of the Anti-de Sitter Group

5.4.2 Character Formulae of Positive Energy UIRs Here we apply the character formulae of the previous subsection to the positive energy UIRs of so(3, 2). Rac. We have m1 = 1 and m2 = 1/2, i.e., we have a special case of (5.76): s

chLRac = chV D ( 1 – e!1 – e2!3 + e!1 +2!3 ) = e(Ds ) ( 1 – e2!3 )/(1 – e!2 )(1 – e!3 )(1 – e!4 ) = e(Ds ) ( 1 + e!3 )/(1 – e!2 )(1 – e!4 ) ∞

= e(Ds )

n

en!3

n=0

= e(D)

∞

ep!1

p=–n n

e(n–|p|)!3 t′|p| ,

(5.81)

n=0 p=–n

where t′ =

⎧ ⎨e(!4 )

for p ≥ 0,

⎩e(! ) 2

for p < 0.

Character formula (5.81) is equivalent to the spectrum description given in [170], and clearly the dimension of each weight space is one – this explains the terminology of singleton. Di. We have m1 = 2 and m2 = – 1/2, i.e., again a special case of (5.76): s

chLDi = chV D ( 1 – e2!1 – e!3 + e2!1 +!3 ) = e(Ds ) (1 – e2!1 )/(1 – e!1 )(1 – e!2 )(1 – e!4 ) = e(Ds ) (1 + e!1 )/(1 – e!2 )(1 – e!4 ) = e(Ds )

∞

en!3

∞

n=0

ep!1

p=–n

n=0

= e(Ds )

n+1

en!2

2n+1

er!1 .

(5.82)

r=0

Character formula (5.82) is equivalent to the singleton spectrum description given in [170]. Spin zero. Next we consider the case s0 = 0, E0 ≥ 1 (cf. (5.33), (5.34), and the text in-between). We have only the singular vector v!1 ,1 . This is clear for E0 > 1, while for E0 = 1 one should note that for s0 = 0 the singular vectors v!3 ,1 in (5.34c) and v!4 ,1 in

5.4 Character Formulae

187

(5.34d) are descendants of v!1 ,1 . In fact, for E0 > 1 the Verma module is V D1 from a multiplet of subtype R1 for parameter m1 = 1, while for E0 = 1 the Verma module is 2 V D1 from a multiplet of subtype L2 for parameter m = 1. Thus, the character formula is (5.72). Spin 1/2. Analogously for s0 = 1/2 and E0 ≥ 3/2, we have only the singular vector v!1 ,2 . This is clear for E0 > 3/2, while for E0 = 3/2 one should note that for s0 = 1/2 the singular vector v!4 ,1 in (5.34d) is descendant of v!1 ,2 . In fact, for E0 > 3/2 the Verma module is V D1 from a multiplet of subtype R1 for parameter m1 = 2, while for E0 = 3/2 12 the Verma module is V D1 from a multiplet of subtype L1 for parameter m = 1. Thus, the character formula is (5.72) as in the previous case. Higher spins. Analogously for s0 ≥ 1 and E0 > s0 + 1 (cf. (5.33b)), we have only the singular vector v!1 ,m1 , where m1 = 2s0 +1, since m2 , m3 , m4 ∉ N. Thus, the Verma module is V D1 from a multiplet of subtype N1 for parameter m1 ≥ 3, and the character formula is (5.72) as in the previous case. Massless irreps. Finally, we consider the massless representations with E0 = s0 +1 and s0 ≥ 1. We have two singular vectors: v!1 ,m (m = 2s0 + 1), and v!4 ,1 . The signature is (m′1 , m′2 , m′3 , m′4 ) = (2s0 + 1, –2s0 , 1 – 2s0 , 1). This signature appears in the Verma 12 ′+ module V2s of the multiplet F2s0 –1,1 (and also in the ER 72s from the quartet 0 –1,1 0 –1,1 diagram 5.2.5). Thus, the character formula (found ﬁrst in [170]) is a special case of (5.77): chLD12

m–2,1

12

= chV Dm–2,1 ( 1 – em!1 – e!4 + em!1 +!2 ),

(5.83)

m = 2s0 + 1. Thus, the massless UIRs are in one-parameter family of multiplets F2s0 –1,1 . This family contains also the Verma modules V D2s0 –1,1 on the top of the multiplet F2s0 –1,1 . This family of Verma modules has the signatures (m′1 , m′2 , m′3 , m′4 ) = (2s0 – 1, 1, 2s0 + 1, 2s0 ), and thus we have a family of ﬁnite-dimensional irreps of dimension: s0 (4s20 – 1)/3, – s0 = 1, 32 , . . . (found also in the ER 72s from the quartet diagram in Subsection 5.2.5). 0 –1,1 This family contains also the trivial one-dimensional irrep of so(5, C) (and of so(3, 2)) for the lowest possible value s0 = 1.

6 Conformal Case in 4D Summary The conformal group and algebra in four-dimensional space-time is a very important example. From the physics point of view this is natural since four-dimensional space-time is where we live. From the mathematics point of view this is natural since su(2, 2) ≅ so(4, 2) is of rank three and most basic phenomena of representation theory appear. Here the split rank is two and all three nonconjugate nontrivial parabolics are interesting (unlike the case so(3, 2) where two of the three nonconjugate nontrivial parabolics are isomorphic). This Chapter is based mostly on the papers [122, 126, 142, 155, 159, 162, 164]. We give in detail the realizations of the group SU(2, 2) and then of the Lie algebra su(2, 2). Further we give the restricted root system, Bruhat and Iwasawa decompositionsm, the restricted Weyl group W(G, A0 ), and the parabolic subalgebras. We present the complexified Lie algebra, its root system and the corresponding compact and noncompact roots. We give in detail the important subgroups of G pertinent to all parabolics. Then we define the elementary representations (ERs) of SU(2, 2) separately for induction from all parabolic subgroups. The ERs are given in the general induction picture and in the noncompact picture. We present properties of the ERs including the relation between ERs induced from different parabolics. We give explicitly the Knapp–Stein integral invariant operators for the three parabolics. Then we present the explicit construction of invariant differential operators and the multiplet classification of the reducible ERs again for the three parabolic inductions. We pay special attention to the identification of the holomorphic discrete series and the lowest weight representations in relation to the reducible ERs. Finally, we note that this case serves as the main example of the next Chapter where we give the relevant character formulae.

6.1 Preliminaries 6.1.1 Realizations of the Group SU(2, 2) The standard deﬁnition of the conformal group SU(2,2) is (cf. (3.59)) G = SU(2, 2) = {g ∈ GL(4, C) | g † "0 g = "0 ,

12 , det g = 1}. "0 = 0 –12

(6.1)

Here G leaves invariant the Hermitian form 60 (Z, Z ′ ) ≡ Z † "0 Z ′ ,

Z, Z ′ ∈ C4 .

(6.2)

Other realizations of G are used which differ from (6.1) by unitary transformations of "0

189

6.1 Preliminaries

⎛

0 ⎜0 ⎜ "0 ↦ "1 = U1 "0 U1–1 = ⎜ ⎝–1 0 ⎛

1 ⎜ 1 ⎜0 U1 ≡ √ ⎜ 2 ⎝–1 0 "0 ↦ "2 =

⎞ 0 0⎟ ⎟ ⎟, 0⎠ –1

0 1 0 0

–1 0 0 0

0 √ 2 0 0

1 0 1 0

⎞ 0 0⎟ ⎟ ⎟, 0⎠ √ 2

12 , 0

0 12

U2 "0 U2–1 =

(6.3)

(6.4)

12 . 12

1 12 U2 ≡ √ 2 –12

(The last case was used in [414].) In the realizations " = "0 , "1 , "2 the Cartan subalgebra with zero, one, two, respectively, noncompact generators is diagonal. So each realization is natural for one of the three nonconjugate Cartan subalgebras of the Lie algebra of G. Another realization was used in [315, 623] when studying the holomorphic representations of G

"0 ↦ "3

1 U3 ≡ √ 2 The corresponding transformations:

0 –12

= U3 "0 U3–1 = i

Hermitian

forms

6j (Z, Z ′ ) ≡ Z † "j Z ′ ,

12 –i12

are

12 , 0

(6.5)

–i12 . 12 invariant

under

the

respective

6j (Uj Z, Uj Z ′ ) = 60 (Z, Z ′ ).

(6.6)

6.1.2 Lie Algebra of SU(2, 2) Clearly, from the deﬁnition of the group above follows that the Lie algebra G of G

G = su(2, 2) = {X ∈ GL(4, C) | tr X = 0, X † " + "X = 0,

" = "0 , "1 , "2 , "3 }.

(6.7)

190

6 Conformal Case in 4D

Next we deﬁne the Cartan involution (: ( X ≡ " X "–1

(6.8)

which determines the Cartan decomposition G = K ⊕ Q (cf. (3.5)). Explicitly, we have for the basis of G (for " = "2 )

i 0 3, , 2 3, 0

i 0 2 –3,

0 , 3k

i 3k 2 0

3, , 0

i 3a 2 0

, = 0, 1, 2, 3,

0 , –3a

ea 0

k = 1, 2, 3,

0 , a = 1, 2, ea

(6.9)

(6.10)

where 3, are the Pauli matrices

0 0 1 ≡ 12 , 31 = , 1 1 0

–i 1 0 , 33 = , 0 0 –1

1 0 0 1 1 e1 ≡ 2 (30 + 33 ) = , e2 ≡ 2 (30 – 33 ) = 0 0 0

1 30 = 0

0 32 = i

(6.11)

0 . 1

It is easy to see that the seven matrices in (6.9) span the maximal compact subalgebra K which is isomorphic to su(2) ⊕ su(2) ⊕ u(1), while the eight matrices in (6.10) span the noncompact subspace Q. Next we list important subalgebras of G. The abelian subalgebra A0 which is a maximal subspace of Q has dimension two - equal to the split rank of G. We choose for the basis of A0 (for " = "2 )

eˆ a ≡

ea 0

0 , –ea

a = 1, 2.

(6.12)

The centralizer M0 ≅ u(1) of A0 in K then is spanned by the generator

i 33 H= 2 0

0 . 33

(6.13)

The Cartan subalgebra H2 consisting of all diagonal matrices in G for " = "2 is spanned by eˆ 1 , eˆ 2 , H. It is the most noncompact Cartan subalgebra. The other noncompact nonconjugate Cartan subalgebra H1 is diagonal for " = "1 and is spanned by

6.1 Preliminaries

eˆ 1 ,

diag (i/2, –i, i/2, 0),

diag (i/2, 0, i/2, –i).

191

(6.14)

The compact Cartan subalgebra H0 is diagonal for " = "0 and is spanned by

i 1 H0 = 2 0

0 , –1

i 33 H1 = 2 0

0 , 0

i 0 H2 = 2 0

0 . 33

(6.15)

As we know the algebra G = su(2, 2) is isomorphic to so(4, 2), the isomorphism being given explicitly by (for " = "2 )

i 3 Xjk = –:jk 2 0

i 0 1 X06 = 2 1 0

Xk0 X56

0 , 3

i 0 Xk5 = 2 3k

3k , 0 (6.16)

i 3k 0 0 3k , , Xk6 = = 0 –3k 0 2 –3k

i 0 0 1 1 1 , X05 = . = 2 0 –1 2 –1 0 1 2

(6.17)

Indeed, one checks easily that [XAB , XCD ] = 'AC XBD + 'BD XAC – 'AD XBC – 'BC XAD ,

(6.18)

where A, B, C, D = 0, 1, 2, 3, 5, 6, '11 = '22 = '33 = '55 = –'00 = –'66 = 1, 'AB = 0 for A ≠ B. We note that the subalgebra K is spanned by (6.16) (and is explicitly isomorphic to so(3) ⊕ so(3) ⊕ so(2) ⊂ so(4, 2)), while the subspace Q is spanned by (6.17). 6.1.3 Restricted Root System, Bruhat and Iwasawa Decompositions Let A∗0 be the space of linear functionals over A0 . They are determined by their values on eˆ a , a = 1, 2. We deﬁne for + ∈ A∗0 , G+ ≡ {X ∈ G | [ˆea , X] = +(ˆea )X}, BR ≡ {+ ∈

A∗0 | +

≠ 0, G+ ≠ {0}}.

(6.19) (6.20)

It is easy to see that in our case BR = {±+k , k = 1, 2, 3, 4}.

(6.21)

192

6 Conformal Case in 4D

The set of positive restricted roots is chosen to be +1 (ˆe1 , eˆ 2 ) = (0, 2),

+2 (ˆe1 , eˆ 2 ) = (1, –1),

+3 (ˆe1 , eˆ 2 ) = (2, 0),

+4 = (ˆe1 , eˆ 2 ) = (1, 1),

+3 = +1 + 2+2 ,

(6.22)

+4 = +1 + +2 ,

the simple restricted roots with this ordering being 0R = {+1 , +2 }. Now we display the corresponding root spaces (for " = "2 ) using more compact notation Gk± ≡ G±+k

0 ie2 0 0 – , G1 = l.s. , ie2 0 0 0

3+ 3+ 0 0 + G2 = l.s. ,i , 0 –3– 0 3–

3– –3– 0 0 – G2 = l.s. ,i , 0 3+ 0 3+

0 ie1 0 0 + – , G3 = l.s. , G3 = l.s. ie1 0 0 0

0 i3 0 0 a G4+ = l.s. , G4– = l.s. , a = 1, 2, i3a 0 0 0

G1+ = l.s.

(6.23)

3± ≡ 21 (31 ± i32 ) where l.s. stands for the linear span. Clearly, the roots ±+2 , ±+4 have multiplicity two. We use the standard notation for the collection of positive and negative root spaces N0+ ≡ ⊕k Gk+ ,

N0– ≡ ⊕k Gk– .

(6.24)

Obviously (N0+ = N0– . Thus, the basic Bruhat decomposition (3.10) here is G = N0+ ⊕ M0 ⊕ A0 ⊕ N0– .

(6.25)

Next, the Iwasawa decomposition (3.9) here is G = K ⊕ A 0 ⊕ N0 ,

N0 = N0± .

(6.26)

6.1 Preliminaries

193

6.1.4 Restricted Weyl Group W(G, A0 ) For future reference we deﬁne for every +k ∈ B+R a vector Hˆ k ∈ A0 by B(Hˆ k , eˆ e ) = +k (ˆea ), B(X, Y) ≡ tr XY,

a = 1, 2,

(6.27)

X, Y ∈ G,

where B is the Killing form on G. It is easy to see that Hˆ 1 = eˆ 2 ,

Hˆ 2 = 21 (ˆe1 – eˆ 2 ),

Hˆ 3 = Hˆ 1 + 2Hˆ 2 = eˆ 1 ,

(6.28)

1 Hˆ 4 = Hˆ 1 + Hˆ 2 = (ˆe1 + eˆ 2 ). 2

The restricted Weyl reﬂections sk in A0 may be deﬁned as sk (ˆea ) ≡ eˆ a – 2

+k (ˆea ) ˆ Hk, +k (Hˆ k )

(6.29)

which explicitly takes the form s1 (ˆe1 , eˆ 2 ) = (ˆe1 , –ˆe2 ),

s2 (ˆe1 , eˆ 2 ) = (ˆe2 , eˆ 1 ),

s3 (ˆe1 , eˆ 2 ) = (–ˆe1 , eˆ 2 ),

s4 (ˆe1 , eˆ 2 ) = (–ˆe2 , –ˆe1 ).

(6.30)

It is well known that the restricted Weyl reﬂections generate the ﬁnite restricted Weyl group W(G, A0 ) W(G, A0 ) = {id, s1 , . . . , s7 },

(6.31)

with simple reﬂections s1 , s2 so that s3 = s2 s1 s2 , s7 =

s25

=

s26 ,

s 4 = s1 s2 s1 ,

s 5 = s2 s1 ,

s 6 = s1 s2 ,

(6.32)

s7 (ˆe1 , eˆ 2 ) = (–ˆe1 , –ˆe2 ) = ((ˆe1 , (ˆe2 ).

Naturally, the group W(G, A0 ) is isomorphic to the Weyl group of B2 ≅ C2 which seems natural since relations (6.22) are the same as in the root system of the simple Lie algebra B2 = C2 (though in the latter the root spaces are standardly of multiplicity 1). We deﬁne also the induced action of the reﬂections on the restricted roots by the formula s∗k +j ≡ +j ○ sk

acting on(ˆe1 , eˆ 2 ),

(6.33)

194

6 Conformal Case in 4D

from which follows: s∗1 (+1 , +2 , +3 , +4 ) = (–+1 , +4 , +3 , +2 ),

(6.34)

s∗2 (+1 , +2 , +3 , +4 ) = (+3 , –+2 , +1 , +4 ).

6.1.5 Parabolic Subalgebras As we know from the general theory in above constructions the subalgebra P 0 ≡ M 0 ⊕ A 0 ⊕ N0

(6.35)

is a minimal parabolic subalgebra of G. We know that the number of standard parabolic subalgebras is 2 , ( = dim a0 ), including the trivial case when the parabolic is the Lie algebra G itself. Thus, we have two more nontrivial parabolic subalgebras to consider P a ≡ M a ⊕ A a ⊕ Na ,

a = 1, 2.

(6.36)

They are maximal parabolics since dim Aa = 1. They may be indexed as follows: sa X = – X,

X ∈ A a ⊂ A0 ,

(6.37)

A2 = l.s. (ˆe1 + eˆ 2 ).

(6.38)

from which follows that A1 = l.s. eˆ 1 ,

Then Ma ⊃ M0 is the centralizer of Aa in G given explicitly by

0 0 ie2 , ⊃ G1+ ⊕ G1– M1 = l.s. eˆ 2 , H, ie2 0 0

i 3a 3a 0 0 , M2 = l.s. eˆ 1 – eˆ 2 , H, 3a 0 –3a 2 0

0 0

⊃ G2+ ⊕ G2–

(6.39)

(6.40)

It is easy to see (as noted in [351]) that M1 = u(1) ⊕ sl(2, R),

(6.41)

M2 = sl(2, C).

(6.42)

For each Aa , a = 1, 2, we deﬁne the roots BaR of {G, Aa } to be the nonzero restrictions to Aa of the restricted roots. Explicitly we have

6.1 Preliminaries

B1R = { ±+, ±2+}, B2R = {±+′ },

+ = +2 | A1 = +4 | A1 ,

2+ = +3 | A1 ,

′

+ = +1 | A2 = +3 | A2 = +4 | A2 .

195

(6.43) (6.44)

Clearly, ±+, ±+′ have multiplicity four, while 2+ has multiplicity 1. Deﬁning Na± to be the corresponding root spaces we have N1± = G2± ⊕ G3± ⊕ G4± , N2± = G1±

⊕ G3±

⊕ G4± ,

dim N1± = 5,

(6.45)

N1±

(6.46)

dim

= 4.

Now we can give the Bruhat decompositions corresponding to the two maximal parabolics G = Na+ ⊕ Ma ⊕ Aa ⊕ Na– ,

a = 1, 2.

(6.47)

6.1.6 Complexified Lie Algebra For G = su(2, 2) the complexiﬁed Lie algebra is G C = sl(4, C). In previous chapters we have seen a lot of information on the algebras sl(4, C). Yet some general considerations should be modiﬁed in order to be better suited to this partial situation. We start with the Cartan subalgebra HC of G C . Since G C is complex the Cartan subalgebra HC is the complexiﬁcation of any Cartan subalgebra of G, in particular, of H0 = A0 ⊕ HM0 , (HM0 = M0 ) C HC = H0C = AC 0 ⊕ M0 .

(6.48)

It is useful to choose the basis of HC so that the roots of the pair (G C , HC ) have real values on A0 ⊕ iM0 . Also ordering of roots must be compatible with their restriction on G. For these reasons we do not use the standard basis of HC consisting of

33 0

0 , 0

e2 0

0 , –e1

eˆ 2 ,

Hˆ ≡ –iH.

0 0

0 , 33

(6.49)

but rather the basis eˆ 1 ,

(6.50)

Then the root system is B = {±!k , k = 1, . . . , 6 },

(6.51)

196

6 Conformal Case in 4D

ˆ = (1, –1, 1), !1 (ˆe1 , eˆ 2 , H) ˆ = (0, 2, 0), !2 (ˆe1 , eˆ 2 , H)

(6.52)

ˆ = (1, –1, –1), !3 (ˆe1 , eˆ 2 , H) ˆ = (1, 1, 1), !4 = !1 + !2 , !4 (ˆe1 , eˆ 2 , H) ˆ = (1, 1, –1), !5 = !2 + !3 , !5 (ˆe1 , eˆ 2 , H) ˆ = (2, 0, 0), !6 (ˆe1 , eˆ 2 , H)

! 6 = ! 1 + !2 + !3 ,

the simple roots with this ordering being 0 = {!1 , !2 , !3 }. C are complexly spanned by the root vectors X ± : The corresponding root spaces G±! k k

0 ie2 0 + , X2 = , 0 0 0

0 0 0 i3+ + , , X4 = 0 3– 0 0

0 i3– 0 ie1 + , X6 = , 0 0 0 0

3– 0 0 0 – , X2 = , –ie2 0 0 0

0 0 0 0 – , , X4 = 0 3+ –i3– 0

0 0 0 0 – , X6 = , –i3+ 0 –ie1 0

X1+ = X3+ = X5+ = X1– = X3– = X5– =

3+ 0

(6.53)

Xk– = (Xk+ )† . C we have the standard triangular decomposition: Denoting G±C = ⊕6k=1 G±! k

G C = G+C ⊕ HC ⊕ G–C .

(6.54)

It is easy to check that ± , [X1± , X2± ] = ±X4± = ±X12

± [X2± , X3± ] = ±X5± = ±X23 ,

± [X1± , X5± ] = [X4± , X3± ] = ±X6± = ±X13 .

(6.55)

The normalization is chosen so that !k (Zk ) = 2,

Zk ≡ [Xk+ , Xk– ].

(6.56)

6.1 Preliminaries

197

Explicitly, we have

Z1 =

1 e1 2 (ˆ

33 ˆ – eˆ 2 ) + H = 0

Z2 = eˆ 2 ,

0 Z3 = – eˆ 2 ) – Hˆ = 0

e1 Z4 = 21 (ˆe1 + eˆ 2 ) + Hˆ = 0

e2 Z5 = 21 (ˆe1 + eˆ 2 ) – Hˆ = 0 1 e1 2 (ˆ

0 , 0

(6.57)

0 , –33 0 = Z1 + Z2 , –e2 0 = Z2 + Z3 , –e1

Z6 = eˆ 1 = Z1 + Z2 + Z3 . ˆ We note also the Equivalently, Zk may be deﬁned so that B(Zk , X) = !k (X), X = eˆ a , H. relation to the compact Cartan basis given in (6.15) i H0 = (Z4 + Z5 ), 2

H1 =

i Z1 , 2

H2 =

i Z3 . 2

(6.58)

We use these generators to deﬁne the Weyl reﬂections wk (X) ≡ X – 2

!k (X) Zk = X – !k (X) Zk , !k (Zk )

X ∈ H.

(6.59)

We know that the Weyl reﬂections w1 , w2 , w3 corresponding to the simple roots generate the Weyl group W(SL(4, C)) (with 4! elements) with the following relations: wk2 = (w1 w2 )3 = (w2 w3 )3 = (w1 w3 )2 = id.

(6.60)

It is instructive to give the action of the generating elements on all Zk w1 (Z1 , Z2 , Z3 , Z4 , Z5 , Z6 ) = (–Z1 , Z4 , Z3 , Z2 , Z6 , Z5 ),

(6.61)

w2 (Z1 , Z2 , Z3 , Z4 , Z5 , Z6 ) = (Z4 , –Z2 , Z5 , Z1 , Z3 , Z6 ), w3 (Z1 , Z2 , Z3 , Z4 , Z5 , Z6 ) = (Z1 , Z5 , –Z3 , Z6 , Z2 , Z4 ) to see how the Weyl group is permuting the latter (plus some sign changes). We deﬁne also the induced action of the reﬂections on the roots by the formula: wk∗ !j ≡ !j ○ wk

acting on Z

(6.62)

198

6 Conformal Case in 4D

from which follows: w1∗ (!1 , !2 , !3 , !4 , !5 , !6 ) = (–!1 , !4 , !3 , !2 , !6 , !5 ),

(6.63)

w2∗ (!1 , !2 , !3 , !4 , !5 , !6 ) = (!4 , –!2 , !5 , !1 , !3 , !6 ), w3∗ (!1 , !2 , !3 , !4 , !5 , !6 ) = (!1 , !5 , –!3 , !6 , !2 , !4 ), i.e., as expected, the same action as on Z . 6.1.7 Compact and Noncompact Roots In the previous subsections matters have been arranged so that the root systems BR and B are compatible +1 = !2 | A0 ,

+2 = !1 | A0 = !3 | A0 ,

+3 = !6 | A0 ,

+4 = !4 | A0 = !5 | A0 ,

(6.64)

recalling that A0 is spanned by eˆ 1 , eˆ 2 . Also the simple roots are compatible 0R = 0 | A0 .

(6.65)

Accordingly, we have that under complexiﬁcation G+C1 = G!C2 ,

G+C2 = G!C1 ⊕ G!C3 ,

G+C3 = G!C6 ,

G+C4 = G!C4 ⊕ G!C5 ,

(6.66)

and consequently (N0± )C = G±C .

(6.67)

The roots ±!1 , ±!3 are called compact (or K-compact) since the root vectors X1± , X3± (6.53) are in KC . The rest of the roots are called noncompact. With respect to the parabolic P0 the roots ±!2 , ±!6 are real since they are zero when restricted to the Cartan subalgebra HM0 of M0 - coinciding with M0 spanned by H = iHˆ !2 | M0 = !6 | M0 = 0.

(6.68)

There are no roots that vanish on A0 (that would be compact), thus, the rest of the roots: ±!1 , ±!3 , ±!4 , ±!5 are complex. Next we consider the parabolics Pa , a = 1, 2. With respect to P1 the root !6 is real since it is zero when restricted to the Cartan subalgebra HM of M1 spanned by eˆ 2 , H = iHˆ 1

!6 | HM1 = 0.

(6.69)

6.1 Preliminaries

199

The root !2 is M1 -compact since it vanishes on A1 (spanned by eˆ 1 ) !2 | eˆ 1 = 0. The rest are complex, i.e., the set of complex roots is the same as for P0 . With respect to P2 there are no real roots, i.e., that vanish on the Cartan subalgebra ˆ The roots ±!1 , ±!3 , are M2 -compact since they HM2 of M2 spanned by eˆ 1 – eˆ 2 , H = iH. vanish on A2 (spanned by eˆ 1 + eˆ 2 ) : !1 | eˆ 1 +ˆe2 = !3 | eˆ 1 +ˆe2 = 0. The rest are complex. Note that the set of M2 -compact roots coincides with the set of K-compact roots. This is very important from representation theory point of view as we shall see below. 6.1.8 Important Subgroups of G We shall usually write the elements g of G as

! " g= ˜ (6.70) # $˜ ˜ ˜ where !, ", #, $ are 2 × 2 complex matrices constrained by the deﬁning conditions (6.1). ˜ ˜ ˜" = "2 we have Explicitly,˜ for !† # + #† ! = 0, ˜ ˜ ˜ ˜

"† $ + $† " = 0, !† $ + #† " = 12 . ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ The maximal compact subgroup K of G is given for " = "0 as follows: K ≡ {g ∈ G | g † = g –1 }

*1 0 = , *i ∈ U(2), det *1 *2 = 1 , ˜ 0 *2 ˜ ˜ ˜ ˜ K ≅ S(U(2) × U(2)) ≅ U(1) × SU(2) × SU(2)

(6.71)

(6.72)

and its Lie algebra is K. Further we deﬁne:

! A0 = exp A0 = {a = ˜ 0

0 , !–1 ˜

! = e1 es + e2 et , s, t ∈ R}, ˜

xz ix (xz† )–1 + + N0 = {n0 = ˜ ˜ ˜† –1 , xz = 30 + z3+ , z ∈ C, 0 (xz ) ˜ ˜

3

x+ –iu x= , x, ∈ R, x, 3, = iu¯ x– ˜ ,=0 x± = x0 ± x3 , u = x2 + ix1 }

(6.73)

200

6 Conformal Case in 4D

N0–

=

{n–0 =

b˜ w ib˜ b˜ w

b˜ = b0 30 –

0 †

(b˜ w )–1

3

k=1

¯ –, w ∈ C , b˜ w = 30 – w3

b– bk 3k = –i¯v

iv , b, ∈ R, b+

b± = b0 ± b3 , v = b2 + ib1 }. Let M0 be the centralizer of A0 in K. Then M0 = {m = #3N 4 | N = 0, 1, 4 ∈ T} = T ⋊ {14 , #3 },

33 0 ˆ = 2H, #3 ≡ 0 33

4(>) 0 ˆ = exp i>H, T = {4(>) = ˜ 0 4(>) ˜ 4(>) = diag(ei>/2 , e–i>/2 )} ≅ SO(2) ˜

(6.74)

and the Lie algebra of M0 (and of T) is M0 . We recall that if M0′ is the normalizer of A0 in K then M0′ /M0 is equal to the restricted Weyl group W(G, A0 ) which we displayed in subsection 6.1.4. In complete parallel to the general algebraic discussion in Section 3.3 P0 = M0 A0 N0 is a minimal parabolic subgroup of G and a standard parabolic subgroup of G is any closed subgroup of G containing P0 . The parabolic subgroups may be displayed in the following way/ Let J = {s1 , s2 } be the set of generating elements of W(G, A0 ). Then to each subset E ⊂ J corresponds a parabolic subgroup of G PE = ∪ P0 3(8)P0 = ME AE NE , 8∈E

(6.75)

where 3(8) ∈ G is a representation of J in G (to be given below). Explicitly, we obtain in parallel to the algebraic description:

P = P0 ,

P{sa } = P0 3(sa )P0 , a = 1, 2,

PJ = G,

AJ = NJ = {1}.

(6.76)

For the two nontrivial cases we obtain explicitly: Pa ≡ P{sa } = Ma Aa Na ,

a = 1, 2,

(6.77)

6.1 Preliminaries

M1 = T × SL(2, R)

= {m = 4(>) rˆ =

ei>/2 e

+e–i>/2 e

1+ –i-e–i>/2 e2

i,e–i>/2 e

2

(6.78a)

2

ei>/2 e1 + 1e–i>/2 e2

,

, ∈ SL(2, R), +, ,, -, 1 ∈ R, +1 – ,- = 1}, 1

e s e 1 + e2 0 A1 = exp(A1 ) = {a1 = , s ∈ R}, 0 e–s e1 + e2

xz ix(x– = 0) (xz† )–1 + + ∈ N0+ }, N1 = {n1 = ˜ ˜ ˜ 0 (xz† )–1 ˜

0 b˜ w – – N1 = {n1 = ∈ N0– }, ˜ + = 0) b˜ w (b˜ † )–1 ib(b

201

+ rˆ ↦ -

(6.78b) (6.78c) (6.78d)

w

M2 = SL(2, C) ⋊ {14 , #3 }

l 33N 0 , = {m2 = ˜ 0 (l† )–1 33N ˜ l ∈ SL(2, C), N = 0, 1}

√ ˜ | a | 12 A2 = exp(A2 ) = {a2 = 0

12 ix + N2+ = {n+2 = ˜ = n0 | z=0 }, 0 12

12 0 – – N2 = {n2 = ˜ = n–0 | w=0 }. ib 12

(6.79a)

0 √1 |a|

12

, | a | ∈ R+ },

(6.79b) (6.79c) (6.79d)

We recall that Ma (a = 1, 2) is the centralizer of Aa in G, and let Ma′ , (a=1,2), be the normalizer of Aa in G. Then we have Ma′ /Ma = W(G, Aa ) = {1, sa },

a = 1, 2.

(6.80)

Now we give matrix representatives of the elements of the restricted Weyl group W(G, A0 ). As matrices they belong to the maximal compact subgroup K. Actually, there is some ambiguity and one may display one-parameter representations of the generating elements as done in [122], but we shall display just the ﬁnally chosen representatives

3(s1 ) =

e1 –e2

–e2 , e1

3(s2 ) =

i32 0

0 . i32

(6.81)

202

6 Conformal Case in 4D

It is easy to check that as matrices 3(sk ) eˆ a 3(sk )–1 = sk (ˆea )

(6.82)

With this choice of representatives we have

–e2 3(s3 ) = e1

3– 3(s5 ) = – 3+

e1 , –e2 3+ , 3–

0 –i32 , 3(s4 ) = –i32 0

3+ 3– 3(s6 ) = , 3– 3+

(6.83)

3(s7 ) = 3(s1 )3(s2 )3(s1 )3(s2 ) = 3(s2 )3(s1 )3(s2 )3(s1 )

0 12 = "2 = 12 0 which will be convenient for the study of the Knapp–Stein (KS) intertwining operators. Next we give matrix representatives of the elements of the restricted Weyl group W(G, Aa ), a = 1, 2. Their nontrivial elements are s1 and s2 , respectively, and we could choose the matrices in (6.81). However, we have more choices since we only need to fulﬁll (cf. (6.38)): 31 (s1 ) eˆ 1 31 (s1 )–1 = – eˆ 1 ,

(6.84a)

32 (s2 ) (ˆe1 + eˆ 2 ) 32 (s2 ) = – (ˆe1 + eˆ 2 ).

(6.84b)

–1

The most convenient choice (expected on general grounds) is: 31 (s1 ) = "2 ,

32 (s2 ) = "2

(6.85) C

Finally, we write representatives of the Weyl group W(G C , cˆ ). Proceeding analogously we ﬁnd the following representations for the generating elements:

31 9(w1 ) = 0

0 , 12

9(w2 ) =

e1 –e2

–e2 , e1

9(w3 ) =

12 0

0 . 31

(6.86)

The representation of the longest element wmax is again related to "2 9(w1 )9(w2 )9(w3 )9(w1 )9(w2 )9(w1 ) = – "2

(6.87)

where we have chosen a reduced expression of wmax = w1 w2 w3 w1 w2 w1 .

6.2 Elementary Representations of SU(2, 2) We display the ERs corresponding to the three nontrivial parabolic subgroups Pk , k = 0, 1, 2.

6.2 Elementary Representations of SU(2, 2)

203

6.2.1 ERs from the Minimal Parabolic Subgroup P0 Considering P0 = M0 A0 N0 we ﬁrst introduce the inducing representations of M0 and A0 . The UIRs of M0 are one-dimensional, belonging to the discrete series and being parametrized by (n, %), where n ∈ Z is indexing a character of T and % = 0, 1 indexes a character of {1, #3 }. Explicitly, the character of M0 is given by in>/2 (–1)%N , Dm,% 0 (m) = e

m = 4(>) #3N .

(6.88)

The (nonunitary) characters of A0 are indexed by two complex numbers c1 , c2 and are given by (with a as in (6.73): c ,c

D01 2 (a) = e–s(3+c1 )–t(1+c2 ) .

(6.89)

The numbers c1 , c2 are viewed as the values of a linear functional + over the basis elements of the Lie algebra A0 of A0 +(ˆea ) = ca , a = 1, 2.

(6.90)

In (6.89) we have added to + the half-sum with multiplicities of the positive restricted roots of the system (G, A0 ) 10 = 21

4

mk +k ,

m1 = m3 = 1, m2 = m4 = 2,

l=1

10 (ˆe1 , eˆ 2 ) = (3, 1).

(6.91)

Further, we introduce more compact notation signature of the P0 -induced ERs: 70 = [n, %, c1 , c2 ],

(6.92)

and for the one-dimensional representation of M0 A0 7

c ,c

1 2 D00 (ma) = Dm,% (a). 0 (m)D0

(6.93)

The representation space of the ER 70 is standardly deﬁned as 7

C70 = {F ∈ C∞ (G, C) | F (gman–0 ) = D00 (ma)–1 F (g), g ∈ G, man–0 ∈ M0 A0 N0– }

(6.94)

and its action is given by the left regular action of G

T 70 (g)F = F (g–1 g ′ ).

(6.95)

204

6 Conformal Case in 4D

We record some useful properties of the ERs. Introduce the standard right action of G C in the space C70 (X ⋅ F ) (g) =

d F (g exp tX) | t=0 , dt

X ∈ GC.

(6.96)

ˆ It is easy to see that (H = –iH) X ∈ (N0– )C = G–C ,

X ⋅ F = 0,

eˆ 1 ⋅ F = (3 + c1 ) F , n H ⋅ F = F. 2

(6.97a)

eˆ 2 ⋅ F = (1 + c1 ) F , (6.97b)

Clearly, (6.97) together with the condition F (g#3N ) = (–1)%N F (g) is equivalent to the covariance property of the functions of C70 . Indeed, (6.97a) is equivalent to F (gn–0 ) = 7 F (g) for n–0 ∈ N0– , while (6.97b) is equivalent to F (g4a) = D0 (4a)–1 F (g). Altogether, (6.97) means that the space C70 has an associated lowest weight Verma module over G C since every element of C70 fulﬁls the deﬁning conditions of lowest weight vector. This a realization of the general structure of associated Verma modules introduced in Section 4.5. Finally, we introduce a K-invariant scalar product in C70 by the formula

F0 , F0′ = *0

* K

*

dk F0 (k) F0′ (k),

dk = 1

(6.98)

K

where *0 = 0 4 is introduced for convenience, dk is the normalized Haar measure on K. The representation T 70 is continuous w.r.t. the topology deﬁned by (6.98). It is complete as normed only w.r.t a Fréchet space topology which we could have introduced in line with [159, 620]. The completion of C70 with respect to (6.98) shall be denoted by H70 . The scalar product (6.98) is also G-invariant iff ck = irk , rk ∈ R, k = 1, 2. Then the latter ERs form the principal series of unitary representations of G acting in H70 [351]. 6.2.2 ERs from the Maximal Cuspidal Parabolic Subgroup P1 We consider P1 = M1 A1 N1 . The discrete series representations of M1 = T × SL(2, R) are parametrized by the triple (n, k, :), where n ∈ Z indexes a character of T (as for M0 above), while k ∈ N, : = ±1, parametrize the discrete series of SL(2, R) [24, 250]. The latter shall be realized in the space [248] 1 Vk,: = {6 ∈ C∞ (R) | 6(x)

→ |x| → ∞

∞ 1

xk+1

j=0

j 1 aj – ; x

6(x + i') is analytic in & = x + i' for :' > 0,

(6.99)

6.2 Elementary Representations of SU(2, 2)

205

by the formula $x – " k –k–1 6 T1 (g)6 (x) = (! – #x) , ! – #x

! g= #

" ∈ SL(2, R). $

(6.100)

The asymptotic behavior 6(x) in (6.99) guarantees the smoothness of the RHS of (6.100) for x → 0 in the case of inversions: ! = $ = 0, " = –# = 1. The (nonunitary) characters of A1 are obtained from those of A0 by the obvious restriction and are given by D-1 (a) = e–s(3+-) ,

- ∈ C,

(6.101)

where 11 (ˆe1 ) = 3, 11 = 10 | A1 , recalling that eˆ 1 spans the Lie algebra A1 of A1 . The signature of the ERs induced from P1 is denoted by 71 = [n, k, :, -]

(6.102)

and the (inﬁnite-dimensional) inducing representation of M1 A1 is denoted by 7

D1 1 (ma) = ein>/2 D-1 (a)T1k,: (ˆr).

(6.103)

The representation space of the ER 71 is standardly deﬁned as C71 = {F1 ∈ C∞ (G × R, C) | F1 (g) ∈ Vk1 , F1 (gman–1 , x) 71 !x + " –1 –in>/2 s(3+-) = D0 (ma) F1 (g) (g, x) = e e F1 g, , $ + #x g ∈ G, man–1 ∈ M0 A0 N1– m = 4(>)ˆr, y ∈ R}

(6.104)

and its action T 71 is given by the left regular action of G as in (6.95). Finally, we introduce a K-invariant scalar product in C71 by the formula

F1 , F1′ = *1

*

dk F1 (k) F1′ (k)1 ,

(6.105)

K

where *1 = 0 3 /2 is introduced for convenience, ⋅, ⋅1 is the SL(2, R)-invariant scalar product in the space Vk1 (cf. [250] and [159] App. B). The representation T 71 is continuous w.r.t. the topology deﬁned by (6.105). The completion of C71 with respect to (6.105) shall be denoted by H71 . The scalar product (6.105) is also G-invariant iff - = i,, , ∈ R. The latter ERs are unitary representations of G acting in H71 [351]. 6.2.3 ERs from the Maximal Noncuspidal Parabolic Subgroup P2 We consider P2 = M2 A2 N2 .

206

6 Conformal Case in 4D

The most general representations of SL(2, C) may be indexed by two complex numbers n1 , n2 such that n1 – n2 ∈ Z. The representation spaces are given by [250]: Vn21 ,n2 = {6 ∈ C∞ (C) | 6(z)

→ | z | → ∞

zn1 z¯ n2

∞

ajk z–j z¯ –k }.

(6.106)

j,k=0

(Note that our n1 , n2 differ by one w.r.t. those of [250].) For the characters of {1, #3 } we take as in the P0 -case the parameter % = 0, 1. Thus, the representation of M2 = SL(2, C)⋊ {1, #3 } we take

n ,n2 ,%

T2 1

(m2 ) 6 (z) = (–1)%N (! – #z)n1 (! – #z)n2 $z – " × 6 (–1)N , m2 = l #3N . ! – #z ˜

(6.107)

The (nonunitary) characters of A2 are obtained from those of A0 by the obvious restriction and are given by Dc2 (a) = | a | –(2+c) ,

c ∈ C,

(6.108)

where 12 (ˆe1 + eˆ 2 ) = 4, 12 = 10 | A2 , recalling that eˆ 1 + eˆ 2 spans the Lie algebra A2 of A2 . The signature of the ERs induced from P2 is denoted by 72 = [n1 , n2 , %, c]

(6.109)

and the (inﬁnite-dimensional) inducing representation of M2 A2 is denoted by 7

n ,n2 ,%

D22 (ma) = Dc2 (a)T2 1

(m).

(6.110)

The representation space of the ER 70 is standardly deﬁned as C72 = {F2 ∈ C∞ (G × C, C) | F2 (g) ∈ Vn21 ,n2 , F2 (gman–2 , z) n ,n ,% = Dc2 (a)–1 T2 1 2 (m)–1 F2 (g, z) = (–1)%N | a | 2+c ($ + (–1)N #z)n1 (! – (–1)N #z)n2

N N !z + (–1) " , × F2 g, (–1) $ + (–1)N #z

l 33N 0 – – g ∈ G, man2 ∈ M2 A2 N2 , m = ˜ , 0 (l† )–1 33N ˜ l ∈ SL(2, C), N = 0, 1} ˜ and its action T 72 is given by the left regular action of G as in (6.95).

(6.111)

6.2 Elementary Representations of SU(2, 2)

207

Finally, we introduce a K-invariant scalar product in C72 by the formula

F2 , F2′ = *2

*

dk F2 (k), F2′ (k) 2 ,

(6.112)

K

where *2 = 0 3 /8 is introduced for convenience, ⋅, ⋅2 is a SU(2) ⋊ {1, #3 } invariant scalar product in Vn21 ,n2 : 6, 6′ 2 =

*

6(z)6′ (z) ,∗ (z)dzd¯z

,∗ (z) = (1 + | z | 2 )(*+¯*)/2 ,

(6.113)

* = n 1 + n2 .

Note that for * = i3, 3 ∈ R, (then ,∗ (z) = 1), (6.113) is an M2 -invariant scalar product and this gives rise to the principal series of unitary irreducible representations of SL(2, C). The representation T 72 is continuous w.r.t. the topology deﬁned by (6.112). The completion of C72 with respect to (6.112) shall be denoted by H72 . The scalar product (6.112) is also G-invariant iff c = i1, * = i3, 1, 3 ∈ R. The latter ERs are unitary representations of G acting in H72 , equivalent to the principal series, as we shall show below.

6.2.4 Noncompact Picture of the ERs Here we shall consider the so-called the equivalent noncompact picture of the ERs. The representation spaces C7k consists of C∞ -functions fk over Nk+ . They may be viewed as restrictions from G to Nk+ of the functions from C7k : fk ∈ C∞ (Nk+ , C).

(6.114)

They have special asymptotic properties (cf. [162], (2.33), (2.42), and (2.48)). Accordingly, the representation action on C7k is determined by the equivalence. We consider the three nontrivial parabolics in turn. 6.2.4.1 Parabolic P0 In the case of 70 we have (T 70 (g)f0 )(x, z) = D70 (mg ag ) f0 (xg , zg ) ˜ ˜ = e–sg (3+c1 )–tg (1+c2 )+in>g /2 (–1)%Ng f0 (xg , zg ) ˜

(6.115)

where mg , ag , xg , zg are deﬁned by the Bruhat decomposition ˜ N0+ N0– A0 M0 of g –1 n+0 –1 g –1 n+0 (x, z) = n+0 (xg , zg )(n–g )–1 a–1 g mg . ˜ ˜

(6.116)

208

6 Conformal Case in 4D

The cases when g –1 n+0 belongs to lower-dimensional subspace and (6.116) does not hold are obtained as limiting cases of (6.115), and the smoothness of of (6.115) is guaranteed by the asymptotic properties of f0 , cf. [162]. The equivalence of the representations T 70 and T 70 is given by B0 : C 70 → C70 ,

(B0 F0 )(x, z) ≡ F0 (n+0 (x, z)), ˜ ˜ 70 –1 (B–1 when g = n+0 (x, z)n–0 a0 m0 , 0 f0 )(g) ≡ D (m0 a0 ) f0 (x, z), ˜ ˜ 70 –1 70 (B–1 0 f0 )(g) ≡ D (m0 a0 ) (T (3(s))f0 )(xs , zs ), ˜ when g = 3(s)n+0s (xs , zs )n–0 a0 m0 . ˜

(6.117)

In the last case of (6.117) the elements g belong to submanifolds of G of lower dimensionality which can not be represented as N0+ N0– A0 M0 . The elements of these submanifolds are decomposed as 3(s)n+0s n–0 am, where 3(s) is a matrix representation of the elements s ∈ W(G, A0 ). The latter cases were described explicitly in Proposition 5.2 of [122]. 6.2.4.2 Parabolic P1 In the case of 71 we have (T 71 (g)f1 )(n+1 , x) = (D71 (mg ag ) f1 )(n+1g , x) D-1 (ag ) (T1k (ˆrg ) f1 )(n+1g , x)

=e

in>g /2

=e

in>g /2 –sg (3+-)

e

!g – x#g

–k–1

f1

n+1g ,

(6.118) x$g – "g , !g – x#g

where mg , ag , n+1g are deﬁned by the Bruhat decomposition N1+ N1– A1 M1 of g –1 n+1 –1 g –1 n+1 = n+1g (n–1g )–1 a–1 g mg .

(6.119)

The equivalence of the representations T 71 and T 71 is given by B1 : C 71 → C71 ,

(B1 F1 )(x, z) ≡ F1 (n+0 (x(x– = 0), z)) ˜ ˜ 71 –1 (B–1 1 f1 )(g) ≡ D (ma) f1 (x(x– = 0), z), ˜ when g = n+1 (x(x– = 0), z)n–1 am, ˜ –1 7 71 1 (B–1 1 f1 )(g) ≡ D (ma) (T (3(s))f1 )(xs , zs ), ˜ when g = 3(s)n+1s n–1 am, 3(s) = "2 ,

(6.120)

where in the last case of (6.120) the element g (resp. n+1s ) belongs to a 13-dimensional submanifold of G (resp. to a four-dimensional submanifold of N0+ ), the explicit ¯ x– = 0, u, z). parametrization being n+1s = n+0 (x+ = i(u¯z – uz),

6.2 Elementary Representations of SU(2, 2)

209

6.2.4.3 Parabolic P2 In the case of 72 we have (T 72 (g)f2 )(x, z) = (D72 (mg ag ) f2 )(xg , z) ˜ ˜ n ,n ,% = Dc2 (ag ) (T2 1 2 (mg ) f2 )(xg , z) ˜ n n = (–1)%N | ag | –2–c !g – z#g 1 !g – z#g 2 z$g – "g × f2 xg , (–1)N , !g – z#g ˜

(6.121)

where mg , ag , n+2g are deﬁned by the Bruhat decomposition N2+ N2– A2 M2 of g –1 n+2 –1 g –1 n+2 = n+2g (n–2g )–1 a–1 g mg .

(6.122)

The equivalence of the representations T 72 and T 72 is given by B2 : C 72 → C72 ,

(B2 F2 )(x) ≡ F2 (n+0 (x, z = 0)), ˜ ˜ 72 –1 (B–1 2 f2 )(g) ≡ D (m2 a2 ) f2 (x), ˜ when g = n+2 (x)n–2 a2 m2 , ˜ 72 –1 72 f )(g) ≡ D (m (B–1 2 2 a2 ) (T (3(s))f2 )(xs ), 2 ˜ when g = 3(s)n+2s n–2 a2 m2 , 3(s) = "2 ,

(6.123)

where in the last case of (6.123) the element g (resp. n+2s ) belongs to a 14-dimensional submanifold of G (resp. to a three-dimensional submanifold (a cone) of N2+ ), the explicit parametrization being n+2s = n+2 (x) | x,2 =0 . ˜ 6.2.5 Properties of ERs 6.2.5.1 Some Relations between the ERs Induced from Different Parabolics First we give the following Proposition announced in [122] and proved in [162]. Proposition: The ERs T 70 and T 72 deﬁned in 6.2.4.1, 6.2.4.3, respectively are equivalent whenever the representation parameters 70 = [n, %, c1 , c2 ],

72 = [n1 , n2 , %′ , c]

are related by the formulae % = %′ ,

n = n 1 – n2 ,

c1 = c – 21 (n2 + n2 ),

c2 = c + 21 (n2 + n2 ).

(6.124)

210

6 Conformal Case in 4D

The operators realizing the equivalence are: A02 : C72 → C70 ,

A–1 02 : C70 → C72 ,

(A02 f2 ) (n+0 (x, z)) ≡ f2 (x; z), ˜ ˜ –1 A02 f0 (x; z) ≡ f0 (n+0 (x, z)), ˜

˜

(6.125)

f2 ∈ C72 , f2 ∈ C72 .

♢

We omit the Proof given in [162]. Next we give a result on the relation between the ERs induced from P0 and P1 also announced in [122] and proved in [162]. ′

Proposition: The ERs T 71 and T 71 deﬁned in 6.2.4.2, where 71 = [n′ , k, : = 0, -], 71′ = [n′ , k, : = 1, -], are partially equivalent to the ER T 70 , 70 = [n, %, c1 , c2 ], whenever the representation parameters are related by the formulae n = n′ ,

% = (k + 1)(mod2),

c1 = -,

c2 = k

(6.126)

The operators realizing the partial equivalence are embeddings given by A01 : C71 → C70 ,

A′01 : C7′ → C70 ,

(6.127)

1

(A01 f1 ) (n+0 (x+ , x– , u; z)) ≡ f1 (n+1 (x+ – x– | z | 2 , u – ix– z, z), y = x– ) ′ ′ + A01 f1 (n0 (x+ , x– , u; z)) ≡ f1′ (n+1 (x+ – x– | z | 2 , u – ix– z, z), y = x– ) and C71 , C7′ , are embedded in C70 with trivial intersection. 1

f1 ∈ C71 f1′ ∈ C7′

1

♢

We omit the Proof given in [162]. Remark 1: Note that the image of the operator A01 , A′01 , respectively, consists of functions f0 (n+0 (x+ , x– , u, z)) which are analytic in y = x– + i', ' > 0 for A01 , ' < 0 for A′01 , respectively. 6.2.5.2 Casimirs It is well known that the ERs are operator irreducible (cf. [289, 352, 353, 614]). Thus, the Casimir operators are scalar multiples of the identity operator. The inﬁnitesimal generators and the Casimir operators were given in detail in [162]. Here we give only a summary. The value of the second-order Casimir in the ER 70 is 1 C2 ( 70 ) = (c21 + c22 + 21 n2 ) – 5. 2

(6.128)

The values for the P1 and P2 induced ERs are obtained by substituting (6.126) and (6.124) 1 2 1 2 1 2 C 2 ( 71 ) = (6.129) - + k + n – 5, C2 ( 72 ) = (n21 + n22 ) + c2 – 5. 2 2 2

6.2 Elementary Representations of SU(2, 2)

211

Note that the arbitrary additive constant is chosen to be –5 so that the 2nd Casimir has value zero on the trivial representation. Using the explicit expressions of Yao [634] for the third- and fourth-order Casimirs we have [162] 1 n(c21 – c22 ), 8 1 4 1 2 1 C 4 ( 70 ) = n + n (c1 c2 – 1) + (c21 + c22 )2 + 2c1 c2 – 31. 16 2 4

C 3 ( 70 ) =

(6.130)

6.2.6 Integral Invariant Operators In Subsection 4.3.3 we recalled the well-known fact [352, 353] that to every nontrivial element of a restricted Weyl group W(G, A) there corresponds an integral intertwining KS operator which establishes (partial) equivalence between certain ERs. In this subsection we deﬁne these operators for SU(2, 2) and specify the ERs they intertwine. The general deﬁnition of the integral intertwining operator is (4.10) * A7a f (g) ≡ #s ( 7a ) f g3(s)n+as dn+as (6.131) + Nas

+, where f ∈ C7a , a = 0, 1, 2, s ∈ W(G, Aa ), 3(s) is a matrix representation of s, n+as ∈ Nas + the latter is the submanifold of Na that is not invariant under s, i.e., + Nas ≡ 3(s)–1 Na– 3(s) ∩ Na+ ,

(6.132)

+. dn+as is the Haar measure on Nas

6.2.6.1 P0 We consider ﬁrst the KS operators between P0 -induced ERs. We enumerate the seven nontrivial elements sk of the restricted Weyl group W(G, A0 ) as in (6.32) and denote + ≡ N + . Then explicitly we have N0k 0sk + N01 = {n+0 (x+ = 0, x– , u = 0, z = 0)},

dn+01 = dx– ,

+ = {n+0 (x+ = 0, x– = 0, u = 0, z)}, N02

dn+02 = dz d¯z,

+ = {n+0 (x+ , x– = 0, u, z)}, N03

dn+03 = dx+ du du¯ dz d¯z,

+ = {n+0 (x+ , x– , u, z = 0)}, N04

¯ dn+04 = dx+ dx– du du,

+ = {n+0 (x+ , x– = 0, u = 0, z)}, N05

dn+05 = dx+ dz d¯z,

+ = {n+0 (x+ = 0, x– , u, z = 0)}, N06

¯ dn+06 = dx– du du,

+ = N0+ , N07

dn+07 = dn+0 = dx+ dx– du du¯ dz d¯z,

(6.133)

212

6 Conformal Case in 4D

where we have used the following matrix representations for the generating elements:

1 + i e1 3(s1 ) = √ 2 –e2

–e2 e1

∈ K,

i32 0

3(s2 ) =

0 i32

∈ K.

(6.134)

Next we recall from [122], formulae (6.5), the action of the corresponding operators, i.e., the ERs which are intertwined with the initial 70 : 70 = [n, %, c1 , c2 ]

(6.135)

A(s1 )70 = s1 70 = [n, %, c1 , –c2 ],

c2 ≠ 0,

A(s2 )70 = s2 70 = [–n, (% + n)(2) , c2 , c1 ], A(s3 )70 = s3 70 = [n, %, –c1 , c2 ],

n ≠ 0, c1 ≠ c2 ,

c1 ≠ 0,

A(s4 )70 = s4 70 = [–n, (% + n)(2) , –c2 , –c1 ],

n ≠ 0, c1 ≠ –c2 ,

A(s5 )70 = s5 70 = [–n, (% + n)(2) , c2 , –c1 ],

n ≠ 0 or c1 ≠ 0 or c2 ≠ 0,

A(s6 )70 = s6 70 = [–n, (% + n)(2) , –c2 , c1 ],

n ≠ 0 or c1 ≠ 0 or c2 ≠ 0,

A(s7 )70 = s7 70 = [n, %, –c1 , –c2 ],

c1 ≠ 0 or c2 ≠ 0,

where x(2) ≡ x(mod 2). The exclusion conditions are to avoid the trivial cases, e.g., when s1 70 = 70 , when the operator would be identity. From the explicit expressions of the Haar measures it is clear that it is more convenient to present the explicit expressions for the operators A70 in the noncompact picture which we do following [162] A70 (s1 ) f (x+ , x– , u, z) = M1 ( 70 )

*

sign% (x–′ – x– ) f (x+ , x–′ , u, z) dx–′ . | x–′ – x– | 1–c2

(6.136)

From the action of A70 (s1 ) we expect that As1 70 (s1 ) A70 (s1 ) should be proportional to the identity operator. Requiring As1 70 (s1 ) A70 (s1 ) f = f

(6.137)

one obtains

M1 ( 70 ) M1 (s1 70 ) =

(–1)% 0

A

1+%+c2 2 2 A %+c 2

% ± c2 ≠ –k,

1+%–c2 2 %–c2 A 2

A

k = 0, 1, . . . .

(6.138a) (6.138b)

6.2 Elementary Representations of SU(2, 2)

213

Thus, (6.137) holds except in the excluded cases (6.138b) in which cases As1 70 (s1 ) is not the inverse of A70 (s1 ). In these cases the ERs are reducible and some of the KS operators degenerate to differential operators. The reducible ERs are classiﬁed in the next section. Analogously, we have

A70 (s2 ) f (x+ , x– , u, z) = M2 ( 70 )

* C

M2 ( 70 ) M2 (s2 70 ) =

(z –

f (x+ , x– , u, z′ ) dz′ d¯z′ ′ 1+(c z ) 2 –c1 +n)/2 (¯z – z¯ ′ )1+(c2 –c1 –n)/2

(–1)n 2 n – (c1 – c2 )2 , (40)2

c1 – c2 ± n ≠ 0.

(6.139)

(6.140)

Again in the excluded cases given in (6.140) the operator As1 70 (s2 ) is not the inverse of A70 (s2 ). Further, one may we use the fact that s1 , s2 are the generating reﬂections and thus, the operators for s3 , . . . , s7 may be obtained by composing the operators for s1 , s2 . The explicit expressions are given in [162], Section 4. We present only A70 (s7 ) furthermore using directly the deﬁnition (6.131): A70 (s7 ) f (x1+ , x1– , u1 , z1 ) = M7 ( 70 ) × sign% (–x12M )

*

f (x2+ , x2– , u2 , z2 ) 2 | 1–c2 | x12,

q(n+1 , n+2 )n dx2+ dx2– du2 du¯ 2 dz2 d¯z2 , | q(n+1 , n+2 ) | n+2–c1 +c2

(6.141)

where x12 = x1 – x2 ,

2 x12M ≡ (x12 ), (x12 ), ,

x12± ≡ x1± – x2± , q(n+1 , n+2 )

u12 ≡ u1 – u2 ,

(6.142)

≡ –x12+ + iz1 u¯ 12 – i¯z2 u12 – z1 z¯ 2 x12–

1 1 –x12+ –i u12 , = – (1, z1 ) 3 (x12 ), = (1, z1 ) z¯ 2 z¯ 2 i u¯ 12 –x12–

A nice feature of (6.141) is that the LHS variables are not present in the function f under the integral. This allows to extract reliable information on the asymptotic behavior of the LHS which was used in [162]. 6.2.6.2 P1 Next, we consider the operators between P1 -induced ERs. Actually, the restricted Weyl group W(G, A1 ) has only one nontrivial element, thus, there is only one intertwining operator. We specialize the deﬁnition (6.131): * A71 f (g) ≡ #1 ( 71 ) f (g3(s)n+1 ) dn+1 , (6.143) N1+

214

6 Conformal Case in 4D

where

dn+1 = dx+ du du¯ dz d¯z

and we make the choice 3(s) =

0 12

12 . The intertwin0

ing properties are [162]: s 71 = [n, k, :, –-]

71 = [n, k, :, -]

for

(6.144)

Explicitly, the operator A71 is given in (4.35) of [162]: r(n+1 , n+2 ; y) f x2+ , u2 , z2 ; s(n+1 , n+2 ; y) + + n w(n1 , n2 ) 1 × dx2+ du2 du¯ 2 dz2 d¯z2 + + + n+2–-+k | w(n1 , n2 ) | | s(n1 , n+2 ; y) | 1+k *

(A71 f )(x1+ , u1 , z1 ; y) = M1 ( 71 )

(6.145)

where w(n+1 , n+2 ) ≡ i(x12+ – iz1 u¯ 12 + i¯z2 u12 )

1 x12+ i u12 = i(1, z1 ) = – i (q(n+1 , n+2 ))x12– = 0 z¯ 2 –i u¯ 12 0

(6.146)

r(n+1 , n+2 ) ≡ | u12 | 2 y – i w(n+1 , n+2 ), s(n+1 , n+2 ) ≡ | z12 | 2 + i( w(n+1 , n+2 )y.

(6.147)

Furthermore, we have M1 (s 71 )M1 ( 71 ) =

0- % 2 2 2 2 [– (n + k) ] [– (n – k) ] tg . 29 0 5 2

(6.148)

The last formula agrees with the one derived by use of the Plancherel formula in [351], p.52 with the replacements: - ↦ 3z, k ↦ k – 1. The advantage of (6.148) is that all numerical factors are ﬁxed. 6.2.6.3 P2 Finally, we consider the operators between P2 -induced ERs. The restricted Weyl group W(G, A2 ) has only one nontrivial element, thus, there is only one intertwining operator. We specialize the deﬁnition (6.131): * A72 f (g) ≡ #2 ( 72 ) f (g3(s)n+2 ) (dn+ )2 (6.149) N2+

where dn+2 = dx+ dx– du du¯ (= d4 x, the latter used standardly space-time) and

for 4D 0 12 . The intertwining we make the same choice as in the previous case: 3(s) = 12 0 properties are s 72 = [n1 , n2 , %, –c]

for

72 = [n1 , n2 , %, c].

(6.150)

215

6.3 Invariant Differential Operators and Multiplet Classification of the Reducible ERs

Explicitly, the operator A72 is given in (4.51) of [162]: *

dx2+ dx2– du2 du¯ 2 (–1)%N 2 | x12M | 2–c n1

x12– + (–1)N izu¯ 12 × 2 | x12M | 1/2

n2 x12– – (–1)N i¯zu12 × 2 | x12M | 1/2

zx12+ – (–1)N iu12 × f x2+ , x2– , u2 , u¯ 2 ; x12– + (–1)N izu¯ 12

(A72 f )(x1+ , x1– , u1 , u¯ 1 ; z1 ) = M2 ( 72 )

(6.151)

2 ). Note that due to our formalism the integral kernel in (6.151) where (–1)N = sign(–x12M seems to be different from the standard conformal two-point function, cf., e.g., [518], yet they are equivalent, cf. [162]. Finally, we have in (4.59) of [162]:

M2 (s 72 )M1 ( 72 ) = × ×

c2 – n21 n22 40 4

(6.152)

A( 21 (1 + % + c + (n1 + n2 )/2)) A( 21 (1 + % – c – (n1 + n2 )/2)) A( 21 (% + c + (n1 + n2 )/2)) A( 21 (% – c – (n1 + n2 )/2)) A( 21 (1 + %′ + c – (n1 + n2 )/2)) A( 21 (1 + %′ – c + (n1 + n2 )/2)) A( 21 (%′ + c – (n1 + n2 )/2)) A( 21 (%′ – c + (n1 + n2 )/2))

where %′ = (% + n1 – n2 )(2) .

6.3 Invariant Differential Operators and Multiplet Classification of the Reducible ERs 6.3.1 Explicit Expressions for the Invariant Differential Operators Following the general procedure of Chapter 3, we need to provide the explicit expressions for the invariant differential operators corresponding to the positive roots of sl(4, C) = su(2, 2)C . These expressions were derived in [122, 164, 518]. We present ﬁrst the operators as acting on the functions f0 (x, z, z¯ ) in the P0 -noncompact picture ˜ (cf. Subsection 6.2.4.1; restoring the explicit z¯ dependence which was suppressed there to avoid notational clutter).

216

6 Conformal Case in 4D

The explicit expressions for the operators Dm" from (4.35) and (4.36) corresponding to the simple roots of sl(4, C) are [122, 126]: Dm!1 =

∂ ∂z

m ≡ I1m

(6.153)

m Dm!2 = z¯ z∂+ + z∂v + z¯ ∂v¯ + ∂– m

z¯ ∂ + ∂v = (z, 1) 1 ∂v¯ ∂–

m

z¯ = (z, 1) 3, ∂, ≡ I2m 1 m ∂ Dm!3 = ≡ I3m , ∂z¯ where we have made judicious choice of the arbitrary nonzero multiplicative constants in the above expressions coming from the simple root singular vectors (Xk– )m v0 . For the two nonsimple nonmaximal roots using the expressions for the corresponding singular vectors [122, 126], we have m

Dmij !ij =

ij

aimij k (Ii )mij –k (Ij )mij (Ii )k

(6.154)

k=0 mij

=

j

am k (Ij )mij –k (Ii )mij (Ij )k , ij

(6.155)

k=0

(–1)k mij , = i, j, m – k k (ij) = (12), (23), mij = mi + mj ∈ N, mi , mj ∉ Z+ .

amij k = a

If mij = mi + mj and mi , mj ∈ Z+ then the expressions simplify Dmij !ij = ci (Ii )mj (Ij )mij (Ii )mi mi

mij

= cj (Ij ) (Ii )

mj

(Ij ) ,

(6.156) mij ∈ N, mi , mj ∈ Z+ .

For the highest root !˜ = !13 = !1 + !2 + !3 , we have

Dm13 !13 =

m13 m13

j=0 k=0

am13 jk (I1 )m13 –j (I3 )m13 –k (I2 )m13 (I1 )j (I3 )k ,

6.3 Invariant Differential Operators and Multiplet Classification of the Reducible ERs

am13 jk = am13

(–1)j+k (m1 – j)(m3 – k)

m13 j

m13 , k

217

(6.157)

m13 = m1 + m2 + m3 ∈ N, m1 , m2 , m3 ∉ Z+ . When some mj ∈ N then the above formula is simpliﬁed. There are many such occurrences which we leave to the reader. We only record when all mj ∈ N: Dm13 !13 = (I1 )m23 (I3 )m12 (I2 )m13 (I3 )m3 (I1 )m1

(6.158)

= (I1 )m23 (I2 )m3 (I3 )m13 (I2 )m12 (I1 )m1 = (I3 )m12 (I2 )m1 (I1 )m13 (I2 )m23 (I1 )m3 , recalling that [I1 , I3 ] = 0. 6.3.2 Multiplet Classification: Case P0 Following our general philosophy, we classify the reducible ERs via the multiplet classiﬁcation. We start with the ERs induced from the minimal parabolic P0 . In the case of P0 , there are no M-compact (imaginary) roots, thus the invariant differential operators corresponding to all positive roots of G C = sl(4, C) are important for the multiplet classiﬁcation of the reducible ERs. Accordingly, the main multiplet is 1-to-1 with the Weyl group of sl(4, C) and has 24 members ( | W(sl(4) | = 4! = 24). Along the general picture, the multiplets are parametrized by the ﬁnitedimensional irreps of su(2, 2). We write as ﬁrst member D0 of the multiplet, the one with dominant highest weight; thus, its Dynkin labels m1 , m2 , m3 are positive integers. For ﬁxed Dynkin labels in the ER D0 is contained the ﬁnite-dimensional irrep denoted Em1 ,m2 ,m3 of dimension: m1 m2 m3 m12 m23 m13 /12 [122]. The relation between the Dynkin labels m1 , m2 , m3 of D0 and its P0 signature 70 = [n, %, c1 , c2 ] is as follows [122]: 1 1 m1 = (c2 – c1 + n), m2 = – c2 , m1 = (c2 – c1 – n), 2 2 n = m1 – m3 , c1 = – m1 – m2 – m3 , c2 = – m2 .

(6.159)

Generically, the discrete representation parameter % is not determined through the Dynkin labels which are pertinent to the Lie algebra. On the other hand, when the representations are reducible, it is ﬁxed: % = (m13 – 1) (mod 2) = (c1 + 1) (mod 2). Thus, further we omit % in the signatures when classifying the reducible ERs.

(6.160)

218

6 Conformal Case in 4D

It will be convenient to parametrize the reducible ERs in the multiplet by the Harish-Chandra parameters: m1 , m2 , m3 , m12 , m23 , m13 , corresponding to the sl(4) positive roots: !1 , !2 , !3 , !12 , !23 , !13 . Thus, the multiplet is given by D0 = (m1 , m2 , m3 ; m12 , m23 , m13 ),

(6.161)

D1 = (–m1 , m12 , m3 ; m2 , m13 , m23 ), D2 = (m12 , –m2 , m23 ; m1 , m3 , m13 ), D3 = (m1 , m23 , –m3 ; m13 , m2 , m12 ), D12 = (m2 , –m12 , m13 ; –m1 , m3 , m23 ), D13 = (–m1 , m13 , –m3 ; m23 , m12 , m2 ), D21 = (–m12 , m1 , m23 ; –m2 , m13 , m3 ), D23 = (m12 , m3 , –m23 ; m13 , –m2 , m1 ), D32 = (m13 , –m23 , m2 ; m1 , –m3 , m12 ), D121 = (–m2 , –m1 , m13 ; –m12 , m23 , m3 ), D123 = (m2 , m3 , –m13 ; m23 , –m12 , –m1 ), D132 = (m23 , –m13 , m12 ; –m1 , –m3 , m2 ), D213 = (–m12 , m13 , –m23 ; m3 , m1 , –m2 ), D323 = (m13 , –m3 , –m2 ; m12 , –m23 , m1 ), D321 = (–m13 , m1 , m2 ; –m23 , m12 , –m3 ), D1213 = (–m2 , m23 , –m13 ; m3 , –m1 , –m12 ), D1323 = (m23 , –m3 , –m12 ; m2 , –m13 , –m1 ), D1321 = (–m23 , –m1 , m12 ; –m13 , m2 , –m3 ), D2132 = (m3 , –m13 , m1 ; –m12 , –m23 , –m2 ), D3213 = (–m13 , m12 , –m2 ; –m3 , m1 , –m23 ), D12132 = (m3 , –m23 , –m1 ; –m2 , –m13 , –m12 ), D13213 = (–m23 , m2 , –m12 ; –m3 , –m1 , –m13 , ) D32132 = (–m3 , –m12 , m1 ; –m13 , –m2 , –m23 ), D213213 = (–m3 , –m2 , –m1 ; –m23 , –m12 , –m13 ), where Di1 i2 ...it = wit ⋯wi2 wi1 D0 . Here wj , j = 1, 2, 3, are the three simple sl(4) reﬂections. The multiplet is given in the following ﬁgure:

219

6.3 Invariant Differential Operators and Multiplet Classification of the Reducible ERs

Λ0 11

22

Λ1

Λ21

Λ12 21

Λ2

Λ3 11

33

122

133

33

Λ121

12

232 Λ13

121 233

133

Λ323

32

121

132

Λ32

132 32

Λ132

123 Λ2132 Λ12132

Λ3213 123

13

122 Λ13213

31

Λ321

23

Λ1321

Λ1213

131

12 231

Λ1323

232

33

131

Λ213

Λ123

21

Λ23

233

22

31

Λ32132

13

Λ132132

(6.162) where the embedding weights m", and accordingly the invariant differential operators, are encoded as follows: jk ∼ Dmj !k and ijk ∼ Dmij !k . As expected the diagram of the multiplet is composed from quartets (created by the commuting reﬂections w1 and w3 ) and sextets created either by the pair w1 , w2 or by the pair w2 , w3 . To make the diagram simpler, we give the sextets of the multiplet using only the embeddings due to simple reﬂections (as in 2.11.2.4) from the A2 = sl(3) case). Consequently, in the embedding weights m", only cases involving the simple roots occur: " = !1 , !2 , !3 . It is important to know the action of the KS integral operators inside the above multiplet. We shall see that the multiplet splits in three orbits under the action of the restricted Weyl group, each orbit with eights reducible ERs [122]. Using (6.135) we display the action of the seven operators on the dominant weight member

D0 = D0 ( 70 ) = (m1 , m2 , m3 ); A(s1 )D0 = D2 = (m12 , –m2 , m23 ); A(s2 )D0 = D13 = (–m1 , m13 , –m3 ),

m2 ≠ 0; m1 , m3 ≠ 0;

220

6 Conformal Case in 4D

A(s3 )D0 = D13213 = (–m23 , m2 , –m12 ); A(s4 )D0 = D2132 = (m3 , –m13 , m1 ); A(s5 )D0 = D132 = (m23 , –m13 , m12 ); A(s6 )D0 = D213 = (–m12 , m13 , –m23 ); A(s7 )D0 = D213213 = (–m3 , –m2 , –m1 ).

(6.163)

The above actions are consistent with s1 ≅ w2 , s2 ≅ w1 w3 = w3 w1 , s3 = s2 s1 s2 , s4 = s1 s2 s1 , s5 = s2 s1 , s6 = s1 s2 , and s7 = s25 = s26 (cf. (6.32)). Formulae (6.163) display one orbit isomorphic to the restricted Weyl group (as we noted, this is the so(5) Weyl group): Λ2

S2

Λ213

S1

Λ2132

S1

S2

•

Λ0

Λ213213

S2

S1 Λ13

S1

Λ132

S2

Λ13213

(6.164)

where we have inserted a bullet signifying the symmetry w.r.t. the maximal element s7 of the restricted Weyl group. These means that the ERs which are symmetric w.r.t. the bullet are related by the A(s7 ) operator. (There are obviously four pairs of ERs related in this way.) Note that the KS operator A(s1 ) from D0 to D2 has degenerated to the differential operator Dm2 !2 . Analogously, in the other three cases the KS operators A(s1 ): from D13 to D132 , from D213 to D2132 , and from D13213 to D213213 have degenerated to Dm13 !2 , Dm13 !2 , and Dm2 !2 , respectively. Another orbit starting from D1 playing the role of D0 , and given in the same order as (6.163), is D1 , D12 , D3 , D3213 , D12132 , D32 , D1213 , D21321 .

(6.165)

The third orbit starting from D21 playing the role of D0 , and given in the same order as (6.163), is D21 , D121 , D23 , D321 , D1323 , D323 , D123 , D1321 .

(6.166)

Of course, the phenomenon that we noted in (6.164) about the degeneration of the operator A(s1 ) is valid for both (6.165) and (6.165). Naturally, it is true for the overall main diagram (6.162) and for all its reductions below.

6.3 Invariant Differential Operators and Multiplet Classification of the Reducible ERs

221

P0 reduced multiplets – Symmetrically reduced multiplets. These depend on two parameters, e.g., m1 , m3 , and may be obtained by formally setting m2 = 0 in (6.161): D0 = (m1 , 0, m3 ; m1 , m3 , m1,3 ),

(6.167)

D1 = (–m1 , m1 , m3 ; 0, m1,3 , m3 ), D3 = (m1 , m3 , –m3 ; m1,3 , 0, m1 ), D12 = (0, –m1 , m1,3 ; –m1 , m3 , m3 ), D13 = (–m1 , m1,3 , –m3 ; m3 , m1 , 0), D32 = (m1,3 , –m3 , 0 ; m1 , –m3 , m1 ), D123 = (0, m3 , –m1,3 ; m3 , –m1 , –m1 ), D132 = (m3 , –m1,3 , m1 ; –m1 , –m3 , 0), D321 = (–m1,3 , m1 , 0 ; –m3 , m1 , –m3 ), D1323 = (m3 , –m3 , –m1 ; 0, –m1,3 , –m1 ), D1321 = (–m3 , –m1 , m1 ; –m1,3 , 0, –m3 ), D13213 = (–m3 , 0, –m1 ; –m3 , –m1 , –m1,3 ). The multiplet is presented in the following diagram: Λ0

11

33

Λ1 Λ12

12

Λ3 33

1,33

Λ13

11

32 Λ32

1,32 Λ123

1,31 Λ132

32

Λ1323

13

31

31

13 Λ13213

Λ321 Λ1321

12

(6.168)

where 1, 3j ≡ m1,3 !j , m1,3 ≡ m1 + m3 , and the bullet shows symmetry w.r.t. the maximal restricted Weyl group element s7 . Here it is important to note that between D13 and D132 act the KS operators A(s1 ) (and with coinciding action A(s7 )), but the operator A(s1 ) from D13 to D132 has degenerated to the differential operator Dm1,3 !2 indicated on the ﬁgure. The same phenomenon happens between the pairs: D1 , D12 ; D123 , D1323 ; D3 , D32 and D321 , D1321 .

222

6 Conformal Case in 4D

There are two orbits of the restricted Weyl group: the ﬁrst orbit has four members and may be seen as reduction of the orbit (6.163) of the main multiplet:

Λ0

S2

Λ13

S1

S2

Λ132

Λ13213

(6.169)

The other orbit has eight members and may be seen as coincidence of the second (6.165) and third (6.166) orbits of the main multiplet. To display it one starts from D1 , playing the role of D0 as in (6.164): Λ12

S2

Λ123

S1

Λ1323

S1

S2

•

Λ1

Λ1321

S2

S1 Λ3

S1

Λ32

S2

Λ321

(6.170)

Of course, the phenomenon that we noted in (6.168) about the degeneration of the operator A(s1 ) is valid for both (6.169) and (6.170). –

Asymmetrically reduced multiplets. These depend on two parameters, e.g., m1 , m2 , and may be obtained by formally setting m3 = 0 in (6.161): D0 = (m1 , m2 , 0 ; m12 , m2 , m12 ), D1 = (–m1 , m12 , 0 ; m2 , m12 , m2 ), D2 = (m12 , –m2 , m2 ; m1 , 0, m12 ), D12 = (m2 , –m12 , m12 ; –m1 , 0, m2 ), D21 = (–m12 , m1 , m2 ; –m2 , m12 , 0), D23 = (m12 , 0, –m2 ; m12 , –m2 , m1 ), D121 = (–m2 , –m1 , m12 ; –m12 , m2 , 0), D123 = (m2 , 0, –m12 ; m2 , –m12 , –m1 ), D213 = (–m12 , m12 , –m2 ; 0, m1 , –m2 ), D1213 = (–m2 , m2 , –m12 ; 0, –m1 , –m12 ), D2132 = (0, –m12 , m1 ; –m12 , –m2 , –m2 ), D12132 = (0, –m2 , –m1 ; –m2 , –m12 , –m12 ).

(6.171)

6.3 Invariant Differential Operators and Multiplet Classification of the Reducible ERs

223

The multiplet is presented in the following diagram: Λ0 11

22 Λ2

Λ1

23

122

Λ12

Λ23

121

121 12 •

21

Λ21

23

123

Λ213

Λ121 Λ123

122

123

21

Λ2132

Λ1213 22

13 Λ12132

(6.172)

Also here there are two orbits of the restricted Weyl group: the ﬁrst orbit has four members and may be seen as reduction of the orbit (6.166) of the main multiplet. The four members are D23 , D21 , D121 , and D123 corresponding to the members D0 , D13 , D132 , and D13213 , respectively, of the four-member orbit above and being representable by the same ﬁgure (6.169). The other orbit has eight members and may be seen as coincidence of the ﬁrst (6.163) and the second (6.165) of the main multiplet. It may be depicted as starting from D2 , playing the role of D0 and given in the same order as (6.163): D2 , D0 , D213 , D12132 , D12 , D2132 , D1 , D1213 .

(6.173)

Of course, the phenomenon about the degeneration of the operator A(s1 ) is valid here also. There is a conjugate reduced multiplet in which the role of reﬂections s1 and s3 are interchanged, also in the signatures and the diagram indices 1 and 3 are interchanged, the diagram would be mirror-reﬂected. – Symmetrically doubly reduced multiplets. These depend on one parameter, e.g., m2 , and may be obtained by formally setting m1 = m3 = 0 in (6.161): D0 = (0, m2 , 0 ; m2 , m2 , m2 ), D2 = (m2 , –m2 , m2 ; 0, 0, m2 ), D21 = (–m2 , 0, m2 ; –m2 , m2 , 0), D23 = (m2 , 0, –m2 ; m2 , –m2 , 0), D213 = (–m2 , m2 , –m2 ; 0, 0, –m2 ), D2132 = (0, –m2 , 0 ; –m2 , –m2 , –m2 ).

(6.174)

224

6 Conformal Case in 4D

The multiplet is presented in the following diagram: Λ0 22 Λ2

21

Λ21

23

Λ23

23

•

21

Λ213 22 Λ2132

(6.175)

There are two orbits: one with four members represented as follows: Λ0

S1

Λ2

S2

Λ213

s2

Λ23

Λ2132

S1

(6.176)

and the other with two members: Λ21

(6.177)

Of course, the phenomenon that we noted in (6.168) about the degeneration of the operator A(s1 ) is valid for (6.175) and (6.176). –

Asymmetrically doubly reduced multiplets. These depend on one parameter, e.g., m1 , and may be obtained by formally setting m2 = m3 = 0 in (6.161): D0 = (m1 , 0, 0 ; m1 , 0, m1 ),

(6.178)

D1 = (–m1 , m1 , 0 ; 0, m1 , 0), D12 = (0, –m1 , m1 ; –m1 , 0, 0), D123 = (0, 0, –m1 ; 0, –m1 , –m1 ).

The multiplet is presented in the following diagram: Λ0

11

Λ1

12

Λ12

13

Λ123

(6.179)

6.3 Invariant Differential Operators and Multiplet Classification of the Reducible ERs

225

This multiplet is also an orbit of the restricted Weyl group: Λ0

s2

Λ1

s1

Λ12

s2

Λ123

(6.180)

Actually, one should combine the two diagrams since all the integral operators acting from left to right are degenerate and are represented by the differential operators on the upper picture. Thus, ﬁnally, we have s2

11

s1

Λ0

12

Λ1

s2

13

Λ12

Λ123

(6.181)

There is a conjugate reduced multiplet in which the role of reﬂections s1 and s3 are interchanged, as explained above.

6.3.3 Multiplet Classification: Case P2 Next we consider the ERs induced from the maximal noncuspidal parabolic P2 = M2 ⊕ A2 ⊕ N2 when the inducing M2 -representations are ﬁnite-dimensional. These are mostly used in applications and are K-ﬁnite due to the following fact: C K C = MC 2 ⊕ A2 , C

C

(6.182) C

C

K = su(2) ⊕ su(2) ⊕ u(1) = so(3, C) ⊕ so(3, C) ⊕ so(2, C), C MC 2 = so(3, 1) = so(4, C) = so(3, C) ⊕ so(3, C), C AC 2 = so(1, 1) = so(2, C).

It is easy to ﬁnd the ﬁnite-dimensional signatures of the P2 -induced reducible ERs these are the signatures of the P0 -induced reducible ERs with positive m1 , m3 entries. Thus, we use reduced functions which are polynomials in the variables z, z¯ of degrees m1 – 1, m3 – 1, respectively, denoted also by n1 , n2 , or 2j1 , 2j2 . These then carry ﬁnitedimensional irreducible representations of the Lorentz algebra M2 of dimension m1 m3 . Let us stress that this is an indexless notation on which all Lorentz components of the ﬁelds are gathered together by the polynomial dependence in z, z¯ . Thus, looking at the 24-member multiplet (6.161) we recover the well-known fact that the P2 multiplets are sextets [122, 164, 518]: D0 = (m1 , m2 , m3 ; m12 , m23 , m13 ) = 7– , ′–

D2 = (m12 , –m2 , m23 ; m1 , m3 , m13 ) = 7 , D12 = (m2 , –m12 , m13 ; –m1 , m3 , m23 ) = 7′′– , D32 = (m13 , –m23 , m2 ; m1 , –m3 , m12 ) = 7′′+ ,

(6.183)

226

6 Conformal Case in 4D

D132 = (m23 , –m13 , m12 ; –m1 , –m3 , m2 ) = 7′+ , D2132 = (m3 , –m13 , m1 ; –m12 , –m23 , –m2 ) = 7+ , where we use speciﬁc notation introduced ﬁrst in [164] and for Lorentz symmetric traceless tensors, (i.e., for equal m1 , m3 entries, thus, excluding 7′′± ) in [159]. It is useful to arrange the sextet as in [164] with signatures [n1 , n2 ; d] explicitly given by – 7p-n = [p – 1, n – 1; 2 – - – 21 (p + n)], + 7p-n ′– 7p-n ′+ 7p-n ′′– 7p-n ′′+ 7p-n

= = = = =

1 2 (p

[n – 1, p – 1; 2 + - + [p + - – 1, n + - – 1; 2 – [n + - – 1, p + - – 1; 2 + [- – 1, p + n + - – 1; 2 + [p + n + - – 1, - – 1; 2 +

(6.184)

+ n)], 1 2 (p + n)], 1 2 (p + n)], 1 2 (p – n)], 1 2 (n – p)],

where p, -, n are positive integers which are exactly the Dynkin labels m1 , m2 , m3 (in – with dominant highest weight. The sextets are given the same order) of the ER D0 = 7p-n in the following ﬁgure: Λ−pνn

Λ+pνn

ν2

ν13

– Λʹpνn

p12

+ Λʹpνn

n23

– Λʺpνn

n23

p12

+ Λʺpνn

(6.185)

6.3 Invariant Differential Operators and Multiplet Classification of the Reducible ERs

227

where the differential operators are denoted as -2 ∼ D-!2 , qij ∼ Dq!ij , (q = p, -, n). Note that for all pairs of ERs/GVMs with signature distinguished by ± the sum of the ds of the two ERs/GVMs equals 4 – the dimension of Minkowski space-time. There is only one type of KS operator connecting the ± pairs depicted by the dashed two-way horizontal arrows. We note that the n1 , n2 entries in the sextet signatures are nonnegative integers, i.e., they correspond to ﬁnite-dimensional irreps of the inducing group M2 . This explains why the invariant differential operators corresponding to the compact roots play no role in the embedding diagram. Actually, these operators are deﬁned on the ERs of the sextets, as one can see from the P0 embedding diagram (6.162). However, the sextet ERs are in the kernel of the compact invariant differential operators, and thus, it makes no sense to display them. This simpliﬁes also the noncompact invariant differential operators. To explain this we ﬁrst note that the ﬁnite-dimensional irreps of M2 have two realizations [250]: polynomials in z, z¯ of degrees n1 , n2 , and homogeneous polynomials in z1 , z2 of degree n1 and in z¯ 1 , z¯ 2 of degree n2 . (The two realizations are related via z = z1 /z2 , z¯ = z¯ 1 /¯z2 .) In the second realization the functions of the P2 -induced representations will be F (x; z1 , z2 ; z¯ 1 , z¯ 2 ) with the prescribed homogeneity properties. On these functions the ˜ noncompact invariant differential operators become [122, 164, 518]: Dm!2 = (I2 )m = (z¯ 1 z1 ∂+ + z1 z¯ 2 ∂v + z¯ 1 z2 ∂v¯ + z¯ 2 z2 ∂– )m

m

z1 , = (¯z1 , z¯ 2 ) 3 ∂, , z2

m ∂z1 m , , Dm!12 = (I12 ) = (¯z1 , z¯ 2 ) 3 ∂, % ∂z2

m

z1 m , , Dm!23 = (I23 ) = (∂z¯ 1 , ∂z¯ 2 ) %3 ∂, z2

m

∂z1 m , , Dm!13 = (I13 ) = (∂z¯ 1 , ∂z¯ 2 ) 3 ∂, ∂z2

(6.186a)

(6.186b)

(6.186c)

(6.186d)

where 3, are the Pauli matrices, % = i32 . Remark 2: Note that for n1 = n2 = ∈ Z+ , the Lorentz irreps are represented by symmetric traceless tensors of rank : T,1 ,...,, (x), which are customarily replaced by ˜ homogeneous polynomials [159]: T(x, . ) = T,1 ,...,, (x) . ,1 , . . . , . , , where . , ., = 0 (to ˜ ˜ ensure tracelessness). On these functions the operator I2 acts as . , ∂, which is actu,

ally consistent with (6.186) if we deﬁne: .s = (¯z1 , z¯ 2 ) 3,

z1 , , and checking .s (.s ), = 0. z2 ,

Analogously, but more involved is to rewrite the operator I13 as ∂. ∂, . In fact, this is possible only if we extend the variable . outside the cone . , ., = 0 and then require

228

6 Conformal Case in 4D

that the functions T(x, . ) are harmonic in . (to ensure tracelessness). This is explained ˜ in detail in [159]. ♢

As for P0 , the P2 sextets are parametrized by the ﬁnite-dimensional irrep of su(2, 2). In – (which is D in the P case) contains the ﬁnite-dimensional ireach sextet the ER 7p-n 0 0 rep denoted here Ep-n of dimension: p-n(p+-)(n+-)(p+-+n)/12 [122]. As a consequence, the sextets hold also the discrete series representations. Besides in the sextets the reducible ERs occur in some doublets. The doublets are ± , 7± , 7± , and the expression for their signatures can of three kinds, denoted by 1 7p2 pn 3 -n ′± , by setting, n = 0, - = 0, p = 0, respectively, i.e., be obtained from the signatures of 7p-n – 1 7p-

= [p + - – 1, - – 1; 2 – 21 p],

+ 1 7p– 2 7pn + 2 7pn – 3 7-n + 3 7-n

1 2 p],

= [- – 1, p + - – 1; 2 +

(6.187)

= [p – 1, n – 1; 2 – 21 (p + n)], = [n – 1, p – 1; 2 + 21 (p + n)], = [- – 1, n + - – 1; 2 – 21 n], = [n + - – 1, - – 1; 2 + 21 n].

The doublets are as follows: − 1Λpν

/

− 2Λpn

/

− 3Λvn

/

(I12)p

Dp,n

(I23)n

+ 1Λpν

+ 2Λpn

+ 3Λvn

(6.188)

Note that here the invariant operators are deformations of the horizontal KS integral operators from the sextet picture. Thus, those from 7+ to 7– are still integral operators, while those from 7– to 7+ are differential operators via degeneration of the KS integral operators. Yet in the ﬁrst and third case these are differential operators inherited from – to 7+ are obtained due to genuine degenerathe sextets, only the operators from 2 7pn 2 pn tion of the KS integral operators. Furthermore, in the case p = n that operator becomes a degree of the d’Alembert operator: Dn,n = ◻n which results also from a subsingular vector as explained in the next Chapter.

(6.189)

6.3 Invariant Differential Operators and Multiplet Classification of the Reducible ERs

229

′– to 7′′– in the limit - → 0 becomes the Note that the sextet operator Dp!12 from 7p-n p-n ′– → 7– since the P -ERs are composition Dp!2 Dp!1 which composition is zero on 7p-n 2 pn 2 M2 -ﬁnite-dimensional (as explained in detail before). The same holds for the sextet ′– to 7′′+ in the limit - → 0. operator Dn!23 from 7p-n p-n Finally, we should mention the special representations

7s = [- – 1, - – 1; 2].

(6.190)

These singlets could be omitted here since they are not related to other reducible P2 -induced ERs. Their importance was understood in the framework of conformal supersymmetry, i.e., in the multiplet classiﬁcation for the superconformal algebra su(2, 2/N) given in [165]. It turns out that the inﬁnite multiplets of su(2, 2/N) have as building blocks all mentioned above sextets, doublets and singlets. This will be discussed in more detail in Volume 2. 6.3.4 Holomorphic Discrete Series and Lowest Weight Representations According to the results of Harish-Chandra the holomorphic discrete series happen when the numbers m! are negative integers for all noncompact roots. Thus, we see from (6.167) and (6.184) that the holomorphic discrete series are contained in the ERs + . The limits of the holomorphic discrete series happens when some of the noncom7p-n pact numbers m! become zero, while the rest of the noncompact numbers m! remain + (obtained also from negative. We see that these limits are contained in the ERs 2 7pn + 7p-n for - =0). Next we discuss how the lowest weight positive energy representations ﬁt in the multiplets, and when they are inﬁnitesimally equivalent to holomorphic discrete series. There are two basic cases of positive energy representations [414]: 1)j1 j2 ≠ 0,

2)j1 j2 = 0,

where jk = nk /2 in the notation of [414]. 6.3.4.1 j1 j2 ≠ 0 In case (1) the positive energy representations fulﬁll the condition [414]: d ≥ 2 + j1 + j2 ,

j1 j2 ≠ 0.

For d > 2 + j1 + j2 the GVMs are irreducible and unitary. The point d = d0 ≡ 2 + j1 + j2 is the ﬁrst reduction point. In our picture it is realized ′+ , so that j = 1 n, j = 1 p. in the GVM with signature 7p1n 1 2 2 2 The point d = d0 + 1 is a limit of discrete series, while the integer points with d ≥ d0 + 2 are the holomorphic discrete series. Indeed, the former are contained in

230

6 Conformal Case in 4D

+ 2 7pn ,

+ . In both cases we while the latter are contained in the ERs with signature 7p-n 1 1 1 have j1 = 2 (n – 1), j2 = 2 (p – 1), d = 2 + - + 2 (n + p), where n, p > 1 and - = 0, - ∈ N, distinguishes the two cases. Thus, these cases correspond to c0 = 1 (see above) and A(+0 ) = 1 in the terminology of [194]. Here and below the unitarity parameter z of [194] is related to ours as z = – d + d0 + A(+0 ).

6.3.4.2 j1 j2 = 0 In case (2) the positive energy representations fulﬁll the condition [414]: d ≥ 1 + j 1 + j2 ,

j1 j2 = 0.

For d > 1 + j1 + j2 the GVMs are irreducible and unitary. The point d00 = 1 + j1 + j2 is the ﬁrst reduction point. These are the massless representations of so(4, 2). ′′+ , with j = For j1 + j2 ≥ 1 the FRP is realized in the ERs/GVMs with signatures: 711n 1 1 1 ′′– (n + 1) ≥ 1, j = 0, and 7 , with j = 0, j = (p + 1) ≥ 1. 2 1 2 p11 2 2 – , with j = 1 , For j1 +j2 = 21 the FRP is realized in the ERs/GVMs with signatures: 3 711 1 2 1 – j2 = 0, and 1 711 , with j1 = 0, j2 = 2 . The two conjugated representations are two-component massless Weyl spinors. + , 7+ mentioned below, and the corresThey are partially equivalent to the ERs 3 711 1 11 ponding KS operators from these FRPs degenerate to the two well-known ﬁrst-order conjugated Weyl equations. –. For j1 = j2 = 0 the FRP is realized in the ER with signature: 2 711 +, The above massless scalar representation is partially equivalent to the scalar ER 2 711 mentioned below. The two ERs are related by KS integral operators, however, the oper– to 7+ degenerates to the d’Alembert operator. That d’Alembert operator ator from 2 711 2 11 arises also as a conditionally invariant differential operator due to the presence of a subsingular vector in the corresponding Verma module with signature (m1 , m2 , m3 ) = (1, 0, 1) (cf. the next chapter). As we shall see, these cases correspond to c0 = 1 (see above). and A(+0 ) = 2 in the terminology of [194] (the FRP is z = A(+0 ) = 2). The point next to the FRP (z = 1 by [194]) with d = d00 + 1 = 2 + j1 + j2 is part of the analytic continuation of the discrete series. + , so that j = 1 n, j = 0, and 7+ , so that j = 0, For j1 + j2 ≥ 21 it ﬁts the ERs 3 71n 1 2 1 p1 1 2 1 j2 = 2 p. For j1 = j2 = 0 it is realized in the singlet ER with signature: 71s . + The next point with d = d00 + 2 = 3 + j1 + j2 , (z = 0 by [194]), ﬁts the ERs 2 7pn 1 with either p = 1 or n = 1, which contain limits of discrete series (with j1 = 2 (n – 1), j2 = 21 (p – 1), as above for j1 j2 ≠ 0). Finally, the cases with integer d ≥ d00 + 3 = 4 + j1 + j2 (z < 0 by [194]) are realized by + which contain the holomorphic discrete series (as above for j j ≠ 0). the ERs 7p-n 1 2

6.3 Invariant Differential Operators and Multiplet Classification of the Reducible ERs

231

All this is illustrated on the two ﬁgures below.

d = k + j1 + j2

HDS

.. .. .. .. .

d = k + j1 + j2

HDS

.. .. .. .. . d = 4 + j1 + j2

HDS

d = 4 + j1 + j2

HDS

d = 3 + j1 + j2

LHDS

d = 3 + j1 + j2

LHDS

d = 2 + j1 + j2

FRP

d = 2 + j1 + j2

j1 j2 = 0 d = 1 + j1 + j2

FRP

j1 j2 = 0

(6.191)

Remark 3: Above we described the occurrences of the holomorphic discrete series. But it is a general feature that we have seen in the cases so(2, 1) and so(3, 2), that the conjugate antiholomorphic discrete series is contained together with the holomorphic + . This is discrete series (as nonintersecting subspaces) in the reducible ER, here, 7p-n expected if we take into account that every two P1 -induced ERs (differing only by the value : = ±1) are embedded (as nonintersecting subspaces) in a P2 -induced ER. The antiholomorphic discrete series irreps correspond to negative energy representations with negative conformal weight bounded from above. The same phenomena apply to + . the limits of discrete series reducible ERs 2 7pn ♢

6.3.5 Multiplet Classification: Case P1 6.3.5.1 Main P1 Multiplets The signatures of the ERs are 71 = {n′ , k, :, -′ },

(6.192)

232

6 Conformal Case in 4D

where n′ ∈ Z is a character of SO(2), -′ ∈ C is a character of A1 , k, : ﬁx a discrete series representation of SL(2, R), k ∈ N, : = ±1, or a limit thereof when k = 0. The relation with the sl(4) Dynkin labels is as follows [122]: m1 = 21 (k – -′ + n′ ),

m2 = – k,

m3 = 21 (k – -′ – n′ ).

(6.193)

Since relations (6.193) do not involve the discrete parameter : we omit it in the signatures when classifying the reducible ERs. The main multiplet of reducible ERs contains 12 members which we parametrize as part of the main sl(4) multiplet with 24 members. Thus, the 12-plet has the following signatures: D2 = (m12 , –m2 , m23 ) = {n′ = m1 – m3 , k = m2 , -′ = – m13 }, D12 = (m2 , –m12 , m13 ) = {–m1 – m3 , m12 , –m23 }, D32 = (m13 , –m23 , m2 ) = {m1 + m3 , m23 , –m12 }, D121 = (–m2 , –m1 , m13 ) = {–m13 – m2 , m1 , –m3 }, D132 = (m23 , –m13 , m12 ) = {m3 – m1 , m13 , –m2 }, D232 = (m13 , –m3 , –m2 ) = {m13 + m2 , m3 , –m1 }, D1232 = (m23 , –m3 , –m12 ) = {m13 + m2 , m3 , m1 }, D1321 = (–m23 , –m1 , m12 ) = {–m13 – m2 , m1 , m3 }, D2132 = (m3 , –m13 , m1 ) = {m3 – m1 , m13 , m2 }, D12132 = (m3 , –m23 , –m1 ) = {m1 + m3 , m23 , m12 }, D21321 = (–m3 , –m12 , m1 ) = {–m1 – m3 , m12 , m23 }, D121321 = (–m3 , –m2 , –m1 ) = {m1 – m3 , m2 , m13 },

(6.194)

where D0 is as in (6.161) the ER/GVM ﬁxing the sl(4) 24-plet. We have given the signature in both the sl(4) signature notation (⋅, ⋅, ⋅) and in the P1 -induced notation (6.192). We would like to follow connections also with ERs induced from the maximal noncuspidal parabolic P2 . Thus, below we shall replace the mk notation with the equivalent one: (p, -, n) = (m1 , m2 , m3 ). Thus, we give the same 12-plet in p, -, n parametrization and adding the HarishChandra (HC) parameters [279]. We remind that the K-noncompact HC parameters are: m2 , m12 , m23 , m13 . Thus, the 12-plet is given now as ′– , D–0 = D2 = (p + -, –-, n + - ; p, n, p + - + n) = 7p-n ′′– D–a = D12 = (-, –p – -, p + - + n ; –p, n, n + -) = 7p-n , ′′+ D–b = D32 = (p + - + n, –n – -, - ; p, –n, p + -) = 7p-n ,

6.3 Invariant Differential Operators and Multiplet Classification of the Reducible ERs

233

D–c = D121 = (–-, –p, p + - + n ; –p – -, n + -, n), ′+ D–d = D132 = (n + -, –(p + - + n), p + - ; –p, –n, -) = 7p-n ,

D–e = D232 = (p + - + n, –n, –- ; p + -, –n – -, p), D+e = D1232 = (n + -, –n, –p – - ; -, –p – - – n, –p), + D+d = D2132 = (n, –(p + - + n), p ; –p – -, –- – n, –-) = 7p-n ,

D+c = D1321 = (–n – -, –p, p + - ; –p – - – n, -, –n), D+b = D12132 = (n, –n – -, –p ; –-, –p – - – n, –p – -), D+a = D21321 = (–n, –p – -, p ; –p – - – n, –-, –- – n), D+0 = D121321 = (–n, –-, –p ; –- – n, –p – -, –p – - – n),

(6.195)

′ ′′

( , )± coinciding by signatures with ERs where we have also indicated the ﬁve cases 7p-n induced from the maximal noncuspidal parabolic. The notations D±0 , D±a , etc, are used in the ﬁgure below where we present this 12-plet.

Λ–a Λc–

ν1

Λ–0

p12

n23

n23

n13

Λ–d

Λb–

p12

ν3 Λe–

ν13 Λ+c

p13 Λ+d

ν23

Λa+

Λe+

n1

p3

p3

n1 Λ+0

Λ+b

ν12

(6.196)

As before, only the M-noncompact roots are involved, i.e., the M1 -compact root !2 is not relevant for the invariant differential operators. + Some remarks on the ERs content: As in the general situation the ERs D+d = 7p-n in (6.195) contain holomorphic discrete series representations when the discrete parameter : = 1, and the antiholomorphic discrete series representations when the discrete parameter : = –1. The same phenomenon is true in all cases below: we shall mention only the holomorphic discrete series, the antiholomorphic will be understood. ′′– and D– = 7′′+ are contained the massless representations with In the ERs D–a = 7p11 11n b conformal weight d = 1 + j with spin j ≥ 1, where j = (p + 1)/2 and j = (n + 1)/2, resp. (the three with lower spin are considered below). Our diagrams account also for the KS [352, 353] integral operator relevant for P1 induction. It acts on the signatures as the highest sl(4) root !13 , [122, 162]. Thus, on the P1 -signature it acts by changing the sign of the last entry in the {} -notation (-′ ). On the

234

6 Conformal Case in 4D

ﬁgure the KS operators intertwine the ERs symmetric w.r.t. the dashed line. Of course, the KS from D–c , D–d , D–e to D+c , D+d , D+e , resp., degenerated to differential operators of degrees n, -, p, resp., as shown on Fig. (6.196). The KS operators from D+c , D+d , D+e to D–c , D–d , D–e , resp., remain integral operators. The same remarks about the KS integral operators will be true verbatim for all further ﬁgures below and will not be repeated.

6.3.5.2 Symmetrically Reduced Multiplets We have several types of reduced multiplets. – The ﬁrst case is a septuplet depending on two parameters, and the signatures may be obtained by setting formally - = 0 in (6.194): ′– – D2 = (p, 0, n) = {n′ = p – n, k = 0, -′ = – p – n} = 7p0n = 2 7pn , ′′– , D12 = (0, –p, p + n) = {–p – n, p, –n} = 7p0n ′′+ , D32 = (p + n, –n, 0) = {p + n, n, –p} = 7p0n ′+ + + = 7p0n = 2 7pn , D132 = (n, –p – n, p) = {n – p, p + n, 0} = 7p0n

D1232 = (n, –n, –p) = {p + n, n, p}, D1321 = (–n, –p, p) = {–p – n, p, n}, D121321 = (–n, 0, –p) = {p – n, 0, p + n},

(6.197)

or adding the HC parameters – D–0 = D2 = (p, 0, n ; p, n, p + n) = 2 7pn ,

D–a = D12 = (0, –p, p + n ; –p, n, n), D–b = D32 = (p + n, –n, 0 ; p, –n, p), + , Dc = D132 = (n, –p – n, p ; –p, –n, 0) = 2 7pn

D+b = D1232 = (n, –n, –p ; 0, –n – p, –p), D+a = D3212 = (–n, –p, p ; –n – p, 0, –n), D+0 = D213213 = (–n, 0, –p ; –n, –p, –n – p).

(6.198)

± form a doublet w.r.t. induction from the maximal The ﬁrst and fourth entries 2 7pn noncuspidal parabolic. – is induced from a limit of sl(2, R) holomorphic discrete The ﬁrst entry 2 7pn series. For p = n = 1 it contains the scalar massless representation with conformal weight d = 1. + contains limits of holomorphic discrete series. The fourth entry 2 7pn

6.3 Invariant Differential Operators and Multiplet Classification of the Reducible ERs

235

This septuplet is shown on the ﬁgure below. –

Λ0 –

p12

n23 –

Λa

Λb p12

n23 Λd

p3

n1 + Λa

+

Λb

p3

n1 +

Λ0

(6.199)

6.3.5.3 Asymmetrically Reduced Multiplets – The second case is also a septuplet depending on two parameters, the signatures may be obtained by setting formally p = 0 in (6.194) ′– ′′– – D–0n = D2 = (-, –-, n + - ; 0, n, - + n) = 70-n = 70-n = 1 7-n , ′′+ ′+ + D–b = D32 = (- + n, –n – -, - ; 0, –n, -) = 70-n = 70-n = 1 7-n ,

D–d = D212 = (–-, 0, - + n ; –-, n + -, n), De = D232 = (- + n, –n, –- ; -, –n – -, 0), D+d = D3212 = (–n – -, 0, - ; –- – n, -, –n), + D+b = D2132 = (n, –(- + n), 0 ; –-, –- – n, –-) = 70-n ,

D+0n = D23212 = (–n, –-, 0 ; –- – n, –-, –- – n).

(6.200)

± form a doublet w.r.t. induction from the maximal nonThe ﬁrst two entries 1 7-n ± contains one (of the two) spin 1/2 massless cuspidal parabolic. The ER 1 711 representations with conformal weight d = 3/2. The third and ﬁfth entries are induced from limits of discrete series. The conjugate septuplet may be obtained by setting n = 0: ′– ′′+ – D–0p = D2 = (p + -, –-, - ; p, 0, p + -) = 7p-0 = 7p-0 = 3 7p, ′′– ′+ + = 7p-0 = 3 7p, D–a = D12 = (-, –p – -, p + - ; –p, 0, -) = 7p-0

D–e = D232 = (p + -, 0, –- ; p + -, –-, p), Dd = D212 = (–-, –p, p + - ; –p – -, -, 0), D+e = D1232 = (-, 0, –p – - ; -, –p – -, –p), + , D+a = D23212 = (0, –p – -, p ; –p – -, –-, –-, ) = 7p-0

D+0p = D213213 = (0, –-, –p ; –-, –p – -, –p – -).

(6.201)

236

6 Conformal Case in 4D

The interpretation of the members is exactly as for the last case, e.g., the ﬁrst ± form a doublet w.r.t. induction from the maximal noncuspidal two entries 3 7p± contains the other spin 1/2 massless representation with parabolic. The ER 3 711 conformal weight d = 3/2. The last two septuplets are shown on the two ﬁgures below.

Λ–a

p12

Λ–0p

Λ–0n ν3

Λ–e

ν1

Λ–c

n23

Λ–b

ν1

ν3

Λc

Λe

n13

p13 ν23 Λ+a

Λ+e

p3 Λ+0p

ν12

Λ+c

ν23

Λ+0n

n1

Λ+b

ν12

(6.202)

6.3.5.4 Symmetrically Doubly Reduced Multiplets – The next case is a quartet depending on one parameter with signatures may be obtained by setting formally p = n = 0 in (6.194): ′– ′′– ′′+ ′+ D–0 = D2 = (-, –-, - ; 0, 0, -) = 70-0 = 70-0 = 70-0 = 70-0 = 7-s ,

Dd = D212 = (–-, 0, - ; –-, -, 0), De = D232 = (-, 0, –- ; -, –-, 0), + D+0 = D2132 = (0, –-, 0 ; –-, –-, –-) = 70-0 .

(6.203)

The ﬁrst entry is a singlet w.r.t. induction from the maximal noncuspidal parabolic. For - = 1 it is a Lorentz scalar positive energy representation with conformal weight d = 2, i.e., above the unitarity threshold d = 1, but below the limit of holomorphic discrete series d = 3. The second and third entries are induced from limits of discrete series. This quartet is shown on the ﬁgure below. –

Λ0 ν3

ν1 Λc

Λe ν23

ν12 +

Λ0

(6.204)

6.3 Invariant Differential Operators and Multiplet Classification of the Reducible ERs

237

6.3.5.5 Asymmetrically Doubly Reduced Multiplets – The next two conjugate cases are triplets depending on one parameter. The ﬁrst be obtained by setting formally p = - = 0 in (6.194) ′– ′′– D–0p- = D2 = (0, 0, n ; 0, n, n) = 700n = 700n , ′′+ ′+ + Db = D32 = (n, –n, 0 ; 0, –n, 0) = 700n = 700n = 700n ,

D–0p- = D3212 = (–n, 0, 0 ; –n, 0, –n).

(6.205)

The ﬁrst and third entries are induced from limits of discrete series. –

The conjugate case ′– ′′+ D–0n- = D2 = (p, 0, 0 ; p, 0, p) = 7p00 = 7p00 , ′′– ′+ + Da = D12 = (0, –p, p ; –p, 0, 0) = 7p00 = 7p00 = 7p00 ,

D+0n- = D121321 = (0, 0, –p ; 0, –p, –p). The two conjugate triplets are shown below. −

Λ0p

−

Λ0n

p12

n23

Λa

Λb

p3

n1

+

Λ0p

+

Λ0n

(6.206)

7 Kazhdan–Lusztig Polynomials, Subsingular Vectors, and Conditionally Invariant Equations Summary This chapter is based mostly on [42, 135, 137, 332]. The main theme of our book is the relation between invariant differential operators and singular vectors of Verma modules. On the other hand, from the Bernstein–Gel’fand–Gel’fand (BGG) paper [42] we know about another interesting feature of Verma modules: subsingular vectors. The latter happen only for reducible Verma modules, and they are associated in a certain sense to some singular vector(s). By itself a subsingular vector vsu of a reducible Verma module V D is not singular vector, it becomes a singular vector in the factor-module V D /IsD , where IsD is the submodule generated by the singular vector(s) associated to the subsingular vector vsu . What is more important from our point of view is that to each subsingular vector there corresponds a conditionally invariant equation. This equation is defined on the elementary representation C7(D) and becomes invariant only on the kernel(s) of the invariant equation(s) corresponding to the singular vector(s) associated with vsu . What makes this story even more important is that it turns out that a Verma module V D possesses a subsingular vector only if the integer number Pw,wD (1) is bigger than 1. (Here Pw,wD (u) are the Kazhdan–Lusztig polynomials [332] in a variable u, and w and wD are elements of the Weyl group of GC , which have been explained in detail in the chapter.) The next important point is D, that the numbers Pw,wD (1) enter the character formulae of the irreducible factor-modules LD = V D /Im D is the maximally invariant submodule of V D . For instance, if D is a dominant highest (or lowwhere Im est) weight then all numbers Pw,wD (1) are equal to 1, and we have the classical Weyl character formula. In the present chapter, we present the systematic study of subsingular vectors of Verma modules over simple Lie algebras. We give an exact and constructive definition of the notion of subsingular vector (Section 7.1). This is illustrated with two explicit examples of subsingular vectors occurring for sl(4, C) (one of them is summarized in Proposition 1). We recall the Kazhdan–Lusztig polynomials (Section 7.2) more precisely and in detail, yet in a simplified exposition different from the original one [332] and sufficient for our representation-theoretic purposes. Our other aim is to provide explicit formulae not only for subsingular vectors but also for related character formulae. For this purpose, we give explicit character formulae (Section 7.3), including all occurrences of nontrivial Kazhdan–Lusztig polynomials for sl(4, C) (Proposition 2). We show also that a nontrivial Kazhdan–Lusztig polynomial does not necessarily guarantee the existence of a subsingular vector (Section 7.4). Thus, we make a conjecture when exactly an integer Pw,wD (1) bigger than 1 leads to a subsingular vector. We give also a systematic discussion of the relation between subsingular vectors of Verma modules over semisimple Lie algebras G and differential equations which are conditionally G-invariant (Section 7.5). We treat in detail the conformal algebra su(2, 2) and its complexification sl(4, C). This example is natural since rank 3 is the lowest possible where these phenomena happen. The conditionally invariant equations are the d’Alembert equation, and a new equation arising from a subsingular vector proposed by Bernstein, Gel’fand, and Gel’fand.

7.1 Subsingular Vectors

239

7.1 Subsingular Vectors 7.1.1 Preliminaries Let G be a semisimple Lie algebra with Chevalley generators Xi± ≡ X!±i , Hi ∈ H, i = 1, . . . , r of G. A lowest weight module (LWM) M D over sl(G) is given by the lowest weight D ∈ H∗ ∗ (H is the dual of H) and a lowest weight vector v0 so that Xv0 = 0 if X ∈ G – , Hv0 = D(H)v0 if H ∈ H. In particular, we use the Verma modules V D over sl(G) which are the LWMs such that V D ≅ Uq (G + ) ⊗ v0 . Note that the Dynkin labels mi = mi (D) ≐ (1 – D)(Hi ) = 1 – D(Hi ) = 1 – (D, !∨i ),

i = 1, . . . , r

(7.1)

(1(Hk ) = (1, !k ) = 1) completely determine the lowest weight D and shall be used also for the characterization of the LWM. The collection of these numbers shall be called the signature of D and denoted 7(D) or just 7: 7 = 7(D) ≐ (m1 , . . . , mr ).

(7.2)

Remark 1: Note that in this chapter we are using lowest weight modules instead of highest weight modules, as in most other chapters. Thus, some formulae look different but the conjugated contents are the same. ♢ Further, we shall also use the following notions: The signature 7 = (m1 , . . . , mr ) is called dominant, semidominant, antidominant, and anti-semidominant, resp. if mk ∈ N, mk ∈ Z+ , mk ∈ – N, mk ∈ Z– resp. Analogously, we shall also use the Harish-Chandra parameters corresponding to arbitrary positive roots: m! = m! (D) ≐ (1 – D)(H! ) = (1 – D, !∨ ),

! ∈ B+ ,

(7.3)

where H! ∈ H corresponds to the root ! by the isomorphism H ≅ H∗ . Naturally, m!i = mi . A Verma module V D is reducible [42] iff at least one of the numbers m! is a positive integer: m! ∈ N.

(7.4)

Whenever (7.4) is fulﬁlled there exists a singular vector vsm! in V D such that vsm! ∉ Cv0 , Xvsm! = 0, ∀X ∈ G – , and Hvsm! = (D + m! !)(H) vsm! , ∀H ∈ H. The space I m! = U(G + ) vsm! is a proper submodule of V D isomorphic to the Verma module V D+m! ! with a shifted lowest weight D + m! ! [126]. Clearly, this implies that V D and V D+m! ! have the same values of the Casimir operators.

240

7 KL Polynomials and Conditionally Invariant Equations

It is important that one can ﬁnd explicit formulae for the singular vectors. The singular vector introduced above is given by (cf. Section 4.8, [126, 127]) vs = v!,m! = P !,m! (X1+ , . . . , Xr+ )v0 ,

(7.5)

where P !,m! is a homogeneous polynomial in its variables (cf. (2.212)). Section 4.8 contains all explicit singular vectors needed in the present chapter. Note that the modules cited above are highest weight modules and the singular vectors are polynomials in Xi– ; the translation of those formulae to the LWM setting used in this chapter is straightforward.

7.1.2 Definition Certainly, (7.4) may be fulﬁlled for several positive roots (even for all of them). Let us denote I˜D ≡ ∪I m! , where the union is over all positive roots for which (7.4) is fulﬁlled. Clearly, I˜D is a proper submodule of V D . Let us also denote F˜ D ≡ V D /I˜D .

The Verma module V D contains a unique proper maximal submodule I D (⊇ I D ) [42]. Among the LWM with lowest weight D there is a unique irreducible one, denoted by LD , i.e., LD = V D /I D . (If V D is irreducible then LD = V D .) It may happen that the maximal submodule I D coincides with the submodule I˜D generated by all singular vectors. This is, e.g., the case for all Verma modules if rank G ≤ 2, or when (7.4) is fulﬁlled for all simple roots (and, as a consequence, for all positive roots). Here we are interested in the cases when I˜D is a proper submodule of I D . Now we introduce the most important notion in this chapter: Definition 1: Let V D be a reducible Verma module. A vector vsu ∈ V D is called a subsingular vector if vsu ∉ I˜D and the following holds: X vsu ∈ I˜D ,

∀X ∈ G – .

♢

(7.6)

Remark 2: The image of a subsingular vector in the factor-module F˜ D is a singular vector of F˜ D . For shortness, we shall say the subsingular vector “becomes” a singular vector in the corresponding factor-module. ♢ We need to be more explicit even in the general case. First of all, it is clear that it is enough for a vector to be subsingular if (7.6) holds for the negative simple root vectors Xi– . We can rewrite (7.6) in the following way: Xi– vsu =

Qi! v!,m! ,

(7.7)

!∈Bi

where Qi! are homogeneous polynomials such that the RHS is a homogeneous polynomial, and Bi is a subset of BD ⊂ B+ such that ! ∈ Bi iff Qi! is a nonzero polynomial. Let us denote by Bsu the union of Bi : Bsu ≡ ∪ri=1 Bi . We shall call Bsu the set of associated roots

7.1 Subsingular Vectors

241

to the subsingular vector vsu . The corresponding set of singular vectors {v!,m! | ! ∈ Bsu } will be called singular vectors associated to the subsingular vector vsu . Clearly, Bsu is a subset of BD and in general a proper subset. Let Isu ≡ ∪!∈Bsu I ! (⊆ I˜D ) and Fsu ≡ V D /Isu ; then vsu becomes a singular vector in Fsu , i.e., when we factorize all singular vectors associated with it. D so that Clearly, vsu and I˜D generate a submodule Isu D ⊆ ID ⊂ V D, Isu ⊆ I˜D ⊂ Isu

(7.8)

D coincides where all embeddings are proper. Of course, there is no claim that Isu with I D . Further we shall use also the following notions. The singular vector v1 is called descendant of the singular vector v2 ∉ Cv1 if there exists a homogeneous polynomial P12 in Xi+ so that v1 = P12 v2 . Clearly, in this case we have I 1 ⊂ I 2 , where I k is the submodule generated by vk . If a singular vector vs′ is descendant of another singular vector then vs′ is called a composite singular vector. Clearly, if two singular vectors v1 and v2 belong to BD (Bi , Bsu ) and v1 is descendant of v2 , then we can omit v1 from the set BD (Bi , Bsu ). We restrict now to G = sl(4, C). The simple roots are !1 , !2 , !3 with nonzero scalar products: (!j , !j ) = 2, j = 1, 2, 3, (!1 , !2 ) = (!2 , !3 ) = – 1. The Chevalley generators are denoted by Xi± , i = 1, 2, 3. The nonsimple roots are: !12 = !1 + !2 , !23 = !2 + !3 , !13 = !1 + !2 + !3 , and the corresponding Cartan–Weyl generators were given in (2.26). All commutation relations between the generators which we shall use follow from these deﬁnitions and (5.1). For the six positive roots of the root system of sl(4, C) one has from (7.1), (7.3) (see [126]):

m1 = 1 – D(H1 ),

(7.9a)

m2 = 1 – D(H2 ),

(7.9b)

m3 = 1 – D(H3 ),

(7.9c)

m12 = 2 – D(H12 ) = m1 + m2 ,

(7.9d)

m23 = 2 – D(H23 ) = m2 + m3 ,

(7.9e)

m13 = 3 – D(H13 ) = m1 + m2 + m3 .

(7.9f)

Thus the signature here is: 7 = (m1 , m2 , m3 ). For further reference we give the value of the sl(4, C) second-order Casimir operator (cf. Subsection 6.2.5.2) in terms of the above notation: 1 1 C2 = m213 + m22 + (m1 – m3 )2 – 5, (7.10) 2 2 which is normalized to take zero value on the trivial irrep when mk = 1 (and thus on all representations partially equivalent to it).

242

7 KL Polynomials and Conditionally Invariant Equations

7.1.3 Bernstein–Gel’fand–Gel’fand Example The ﬁrst example of a subsingular vector was given for the algebra sl(4, C) in the seminal paper [42]. It occurs for D(H1 ) = D(H3 ) = 1, D(H2 ) = 0, i.e., 7 = (m1 , m2 , m3 ) = (0, 1, 0). Thus there are four positive m! ∈ N from (7.9): m2 = m12 = m23 = m13 = 1. Correspondingly, there are four singular vectors: v2 = X2+ v0 ,

m2 = 1,

(7.11a)

′ = X1+ X2+ v0 , v12

m12 = 1,

′ = X3+ X2+ v0 , v23

m23 = 1,

′ = X1+ X3+ X2+ v0 , v13

m13 = 1.

(7.11b)

However, only the singular vector v2 is relevant, the others are descendants of v2 . Indeed, we have the following embeddings: ′ ⊂ I′ , I13 12

∩

∩

(7.12)

′ ⊂ I = I˜ D ⊂ I D , I23 2

where I2 , Iij′ , are the submodules generated by v2 , vij′ , resp. All embeddings are proper, including the last one since there is the following subsingular vector: (7.13a) vbgg = X1+ X2+ X3+ – X3+ X2+ X1+ v0 . It is easy to see ′ , X1– vbgg = v23

X2– vbgg = 0, ′ , X3– vbgg = – v12

(7.14)

thus, indeed, (7.6) is fulﬁlled. Formula (7.13a) is in the unordered Chevalley basis. An expression in the ordered PBW basis is: + + + + + X3+ X12 + X23 X1 v 0 , (7.13b) vbgg = X13 which is exactly equal to (7.13a). A third expression coinciding with (7.13a,b) is: + + + + X3 + X23 X1 v 0 . (7.13c) vbgg = X12 Note that we have translated the result of [42] into our LWM setting and that the actual expression for vbgg in [42] is not correct. (Also formulae (7.11b) and (7.12) are not given

7.1 Subsingular Vectors

243

in [42].) Clearly, vbgg becomes a singular vector in the factor–module F2 = V D /I2 . Let |2 bgg denote the lowest weight vector of F2 . Then the singular vectors in (7.11) become null-conditions, the relevant one (7.11a) giving: X2+ | 2 bgg = 0.

(7.15)

If we factor out also vbgg we have the following null conditions in the resulting irreducible module with lowest weight vector | bgg: X2+ | bgg = 0, X1+ X2+ X3+ – X3+ X2+ X1+ | bgg = 0.

(7.16a) (7.16b)

7.1.4 The Other Archetypal sl(4, C) Example 7.1.4.1 Consider ﬁrst an arbitrary Verma module V D and the following vector: vf = Pv0 .

(7.17)

where P is a homogeneous polynomial in U(G + ): + + + + X2 – X12 X23 . P = X13

(7.18)

Below we shall need the following technical result: X2– vf = (D(H2 ) + 1)X2+ X1+ – D(H2 )X1+ X2+ X3+ v0 + X3+ (D(H2 ) – 1)X1+ X2+ – D(H2 )X2+ X1+ v0

(7.19a)

+

= (D(H2 ) + 1)X2+ X3+ – D(H2 )X3+ X2 X1+ v0 + X1+ –(D(H2 ) – 1)X1+ X2+ + D(H2 )X2+ X1+ v0 .

(7.19b)

For future reference we note several equivalent forms of the polynomial P: + + + + X2 – X12 X23 P = X13 + + + + = X13 X2 – X23 X12 = X1+ X2+ X3+ X2+ + X2+ X3+ X2+ X1+ – 2X2+ X1+ X3+ X2+

= X3+ X2+ X1+ X2+ + X2+ X1+ X2+ X3+ – 2X2+ X1+ X3+ X2+

(7.20a) (7.20b) (7.20c) (7.20d)

244

7 KL Polynomials and Conditionally Invariant Equations

and two forms valid if a ≡ D(H2 ) ≠ 1: 1 + + X X – 2X2+ X3+ (a – 1)X1+ X2+ – aX2+ X1+ a–1 3 2 1 + X + (a – 1)X1+ X2+ – aX2+ X1+ X3+ a–1 2 1 + + X1 X2 – 2X2+ X1+ (a – 1)X3+ X2+ – aX2+ X3+ = a–1 1 + X2+ (a – 1)X3+ X2+ – aX2+ X3+ X1+ . a–1

P=

(7.21a)

(7.21b)

The need for the introduction of these special forms will become clear below. 7.1.4.2 Consider now a Verma module V D with lowest weight D satisfying the conditions: D(H3 ) = 0 ⇐⇒ m3 = 1,

(7.22a)

D(H1 + H2 ) = 1 ⇐⇒ m12 = 1,

(7.22b)

71 (a) = 7(D) = (a, 1 – a, 1),

a ∈ C,

(7.22c)

cf. (7.9c,d). From these conditions follow that there are two singular vectors which are explicitly given by [122]: v3 = X3+ v0 , m3 = 1, v12 = (a – 1)X1+ X2+ – aX2+ X1+ v0 ,

(7.23a) m12 = 1.

(7.23b)

There is also a singular vector corresponding to m13 = 2 [122], which, however, here is a composite one: (2) = X3+ (a – 1)(a – 2)(X1+ )2 (X2+ )2 – 2a(a – 2)X1+ (X2+ )2 X1+ v13 + a(a – 1)(X2+ )2 (X1+ )2 X3+ v0 , m13 = 2, m1 = a, m3 = 1.

(7.24)

This vector gives no new condition since factoring out the submodule I m3 we factor out also the submodule I m13 (it is a submodule of Im3 ). Denote by I1 the submodule generated by these singular vectors, I1 = I m3 ∪I m12 , and by F1 = V D /I1 the corresponding factor–module. Let 3 | 1 denote the lowest weight vector of F1 . Then the expressions in (7.23) become null-conditions, namely we have | 1 = 0, m3 = 1, X3+ 3 | 1 = 0, (a – 1)X1+ X2+ – aX2+ X1+ 3

(7.25a) m12 = 1.

(7.25b)

7.1 Subsingular Vectors

245

For the Verma module under consideration the vector (7.17) is a subsingular vector iff a = 1, as we shall see in Subsection 7.1.4.7. However, if a ≠ 1 then vf itself belongs to the maximal invariant submodule I˜D generated by the singular vectors, and thus is not a subsingular vector by deﬁnition. Explicitly we note that if a ≠ 1 then using (7.21a) we have vf =

1 + + X3 X2 – 2X2+ X3+ v12 a–1 1 + X + (a – 1)X1+ X2+ – aX2+ X1+ v3 . a–1 2

(7.26)

Thus, indeed, vf belongs to I˜D = I m12 ∪ I m3 , for D from (7.22). We now write down systematically all situations. 7.1.4.3 If a ∉ Z there are no other nondescendant singular vectors, besides (7.23) and the maximal invariant submodule is: I D = I1′ = I !3 ∪ I !12 . We denote by L′1 = V D /I1′ the corresponding irreducible factor–module, and by | 1′ the lowest weight vector of L′1 . Then the expressions in (7.23) become null-conditions, namely we have X3+ | 1′ = 0, (a – 1)X1+ X2+ – aX2+ X1+ | 1′ = 0.

(7.27a) (7.27b)

7.1.4.4 If a ∈ –N then in addition to (7.23) there is one more singular vector [122] corresponding to m2 = 1 – a ∈ N + 1: 1–a v0 v2 = X2+

(7.28)

and two descendants corresponding to m23 = 2 – a, m13 = 2. Thus the maximal invariant submodule is: I D = I1′′ = I !3 ∪I !12 ∪I !2 , L′′1 = V D /I1′′ is the irreducible factor–module, | 1′′ is the lowest weight vector of L′′1 . Then the null-conditions are X3+ | 1′′ = 0, (a – 1)X1+ X2+ – aX2+ X1+ | 1′′ = 0,

X2+

1–a

| 1′′ = 0.

(7.29a) (7.29b) (7.29c)

246

7 KL Polynomials and Conditionally Invariant Equations

7.1.4.5 If a = 0 then there is a singular vector corresponding to m2 = 1 and given by (7.28) with a = 0. Here also (7.23b) is descendant and the maximal invariant submodule is generated by the singular vectors (7.23a) and (7.28), I D = I1′′′ = I !3 ∪ I !2 . We denote by ′′′ D ′′′ ′′′ L′′′ 1 = V /I1 the irreducible factor–module; | 1 the lowest weight vector of L1 . Then the null-conditions are X3+ | 1′′′ = 0,

(7.30a)

X2+ | 1′′′ = 0.

(7.30b)

7.1.4.6 If a ∈ N + 1 then there exists another singular vector [122]: a v1 = X1+ v0 .

(7.31)

D IV Thus the maximal invariant submodule is: I D = I1IV = I !3 ∪ I !12 ∪ I !1 , LIV 1 = V /I1 is IV IV the irreducible factor–module, | 1 is the lowest weight vector of L1 . Then the nullconditions are

X3+ | 1IV = 0, (a – 1)X1+ X2+ – aX2+ X1+ | 1IV = 0,

X1+

a

| 1IV = 0.

(7.32a) (7.32b) (7.32c)

7.1.4.7 Finally, if a = 1 then the nondescendant singular vectors are v3 = X3+ v0 , cf. (7.23a), and v1 = X1+ v0 , cf. (7.31) with a = 1, while (7.23b) is descendant of (7.31), and there appears ′ , cf. (7.11b), corresponding to m also a singular vector v23 23 = 1 which is descendant to (7.23a). Here we have also the subsingular vector vf , cf. (7.17), from the latter the essential one simplifying here to: X2– vf = 2X2+ X1+ – X1+ X2+ v3 – X3+ X2+ v1 .

(7.33)

Now it remained from the proof above to show that vf can not be represented as a linear combination of descendants of v1 and v3 , and thus does not belong to I˜D , which is easy to see also by inspecting (7.20). We denote by I˜D = I !1 ∪ I !3 the submodule generated by these singular vectors, by F1 = V D /I˜D the factor–module, by 2 | 0 the lowest weight vector of F1 . We have the following null conditions in F1 : X3+ 2 | 0 = 0, +2 X1 | 0 = 0.

(7.34a) (7.34b)

7.1 Subsingular Vectors

247

The vector vf becomes a singular vector in F1 which we denote as v f1 = P 2 | 0.

(7.35)

Factoring out the submodule built on vf1 we obtain the irreducible factor–module L1 = V D /I1D . We denote by | 0 the lowest weight vector of L1 . Then the nullconditions are X3+ | 0 = 0,

(7.36a)

X1+ | 0

(7.36b)

= 0, + + + + + + X3 X2 – 2X2 X3 X1 X2 | 0 = 0,

(7.36c)

where for (7.36c) we have used (7.36a) and (7.20d). An equivalent condition to (7.36c) is: (7.36c′ ) X1+ X2+ – 2X2+ X1+ X3+ X2+ | 0 = 0 where we have used (7.36b) and (7.20c). 7.1.4.8 Analogously consider a Verma module V D with lowest weight D satisfying the conditions: D(H1 ) = 0 ⇐⇒ m1 = 1,

(7.37a)

D(H2 + H3 ) = 1 ⇐⇒ m23 = 1,

(7.37b)

73 (a) = 7(D) = (1, 1 – a, a),

a ∈ C,

(7.37c)

cf. (7.9a,d). From these conditions follow that there are two singular vectors which are explicitly given by [122]: m1 = 1, = (a – 1)X3+ X2+ – aX2+ X3+ v0 ,

v23

v1 = X1+ v0 ,

(7.38a) m23 = 1.

(7.38b)

Denote by I3 the submodule generated by these singular vectors, I3 = I m1 ∪ I m23 , and by F3 = V D /I3 the corresponding factor–module. Let 2 | 3 denote the lowest weight vector of F3 . Then the expressions in (7.38) become null-conditions, namely we have

| 3 = 0, X1+ 2

m1 = 1,

| 3 = 0, (a – 1)X3+ X2+ – aX2+ X3+ 2

(7.39a) m23 = 1.

(7.39b)

For the Verma module under consideration and iff a = 1 the vector (7.17) is the subsingular vector considered in Subsection 7.1.4.7. However, if a ≠ 1 then vf itself belongs to the maximal invariant submodule I˜D generated by the singular vectors, and thus is

248

7 KL Polynomials and Conditionally Invariant Equations

not a subsingular vector by deﬁnition. Explicitly we note that if a ≠ 1 then using (7.21b) we have vf =

1 + + X1 X2 – 2X2+ X1+ v23 a–1 1 + X + (a – 1)X3+ X2+ – aX2+ X3+ v1 . a–1 2

(7.40)

Thus, indeed, vf belongs to I˜D = I m23 ∪ I m1 , for D from 7.37. 7.1.4.9 Note that a lowest weight can satisfy both (7.22) and (7.37) which happens exactly for the special case a = D(H2 ) = 1 and D(H1 ) = D(H3 ) = 0. In this case there are four singular vector given in (7.23) and (7.38). However, v12 , (cf. (7.23b)), becomes a descendant of the singular vector v1 , (cf. (7.38a)), while v23 , (cf. (7.38b)), becomes a descendant of the singular vector v3 , (cf. (7.23a). Then in the Verma module will have both intersecting submodules I1 and I3 : I1 ∩ I3 = I m12 ∪ I m23 ,

a = D(H2 ) = 1,

D(H1 ) = D(H3 ) = 0

7 = 71 (1) = 73 (1) = (1, 0, 1),

(7.41)

showing which we have used: I m12 ⊂ I m1 ,

I m23 ⊂ I m3 ,

D(H2 ) = 1, D(H1 ) = D(H3 ) = 0.

(7.42)

To summarize, we have shown: Proposition 1: Let the Verma module V D satisfy at least one of the two sets of conditions (7.22), (7.37), i.e., it has signature 7(D) = 71 (a) = (a, 1 – a, 1) or 7(D) = 73 (a) = (1, 1 – a, a). Then there exists a subsingular vector given by (7.17) iff a = D(H2 ) = 1, i.e., 7(D) = 71 (1) = 73 (1) = (1, 0, 1). It becomes a singular vector for the factor–module F D = V D /(I1 ∪ I3 ). ♢ Conditions (7.32) and (7.36) ((7.36c,c’) in a different, but equivalent form) were given ﬁrst in [160]. The corresponding irreps (for a ∈ N) were shown [160] to be a construction of the irreducible massless representations of a q - conformal algebra (with | q | = 1) characterized by the helicity h = (a – 1)/2 ∈ (1/2)Z+ .

7.2 Kazhdan–Lusztig Polynomials The Kazhdan–Lusztig polynomials (KLP) were introduced in [332] in the construction of an alternative basis in Hecke algebras. Here we shall introduce them using only a recursive procedure, (also contained in [332]), independent of the Hecke algebra context.

7.2 Kazhdan–Lusztig Polynomials

249

Let W be a Coxeter group and S be the set of simple reﬂections; let e is the unit element of W. Next we need the Bruhat order on W, (cf., e.g., [120]): Let w, w′ ∈ W. One writes ′ w ← w if (w) = (w′ ) + 1 and w = sw′ , with s a simple reﬂection. One writes w′ ≤ w if either w′ = w or exist elements wk ∈ W, k = 1, . . . , n, so that w′ = w1 ← w2 ← ⋅ ⋅ ⋅ ← wn = w. Thus, the Bruhat order w′ ≤ w is a partial ordering of W. We shall also write w′ < w if w′ ≤ w and w′ ≠ w. Let y, w ∈ W and y ≤ w in the Bruhat order on W. The KLP Py,w (u) in an indeterminate u may be introduced in a recursive procedure with the following initial conditions: Pe,e = 1, Py,w = 0,

(7.43a)

y ≰ w.

(7.43b)

In the recursive procedure [332] is used the following additional relation in W depending on the KLP: y ≺ w,

1 if y < w, %y = –%w , deg Py,w = ((w) – (y) – 1) 2

(7.44)

where (x), x ∈ W is the length function on W, %x ≡ (–1)(x) is the sign function on W. Suppose now that the Py,w′ are known for all y ≤ w′ with (w′ ) < 0 . Let (w) = 0 and write w = sv with s ∈ S, (thus (v) = 0 ). Then Py,w may be determined recursively from the following relation ((2.2.c) of [332]): Py,w = u1–c Psy,v + uc Py,v

+ ,(z, v) u–1/2 u1/2 z w Py,z ,

y ≤ w,

(7.45)

z y≤z≺v sz y x∈W

where ,(z, v) is the coefﬁcient of the highest power of u in Pz,v . Note that from (7.43a) and (7.45) follows that Py,w′ are polynomials in u. Indeed each term on the RHS adds nonnegative integer powers of u (for the summation term note that (w) – (z) ∈ 2N). Consider (7.45) ﬁrst for y = w. On the RHS in the second term Py,v = 0 since y = w = sv > v. For the same reason the summation term is empty. Then c = 1 since sy = sw = v < w = y and the RHS is Psy,v = Pv,v , while the LHS is Py,w = Psv,sv . Thus, (7.45) becomes: Psv,sv = Pv,v . Continuing in the same way one obtains, using (7.43a): Pw,w = 1

(7.46)

250

7 KL Polynomials and Conditionally Invariant Equations

Next we consider (7.45) for (w) = (y) + 1. Let ﬁrst w = sy, as reduced expressions with s ∈ S. Then in (7.45) sy > y = v and because of this one has c = 0, in the ﬁrst term in the RHS Psy,v is zero, and the summation is empty. Thus, we are left with Py,w = Py,v = Py,y = 1. Let now w = ys′ , as reduced expressions with s′ ∈ S. Write y = sy′ , then w = sy′ s′ , v = y′ s′ . Then in (7.45) c = 1 since sy < y and the summation is empty since sy′ is not comparable with y′ s′ . For the latter reason the second term Psy′ ,y′ s′ is zero. Thus (7.45) is reduced to Py,ys′ = Py′ ,y′ s′ . Continuing in the same way we can show that Py,ys′ = Pe,s′ , which is equal to 1 by the already shown ﬁrst case w = sy, (replacing y = e, s ↦ s′ ). Thus, we have shown [332]: Py,w = 1,

if (w) = (y) + 1.

(7.47)

From (7.44) and (7.47) follows: y ≺ w,

if (w) = (y) + 1.

(7.48)

Analogously using (7.45) may be shown: Py,w (0) = 1,

∀y ≤ w.

(7.49)

For this note that for u = 0 only one of the ﬁrst two terms in the RHS of (7.45) survives (depending on the value of c), while the summation term vanishes since the leading power is positive (as z < w). Thus (7.45) is reduced to a simple recursive relation which is used repeatedly until (7.49) is achieved. An important property of the KLP which may be proved also using recursively (7.45) is [332]: deg Py,w ≤ 21 ((w) – (y) – 1),

y < w.

(7.50)

Thus ≺ is related to the polynomials of maximal possible degree. (Note that in the exposition of [332] this together with (7.46) is one of the deﬁning conditions.) For illustration let us consider also the example w = sys′ , (as reduced expressions) with s, s′ ∈ S, thus, (w) = (y) + 2. Then in (7.45) c = 0 since sy > y and the summation is empty, since the only possible z = y does not fulﬁll sz < z. Then the ﬁrst term in (7.45) Psy,ys′ is zero since sy is not comparable with ys′ . Thus, only the second term contributes and one has Py,sys′ = Py,ys′ , which is equal to 1 because of (7.47). Analogously one may consider the other possibilities for lengths differing by two and obtain [332]: Py,w = 1,

if (w) = (y) + 2.

(7.51)

Let us mention also the following relation (cf. (2.3.g) of [332]): Py,w = Psy,w ,

if y < w, and s ∈ S is such that sw < w, sy > y.

(7.52)

251

7.3 Characters of LWM and Nontrivial KL Polynomials

From this follows (cf. Lemma 2.6.vi of [332]) if W is ﬁnite with longest element w0 : Py,w0 = 1,

∀y ∈ W.

(7.53)

There is a conjecture in [332] that all coefﬁcients in the KLP are nonnegative integers. This conjecture is already proved in many important cases, e.g., when W is a Weyl and afﬁne Weyl group [333]; for more information cf. Humphreys [306]. Note that if W is a dihedral group with S = (s, t) then Py,w = 1 for all y ≤ w [306, 332]. The ﬁrst nontrivial case is the Weyl group W of A3 = sl(4, C). Denote S = (w1 , w2 , w3 ) with: w12 = w22 = w32 = e, (w1 w2 )3 = e, (w2 w3 )3 = e, w1 w3 = w3 w1 . There are only two cases in which y ≺ w and degPy,w > 0, namely [332]: Pw2 ,w2 w1 w3 w2 = 1 + u,

(7.54a)

Pw1 w3 ,w1 w3 w2 w1 w3 = 1 + u.

(7.54b)

There are also several cases in which y < w and degPy,w > 0, however, y ≺/ w, namely: Pe,w2 w1 w3 w2 = 1 + u, Pt,w1 w3 w2 w1 w3 = 1 + u,

(7.55a) t = e, w1 , w3 .

(7.55b)

Note that (7.55a) may be obtained from (7.54a) using (7.52) with y = e, s = w2 , w = w2 w1 w3 w2 . In the same way (7.55b) may be obtained from (7.54b) using (7.52) ﬁrst with y = e, s = w1 , w3 , w = w1 w3 w2 w1 w3 , and then with y = w1 , w3 , s = w3 , w1 , resp., w = w1 w3 w2 w1 w3 .

7.3 Characters of LWM and Nontrivial KL Polynomials 7.3.1 Preliminaries on Characters of Lowest Weight Modules The character theory was developed in Section 2.11 for highest weight modules. Thus, we are very brief here just recording the notions. Let G be any simple Lie algebra. Let A, (resp. A+ ), be the set of all integral, (resp. integral dominant), elements of H∗ , i.e., + ∈ H∗ such that (+, !∨i ) ∈ Z, (resp. Z+ ), for all simple roots !i . For each invariant subspace V ⊂ U(G + )v0 ≅ V D we have the following decomposition: V = ⊕ V, , ,∈A+

(Note that V0 = Cv0 .)

V, = {u ∈ V | Hu = (+ + ,)(H)u, ∀H ∈ H}

(7.56)

252

7 KL Polynomials and Conditionally Invariant Equations

The character of V is the lowest weight analog of (2.274): chV =

(dim V, )e(D + ,) = e(D)

,∈A+

(dim V, ) e(,)

(7.57)

,∈A+

Analogously to (2.275) the character of a lowest weight Verma module is: chV D = e(D)

P(,)e(,) = e(D)

,∈A+

'

(1 – e(!))–1

(7.58)

!∈B+

(in the present situation mult ! = 1). The classical Weyl character formula for the ﬁnite-dimensional irreducible LWM over G, i.e., when D ∈ –A+ , has the form (compare with (2.281a): chLD =

(–1)(w) chV w⋅D ,

D ∈ –A+ ,

(7.59)

w∈W

where the dot ⋅ action is deﬁned by w ⋅ + = w(+ – 1) + 1. If q is not a root of unity this formula holds for the ﬁnite-dimensional irreducible LWM over G (this can be deduced from the results of [401]). For future reference we note: s! ⋅ D = D + m! !,

(7.60)

which follows easily using the deﬁnition (7.3). A more general character formula involves the Kazhdan–Lusztig polynomials Py,w (u), y, w ∈ W [332], which we have recalled in Section 7.2. Namely, the character formula of the irreducible LWM when D ∈ A is ([332], [401]) chLD =

–1

(–1)(wD w) Pw,wD (1)chV w⋅(wD ⋅D) ,

D ∈ A,

(7.61)

w∈W w≤wD

where wD is a unique element of W with minimal length such that the signature of D˜ = wD–1 ⋅ D is anti-semidominant: ˜ 2, m ˜ 3 ), ˜ 1, m 7˜ = (m

˜ K ) ∈ Z– . ˜ k = 1 – D(H m

(7.62)

Note that Py,w (1) ∈ N for y ≤ w. Note that (7.59) follows simply from (7.61). Indeed, the signature 7(D) of the ﬁnitedimensional irreducible LWM LD is dominant: mk = 1 – D(Hk ) ∈ N, (since D ∈ –A+ ), and hence wD = w0 , where w0 is the maximal length element of W, since the signature of w–1 ⋅ D is anti-dominant only for w = w0 . Then one notes Pw,w0 = 1 and w ≤ w0 ∀w ∈ W. Thus the summation range coincides and it remains to make the change of summation w = w′ w0 and to note that (w0 w′ w0 ) = (w′ ).

7.3 Characters of LWM and Nontrivial KL Polynomials

253

7.3.2 Case sl(4, C) We restrict now to G = sl(4, C), which is exactly the simplest case in which nontrivial KLP appear. These are given by (cf. [332] and Section 7.2): Pt,w2 w1 w3 w2 (u) = 1 + u, Pt,w1 w3 w2 w1 w3 (u) = 1 + u,

t = e, w2 ,

t = e, w1 , w3 , w1 w3 .

(7.63a) (7.63b)

Thus, there are only two kinds of LWM LD whose character formula contain these nontrivial KLP. We shall parametrize these LWM using the parametrization of the LWM D0 with semidominant signatures 70 = (m1 , m2 , m3 ), mk ∈ Z+ . Namely, for each such signature there would be at most one of those two kinds of LWM. Before making the precise statement we denote by D˜ 0 the lowest weight anti-semidominant signature 7˜ 0 = (–m1 , –m2 , –m3 ), which for ﬁxed mk is equal to w0 ⋅ D0 , where w0 is the maximal length element of W with a reduced expression, e.g., w0 = w1 w3 w2 w1 w3 w2 . Then we introduce notation for the two special elements of W which appear in (7.49): w0′ = w2 w1 w3 w2 ,

(7.64a)

w0′′ = w1 w3 w2 w1 w3 .

(7.64b)

Now we can state: Lemma 1: Fix a semidominant signature 70 = (m1 , m2 , m3 ), mk ∈ Z+ , with m13 = m1 + m2 + m3 > 0. Let D′ , D′′ be the lowest weights with signatures (resp.): 7′ = (m′1 , m′2 , m′3 ) = (–m1 , m13 , –m3 ), 7′′ = (m′′1 , m′′2 , m′′3 ) = (m12 , –m2 , m23 ),

m′i = 1 – D′ (Hi ),

(7.65)

m′′i = 1 – D′′ (Hi ),

with m12 = m1 + m2 , m23 = m2 + m3 . ′ ′′ ˜ (i) The Verma modules V D0 , V D0 , V D , V D enter the following embedding diagrams depending on the values of mk : ˜

′

V D0 ⊂ V D ∩ V

D′′

∩

m1 + m3 , m2 > 0

(7.66a)

m1 = m3 = 0, m2 > 0

(7.66b)

⊂ V D0

˜

′

V D0 ⊂ V D ∩ V

D′′

⊂ V D0

254

7 KL Polynomials and Conditionally Invariant Equations

˜

′

V D0 ⊂ V D ∩ V

D′′

∩

m1 + m3 > 0, m2 = 0

(7.66c)

= V D0

(ii) The elements wD′ , wD′′ of minimal length which correspond to D′ , D′′ in (7.61) are given as follows: ⎧ ′ w0 for m1 , m3 ≥ 0, m2 > 0 ⎪ ⎪ ⎨ w2 w1 w3 for m1 , m3 > 0, m2 = 0 wD′ = ⎪ ww for m1 > 0, m2 = m3 = 0 ⎪ ⎩ 2 3 w2 w1 for m3 > 0, m1 = m2 = 0 ⎧ ′′ w0 for m1 , m3 > 0, m2 ≥ 0 ⎪ ⎪ ⎪ ⎪ ⎪ w w w w for m1 = 0, m2 , m3 > 0 1 3 2 3 ⎪ ⎪ ⎨ w1 w3 w2 w1 for m3 = 0, m1 , m2 > 0 wD′′ = ⎪ w1 w3 w2 for m1 = m3 = 0, m2 > 0 ⎪ ⎪ ⎪ ⎪ w w w for m1 = m2 = 0, m3 > 0 ⎪ 1 2 3 ⎪ ⎩ w3 w2 w1 for m2 = m3 = 0, m1 > 0

(7.67a)

(7.67b)

Proof. (i) Consider ﬁrst the case of dominant signature 70 = (m1 , m2 , m3 ), mk ∈ N. The embedding picture (7.66a) is shown as follows. We recall that whenever (7.4) is fulﬁlled for some ! ∈ B+ the Verma module V D is reducible and contains as a submodule the Verma module V D+m! ! = V s! ⋅D , cf. also (7.60). Now we can check that D˜ 0 = w0′ ⋅ D′ = w2 ⋅ (w1 ⋅ (w3 ⋅ (w2 ⋅ D′ ))) = w0′′ ⋅ D′′ = w1 ⋅ (w3 ⋅ (w2 ⋅ (w1 ⋅ (w3 ⋅ D′′ )))).

(7.68)

This shows the embeddings w.r.t. D˜ 0 , (though omitting the intermediate embeddings evident in (7.68)). For the embeddings w.r.t. D0 it remains to note: D′ = w1 ⋅ (w3 ⋅ D0 ) = w3 ⋅ (w1 ⋅ D0 ) = D0 + m1 !1 + m3 !3 , D′′ = w2 ⋅ D0 = D0 + m2 !2 .

(7.69)

To show the embeddings in (7.66) in the cases when some of the parameters mk are zero one has to observe when D′ or D′′ coincides with D0 . The coincidences are clearly seen from (7.69) and so we easily obtain the rest of (7.66). Finally, we note that D˜ 0 may coincide with either of D′ or D′′ only if all mk = 0 (cf. the signatures), which we have excluded. (ii) Consider again the case of dominant signature 70 = (m1 , m2 , m3 ), mk ∈ N. We need to know the signatures of w ⋅ L˜ 0 which we denote by 7(w ⋅ D˜ 0 ) for the elements w which occur in the statement of the Lemma. We have

7.3 Characters of LWM and Nontrivial KL Polynomials

255

7(w2 w1 w3 ⋅ D˜ 0 ) = (–m12 , m13 , –m23 ),

(7.70a)

7(w2 w3 ⋅ D˜ 0 ) = (–m13 , m12 , –m2 ),

(7.70b)

7(w2 w1 ⋅ D˜ 0 ) = (–m2 , m23 , –m13 ),

(7.70c)

7(w1 w3 w2 w3 ⋅ D˜ 0 ) = (m13 , –m23 , m2 ),

(7.71a)

7(w1 w3 w2 w1 ⋅ D˜ 0 ) = (m2 , –m12 , m13 ),

(7.71b)

7(w1 w3 w2 ⋅ D˜ 0 ) = (m23 , –m13 , m12 ),

(7.71c)

7(w1 w2 w3 ⋅ D˜ 0 ) = (m13 , –m3 , –m2 ),

(7.71d)

7(w3 w2 w1 ⋅ D˜ 0 ) = (–m2 , –m1 , m13 ).

(7.71e)

Clearly, 7′ and 7′′ do not coincide with any of these signatures when all mk > 0. Next we consider the cases when some of the parameters mk are zero. From (7.70) and (7.71) we obtain ﬁrst that 7′ , 7′′ , resp., does not coincide with any of these signatures when m2 > 0, m1 , m3 > 0, resp. Further from (7.70) and (7.71) we obtain: 7(w2 w1 w3 ⋅ D˜ 0 ) = 7′ ,

m1 + m3 > 0, m2 = 0,

(7.72a)

7(w2 w3 ⋅ D˜ 0 ) = 7′ ,

m1 > 0, m2 = m3 = 0,

(7.72b)

7(w2 w1 ⋅ D˜ 0 ) = 7′ ,

m3 > 0, m1 = m2 = 0,

(7.72c)

7(w1 w3 w2 w3 ⋅ D˜ 0 ) = 7′′ ,

m1 = 0, m2 , m3 > 0,

(7.72d)

7(w1 w3 w2 w1 ⋅ D˜ 0 ) = 7′′ ,

m3 = 0, m1 , m2 > 0,

(7.72e)

7(w1 w3 w2 ⋅ D˜ 0 ) = 7′′ ,

m1 = m3 = 0, m2 > 0,

(7.72f)

7(w1 w2 w3 ⋅ D˜ 0 ) = 7′′ ,

m1 = m2 = 0, m3 > 0,

(7.72g)

7(w3 w2 w1 ⋅ D˜ 0 ) = 7′′ ,

m2 = m3 = 0, m1 > 0.

(7.72h)

From (7.72) statement ii) of the Lemma follows by just noting the lengths of the elements, which are 3, 2, 2 in (7.72a,b,c), respectively, with (w0′ ) = 4, and 4, 4, 3, 3, 3 in (7.72d,e,f,g,h), respectively, with (w0′′ ) = 5. ∎ Using the above result we shall give explicitly all character formulae in the case sl(4, C) which involve nontrivial KLP, i.e., chLD′ , chLD′′ , resp., when m2 > 0, m1 , m3 > 0, resp. For the simpliﬁcation of the character formulae we use notation for the formal exponents corresponding to the simple roots: tj = e(!j ), j = 1, 2, 3; then for the three nonsimple roots !12 = !1 + !2 , !23 = !2 + !3 , !13 = !1 + !2 + !3 we have t12 = e(!12 ) = t1 t2 ,

256

7 KL Polynomials and Conditionally Invariant Equations

t23 = e(!23 ) = t2 t3 , t13 = e(!13 ) = t1 t2 t3 . In terms of these the character formula for the Verma module is: chV D = e(D)/(1 – t1 )(1 – t2 )(1 – t3 )(1 – t12 )(1 – t23 )(1 – t13 ).

(7.73)

Now we can prove: Proposition 2: The character formulae of the irreducible LWM LD′ , LD′′ with signatures in (7.65) are explicitly given as follows: ′ m m m m chLD′ = chV D 1 – t2 13 – t1223 – t2312 – t132 + m m

m

m

m

m m

m

+ t132 t2 13 + t2 13 t3 12 + t1 2 t2312 + t1223 t3 2 + m23 m13 t2

m m

m

m

m

m

– t122 t2 13 t3 12 – t1 23 t2 13 t232 – m m m m m – 2 t1 23 t2 13 t3 12 + 2 t1223 t2312 ,

+ t1

(7.74)

m1 , m3 ∈ Z+ , m2 ∈ N ′′

chLD′′ = chV D

m12

1 – t1

m m

m

m

m

m

– t3 23 – t121 – t233 + m

m m

m

m

+ t121 t233 + t1 13 t233 + t121 t3 13 + t1 12 t3 23 + m

m

m

m

m

m

m

+ t1 12 t2 1 + t2 3 t3 23 – 2 t1 12 t2 13 t3 23 – m m

m

m m

m

m

m

m

– t121 t2 3 t3 13 – t1 12 t2 1 t3 13 – t1 13 t2 3 t3 23 – m

m m

m

m

m

m

– t1 13 t2 1 t233 + 2 t1 12 t2313 + 2 t1213 t3 23 + m m +m m m + t1 13 t2 1 3 t3 13 – 2 t1313 ,

(7.75)

m1 , m3 ∈ N, m2 ∈ Z+ . Proof. Let ﬁrst mk > 0, k = 1, 2, 3. The proof amounts to the correct identiﬁcation of the ′ ′′ Verma modules which are submodules of V D , V D . In particular, the Verma modules which contribute two times because of (7.49) have the following weights (expressed in terms of D˜ 0 and t from (7.49)): D′t = t ⋅ D˜ 0 ,

t = e, w2 ,

(7.76a)

D′′t = t ⋅ D˜ 0 ,

t = e, w1 , w3 , w1 w3 ,

(7.76b)

i.e., D′e = D′′e = D˜ 0 . Their weight differences (which enter the character formulae) are D′w2 – D′ = m23 !1 + m13 !2 + m12 !3 , D˜ 0 – D′ = m23 !12 + m12 !23 ,

(7.77a) (7.77b)

7.3 Characters of LWM and Nontrivial KL Polynomials

D′′w1 w3 – D′′ = m12 !1 + m13 !2 + m23 !3 ,

257

(7.78a)

D′′w1 – D′′ = m12 !1 + m13 !23 ,

(7.78b)

D′′w3

(7.78c)

′′

– D = m13 !12 + m23 !3 ,

D˜ 0 – D′′ = m13 !13 ,

(7.78d)

where !12 = !1 + !2 , !23 = !2 + !3 , !13 = !1 + !2 + !3 . The extension to the cases when some mk are zero is done as in the Lemma. ∎ For future reference we record the signatures of the lowest weights in (7.76) besides D˜ 0 (in evident notation): 7w′ 2 = (–m23 , m2 , –m12 ),

(7.79a)

7w′′1

= (m3 , –m23 , –m1 ),

(7.79b)

7w′′3 = (–m3 , –m12 , m1 ),

(7.79c)

7w′′1 w3 = (m3 , –m13 , m1 ).

(7.79d)

Note that the character formulae (7.74), (7.76) are simpliﬁed when some mk are zero (as speciﬁed). In particular, (7.74) has six terms when m1 m3 = 0, m1 + m3 > 0, and four terms when m1 = m3 = 0, while (7.76) has eight terms when m2 = 0: ′ m m m m chLD′ = chV D 1 – t2 23 – t232 + t2 23 t3 2 m m m m – t1223 t3 2 + t1223 t232 , m1 = 0; ′ m m m m chLD′ = chV D 1 – t2 12 – t122 + t1 2 t2 12 m m m m – t1 2 t2312 + t122 t2312 , m3 = 0; ′ m m chLD′ = chV D 1 – t2 2 1 – t132 ,

chLD′′

m1 = m3 = 0; ′′ m m 1 – t1 1 1 – t3 3 = chV D m m m m – t121 t233 1 – t1 3 1 – t3 1 ,

(7.80a)

(7.80b)

(7.80c)

m2 = 0.

(7.80d)

We record also the factorization of (7.80d) for m1 = m3 : ′′ chLD′′ = chV D 1 – t1m 1 – t3m 1 – (t12 t23 )m , m2 = 0 , m1 = m3 = m, For m = 1 formula (7.80e) appeared in [160].

7′′ = (m, 0, m).

(7.80e)

258

7 KL Polynomials and Conditionally Invariant Equations

Remark 3: The above Proposition gives (for sl(4, C)) an exhaustive parametrization of all occurrences of nontrivial KLP in character formulae. It shows also, together with Lemma 1, the relation between the two principal situations, since, a priori there is no ′ reason to use the same numbers in the signatures of D′ , D′′ . The Verma modules V D , ′′ V D for ﬁxed mk have the same values of their Casimir operators, since they belong to the same embedding schemes (7.66). We also stress that unless all mk > 0 there are no KL multiplicities, (cf. (7.80), each character of a Verma module is contributing once. We also have ′

′′

Corollary: The Verma modules V D , V D for ﬁxed mk have the following maximal common submodule generated by singular vectors if m2 > 0 and m1 + m3 > 0: ′

′′

′

′

′

′

V D ∩ V D ⊃ V D1 ∪ V D2 ∪ V D3 ∪ V D4 D′1 = w12 ⋅ D′ = D′ + m23 !12 = w1 ⋅ (w23 ⋅ D′′ ) = D′′ + m13 !1 + m3 !23 , D′2 = w2 ⋅ D′ = D′ + m13 !2 = w23 ⋅ (w12 ⋅ D′′ ) = D′′ + m1 !12 + m3 !23 = w1 ⋅ (w3 ⋅ (w2 ⋅ D˜ 0 )), D′3 = w23 ⋅ D′ = D′ + m12 !23 = w3 ⋅ (w12 ⋅ D′′ ) = D′′ + m1 !12 + m13 !3 , D′4 = w13 ⋅ D′ = D′ + m2 !13 = w3 ⋅ (w1 ⋅ D′′ ) = D′′ + m12 !1 + m23 !3 = w2 ⋅ (w1 ⋅ (w3 ⋅ D˜ 0 ))

(7.81)

where wjk ≡ w!jk , and we have also recalled that D′2 , D′4 have appeared already in (7.71c) and (7.70a), respectively. ′

Proof. From the signature of V D it follows that its maximal submodule generated by singular vectors is ′ ′ ′ ′ ′ I˜D = I m12 ∪ I m2 ∪ I m23 ∪ I m13 ′

′

′

′

= V D1 ∪ V D2 ∪ V D3 ∪ V D4 .

(7.82) ′

It remains to notice that none of the four Verma modules V Dk is a submodule of the ′ ′′ other three, and that all of them are proper submodules of both V D and V D . This is ensured by the restrictions on the parameters mk . ∎

7.3 Characters of LWM and Nontrivial KL Polynomials

259

′

Remark 4: The contributions of the Verma modules V Dk in the character formulae of LD′ and LD′′ are the terms (with minus sign) on the ﬁrst line of (7.74) and the terms on the second line of (7.76). ♢ 7.3.3 Related Character Formulae One may wonder about the character formulae of LD′ , LD′′ in the excluded cases m2 = 0, m1 m3 = 0, resp. Certainly, formulae (7.74) and (7.76) are not valid in these cases, and the formal substitution of m2 = 0 in (7.74) and of m1 = 0 or m3 = 0 in (7.76) gives zero. We shall give these formulae only for LD′′ since we shall need them below. We ﬁrst introduce notation for the lowest weights with signatures which appeared already in (7.71a,b): !˜ = w1 w3 w2 w3 ⋅ D˜ 0 = w2 w3 ⋅ D0 ,

(7.83a)

b˜ = w1 w3 w2 w1 ⋅ D˜ 0 = w2 w1 ⋅ D0 .

(7.83b)

Next we need the following: Lemma 2: For ﬁxed mk ∈ Z+ there exist the following embedding diagrams: ′

V D2 ⊂ V !˜ ∩

∩

˜ Vb

⊂V

′

˜

m2 ∈ Z+ , m1 , m3 ∈ N

(7.84a)

D′′ ′′

V D2 = V b ⊂ V !˜ = V D , m2 ∈ Z+ , m1 ∈ N, m3 = 0 ˜

′

(7.84b)

′′

V D2 = V !˜ ⊂ V b = V D , m2 ∈ Z+ , m3 ∈ N, m1 = 0 ˜

′

(7.84c)

′′

V D2 = V !˜ = V b = V D , m2 ∈ Z+ , m1 = m3 = 0,

(7.84d)

where D′2 was given in (7.71c) and (7.81). The character formulae are m m m chL!˜ = chV !˜ 1 – t1 13 – t3 2 – t121 m

m

m

m

m

m

m m

+ t1 13 t2 1 + t1 13 t3 2 + t1212 t3 2 + t121 t3 12

m m m m m m m m m – t1 13 t2 1 t3 12 – t1 13 t2 12 t3 2 – t1312 + t1 13 t2312 , m1 , m2 ∈ N, m3 ∈ Z+ ,

(7.85a)

260

7 KL Polynomials and Conditionally Invariant Equations

˜ m m m chLb˜ = chV b 1 – t1 2 – t3 13 – t233 m

m

m m

m m

m23 m3 t23 m23 m3 m13 m23 m m m2 m23 m13 t1 t2 t3 – t1 t2 t3 – t13 + t1223 t3 13 ,

+ t2 3 t3 13 + t1 2 t3 13 + t1 2 t2323 + t1 –

chLD′

2

m2 , m3 ∈ N, m1 ∈ Z+ , ′ m m m = chV D2 1 – t1 23 – t132 – t3 12 m23 m12 t3

+ t1

m23 m2 t23

+ t1

m m

(7.85b)

m23 m2 m12 t2 t3

+ t122 t3 12 – t1

,

m2 ∈ N, m1 , m3 ∈ Z+ .

(7.85c)

∎

Proof. The proof is analogous to that of Proposition 2 and Lemma 1.

(Formulae (7.85a) and (7.85b), resp., for m1 = m2 = 1, m2 = m3 = 1, resp., have appeared in [160].) Now we have the desired formulae: Corollary: The character formula of LD′′ when m1 m3 = 0 is given by m m m m m chLD′′ = chL!˜ = chV !˜ 1 – t1 12 – t3 2 – t121 + t1 12 t2 1 m m m m m m m + t1 12 t3 2 + t121 t3 12 – t1 12 t2 1 t3 12 , valid when m1 , m2 ∈ N, m3 = 0 ˜ m m m m m chLD′′ = chLb˜ = chV b 1 – t1 2 – t3 23 – t233 + t2 3 t3 23 m m m m m m m + t1 2 t3 23 + t1 23 t233 – t1 23 t2 3 t3 23 ,

(7.86a)

valid when m2 , m3 ∈ N, m1 = 0

(7.86b)

′ m m chLD′′ = chLD′ = chV D2 1 – t1 2 1 – t3 2 , 2

valid when m2 ∈ N, m1 = m3 = 0.

♢

(7.86c)

7.4 KL Polynomials and Subsingular Vectors: A Conjecture The folklore says that a subsingular vector of the Verma module V D is expected whenever Pw,wD (1) > 1 for some w. It is clear from (7.61) that such a subsingular vector should have the following weight: w ⋅ (wD–1 ⋅ D) – D.

(7.87)

Here we shall demonstrate that subsingular vectors may not appear in the above situation. We begin by describing the standard situation in detail. We use again the sl(4, C) case.

7.4 KL Polynomials and Subsingular Vectors: A Conjecture

261

From the character formulae (7.74) and (7.76) follows that the subsingular vectors ′ ′′ are vectors in the Verma modules V D , V D with weights equal to the weight differences given in the proof of Proposition 2, (7.77), (7.78), and which are valid under the restrictions there, namely, m2 > 0 for (7.77), m1 m3 > 0 for (7.78). Thus the simplest subsingular vectors are expected to have the weights: D′w2 – D′ = !13 ,

(7.88a)

D˜ 0 – D′ = !13 + !2 , m2 = 1, m1 = m3 = 0,

(7.88b) 7′ = 70 = (0, 1, 0),

D′′w1 w3 – D′′ = !1 + 2!2 + !3 ,

(7.88c)

(7.89a)

D′′w1 – D′′ = !1 + 2!2 + 2!3 ,

(7.89b)

D′′w3 – D′′ = !1 + 2!2 + 2!3 ,

(7.89c)

D˜ 0 – D′′ = 2!13 , m2 = 0, m1 = m3 = 1,

(7.89d) 7′′ = 70 = (1, 0, 1).

(7.89e)

We have given explicitly the subsingular vectors corresponding to (7.88a) and (7.89a). Case (7.89a) is the vector vbgg (cf. [42] and (7.13)). Case (7.89a) corresponds to the subsingular vector vf (cf. (7.17)). We start checking (7.88) and (7.88). Consider ﬁrst (7.88b). It is easy to see that there is no subsingular vector. Indeed, it is easy to check that any nonzero vector v′ ∉ Cv0 of this weight which satisﬁes the properties: Xk– v′ ∈ I˜D = I2 , k = 1, 2, 3, (for I2 cf. (7.12)), will necessarily be itself in I2 . An example of such a vector is a descendant of vbgg : ′ = X2+ vbgg vbgg

(7.90) D′w

which is a composition using the singular vector vs D′w v0 2 );

D′w

2

D′w

= X2+ v0

2

of the Verma module

the latter singular vector gives the embedding V 2 (with lowest weight vector ′ ˜ V Dw2 ⊃ V L0 . Indeed, substituting (7.13a) we have ′ = vbgg

+ X+ + X+ X+ v X2+ X12 0 3 23 1

+ X+ + X+ X+ X+ v ∈ I . = X12 2 3 23 1 2 0

(7.91)

262

7 KL Polynomials and Conditionally Invariant Equations

In the same way one may check that there are no subsingular vectors of weights (7.89ab, c, d). In particular, there are three descendants corresponding to the embedding scheme: ′′

˜

V D0 ⊂ V Dw1 ∩ V

∩

D′′ w

3

⊂V

(7.92)

D′′ w1 w3

These descendants come from singular vectors of V

D′′ w w X3+ v0 1 3 ,

D′′ w w X1+ X3+ v0 1 3 .

D′′ w1 w3

D′′ w1 w3

which are: X1+ v0

,

The descendants are given explicitly by ′′

v1′′ = X1+ vf = X1+ P v0 = PX1+ v0 ∈ I˜D

′′

v3′′ = X3+ vf = X3+ P v0 = PX3+ v0 ∈ I˜D

′′

′′ = X + X + v = X + X + P v = PX + X + v ∈ I˜ D v13 0 3 1 f 3 1 3 1 0

(7.93)

′′ and we have also demonstrated that they belong also to I˜D = I m1 ∪ I m3 . Thus, from the six nontrivial KLP Py,w in (7.49) only two generate subsingular vectors, namely, those for which (y, w) = (w2 , w2 w1 w3 w2 ) and (y, w) = (w1 w3 , w1 w3 w2 w1 w3 ). It is important to notice that these are exactly the only two cases of pairs of elements of W (for sl(4, C)) which fulﬁll the special relation y ≺ w (cf. Section 7.2). Thus we are able to conclude by formulating the following:

Conjecture: Consider a Verma module V D with D ∈ A. Let w be such that Pw,wD (1) ∈ N + 1, where wD is the unique element of W with minimal length such that the signature of D˜ = wD–1 ⋅ D is anti-semidominant (cf. (7.61) and (7.62)). A subsingular vector exists iff w ≺ wD . ♢

7.5 Conditionally Invariant Differential Equations 7.5.1 Preliminaries In this section, we write down explicitly the conditionally invariant equations related to the subsingular vectors considered in Section 7.1. We work with the induced representations, called elementary representations (ERs) and introduced in Chapter 4, though here we take the conjugate picture with associated lowest weight modules (instead of highest). Thus, instead of (4.32) we have XR > = D(X) ⋅ >, XR > = 0,

X ∈ H,

X ∈ G–.

(7.94a) (7.94b)

7.5 Conditionally Invariant Differential Equations

263

Thus, we work in the picture conjugate to the one in Section 4.6. Thus, to the singular vector vs = v!,m! (cf. (4.34)) of the Verma module V D there corresponds an intertwining differential operator ′

D!,m! : CD → CD ,

D′ = D + m! !,

D′

D!,m! ○ XLD = XL ○ D!,m! ,

∀X ∈ G,

(7.95a) (7.95b)

the operator being given explicitly by D!,m! = P !,m ((X1+ )R , . . . , (Xr+ )R ),

(7.96)

where P !,m is the same polynomial as in (7.5), and (Xi+ )R is the right action (4.17). This operator gives rise to the G-invariant equation: ˆ => ˆ ′, D!,m! >

ˆ ∈ CD , > ˆ ′ ∈ CD+m! ! . >

(7.97)

7.5.2 Conditionally Invariant Operators In a similar way a subsingular vector vsu produces a differential operator Dsu and equation which are conditionally invariant. The latter means that this invariance hold only on the intersection of the kernels of all intertwining operators D!,m! such that ! and the singular vectors v!,m! are associated with the singular vector in question, i.e., on the space: ˆ ∈ CD | D!,m! > ˆ = 0, Csu = {>

∀! ∈ Bsu }.

(7.98)

D′ = w ⋅ (wD–1 ⋅ D),

(7.99a)

Thus, instead of (7.95) we have (cf. (7.87)): ′ Dsu : C˜ D → CD , ′

ˆ = XLD ○ Dsu >, ˆ Dsu ○ XLD >

∀X ∈ G,

ˆ ∈ Csu , ∀>

(7.99b)

where C˜ D is such that Csu ⊂ C˜ D ⊂ CD holds, and we use that Csu is G-invariant: XLD Csu ⊂ Csu ,

∀X ∈ G.

(7.100)

A conditionally invariant equation has nontrivial RHS if we take the situation corresponding to the reducible factor–module F D = V D /I˜D ; the latter is realized when we do not impose in F D the null condition corresponding to the subsingular vector which in F D is a singular vector. A conditionally invariant equation has trivial RHS if we take the situation corresponding to the irreducible factor–module LD = V D /I D , i.e., if we impose in F D the null condition corresponding to the subsingular vector. Below we consider both situations, for which we are prepared by the detailed analysis of Section 7.1.

264

7 KL Polynomials and Conditionally Invariant Equations

7.6 Application to sl(4, C) Further, we restrict to G = sl(4, C), G = sl(4, C). We pass to functions on the ﬂag manifold Y = sl(4, C)/B, where B is the Borel subgroup of sl(4, C) consisting of all upper diagonal matrices. (Equally well one may take the ﬂag manifold sl(4, C)/B′ , where B′ is the Borel subgroup of lower diagonal matrices.) We denote the six local coordinates on Y by x± , v, v¯ , z, z¯ . From the explicit form of the singular vectors, it is clear that we need only the right action of the three simple root generators. Denoting this right action of Xk+ by Rk , we have from (6.153): R1 = ∂z ≡

∂ , ∂z

R2 = z¯ z∂+ + z∂v + z¯ ∂v¯ + ∂– , R3 = ∂z¯ ≡

∂ , ∂ z¯

(7.101)

where ∂± ≡

∂ , ∂x±

∂v ≡

x± ≡ x0 ± x3 ,

∂ , ∂v

∂v¯ ≡

v ≡ x1 – ix2 ,

∂ , ∂ v¯

(7.102)

v¯ ≡ x1 + ix2

while z, z¯ encode the inducing Lorentz representation as explained before. In particular, one may use the following covariant representation for R2 [122] by employing the Pauli matrices3, :

z R2 = z¯ 1 3, ∂ 1

,

(7.103)

(cf. (6.186a)). Note also that under the natural conjugation 9(x± ) = x± ,

9(v) = v¯ ,

9(z) = z¯ ,

(7.104)

Y is also a ﬂag manifold of the conformal group SU(2, 2). Remark 5: As we noted if one wants to treat the case of a real noncompact algebra G0 one has to use also the results for its complexiﬁcation G. The application of these results to G0 has some subtleties [126]. However, in the case at hand when G0 = su(2, 2) and G = sl(4, C) the passage to su(2, 2) is straightforward [122]. Also considering representations of the corresponding groups (which are used here only to provide the representation spaces) involves some subtleties [126], which, however, are not felt in the case under consideration [122]. ♢

7.6 Application to sl(4, C)

265

The reduced function spaces of the ERs in which our equations are deﬁned are complex-valued C∞ functions on the ﬂag manifold. The holomorphic ERs of sl(4, C) are labeled by the signature 7 = (m1 , m2 , m3 ). 7.6.1 Equations Arising from the BGG Example We start with the equations arising from the BGG (Bernstein–Gel’fand–Gel’fand) example of subsingular vector. Substituting (7.101) in (7.15) we obtain the following sl(4, C) and su(2, 2) invariant equation: ˆ = 0, ˆ = z¯ z∂+ + z∂v + z¯ ∂v¯ + ∂– > R2 > (7.105a) while the subsingular vector vbgg gives rise to the following conditionally invariant equation: ˆ = ∂v ∂z¯ – ∂v¯ ∂z + z¯ ∂z¯ – z∂z ∂+ > ˆ => ˆ ′, R1 R 2 R 3 – R 3 R 2 R 1 > (7.105b) ′

ˆ ′ ∈ CD , D′ = D – !13 , the corresponding signatures ˆ ∈ CD and satisﬁes (7.105a), > where > ′ being 7 = (0, 1, 0) and 7 = (–1, 1, –1). (Note that the second Casimir operator has the same value in the two representations: C2 (7) = C2 (7′ ) = – 4 (cf. (7.10)).) If we consider the irreducible factor-module LD , which means that we should use (7.16) instead of (7.15), we have instead of (7.105b): ˆ = 0. ∂v ∂z¯ – ∂v¯ ∂z + z¯ ∂z¯ – z∂z ∂+ > (7.105c) 7.6.2 Equations Arising from the Other Archetypal sl(4, C) Example We pass now to equations arising from the other archetypal sl(4, C) example. We consider the case when the lowest weight satisﬁes conditions (7.22). We shall substitute the operators Rk into the null conditions (7.27), (7.29), (7.25), (7.32), (7.34), and (7.36). In all cases we have the equation arising from the singular vector v3 = X3+ v0 (null conditions (7.27a), (7.29a), (7.25a), (7.32a), (7.34a), and (7.36a)): ˆ = ∂z¯ > ˆ = 0, R3 >

(7.106)

which means that our functions do not depend on the variable z¯ – this is valid for the signature 71 (a) and arbitrary a. (In the conjugate situation with signature 73 (a) our functions do not depend on the variable z.) Further, we have the equations arising from the singular vector v2 , when a ∈ Z– , (null conditions (7.29c) and (7.25b)): ˆ = 0, (R2 )1–a >

a ∈ Z– .

(7.107)

266

7 KL Polynomials and Conditionally Invariant Equations

Next, we have the equations arising from the singular vector v1 , when a ∈ N, (null conditions (7.32c), (7.34b), and (7.36b)): ˆ = 0, (∂z )a >

a ∈ N,

(7.108)

which means that our functions are polynomials in the variable z of degree a – 1. Thus for a = 1 our functions do not depend also on z. Next we write down the equation arising from the singular vector v12 (null conditions (7.27b), (7.29b), and (7.32b)):

ˆ = (a – 1)(∂v + z¯ ∂+ ) – R2 ∂z > ˆ = 0. (a – 1)R1 R2 – aR2 R1 >

(7.109)

It is also valid in all cases, however, for a = 0 it follows from (7.107) and for a = 1 it follows from (7.108). Now, since (7.109) is a ﬁrst degree polynomial in z¯ , on which our functions do not depend, it actually consists of two equations, though not invariant by themselves, i.e., we have

ˆ = 0, (a – 1 – z∂z )∂v – ∂– ∂z >

(7.110a)

ˆ = 0. (a – 1 – z∂z )∂+ – ∂v¯ ∂z >

(7.110b)

Finally, we obtain the conditionally invariant equations corresponding to the subsingular vector vf . Let us denote by Pˆ the polynomial P with Xk+ replaced by Rk . Now we shall obtain this operator in explicit form: ˆ = (R3 R2 – 2R2 R3 ) R1 R2 > ˆ Pˆ >

(7.111a)

ˆ = (z∂+ + ∂v¯ – R2 ∂z¯ ) ∂z R2 > ˆ = (z∂+ + ∂v¯ ) ∂z R2 – R2 ∂z (z∂+ + ∂v¯ ) >

(7.111b)

ˆ = ◻ >, ˆ = (∂v¯ ∂v – ∂– ∂+ ) >

(7.111d)

(7.111c)

where we used (7.106) in passing from (7.111b) to (7.111c). Thus, we have recovered the d’Alembert operator. Note that (7.111) is valid for arbitrary a since we have used only condition (7.106) which is valid for all of our representations. Now for a = 1 if we take only invariant equations arising from the conditions (7.34a) (i.e., we work with the counterpart of the factor-module F1 ), we have the following system of differential equations: ˆ = 0, ∂z¯ >

(7.112a)

ˆ = 0, ∂z >

(7.112b)

◻>ˆ =

ˆ ′, >

(7.112c)

7.6 Application to sl(4, C)

267

′

ˆ ′ ∈ CD , D′ = D – !13 – !2 , the corresponding ˆ ∈ CD and satisﬁes (7.112a,b), > where > signatures being 7 = (1, 0, 1), 7′ = (1, –2, 1). (Note that the second Casimir operator has the same value in the two representations: C2 (7) = C2 (7′ ) = – 3, cf. (7.10).) If we consider the irreducible factor-module L1 , which means that we should use (7.36) instead of (7.34), we have instead of (7.112c): ◻>ˆ = 0,

(7.112d)

ˆ is as in (7.112c) and again satisﬁes (7.112a,b). where > Thus, from the subsingular vector vf we have obtained the d’Alembert equations (7.112c,d) as conditionally sl(4, C) and su(2, 2) invariant equations. Now we pass to the cases when a ≠ 1. In these cases the vector vf is a linear combination of the singular vectors v1 and v12 and it becomes zero when these singular vectors are factorized. Since vf gives rise to the d’Alembert operators for all a we expect that the d’Alembert equation (7.112d) will hold automatically if the invariant equations (7.106) and (7.109) (arising from v1 , v12 ) hold. This is indeed so. We use the two equations (7.110) which are the two components of (7.109). First we take ∂v¯ derivative from (7.110a) and ∂– derivative from (7.110b) and subtracting the two we get: ˆ (a – 1 – z∂z ) ∂– ∂+ – ∂v¯ ∂v > ˆ = 0. = (a – 1 – z∂z ) ◻ >

(7.113a)

This still follows from (7.112d). Analogously, taking ∂+ derivative from (7.110a) and ∂v derivative from (7.110b) and subtracting the two we get: ˆ =0 ˆ = ∂z ◻ > ∂z ∂– ∂+ – ∂v¯ ∂v >

(7.113b)

This also follows from (7.112d). Now, clearly from (7.113a,b) follows: ˆ =0 (a – 1) ◻ >

(7.113c)

which implies the d’Alembert equation if a ≠ 1. Using the conjugate situation with signature 73 (a), we recover the d’Alembert equation on functions which do not depend on z and satisfy

ˆ (a – 1)R3 R2 – aR3 R1 >

ˆ =0 = (a – 1)(∂v¯ + z∂+ ) – R2 ∂z¯ >

(7.114)

268

7 KL Polynomials and Conditionally Invariant Equations

instead of (7.109). Furthermore, the analogs of (7.110a,b), (7.106), and (7.108), respectively, are ˆ = 0, (7.115a) (a – 1 – z¯ ∂z¯ ) ∂v¯ – ∂– ∂z¯ > ˆ = 0, (a – 1 – z¯ ∂z¯ )∂+ – ∂v ∂z¯ > (7.115b) ˆ = 0, ∂z > ˆ = 0, (∂z¯ ) > a

(7.115c) a ∈ N.

(7.115d)

Thus if a ∈ N, then the functions of the irreducible representations are polynomials in z¯ of degree a – 1. If a ∈ Z– our functions satisfy (7.107) as those with signature 71 (a). Finally the d’Alembert equation (7.112d) follows from equations (7.115a,b) (a ≠ 1). We do not need to consider a = 1 since the two signatures coincide. We summarize now the results of this section. In the sl(4, C) setting we have that the d’Alembert equation (7.112d) holds in the representation spaces with signatures 71 (a) = (a, 1 – a, 1), resp., 73 (a) = (1, 1 – a, a), if our functions do not depend on the variable z¯ , resp., z and in addition satisfy (7.112a,b), resp., (7.115a,b). For a = 1 the d’Alembert equations (7.112c,d) are conditionally sl(4, C) and su(2, 2) invariant, while for a ≠ 1 the d’Alembert equation (7.112d) just follows from (7.112a,b), resp., (7.115a,b). If a ∈ N then our functions are polynomials in z, resp., z¯ , of degree a – 1. Summarizing our results in the su(2, 2) setting we again recall that the variables z, z¯ are representing the spin dependence coming from the Lorentz representation [164], [122], [126]. The above result then is restated so in the case a ∉ N: the d’Alembert equation holds if the ﬁelds carry holomorphic (depending only on z) or antiholomorphic (depending only on z¯ ) inﬁnite-dimensional representations of the Lorentz algebra; in addition they satisfy (7.112a,b), resp., (7.115a,b). In the case a ∈ N we restrict to Lorentz representations which are ﬁnite-dimensional; in fact, of dimension a. The case a = 1 is remarkable in one more respect, namely, in this case one may have a nontrivial RHS, cf. (7.112c). It is easy to check that there are no other cases with nontrivial RHS. In fact, for a ≠ 1 (7.112d) follows from (7.112a,b), or (7.115a,b). This can be shown also independently. Indeed, in the ﬁrst case the candidate signatures would be: 71 (a) = (a, 1 – a, 1), 71′ (a) = (a, –1 – a, 1). We know that a necessary condition to have an invariant equation is that the two representations would have the same Casimir operators, in particular, one should have C2 (71 (a)) – C2 (71′ (a)) = 0, where C2 is given in (7.10). Calculating this difference we obtain: C2 (71 (a)) – C2 (71′ (a)) = 2(a – 1) which is not zero unless a = 1.

(7.116)

7.6 Application to sl(4, C)

269

The cases a > 1 are interesting in other contexts, especially, if we consider together the representations with the conjugated signatures 71 (a) and 73 (a) with the same a ∈ N + 1. In particular, in the case a = 2, the two conjugated ﬁelds are two-component mass– = 7 (2) and 7– = 7 (2) (cf. Subsection less Weyl spinors occurring in the ERs 1 711 1 3 11 3 6.3.4.2), while (7.109) and (7.114) are the corresponding two conjugated Weyl equations (cf. the ﬁrst and third map to the right on (6.188) for p = - = n = 1). The cases a = 3 are maybe most interesting. The Lorentz dimension is 6 (= 2a) and the resulting ﬁeld is the Maxwell ﬁeld. This case belongs also to another hierarchy, the Maxwell hierarchy, cf. and was treated in detail in the quantum group setting in [132] (cf. Volume 2). We may write out many other equations with indices, however, one of the main points here is that in this form equations (7.112) and (7.115) are valid for different representation spaces, the different representations manifesting themselves only through the parameter a. Remark 6: It is interesting to note that there are other conditionally invariant equations involving the d’Alembert operator, from which (7.112c) is a partial case (m = 1), namely, ˆ => ˆ ′, ◻m >

m ∈ N,

(7.117)

′

ˆ ′ ∈ CD , D′ = D – m (!13 + !2 ), the corresponding signatures being ˆ ∈ CD , > where > – + 7 = 2 7m,m = (m, 0, m), 7′ = 2 7m,m = (m, –2m, m), using notation from the su(2, 2) doublets. These are produced by subsingular vectors of weights m (!13 + !2 ) (cf. (7.77b)), noting ˆ > ˆ ′ carry irreducible Lorentz representaalso that D′ = D – m(!13 + !2 ). The functions >, tions which are symmetric traceless tensors of rank m–1. (For early examples, namely, (7.117) with m = 2, obtained from other considerations, cf. [45, 245, 556].) ♢ Remark 7: We should note that there are conditionally invariant equations involving the d’Alembert operator, which were obtained from reduction of integral intertwining operators. These equations are also given by (7.117), but the corresponding signa′– ′+ tures are given by 7 = 7m,n,m = (m + n, –n, m + n), 7′ = 7m,n,m = (m + n, –n – 2m, m + n), m, n ∈ N, using the notation from the su(2, 2) sextets. Here also holds: D′ = D – m(!13 + !2 ). Another similar situation (ﬁrst noticed in [244]) is given in our notation – by (7.117) with m ↦ m + n, the corresponding signatures being: 7 = 7m,n,m = (m, n, m), ′ + 7 = 7m,n,m = (m, –n – 2m, m), m, n ∈ N, again using sextet notations. Here D′ = D – (m + n)(!13 + !2 ). Note that setting formally n = 0 both situations from this remark collapse to the situation from the previous remark. ♢ Remark 8: We should note that in (most of) the physical applications (7.117) is not considered conditionally invariant. The reason is that there only representations

270

7 KL Polynomials and Conditionally Invariant Equations

induced from ﬁnite-dimensional Lorentz representations are considered. The fact that these representations are also subspaces of reducible representations is ignored and thus the restriction to these subspaces is not considered to be a condition (cf. [45, 244, 245, 536, 556]). Remark 9: The results of this Chapter are easily extended to Drinfel’d–Jimbo quantum groups (cf. [135, 137]), to afﬁne Kac–Moody algebras, W-algebras and their corresponding quantum algebras. ♢

8 Invariant Differential Operators for Noncompact Lie Algebras Parabolically Related to Conformal Lie Algebras Summary In the present chapter, we continue the main topic of the book on various noncompact semisimple Lie algebras. Our starting points is the class of algebras, which we call “conformal Lie algebras” (CLAs), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduce the new notion of parabolic relation between two noncompact semisimple Lie algebras G and G ′ that have the same complexification and possess maximal parabolic subalgebras with the same complexification [147]. Thus, we consider the orthogonal algebras so(p, q) all of which are parabolically related to the conformal algebra so(n, 2) with p + q = n + 2, the parabolic subalgebras including the Lorentz subalgebra so(n – 1, 1) and its analogs so(p – 1, q – 1). We consider also other examples of CLAs as su(n, n), and its parabolically related sl(2n, R), and for n = 2k – su∗ (4k). Furthermore, we consider sp(n, R) and for n = 2r its parabolically related sp(r, r). Furthermore, we consider so∗ (4n) and its parabolically related so(2n, 2n) on the example of n = 3. Next we consider the exceptional algebra E7(7) which is parabolically related to the CLA E7(–25) . Other interesting examples are E6(6) and E6(2) which are parabolically related to the hermitian symmetric case E6(–14) . We also give a formula for the number of representations in the main multiplets valid for CLAs and all algebras that are parabolically related to them. In all considered cases, we give the multiplets of indecomposable elementary representations including the necessary data for all relevant invariant differential operators. In most cases, we give also the minimal irreps. We should stress that the multiplets are given in the most economic way in pairs of shadow fields. Furthermore we should stress that the classification of all invariant differential operators includes as special cases all possible conservation laws and conserved currents, unitary or not. This chapter is based on [143–154].

8.1 Generalities In the preceding chapters in general and by important interesting examples, we have set the stage for systematic study of invariant differential operators for various noncompact groups. Since the study and description of detailed classiﬁcation should be done group by group we have to decide which groups to study ﬁrst. A natural choice would be noncompact groups that have discrete series of representations. This class is still rather big, thus, one would ﬁrst consider a subclass, namely, the class of Hermitian symmetric spaces. The practical criterion is that in these cases, the maximal compact subalgebra K is of the form: K = so(2) ⊕ K′ .

(8.1)

so(n, 2), sp(n, R), su(m, n), so∗ (2n), E6(–14) , E7(–25) .

(8.2)

The Lie algebras from this class are

272

8 IDOs and Parabolic Relations

These groups/algebras have highest/lowest weight representations, and relatedly holomorphic discrete series representations. The most widely used of these algebras are the conformal algebras so(n, 2) in ndimensional Minkowski space-time. (We studied in detail the cases n = 3, 4 in Chapters 5 and 6.) In that case, there is a maximal Bruhat decomposition [69]: =M ⊕ A ⊕ N ⊕ N , so(n, 2) = P ⊕ N

(8.3)

=n M = so(n – 1, 1), dim A = 1, dim N = dim N that has direct physical meaning, namely, so(n – 1, 1) is the Lorentz algebra of n-dimensional Minkowski space-time, the subalgebra A = so(1, 1) represents the are the algebras of translations, and spedilatations, the conjugated subalgebras N , N cial conformal transformations, both being isomorphic to n-dimensional Minkowski ) is a maximal parabolic space-time. The subalgebra P = M ⊕ A ⊕ N (≅ M ⊕ A ⊕ N subalgebra. There are other special features which are important. In particular, the complexiﬁcation of the maximal compact subgroup is isomorphic to the complexiﬁcation of the ﬁrst two factors of the Bruhat decomposition: KC = so(n, C) ⊕ so(2, C) ≅ so(n – 1, 1)C ⊕ so(1, 1)C = MC ⊕ AC .

(8.4)

In particular, the coincidence of the complexiﬁcation of the semisimple subalgebras: K′C = MC

(8.5)

means that the sets of ﬁnite-dimensional (nonunitary) representations of M are in 1-to-1 correspondence with the ﬁnite-dimensional (unitary) representations of so(n). The latter leads to the fact that the corresponding induced representations are representations of ﬁnite K-type [279]. It turns out that some of the hermitian-symmetric algebras share the abovementioned special properties of so(n, 2). That is why, in view of applications to physics, these algebras should be called conformal Lie algebras (CLAs) (or groups). This subclass consists of so(n, 2), sp(n, R), su(n, n), so∗ (4n), E7(–25) ;

(8.6)

the corresponding analogs of Minkowski space-time V being: Rn–1,1 , Sym(n, R), Herm(n, C), Herm(n, Q), Herm(3, O).

(8.7)

Remark 1: The above class was identiﬁed from different considerations in [418] where they were called “simple space-time symmetries generalizing conformal symmetry.” The above groups are also called “Hermitian symmetric spaces of tube type” [198] or “conformal groups of simple Jordan algebras” [271]. Note that the relation between

8.1 Generalities

273

Jordan algebras and division algebras was known long time ago. For more references on Jordan algebras relevant in our approach cf., e.g., [84, 87, 89, 212–214, 274, 304, 527, 555, 593, 595]. ♢ We have started the study of the above class which will be presented in the following sections. In this study we are mainly interested in noncompact Lie algebras (and groups) that are “parabolically” related to the conformally Lie algebras. Definition 1: Let G, G ′ be two noncompact semisimple Lie algebras with the same complexiﬁcation G C ≅ G ′C . We call them parabolically related algebras if they have parabolic subalgebras P = M ⊕ A ⊕ N , P ′ = M′ ⊕ A′ ⊕ N ′ , such that they have the same complexiﬁcation: P C ≅ P ′C . ♢ Remark 2: Note that from MC ≅ M′C follows P C ≅ P ′C .

♢

Certainly, there are many such parabolic relationships for any given algebra G. Furthermore, two algebras G, G ′ may be parabolically related with different parabolic subalgebras. For example, the exceptional Lie algebras E6(6) and E6(2) are considered below (as related also to E6(–14) ) with maximal parabolics such that MC ≅ M′C ≅ sl(6, C). But these two algebras are related also by another pair of maximal parabolics ˜C≅M ˜ ′C ≅ sl(3, C) ⊕ sl(3, C) ⊕ sl(2, C) (cf. [141], (11.4) and (11.7)). P˜ C , P˜ ′C such that M We summarize the relevant cases in the following table: Table 8.1: Table of Conformal Lie Algebras G with M-Factor Fulfilling (8.5) and the Corresponding Parabolically Related Algebras G ′ G

K′

M dim V

G′

M′

so(n, 2) n≥3

so(n)

so(n – 1, 1)

so(p – 1, q – 1)

su(n, n) n≥3

su(n) ⊕ su(n)

n sl(n, C)R

so(p, q), p+q= =n+2 sl(2n, R)

sl(n, R) ⊕ sl(n, R)

sp(n, R) rank = n ≥ 3

su(n)

n2 sl(n, R)

su∗ (2n), n = 2k sp(r, r), n = 2r

su∗ (2k) ⊕ su∗ (2k) su∗ (2r), n = 2r

so∗ (4n) n≥3

su(2n)

n(n + 1)/2 su∗ (2n)

so(2n, 2n)

sl(2n, R)

E7(–25)

e6

n(2n – 1) E6(–26) 27

E7(7)

E6(6)

below not CLA ! E6(–14)

so(10)

su(5, 1) 21

E6(6) E6(2)

sl(6, R) su(3, 3)

274

8 IDOs and Parabolic Relations

where we have included also the algebra E6(–14) ; we display only the semisimple part K′ of K; sl(n, C)R denotes sl(n, C) as a real Lie algebra, (thus, (sl(n, C)R )C = sl(n, C) ⊕ sl(n, C)); e6 denotes the compact real form of E6 ; and we have imposed restrictions to avoid coincidences or degeneracies due to well-known isomorphisms: so(1, 2) ≅ sp(1, R) ≅ su(1, 1), so(2, 2) ≅ so(1, 2) ⊕ so(1, 2), su(2, 2) ≅ so(4, 2), sp(2, R) ≅ so(3, 2), so∗ (4) ≅ so(3) ⊕ so(2, 1), so∗ (8) ≅ so(6, 2).

8.2 The Pseudo-Orthogonal Algebras so(p,q) 8.2.1 Choice of Parabolic Subalgebra This section is based mostly on [133, 147, 148, 152]. Let G = so(p, q), p ≥ q, p + q > 4. [We shall explain the last restriction at the end of this section. We have considered in previous chapters the cases so(2, 1), so(3, 2), so(4, 2).] For ﬁxed p, q this algebra has at least q maximal parabolic subalgebras [141] (cf. (3.83) and (3.87)). We would like to consider a case, which would relate parabolically all G = so(p, q) for p + q-ﬁxed. Thus, in order in order to include the case q = 1 (where there is only one parabolic which is both minimal and maximal), we choose the case j = 1: M = Mmax = so(p – 1, q – 1). 1

(8.8)

= p + q – 2. dim N = dim N

(8.9)

Then we have

With this choice we get for the conformal algebra exactly the Bruhat decomposition in (8.3). We label the signature of the ERs of G as follows: 7 = {n1 , . . . , nh ; c }, nj ∈ Z/2,

c=d–

|n1 | < n2 < ⋅ ⋅ ⋅ < nh ,

(8.10) p+q–2 2 ,

h ≡ [ p+q–2 2 ],

p + q even,

0 < n 1 < n2 < ⋅ ⋅ ⋅ < n h ,

p + q odd,

where the last entry of 7 labels the characters of A, and the ﬁrst h entries are labels of the ﬁnite-dimensional nonunitary irreps of M ≅ so(p – 1, q – 1). The reason to use the parameter c instead of d will become clear below.

8.2 The Pseudo-Orthogonal Algebras so(p,q)

275

8.2.2 Main Multiplets Following results of [133, 142] we present the main multiplets (which contain the maximal number of ERs with this parabolic) with the explicit parametrization of the ERs in a simple way (helped by the use of the signature entry c): 71± = {: n1 , . . . , nh ; ±nh+1 },

(8.11)

nh < nh+1 , 72±

= {: n1 , . . . , nh–1 , nh+1 ; ±nh }

73± = {: n1 , . . . , nh–2 , nh , nh+1 ; ±nh–1 } ... ± 7h–1 = {: n1 , n2 , n4 , . . . , nh , nh+1 ; ±n3 }

7h± = {: n1 , n3 , . . . , nh , nh+1 ; ±n2 } ± 7h+1 = {: n2 , n3 , . . . , nh , nh+1 ; ±n1 } ±, p + q even := 1, p + q odd

(: = ± is correlated with 7± ). Clearly, the multiplets correspond 1-to-1 to the ﬁnitedimensional irreps of so(p + q, C) with signature {n1 , . . . , nh , nh+1 } and we are able to use previous results due to the parabolic relation between the so(p, q) algebras for p + q-ﬁxed. Note that the two representations in each pair 7± were called shadow ﬁelds in the 1970s, see more on this towards the end of this section. Further, the number of ERs in the corresponding multiplets is equal to 2[ p+q 2 ] = 2(h + 1). This maximal number is equal to the following ratio of numbers of elements of Weyl groups: C |W(G C , HC )| / |W(MC , Hm )|,

(8.12)

C are Cartan subalgebras of G C , MC , resp. where HC , Hm The above formula actually holds for all conformal Lie algebras and those parabolically related to them. More precisely, we have – The number of elements of the main multiplets of a conformal Lie algebra G with M-factor fulﬁlling (8.5) is given by (8.12). The same number holds for any algebra G ′ parabolically related to G w.r.t. M. ♢

Further, we denote by Ci± the representation space with signature 7i± . Below we give the multiplets pictorially for p + q odd and even, separately, and then explain notations and results:

276

8 IDOs and Parabolic Relations

C1−

ε1 – ε2

C1+

d1′

d1

C2+

C2–

ε1 – ε3

d2

ε1 – εh

dh – 1

Ch–

ε1 – εh + 1

ε1 + ε2

ε1 + ε3

dh′ – 1

ε1 + εh

Ch+

dh′

dh

Ch–+ 1

d2′

ε1 dh + 1

ε1 + εh + 1

Ch+ + 1

(8.13)

8.2 The Pseudo-Orthogonal Algebras so(p,q)

C1−

ε1 – ε2

277

C1+

d1′

d1

ε1 + ε2

C2+

C2–

ε1 – ε3

d2

ε1 – εh

dh – 1

Ch–

d2′

ε1 + ε3

dh′ – 1

ε1 + εh

Ch+ ε1 + εh + 1 dh′

ε1 – εh + 1

dh

dh

ε1 – εh + 1

dh′ ε1 + εh + 1 Ch–+ 1

Ch+ + 1

(8.14)

278

8 IDOs and Parabolic Relations

where in both ﬁgures arrows di , d′i are differential operators, dashed arrows are integral operators, in (8.13) p + q = 2h + 3 ≥ 5, odd, %1 ± %k , %1 are the noncompact roots, in (8.14) p + q = 2h + 2 ≥ 6, even, %1 ± %k are the noncompact roots. Note in (8.13) the degeneration of the Knapp–Stein (KS) operator from Ch– to Ch+ into invariant differential operator corresponding to the root %1 (which we have seen in the case so(3, 2)). Here there is only one element of the restricted Weyl group of G and thus, one KS operator. Its action in the context of (8.10), (8.11) is G : C7 → C7′ , 7 = {n1 , . . . , nh ; c },

(8.15) ′

p+q+1

7 = {(–1)

n1 , . . . , nh ; –c }.

These operators intertwine the pairs Ci± (cf. (8.11)): G±i : Ci∓ → Ci± ,

i = 1, . . . , 1 + h.

(8.16)

Remark 3: In the conformal setting (both Euclidean q = 1 and Minkowskian q = 2) the integral kernel of the KS operator is given by the conformal two-point function [159]. ♢ The invariant differential operators correspond to noncompact positive roots of the root system of so(p+q, C), cf. [126]. In the current context, compact roots of so(p+q, C) are those that are roots also of the subalgebra so(p + q – 2, C), the rest of the roots are noncompact. In more detail, we brieﬂy recall the root systems: For p + q = 2h + 3 odd, the positive root system of so(2h + 3, C) may be given by vectors %i ± %j , 1 ≤ i < j ≤ h + 1, %k , 1 ≤ k ≤ h + 1. The noncompact roots may be taken as %1 ± %i , %1 . The root %1 – %i corresponds to the operator di–1 , the root %1 + %i corresponds to the operator d′i–1 . The root %1 has a special position since it intertwines the same ERs that are intertwined by the KS integral operator G+h+1 . The latter means that G+h+1 degenerates to the differential operator dh+1 , and this degenerations determines that dh+1 ∼ ◻ n1 , (for n1 ∈ N), where ◻ is the d’Alembert operator, as we explained explicitly for the case so(3, 2), cf. (5.45). For p + q = 2h + 2 even, the positive root system of so(2h + 2, C) may be given by vectors %i ± %j , 1 ≤ i < j ≤ h + 1, where %i form an orthonormal basis in Rh+1 , i.e., (%i , %j ) = $ij . The noncompact roots may be taken as %1 ± %i , 2 ≤ i ≤ h + 1. The root %1 – %i corresponds to the operator di–1 , the root %1 + %i corresponds to the operator d′i–1 . We summarize by listing the spaces from (8.11) the above differential operators intertwine: – di : Ci– → Ci+1 ,

i = 1, . . . , h;

d′i

i = 1, . . . , h – 1;

+ : Ci+1 → Ci+ , + dh : Ch+1 → Ch+ , + d′h : Ch– → Ch+1 ,

(p + q) – even; (p + q) – even;

8.2 The Pseudo-Orthogonal Algebras so(p,q)

– → Ch+ , d′h : Ch+1

(p + q) – even;

+ → Ch+ , d′h : Ch+1

(p + q) – odd;

– + → Ch+1 , dh+1 : Ch+1

(p + q) – odd.

279

(8.17)

The degrees of these invariant differential operators are given just by the differences of the c entries [133] and they happen to be Harish-Chandra parameters: deg di = deg d′i = nh+2–i – nh+1–i = mh+2–i , deg d′h

= n2 + n1 = m1 ,

i = 1, . . . , h,

(p + q) – even,

deg dh+1 = 2n1 = m′h+1 = m1 ,

(p + q) – odd.

(8.18)

where d′h is omitted from the ﬁrst line for (p + q) even. As we have seen in the examples, matters are arranged so that in every multiplet only the ER with signature 71– contains a ﬁnite-dimensional nonunitary subrepresentation of G in a subspace E. The latter corresponds to the ﬁnite-dimensional unitary irrep of so(p + q) with signature {n1 , . . . , nh , nh+1 }. The subspace E is annihilated by the operator G+1 , and is the image of the operator G–1 . Although the diagrams are valid for arbitrary so(p, q) (p + q ≥ 5) the contents is very different. We comment only on the ER with signature 71+ . In all cases it contains an UIR of so(p, q) realized on an invariant subspace D of the ER 71+ . That subspace is annihilated by the operator G–1 , and is the image of the operator G+1 . (Other ERs contain more UIRs.) If pq ∈ 2N the mentioned UIR is a discrete series representation. Other ERs contain more discrete series UIRs. The number of discrete series is given by the formula [349]: |W(G C , HC )| / |W(KC , HC )|,

(8.19)

where HC is a Cartan subalgebra of both G C and KC . And if q = 2 the invariant subspace D is the direct sum of two subspaces D = D + ⊕ – D , in which are realized a holomorphic discrete series representation and its conjugate antiholomorphic discrete series representation, resp. These are contained only in 71+ . Furthermore, any holomorphic discrete series representation is inﬁnitesimally equivalent to a lowest weight GVM of the conformal algebra so(p, 2), while an antiholomorphic discrete series representation is inﬁnitesimally equivalent to a highest weight GVM. Highest/lowest weight GVMs are related to other pairs besides 71+ . A detailed analysis of these occurrences was given for so(3, 2) in Chapter 5 and for so(4, 2) in Chapter 6 (both ﬁrst in [142]).

280

8 IDOs and Parabolic Relations

8.2.3 Reduced Multiplets and Their Representations Besides the main multiplets which are 1-to-1 with the ﬁnite-dimensional irreps of so(p + q, C), there are other multiplets which we describe below. 8.2.3.1 Reduced Multiplets for p + q Odd We start with the case p + q odd, thus h = 21 (p + q – 3). First we rewrite the main multiplets from (8.11) in the following parametrization: 71± = [m1 , . . . , mh ; ± 21 (m1 + 2m2,h+1 )], 72± 73±

= [m1 , . . . , mh–1 , mh,h+1 ;

± 21 (m1

= [m1 , . . . , mh–2 , mh–1,h , mh+1 ;

(8.20)

+ 2m2,h )],

± 21 (m1

+ 2m2,h–1 )],

... 7i± = [m1 , . . . , mh–i+1 , mh–i+2,h–i+3 , mh+4–i , . . . , mh , mh+1 ; ± 21 (m1 + 2m2,h+2–i )], ... ± 7h–1 = [m1 , m2 , m34 , m5 , . . . , mh , mh+1 ; ± 21 (m1 + 2m2,3 )],

7h± = [m1 , m23 , m4 , . . . , mh , mh+1 ; ± 21 (m1 + 2m2 )], ± 7h+1 = [m1 + 2m2 , m3 , . . . , mh , mh+1 ; ± 21 m1 ],

where the last entry (as before) is the value of c, while mi ∈ N are the Dynkin labels (as in (8.18)): m1 = 2n1 ,

mj = nj – nj–1 j = 2, . . . , h + 1

(8.21)

and we use the shorthand notation: mr,s ≡ mr + ⋅ ⋅ ⋅ + ms , r < s,

mr,r ≡ mr ,

mr,s ≡ 0, r > s.

(8.22)

In this case there are h + 1 reduced multiplets which are doublets and which may be obtained by formally setting one Dynkin label in (8.20) to zero. The signatures of the doublets are compactly given as follows: ± r 71

= [m1 , . . . , mh ; ± 21 (m1 + 2m2,h )], mh+1 = 0 ≅ nh+1 = nh d+ ≥ 2h, d– ≤ 1,

± r 72

= [m1 , . . . , mh ; ± 21 (m1 + 2m2,h–1 )], mh = 0 ≅ nh = nh–1 d+ ≥ 2h – 1, d– ≤ 2,

...

(8.23)

8.2 The Pseudo-Orthogonal Algebras so(p,q)

± r 7j

281

= [m1 , . . . , mh ; ± 21 (m1 + 2m2,h+1–j )], mh–j+2 = 0 ≅ nh–j+2 = nh–j+1 d+ ≥ 2h – j + 1, d– ≤ j, 1 ≤ j ≤ h – 1

... ± r 7h–1

= [m1 , . . . , mh ; ± 21 (m1 + 2m2 )], m3 = 0 ≅ n3 = n2 d+ ≥ h + 2, d– ≤ h – 1,

± r 7h

= [m1 , . . . , mh ; ± 21 m1 ], m 2 = 0 ≅ n2 = n 1 d+ ≥ h + 1, d– ≤ h,

r 7h+1

= [2m1 , m2 , . . . , mh ; 0], m 1 = 0 ≅ n1 = 0 d=h+

1 2

where we have indicated which Dynkin label is set to zero, furthermore we introduced notation d± corresponding to the “±” occurrences: d± = h +

1 2

± |c|.

(8.24)

Naturally, although (8.23) is written compactly as (8.11) no pair is related to any other pair. Inside a ﬁxed pair r 7i± , i = 1, . . . , h, acts the KS integral operator G–i (8.16) (coinciding with G–i+1 for this signature), and a differential operator dˆi of degree 2nh+1–i which is a degeneration of the KS integral operator G+i (coinciding with G+i+1 for this signature). When nh+1–i ∈ N we have for the differential operators dˆi ∼ ◻ nh+1–i . (Actually, for i = h, the degeneration of G+h+1 was present already in the main multiplet.) 8.2.3.2 Special Reduced Multiplets for p + q Odd In addition to the standardly reduced multiplets discussed in the previous subsection, there are special reduced multiplets which may be formally obtained by replacing one or two integer Dynkin labels with a positive half-integer. Again the reduced multiplets are doublets which we present below: – mh+1 ↦ 21 ,, , ∈ 2N – 1 ± s 71

–

= [m1 , . . . , mh ; ± 21 (m1 + 2m2,h + ,)]

(8.25)

mh ↦ 21 ,, mh+1 ↦ 21 ,′ , ,, ,′ ∈ 2N – 1 ± s 72

= [m1 , . . . , mh–1 , 21 (, + ,′ ); ± 21 (m1 + 2m2,h–1 + ,)]

(8.26)

282

–

8 IDOs and Parabolic Relations

mh–1 ↦ 21 ,, mh ↦ 21 ,′ , ,, ,′ ∈ 2N – 1 ± s 73

= [m1 , . . . , mh–2 , 21 (, + ,′ ), mh+1 ; ± 21 (m1 + 2m2,h–2 + ,)]

(8.27)

⋯ –

mh–j+2 ↦ 21 ,, mh–j+3 ↦ 21 ,′ , ,, ,′ ∈ 2N – 1, ± s 7j

2≤j≤h

= [m1 , . . . , mh–j+1 , 21 (, + ,′ ), mh+4–j , . . . , mh , mh+1 ; ± 21 (m1 + 2m2,h+1–j + ,)]

(8.28)

⋯ –

m3 ↦ 21 ,, m4 ↦ 21 ,′ , ,, ,′ ∈ 2N – 1 ± s 7h–1

–

= [m1 , m2 , 21 (, + ,′ ), m5 , . . . , mh , mh+1 ; ± 21 (m1 + 2m2 + ,)]

m2 ↦ 21 ,, m3 ↦ 21 ,′ , ,, ,′ ∈ 2N – 1 ± s 7h

–

(8.29)

= [m1 , 21 (, + ,′ ), m4 , . . . , mh , mh+1 ; ± 21 (m1 + ,)]

(8.30)

m2 ↦ 21 ,, , ∈ 2N – 1 ± s 7h+1

= [m1 + ,, m3 , . . . , mh , mh+1 ; ± 21 m1 ]

(8.31)

In each pair there are the standard KS integral operators G±j between s 7j∓ ; however, the KS operator G+j (8.16) from the reducible s 7k– to s 7k+ degenerates to the differential operator D2|cj |, %1 , where cj is the value of c of the ER s 7j– . Finally, we give a doubly reduced case originating from (8.31) setting m1 = 0: ± rs 7h+1

= [,, m2 , . . . , mh ; 0],

mk ∈ N,

, ∈ 2N – 1.

(8.32)

This is a singlet and the ER is reducible only w.r.t. the M-compact roots. 8.2.3.3 Special Representation Cases for p + q Odd As we mentioned, the holomorphic discrete series representation and its conjugate antiholomorphic discrete series representation are contained in the ER 71+ when q = 2. For uniformity we give the signatures of the holomorphic discrete series (cf. (8.11) and (8.20)) in the current parametrization: 71+ = [m1 , . . . , mh ; d = h + 21 (m1 + 1) + m2,h + -],

-∈N

(8.33)

8.2 The Pseudo-Orthogonal Algebras so(p,q)

283

The next important case are the limits of (holomorphic) discrete series which are contained in the reduced case (8.23): + r 71 = [m1 ,

. . . , mh ; d = h + 21 (m1 + 1) + m2,h ]

(8.34)

(with conformal weight obtained from (8.33) as “limit” for - = 0). Finally, we mention the so called ﬁrst reduction points (FRP). For q = 2 these are the boundary values of d from below of the positive energy UIRs. Most of the FRPs are + , cf. (8.20), which we give with suitable reparametrization: contained in 7h+1 + = [m1 , m2 , . . . , mh ; d = h + 21 m1 – 21 ], 7h+1

m1 ≥ 3.

(8.35)

The FRP cases for m1 = 1, 2 (with the same values of d by specializing m1 ) are found in (8.23): – r 7h r 7h+1

= [1, m2 , . . . , mh ; d = h],

(8.36)

= [2, m2 , . . . , mh ; d = h +

1 2 ].

(8.37)

Finally, we give some discrete unitary points below the FRP which are found in the special reduced ERs (8.30): – s 7h – s 7h+1

= [1, m2 , . . . , mh ; d = h – 21 ],

(8.38)

= [2, m2 , . . . , mh ; d = h].

(8.39)

8.2.3.4 Minimal Irreps for p + q Odd We deﬁne the minimal irreps as involving the lowest dimensional representation of M and – when applicable – the positive energy UIRs with lowest conformal weight. The minimal irreps in this case happen to be subrepresentations of the ERs in the reduced multiplets. Besides the signature we display the equations that are obeyed by the functions of the irrep. First, we give the signatures of the minimal irreps occurring in standardly reduced multiplets: – r 71

= [1, . . . , 1 ; d = 1], – r L1 = {>

– r 72

: D1,%1 –%3 > = 0 ,

(8.40) G+1 > = 0},

= [(1, . . . , 1); d = 2], – r L2 = {>

...

∈

– r C1

∈ r C2– : D1,%1 –%4 > = 0,

G+2 > = 0},

284

8 IDOs and Parabolic Relations

– r 7j

= [1, . . . , 1 ; d = j], – r Lj = {>

∈

– r Cj

1 ≤ j ≤ h – 1, G+j > = 0},

: D1,%1 –%j+2 > = 0, 1 ≤ j ≤ h – 1,

... – r 7h

= [1, . . . , 1 ; dFRP = h], – r Lh = {>

r 7h+1

∈ r Ch– : G+h > = 0},

G+h = D1,%1 ,

= [2, 1, . . . , 1 ; dFRP = h + 21 ] r Lh+1 = {>

– ∈ r Ch+1 : D1,%1 +%h+1 > = 0}

(In the last case there is no KS operator since c = 0.) Next we give the case of special reduced multiplets: – s 71

= [1, . . . , 1 ; d = 21 ], – s L1 = {>

∈

– s C1

(8.41) G+1

: D2h,%1 > = 0 },

h

∼ D2h,%1 ∼ ◻ ,

... – s 7j

= [1, . . . , 1 ; d = j – 21 ],

1≤j≤h

– – s Lj = {> ∈ s Cj : D2(h+1–j),%1 G+j ∼ D2(h+1–j),%1 ∼ ◻h+1–j ,

> = 0 },

... – s 7h

= [1, . . . , 1 ; d = h – 21 ], – s Lh = {>

s 7h+1

∈ s Ch– : D2,%1 > = 0},

G+h ∼ D2,%1 ∼ ◻,

= [2, 1, . . . , 1 ; d = h] – s Lh+1 = {>

– ∈ s Ch+1 : D1,%1 > = 0},

G+h+1 ∼ D1,%1 .

Here all irreps are below the FRP. The “most” minimal representations are the last two cases of (8.41). For h = 1, i.e., so(3, 2) these are the so-called singletons discovered by Dirac [118].

8.2.3.5 Singular Vectors Needed for the Invariant Differential Operators The mostly used case is %1 = !1 + ⋅ ⋅ ⋅ + ! , = h + 1. The corresponding sl(n) singular vector of weight m%1 is given in (4.53). Other cases are: %1 – %j = !1 + ⋅ ⋅ ⋅ + !j–1 . Clearly, one uses again formula (4.53) replacing ↦ j – 1. The last case is %1 + % = !1 + ⋅ ⋅ ⋅ + !–1 + 2! , = h + 1. This is a singular vector for B given in (4.56).

8.2 The Pseudo-Orthogonal Algebras so(p,q)

285

8.2.3.6 Reduced Multiplets for p + q Even Further we consider the case p + q even thus h = 21 (p + q – 2). First we introduce the Dynkin labels parametrization of the multiplets: 71± = [(m1 , . . . , mh )± ; ± 21 (m12 + 2m3,h+1 )], 72± 73±

= =

(8.42)

±

[(m1 , . . . , mh–1 , mh,h+1 ) ; ± 21 (m12 + 2m3,h )] [(m1 , . . . , mh–2 , mh–1,h , mh+1 )± ; ± 21 (m12 + 2m3,h–1 )]

... 7j± = [(m1 , . . . , mh–j+1 , mh–j+2,h–j+3 , mh+4–j , . . . , mh , mh+1 )± ; ± 21 (m12 + 2m3,h+2–j )],

2 ≤ j ≤ h – 1,

... ± 7h–1 = [(m1 , m2 , m34 , m5 , . . . , mh , mh+1 )± ; ± 21 (m12 + 2m3 )]

7h± = [(m1′ 3 , m23 , m4 , . . . , mh , mh+1 )± ; ± 21 m12 ] ± 7h+1 = [(m13 , m3 , . . . , mh , mh+1 )± ; ± 21 (m1 – m2 )]

where the conjugation of the M labels interchanges the ﬁrst two entries: (m1 , . . . , mh )– = (m1 , . . . , mh ),

(8.43)

+

(m1 , m2 , m3 , . . . , mh ) = (m2 , m1 , m3 , . . . , mh ), the last entry (as before) is the value of c, while mi ∈ N are the Dynkin labels (as in (8.18)): m 1 = n1 + n 2 ,

(8.44)

mj = nj – nj–1 , j = 2, . . . , h + 1, ﬁnally, m1′ 3 ≡ m1 + m3 . Then we give the reduced multiplets using the notation r 7k± : ± r 71

= [(m1 , . . . , mh )± ; ± 21 (m12 + 2m3,h )], mh+1 = 0 ≅ nh+1 = nh , d+ ≥ 2h – 1, d– ≤ 1,

± r 72

= [(m1 , . . . , mh–1 , mh+1 )± ; ± 21 (m12 + 2m3,h–1 )], mh = 0 ≅ nh = nh–1 , d+ ≥ 2h – 2, d– ≤ 2,

...

(8.45)

286

8 IDOs and Parabolic Relations

± r 7j

= [(m1 , . . . , mh–j+1 , mh–j+3 , . . . , mh + 1)± ; ± 21 (m12 + 2m3,h+1–j )], mh–j+2 = 0 ≅ nh–j+2 = nh–j+1 , d+ ≥ 2h – j, d– ≤ j,

1 ≤ j ≤ h – 2,

... ± r 7h–1

= [(m1 , m2 , m4 , . . . , mh , mh+1 )± ; ± 21 m12 ], m3 = 0 ≅ n3 = n2 d+ ≥ h + 1, d– ≤ h – 1,

± r 7h

= [(m + m3 , m3 , . . . , mh , mh+1 )± ; ± 21 m], m2 = 0 ≅ n2 = n1 , m1 ↦ m, d+ ≥ h + 21 , d– ≤ h – 21 ,

± r 7h+1

= [(m3 , m + m3 , m4 , . . . , mh , mh+1 )± ; ± 21 m], m1 = 0 ≅ n2 = –n1 , m2 ↦ m, d+ ≥ h + 21 , d– ≤ h – 21 .

Clearly, the signatures r 7j± may be obtained from 7j± by setting the corresponding Dynkin label equal to zero as indicated. Note that the last two cases are conjugate to each other through the M labels, while having the same expressions for c. Although written compactly as (8.11) no pair is related to any other pair. Furthermore inside a ﬁxed pair r 7i± , i = 1, . . . , h + 1, act two operators: a KS integral operator from r 7i+ to r 7i– , and a differential operator from r 7i– to r 7i+ . In more detail: – Let ﬁrst i = 1, . . . , h – 1. Inside a ﬁxed pair r 7i± , acts the KS integral operator G–i (8.16) (coinciding with G–i+1 for this signature), and a differential operator d˜i of degree 2nh+1–i which is a degeneration of the KS integral operator G+i (coinciding with G+i+1 for this signature). When n1 = 0 for this differential operator we have d˜i ∼ ◻ nh+1–i , (nh+1–i ∈ N). –

Inside the ﬁxed pair r 7h± acts the KS integral operator G–h (8.16) (coinciding with G–h+1 for this signature), and the differential operator d′h of degree 2n1 (cf. the previous subsection) which in addition is a degeneration of the KS integral operator G+h (coinciding with G+h+1 for this signature).

–

± acts the KS integral operator G– (8.16) (coinciding with Inside the ﬁxed pair r 7h+1 h+1 G+h for this signature), and the differential operator dh of degree 2n2 which in addition is a degeneration of the KS integral operator G+h+1 (coinciding with G–h for this signature).

Note a last reduction obtained by setting m = 0 when the last two pairs in (8.45) coincide and become further a singlet (being M self-conjugate): r7

s

= [m2 , m2 , m3 , . . . , mh ; 0],

d = h.

(8.46)

8.2 The Pseudo-Orthogonal Algebras so(p,q)

287

8.2.3.7 Special Representation Cases for p + q Even As we mentioned the holomorphic discrete series representation and its conjugate antiholomorphic discrete series representation, are contained in the ER 71+ when q = 2. For uniformity we give the signatures of the holomorphic discrete series in the current parametrization: 71+ = [m1 , . . . , mh ; d = h + 21 m12 + m3,h + -],

-∈N

(8.47)

For arbitrary p + q if pq ∈ 2N the representation r 71+ , contains an UIR called limits of the discrete series representations. And if q = 2 that UIR is the direct sum of two subspaces in which are realized limits of holomorphic discrete series representation and its conjugate limits of antiholomorphic discrete series representation, resp. They are contained in the reduced case (8.45): + r 71 = [m1 ,

. . . , mh ; d = h + 21 m12 + m3,h ]

(8.48)

(with conformal weight obtained from (8.47) as “limit” for - = 0). The latter do not happen in any other doublet. (For so(4, 2) there are more details in Chapter 6 and [142, 518].) Further we discuss the ﬁrst reduction points (FRP). For q = 2 these are the boundary values of d from below of the positive energy UIRs. Most of the FRPs are contained in 7h+ , cf. (8.42), which we give with suitable reparametrization: 7h+ = [m1 , m2 , . . . , mh ; d = h + 21 m12 – 1],

m1 , m2 ≥ 2.

(8.49)

± : Some FRP cases when only one of m1 , m2 is equal to 1 are found in 7h+1 – = [m1 , 1, m3 , . . . , mh ; d = h + 21 (m1 – 3)], 7h+1 + 7h+1

= [1, m2 , . . . , mh ; d = h +

1 2 (m2

– 3)],

m1 ≥ 3,

m2 ≥ 3.

(8.50)

Finally the last three FRP cases (m1 , m2 ) = (1, 1), (2, 1), and (1, 2) are found in – r 7k=h–1,h,h+1 : – r 7h–1

= [1, 1, m3 , . . . , mh ; d = h – 1],

– r 7h – r 7h+1

= [2, 1, m3 , . . . , mh ; d = h – 21 ], = [1, 2, m3 , . . . , mh ; d = h – 21 ].

(8.51)

8.2.3.8 Minimal Irreps for p + q Even The list of ERs containing irreducible minimal subrepresentations is – r 71

= [(1, . . . , 1); d = 1], – r L1 = {>

∈

– r C1

: D1,%1 –%3 > = 0 ,

(8.52) G+1 > = 0},

288

8 IDOs and Parabolic Relations

– r 72

= [(1, . . . , 1); d = 2], – r L2 = {>

∈ r C2– : D1,%1 –%4 > = 0 ,

G+2 > = 0},

... – r 7j

= [(1, . . . , 1); d = j], – r Lj = {>

1 ≤ j ≤ h – 2,

∈ r Cj– : D1,%1 –%j+2 > = 0 ,

G+j > = 0},

... – r 7h–1

= [(1, . . . , 1); dFRP = h – 1], – r Lh–1 = {>

– ∈ r Ch–1 : D1,%1 –%h+1 > = 0,

D1,%1 +%h+1 > = 0,

G+h–1 > = 0}, – r 7h

= [(2, 1, . . . , 1); d–FRP = h – 21 ], – r Lh = {>

– r 7h+1

∈ r Ch– : G+h > = 0},

G+h ∼ D1,%1 –%h+1 ,

= [(1, 2, 1, . . . , 1); dFRP = h – 21 ], – r Lh+1 = {>

– ∈ r Ch+1 : G+h+1 > = 0},

G+h+1 ∼ D1,%1 +%h+1

We see in (8.52) that for h ≥ 3 there are discrete unitary points below the FRPs. Thus, the most famous case so(4, 2) (h = 2) is excluded. 8.2.3.9 Singular Vectors Needed for p + q Even The necessary cases are %1 – %j = !h+3–j + ⋅ ⋅ ⋅ + !h+1 , 2 ≤ j ≤ h + 1, %1 + %h+1 = !1 + !3 + ⋅ ⋅ ⋅ + !h+1 .

(8.53)

These are roots of sl(n) subalgebras (n < h + 1). Thus, we can use formula (4.53) after suitable change of enumeration.

8.2.4 Conservation Laws for so(p,q) We note already the importance of invariant differential equations with trivial RHS, cf. (4.38). Furthermore, in many physical applications in the case of ﬁrst order such differential operators, i.e., for m = m" = 1, equations (4.38) are called conservation laws, and the elements f ∈ ker Dm," are called conserved currents. Below we give them explicitly for so(p, q). For this we use the explicit parametrization of the multiplets (8.11) and the fact that the degrees of the operators there are given by (8.18).

8.2 The Pseudo-Orthogonal Algebras so(p,q)

289

Thus, we are able to list all cases of ﬁrst-order invariant differential operators or conservation laws (we give only the signatures): d1 : {: n1 , n2 , . . . , nh ; –nh – 1} → {: n1 , n2 , . . . , nh–1 , nh + 1; –nh }; d′1 : {n1 , . . . , nh–1 , nh + 1; nh } → {n1 , . . . , nh ; nh + 1}; ...

(8.54)

di : {: n1 , n2 , . . . , nh+1–i , nh+3–i , . . . , nh+1 ; –nh+1–i – 1} → {: n1 , n2 , . . . , nh–i , nh+1–i + 1, nh+3–i , . . . , nh+1 ; –nh+1–i }, i = 2, . . . , h – 1; d′i : {n1 , . . . , nh–i , nh+1–i + 1, nh+3–i , . . . , nh+1 ; nh+1–i } → {n1 , . . . , nh+1–i , nh+3–i , . . . , nh+1 ; nh+1–i + 1}, i = 2, . . . , h – 1; ... dh : {: (n2 – 1), n3 , . . . , nh+1 ; –n2 } → {:n2 , n3 , . . . , nh+1 ; 1 – n2 },

n2 > 21 ;

dh : {n2 , n3 , . . . , nh+1 ; n2 – 1} → {n2 – 1, n3 , . . . , nh+1 ; n2 }, n2 > 21 , d′h

(p + q) – even;

: {n2 – 1, n3 , . . . , nh+1 ; –n2 } → {n2 , n3 , . . . , nh+1 ; 1 – n2 }, n2 > 21 ,

d′h

(p + q) – even;

: {–n2 , n3 , . . . , nh+1 ; n2 – 1} → {1 – n2 , n3 , . . . , nh+1 ; n2 }, n2 > 21 ,

d′h

(p + q) – even;

: {n2 , n3 , . . . , nh+1 ; n2 – 1} → {n2 – 1, n3 , . . . , nh+1 ; n2 }, n2 > 1,

d′h+1

(p + q) – odd;

: {–n2 , n3 , . . . , nh+1 ; – 21 } → {n2 , n3 , . . . , nh+1 ; 21 }, n2 > 1,

(p + q) – odd; p+q

: = –(–1)

.

Some of these conservation laws were given also in [442]. Besides the above cases there are the ERs from the reduced multiplets which also give rise to ﬁrst-order invariant differential operators, resp. conservation laws: d: : 7˜ –: → 7˜ +: ,

7˜ ±:

= {±:

1 2 , n3 , . . . ,

(8.55) nh+1 ; ± 21 },

where : is deﬁned as in (8.11), but here it is not correlated with 7˜ ± . There is also a KS integral operator acting from 7˜ +: to 7˜ –: . The representations 7˜ –: are among the minimal representations.

290

8 IDOs and Parabolic Relations

8.2.5 Remarks on Shadow Fields and History –

We labeled the signature of the ERs in (8.10) as 7 = {n1 , . . . , nh ; c } using the parameter c instead of the conformal weight d = c + p+q–2 2 . This was used already in [159] since the multiplets were given more economically in terms of pairs of ERs in which the parameter c just changes sign. [Also mathematicians use the parameter c due to the fact that in its terms the representation parameter space looks simple: the principal unitary series representation induced from a maximal parabolic is given by c = i1, 1 ∈ R; the supplementary series of unitary representations is given by –s < c < s, s ∈ R, etc.] Otherwise in the current context may use for each KS operators conjugated doublet of shadow ﬁelds: 7+ = [ n1 , . . . , nh ; d ], –

7 = [ (–1)

p+q+1

nj ∈ Z/2,

(8.56)

n1 , . . . , nh ; dshadow = p + q – 2 – d ].

The reason the representations 7± in the 1970s were called “shadow ﬁelds” in the context of the conformal algebra so(n, 2) is that the sum of their conformal weights (cf. (8.9)). equals the dimension n of Minkowski space-time - isomorphic to N or N This continues to be true for general so(p, q): d + dshadow = p + q – 2 = n

(8.57)

and also for all conformal Lie algebras considered in the next sections. Shadow ﬁelds appear all the time in conformal ﬁeld theory. For example, in [138] we showed that in the generic case each ﬁeld on the AdS bulk has two boundary ﬁelds which are shadow ﬁelds being related by a integral KS operator. Later Klebanov–Witten [341] showed that these two boundary ﬁelds are related by a Legendre transform. For a current discussion on shadow ﬁelds we refer to [442]. –

The embedding diagram for p + q even appeared ﬁrst for the Euclidean conformal group in four-dimensional space-time SU ∗ (4) ≅ Spin(5, 1) in [164]. Later it was generalized to the Minkowskian conformal group in four-dimensional space-time SO(4, 2) in [518]. In both cases, the three (= (p + q)/2) doublets (from the previous subsection) were also given together the corresponding degeneration of the KS integral operators. The exposition for general so(p, q) above including Figures (8.13) and (8.14) follows the exposition for Euclidean case so(n + 1, 1) in [133]. Later the results were generalized to the Minkowskian case so(n, 2) [142].

8.2 The Pseudo-Orthogonal Algebras so(p,q)

291

–

Actually, the case of Euclidean conformal group in arbitrary dimensions SO(p, 1) was studied in [159] for representations of M = so(p – 1) which are symmetric traceless tensors. This means that in (8.10) we should set n1 = n2 = ⋅ ⋅ ⋅ = nh–1 = 0, and then only the ﬁrst two pairs 71± , 72± in (8.11) are possible. Thus from the two ﬁgures only the upper quadrants are relevant, and were given in [159], cf. Fig. 1 there.

–

Above we restricted to p+q ≥ 5. The excluded cases are: so(3, 1), so(2, 2) ≅ so(2, 1)⊕ so(2, 1), so(2, 1), and (so(1, 1) is abelian). In the case so(3, 1) the multiplet in general contains only four ERs, and is in fact representable by the diagram in the case of symmetric traceless tensors of so(p, 1), p > 3 (cf. [159], Appendix B, Figure 3). The case so(2, 1) ≅ sl(2, R) is special and must be treated separately, as we did in Subsection 4.7. In that case the multiplets contain only two ERs which may be depicted by the top pair 71± in both ﬁgures above. (Formally, set h = 0 in both ﬁgures.)

8.2.6 Case so(3, 3) ≅ sl(4, R) We consider brieﬂy the case so(3, 3) separately, since it is parabolically related to some cases of the 4D conformal case so(4, 2) considered in detail in Chapter 6. From there we know that so(4, 2) has three nonconjugate parabolics: P0 = so(2) ⊕ A0 (2) ⊕ N0 (6),

(8.58)

P1 = sl(2, R) ⊕ so(2) ⊕ A1 (1) ⊕ N1 (5), P2 = so(3, 1) ⊕ A2 (1) ⊕ N2 (4), where in parenthesis we have recalled the real dimensions of Ak and Nk . The corresponding parabolics of so(3, 3) ≅ sl(4, R) are P0′ = A′0 (3) ⊕ N0′ (6), P1′ = sl(2, R) P2′ = sl(2, R)

⊕ N1′ (5), ⊕ sl(2, R) ⊕ A′2 (1)

(8.59)

⊕ A′1 (2)

⊕ N2′ (4).

It is easy to see that PkC ≅ Pk′C (recalling that so(1, 1)C ≅ so(2)C , so(3, 1)C ≅ so(4)C ≅ sl(2, C) ⊕ sl(2, C)). On the other hand we know that the algebra so(3, 3) has 2r – 1 = 7 nonconjugate nontrivial parabolics. Indeed, it has two more parabolics with the properties of P1′ (cf. Section 3.5). Furthermore, it has two nonconjugate maximal parabolics with properties: P3 = sl(3, R) ⊕ A3 (1) ⊕ N3 (3). The latter are not parabolically related to so(4, 2).

(8.60)

292

8 IDOs and Parabolic Relations

We choose one of the parabolics from (8.60) so that the Dynkin diagram of sl(3) uses Dynkin nodes 1 and 2 of the sl(4) Dynkin diagram. Thus, the sl(4) roots !1 , !2 , !12 form the positive root system of sl(3). (We use the coincidence of the root systems of sl(n, C) and sl(n, R), since the latter is maximally split.) The ERs are induced from ﬁnite-dimensional irreps of sl(3, R). Thus, using the sl(4) signatures (m1 , m2 , m3 ) the signatures of the ERs induced from the parabolic (8.60) (with the above choice of nodes) are 73 = (m1 , m2 , m3 ),

m1 , m2 ∈ N, m3 ∈ C.

(8.61)

These ERs are reducible when at least one of the M-noncompact Harish-Chandra parameters m3 , m23 , m13 is a natural number. It turns out that these reducible ERs are grouped into quadruplets. Thus, the main multiplet of reducible ERs (8.61) contains four members which we parametrize as part of the main sl(4) multiplet with 24 members (6.161), such that the ﬁrst two entries m′1 , m′2 ∈ N. Thus, these quadruplets are given by 70 = (m1 , m2 , m3 ; m12 , m23 , m13 ),

m1 , m2 , m3 ∈ N,

(8.62)

73 = (m1 , m23 , –m3 ; m13 , m2 , m12 ), 723 = (m12 , m3 , –m23 ; m13 , –m2 , m1 ), 7123 = (m2 , m3 , –m13 ; m23 , –m12 , –m1 ). The quadruplet is related by the following chain of invariant differential operators: χ0

m3α3

χ3

m2α23

χ23

m1α13

χ123

Further, there are three reduced multiplets - 73 singlets, each parametrized by two natural numbers m1 , m2 ∈ N: 70 = (m1 , m2 , 0; m12 , m2 , m12 ),

(8.63)

73 = (m1 , m2 , –m2 ; m1,2 , 0, m1 ),

(8.64)

723 = (m1 , m2 , –m12 ; m12 , –m1 , 0),

(8.65)

depending which of the three noncompact Harish-Chandra parameter is zero.

8.3 The Lie Algebra su(n,n) and Parabolically Related This section is based mostly on [145, 147, 150, 151, 153]. Let G = su(n, n), n ≥ 2. The maximal compact subgroup is K ≅ u(1) ⊕ su(n) ⊕ su(n). We choose a maximal parabolic P = M ⊕ A ⊕ N such that A ≅ so(1, 1), M = sl(n, C)R .

8.3 The Lie Algebra su(n,n) and Parabolically Related

293

We label the signature of the ERs of G as follows: 7 = {n1 , . . . , nn–1 , nn+1 , . . . , n2n–1 ; c },

nj ∈ Z+ ,

c = d – 21 n2

(8.66)

where the last entry of 7 labels the characters of A, and the ﬁrst 2n–2 entries are labels of the ﬁnite-dimensional nonunitary irreps of M. The number of ERs in the main multiplets is equal to [147] C

C

|W(G , H )| / |W(M

C

C , Hm )|

=

2n . n

The restricted Weyl reﬂection is given by the KS integral operators [352, 353]: GKS : C7 → C7′ , ′

(8.67) ∗

7 = {(n1 , . . . , nn–1 , nn+1 , . . . , n2n–1 ) ; –c }, (n1 , . . . , nn–1 , nn+1 , . . . , n2n–1 )∗ ≐ (nn+1 , . . . , n2n–1 , n1 , . . . , nn–1 ). Below we give the multiplets for su(n, n) for n = 3, 4. (The case n = 2 was treated separately in detail in Chapter 6.) They are valid also for sl(2n, R) with M-factor sl(n, R) ⊕ sl(n, R), and when n = 2k these are multiplets also for the parabolically related algebra su∗ (4k) with M-factor su∗ (2k) ⊕ su∗ (2k). There are several types of multiplets: the main type, which contains maximal number of ERs/GVMs, the ﬁnite-dimensional and the discrete series representations, and many reduced types of multiplets. The multiplets of the main type are in 1-to-1 correspondence with the ﬁnitedimensional irreps of su(n, n), i.e., they will be labeled by the 2n – 1 positive Dynkin labels mi ∈ N. 8.3.1 Multiplets of su(3, 3) and sl(6, R) 8.3.1.1 Main Multiplets The main multiplet contains 20 ERs/GVMs whose signatures can be given in the following pair-wise manner: 70± = {(m1 , m2 , m4 , m5 )± ; ±m1 } 7a± 7b± 7b±′ 7c± 7c±′ 7c±′′

±

= {(m1 , m23 , m34 , m5 ) ; ±(m1 – m3 )} = {(m12 , m3 , m24 , m5 )± ; ±(m1 – m23 )} = {(m1 , m24 , m3 , m45 )± ; ±(m1 – m34 )} = {(m2 , m3 , m14 , m5 )± ; ±(m1 – m13 )} = {(m12 , m34 , m23 , m45 )± ; ±(m1 – m24 )} = {(m1 , m25 , m3 , m4 )± ; ±(m1 – m35 )}

(8.68)

294

8 IDOs and Parabolic Relations

7d± = {(m2 , m34 , m13 , m45 )± ; ±(m1 – m14 )} 7d±′ = {(m12 , m35 , m23 , m4 )± ; ±(m1 – m25 )} 7e± = {(m2 , m35 , m13 , m4 )± ; ±(m1 – m15 )} where m1 = 21 (m1 + 2m2 + 3m3 + 2m4 + m5 ). The multiplets are given explicitly in the ﬁgure below. Each invariant differential operator is represented by an arrow accompanied by a symbol ijk encoding the root !jk and the number m!jk which is involved in the BGG criterion. We recall that only invariant differential operators which are noncomposite are displayed, and that the data ", m" , which is involved in the embedding V D(7) → V D(chi)–m" " turns out to involve only the Dynkin labels mi corresponding to simple roots, i.e., for each ", m" there exists i = i(", m" , D) ∈ {1, . . . , 2n – 1}, such that m" = mi . Hence the data !jk , m!jk is represented by ijk on the arrows. The pairs 7± are symmetric w.r.t. to the bullet in the middle of the ﬁgure - this represents the Weyl symmetry realized by the KS operators. χ0–

33 χa– 434

223 χb–

χb–′ 434

113 χc–

434 χd–

χc–′

535

113

324 χe+

535 324 214 χc+″

535

223

•

113

223

χc–″

χd–′

113 324

535 χe–

χd+′

535 214

535 χb+′

425

324 χc+′

113

χd+

113

425 214

χ+ a

315 χ0+

425

χb+

χc+

8.3 The Lie Algebra su(n,n) and Parabolically Related

295

As in previously considered examples in every multiplet there is only one ER (here 70– ) which contains a ﬁnite-dimensional nonunitary subrepresentation in a ﬁnite-dimensional subspace E. The latter corresponds to the ﬁnite-dimensional irrep of SU(6). The subspace E is annihilated by the operator G+ , and is the image of the operator G– . The subspace E is annihilated also by the invariant differential operator acting from 7– to 7′– . When all mi = 1 then dim E = 1, and in that case E is also the trivial one-dimensional UIR of the whole algebra G. Furthermore in that case the conformal weight is zero: d = 92 + c = 92 – 21 (m1 + 2m2 + 3m3 + 2m4 + m5 )|m =1 = 0. i Also as before in the conjugate ER 70+ there is a unitary subrepresentation in an inﬁnite-dimensional subspace D. It is annihilated by the operator G– , and is the image of the operator G+ . All the above is valid also for the algebra sl(6, R). However, the latter does not have discrete series representations. On the other hand the algebra su(3, 3) has holomorphic discrete series representations and furthermore highest/lowest weight series representations. Thus, in the case of su(3, 3) the ER 70+ contains both the holomorphic discrete series representation and the conjugate antiholomorphic discrete series. The direct sum of the latter two is realized in the invariant subspace D of the ER 70+ . Note that the corresponding lowest weight GVM is inﬁnitesimally equivalent only to the holomorphic discrete series, while the conjugate highest weight GVM is inﬁnitesimally equivalent to the antiholomorphic discrete series. The conformal weight of the ER 70+ has the restriction d = 92 + c = 92 + 21 (m1 + 2m2 + 3m3 + 2m4 + m5 ) ≥ 9. 8.3.1.2 Reduced Multiplets There are ﬁve types of reduced multiplets, R3a , a = 1, . . . , 5, which may be obtained from the main multiplet by setting formally ma = 0. Multiplets of type R34 , R35 , are conjugate to the multiplets of type R32 , R31 , resp., and are not discussed. The reduced multiplets of type R33 contain 14 ERs/GVMs with signatures: 70± = {(m1 , m2 , m4 , m5 )± ; ±m1 } 7b± 7b±′ 7c± 7c±′′ 7d± 7e±

= = = = = =

(8.69)

{(m12 , 0, m24 , m5 )± ; ±(m1 – m2 )} {(m1 , m24 , 0, m45 )± ; ±(m1 – m4 )} {(m2 , 0, m14 , m5 )± ; ±(m1 – m12 )} {(m1 , m25 , 0, m4 )± ; ±(m1 – m45 )} {(m2 , m4 , m12 , m45 )± ; ±(m1 – m12,4 ) = ∓(m1 – m2,45 )} {(m2 , m45 , m12 , m4 )± ; ±(m1 – m2,4 ) = ± 21 (m1 + m5 )},

here m1 = 21 (m1 +2m2 +2m4 +m5 ). The above is valid for the parabolically related algebras su(3, 3), sl(6, R). Actually, only six of the above ERs: 70± , 7d± , 7e± , are physically relevant being induced from ﬁnite-dimensional irreps of M. They may be called the main type of reduced multiplets since in 70+ are contained the limits of the (anti)holomorphic

296

8 IDOs and Parabolic Relations

discrete series of su(3, 3). Its conformal weight has the restriction d = 92 + 21 (m1 + 2m2 + 2m4 + m5 ) ≥ 152 . When all mk = 1 the ER 7e– contains a minimal irrep of conformal weight d = 7/2. From these ERS the pair 70± forms a standard KS doublet, i.e., 70– ←→ 70+ . The other four ERs are related as shown in the ﬁgure below: χe– 535

113

χd–

535

χd+

113 χe+

The reduced multiplets of type R32 contain 14 ERs/GVMs of which six are physically relevant: 7b± = {(m1 , m3 , m34 , m5 )± ; ±m1,45 }, 7c± = {(m1 , m34 , m3 , m45 )± ; ±(m1 + m5 )}, 7d± = {(m1 , m35 , m3 , m4 )± ; ±(m1 – m5 )}.

(8.70)

These form two conjugated triplets: D–b → D–d → D–d 434

535

●

D+d → D+d → D+b 535

425

(8.71)

When all mk = 1 the ER 7b– contains a minimal irrep of conformal weight d = 3/2. The reduced multiplets of type R31 contain 14 ERs/GVMs of which six are physically relevant: 7c± = {(m2 , m3 , m24 , m5 )± ; ±m45 }, 7d± = {(m2 , m34 , m23 , m45 )± ; ±m5 }, 7e± = {(m2 , m35 , m23 , m4 )± ; ±m5 }.

(8.72)

These are connected by invariant differential operators as shown below:

324

χe–

535

χd–

χc–

χd+

χc+ 425

434 535 χe+

324

When all mk = 1 the ER 7c– contains a minimal irrep of conformal weight d = 5/2.

8.3 The Lie Algebra su(n,n) and Parabolically Related

297

8.3.1.3 Further Reduction of Multiplets There are further reductions of the multiplets denoted by R3ab , a, b = 1, . . . , 5, a < b, which may be obtained from the main multiplet by setting formally ma = mb = 0. From these ten reductions six (for (a, b) = (13), (14), (15), (24), (25), (35)) contain representations of physical interest, i.e., induced from ﬁnite-dimensional irreps of the M subalgebra. Furthermore, the multiplets R325 R335 are conjugate to R314 R313 . The nonconjugate are presented in full detail in [150]. Here we present only the physically relevant ERs. All these four types of multiplets contain 10 ERs/GVMs of which there are two physically relevant in KS doublets with signatures. The reduced multiplets of type R313 contain 10 ERs/GVMs of which the physically relevant is the KS doublet: ± = {(m2 , m4 , m2 , m45 )± ; ± 21 m5 }, 713 ± 714 ± 724 ± 715

= = =

R313 ;

(8.73)

±

{(m2 , m3 , m23 , m5 ) ; ± 21 m5 }, R314 ; {(m1 , m3 , m3 , m5 )± ; ± 21 m1,5 }, R324 ; {(m2 , m34 , m23 , m4 )± ; 0}, R315 .

– in the ﬁrst three cases above contain a minimal irrep of When all mk = 1 the ERs 7... conformal weight d = 4, 4, 7/2, respectively. In these cases the KS operator from 7– to 7+ has degenerated to a differential operator. In the last case the operators between ± are reductions of the KS operators which do not change the conformal weight, but 715 only conjugate the M-signatures.

8.3.1.4 Last Reduction of Multiplets There are further reductions of the multiplets - triple and quadruple, but only one triple reduction contains representations of physical interest. Namely, this is the multiplet R3135 , which may be obtained from the main multiplet by setting formally m1 = m3 = m5 = 0. It contains 7 ERs/GVMs with signatures: 7a± = {(0, m2 , m4 , 0)± ; ±m2,4 },

(8.74)

7b± = {(0, m2,4 , 0, m4 )± ; ±m2 }, 7b±′ = {(m2 , 0, m2,4 , 0)± ; ±m4 }, 7d = {(m2 , m4 , m2 , m4 ); 0}. The only physically relevant ER 7d is a singlet, not in a pair, since it has zero weight c, and the M entries are self-conjugate. These ERs contain minimal irreps (in a more general understanding) characterized by two positive integers which are denoted in this context as m2 , m4 . Each such irrep is the kernel of the two invariant differential operators Dm2 ,!14 and Dm4 ,!25 , which are of order m2 , m4 , resp., corresponding to the noncompact roots !14 , !25 , resp. This is depicted below: D+b ← ● D–d → ● D+b′ . 214

425

(8.75)

298

8 IDOs and Parabolic Relations

8.3.2 Multiplets of su(4, 4), sl(8, R), and su∗ (8) 8.3.2.1 Main Multiplets The main multiplets R4 contain 70 ERs/GVMs with signatures: 70± = {(m1 , m2 , m3 , m5 , m6 , m7 )± ; ±m1 } ± 700 = {(m1 , m2 , m34 , m45 , m6 , m7 )± ; ±(m1 – m4 )} ± 710 = {(m1 , m23 , m4 , m35 , m6 , m7 )± ; ±(m1 – m34 )} ± 701 = {(m1 , m2 , m35 , m4 , m56 , m7 )± ; ±(m1 – m45 )} ± 720 = {(m12 , m3 , m4 , m25 , m6 , m7 )± ; ±(m1 – m24 )} ± 711 = {(m1 , m23 , m45 , m34 , m56 , m7 )± ; ±(m1 – m35 )} ± 702 = {(m1 , m2 , m36 , m4 , m5 , m67 )± ; ±(m1 – m46 )} ± 730 = {(m2 , m3 , m4 , m15 , m6 , m7 )± ; ±(m1 – m14 )} ± 721 = {(m12 , m3 , m45 , m24 , m56 , m7 )± ; ±(m1 – m25 )} ± 712 = {(m1 , m23 , m46 , m34 , m5 , m67 )± ; ±(m1 – m36 )} ± 703 = {(m1 , m2 , m37 , m4 , m5 , m6 )± ; ±(m1 – m47 )} ± 731 = {(m2 , m3 , m45 , m14 , m56 , m7 )± ; ±(m1 – m15 )} ± 722 = {(m12 , m3 , m46 , m24 , m5 , m67 )± ; ±(m1 – m26 )} ± 713 = {(m1 , m23 , m47 , m34 , m5 , m6 )± ; ±(m1 – m37 )} ± 732 = {(m2 , m3 , m46 , m14 , m5 , m67 )± ; ±(m1 – m16 )} ± 723 = {(m12 , m3 , m47 , m24 , m5 , m6 )± ; ±(m1 – m27 )} ± 733 = {(m2 , m3 , m47 , m14 , m5 , m6 )± ; ±(m1 – m17 )} ′± 700 = {(m1 , m24 , m5 , m3 , m46 , m7 )± ; ±(m1 – m35 – m4 )} ′± 710 = {(m12 , m34 , m5 , m23 , m46 , m7 )± ; ±(m1 – m25 – m4 )} ′± 701 = {(m1 , m24 , m56 , m3 , m45 , m67 )± ; ±(m1 – m36 – m4 )} ′± 720 = {(m2 , m34 , m5 , m13 , m46 , m7 )± ; ±(m1 – m15 – m4 )} ′± 711 = {(m12 , m34 , m56 , m23 , m45 , m67 )± ; ±(m1 – m26 – m4 )} ′± 702 = {(m1 , m24 , m57 , m3 , m45 , m6 )± ; ±(m1 – m37 – m4 )} ′′± 720 = {(m13 , m4 , m5 , m2 , m36 , m7 )± ; ±(m1 – m25 – m34 )} ′′± 721 = {(m13 , m4 , m56 , m2 , m35 , m67 )± ; ±(m1 – m26 – m34 )} ′′± 712 = {(m12 , m35 , m6 , m23 , m4 , m57 )± ; ±(m1 – m26 – m45 )} ′′± 702 = {(m1 , m25 , m6 , m3 , m4 , m57 )± ; ±(m1 – m36 – m45 )} ′± 730 = {(m23 , m4 , m5 , m12 , m36 , m7 )± ; ±(m1 – m15 – m34 )} ′± 721 = {(m2 , m34 , m56 , m13 , m45 , m67 )± ; ±(m1 – m16 – m4 )} ′± 712 = {(m12 , m34 , m57 , m23 , m45 , m6 )± ; ±(m1 – m27 – m4 )} ′± 703 = {(m1 , m25 , m67 , m3 , m4 , m56 )± ; ±(m1 – m37 – m45 )} ′± 740 = {(m3 , m4 , m5 , m1 , m26 , m7 )± ; ±(m1 – m15 – m24 )} ′± 731 = {(m23 , m4 , m56 , m12 , m35 , m67 )± ; ±(m1 – m16 – m34 )} ′± 722 = {(m2 , m34 , m57 , m13 , m45 , m6 )± ; ±(m1 – m17 – m4 )} ′′± 722 = {(m13 , m4 , m57 , m2 , m35 , m6 )± ; ±(m1 – m27 – m34 )}

(8.76)

299

8.3 The Lie Algebra su(n,n) and Parabolically Related

where m1 = 21 (m1 +2m2 +3m3 +4m4 +3m5 +2m6 +m7 ). The multiplets are given explicitly by the ﬁgure below: χ0– 44

– χ10

545

224

– χ20

114

545

χ21–

– χ30

545

646

4 35 3 25

20

114 ′– χ40

215

′– χ30

325 ″– 20

646 646 325

6 46 215

646

1

747

536

536

7 1

″+ 12

5 ′+ χ02 426

+ χ03

3 16

+ χ13

215

7

747 316

747 + χ02

3

4

+ 23

426 + χ12

+ χ01

4

5

1 3

5 3

•

′+ 22

1

4

1 7 6

7 2

′– 22

5 3 7

+ 33

5

′+ 11

4

527

224 224 536

5

′+ 21

1 6

′+ χ00

426

7

637 215

+ χ11

1

325

4

10

+ χ21

316

′+ χ31

χ ″+

1

536 χ ′–

747 224

03

637 224

637 114

20

3

′+ χ40

χ ′+ 325

30

′+ χ20

426

637 114

+ χ31

527 114

527 215

527

+ χ00

637 6

+ 32

″– 02

7

22

′+

2 4 26

″+

7 1

″+ 21

1

2 15 637

′– 12

″– 12

″– 02

536

3

+ 22

316

637

7 4

– χ03

334

435

747

−

5

7

+ 12

′+ 01

33

χ13–

224

′– 01 23

114

747

747

435

646

−

21

7

2

02

′– 11

3

215

χ ″+

747 435 2 24

1 ′–

334

– χ12

00

−

747

435

– χ02

22

646

″– 21

″– 22

χ ′– 31

′+ χ03

– 32

1 14

646

χ ′ – 224

224 114

′– 10

– χ01

646

435

646

435

545 334

– χ11

224

114

χ31–

– χ00

334

+ χ30

+ χ20

+ χ10

417 χ0+

8.3.2.2 Main Reduced Multiplets There are four physically relevant and essentially different reductions of multiplets denoted by R44 , R43 , R42 , R41 . Each of them contains 50 ERs/GVMs [151] of which only 20 are physically relevant. We present the latter starting with R44 :

300

8 IDOs and Parabolic Relations

70± = {(m1 , m2 , m3 , m5 , m6 , m7 )± ; ±m1 } ± 711 = {(m1 , m23 , m5 , m3 , m56 , m7 )± ; ±(m1 – m3,5 )} ± 721 = {(m12 , m3 , m5 , m23 , m56 , m7 )± ; ±(m1 – m23,5 )} ± 712 = {(m1 , m23 , m56 , m3 , m5 , m67 )± ; ±(m1 – m3,56 )} ± 731 = {(m2 , m3 , m5 , m13 , m56 , m7 )± ; ±(m1 – m13,5 )} ± 722 = {(m12 , m3 , m56 , m23 , m5 , m67 )± ; ±(m1 – m23,56 )} ± 713 = {(m1 , m23 , m57 , m3 , m5 , m6 )± ; ±(m1 – m3,57 )} ± 732 = {(m2 , m3 , m56 , m13 , m5 , m67 )± ; ±(m1 – m13,56 )} ± 723 = {(m12 , m3 , m57 , m23 , m5 , m6 )± ; ±(m1 – m23,57 )} ± 733 = {(m2 , m3 , m57 , m13 , m5 , m6 )± ; ±(m1 – m13,57 )}

(8.77)

here m1 = 21 (m1 + 2m2 + 3m3 + 3m5 + 2m6 + m7 ). Each multiplet contains a KS doublet 70± , while the rest 18 ERs are related as given in the ﬁgure below: χ21− χ31−

114 646

224

− χ11

646 − χ32

114 215

– χ33

+ χ33

747

χ13

− χ23

224

747 + χ32

637 + χ31

+

747

215 + χ12

637

χ22

6 37

+

χ11

− χ13

114

114

747

+

− χ12

224 747

114 747

+ χ23

− χ22

646

114 + χ21

215

Here in 70+ are contained the limits of the (anti)holomorphic discrete series of su(4, 4). – contains a minimal irrep of conformal weight d = 4. When all mk = 1 the ER 711 Next we give the physically relevant ERs of multiplets R43 : ± 710 = {(m1 , m2 , m4 , m45 , m6 , m7 )± ; ±(m1 – m4 )} ± 711 = {(m1 , m2 , m45 , m4 , m56 , m7 )± ; ±(m1 – m45 )} ± 712 = {(m1 , m2 , m46 , m4 , m5 , m67 )± ; ±(m1 – m46 )} ± 713 = {(m1 , m2 , m47 , m4 , m5 , m6 )± ; ±(m1 – m47 )} ′± 710 = {(m12 , m4 , m5 , m2 , m46 , m7 )± ; ±(m1 – m2,45 – m4 )} ′± 720 = {(m2 , m4 , m5 , m12 , m46 , m7 )± ; ± 21 (m1 – m12,45 – m4 )} ′± 711 = {(m12 , m4 , m56 , m2 , m45 , m67 )± ; ±(m1 – m2,46 – m4 )}

8.3 The Lie Algebra su(n,n) and Parabolically Related

′′± = {(m12 , m45 , m6 , m2 , m4 , m57 )± ; ±(m1 – m2,46 – m45 )} 712 ′± 721 = {(m2 , m4 , m56 , m12 , m45 , m67 )± ; ±(m1 – m12,46 – m4 )} ′± 712 = {(m12 , m4 , m57 , m2 , m45 , m6 )± ; ±(m1 – m2,47 – m4 )}

301

(8.78)

– , 7′– contain here m1 = 21 (m1 + 2m2 + 4m4 + 3m5 + 2m6 + m7 ). When all mk = 1 the ERs 710 10 a minimal irrep of conformal weight d = 5/2, 11/2, respectively. Next R42 : ± 720 = {(m1 , m3 , m4 , m35 , m6 , m7 )± ; ±(m1 – m34 )} ± 721 = {(m1 , m3 , m45 , m34 , m56 , m7 )± ; ±(m1 – m35 )} ± 722 = {(m1 , m3 , m46 , m34 , m5 , m67 )± ; ±(m1 – m36 )} ± 723 = {(m1 , m3 , m47 , m34 , m5 , m6 )± ; ±(m1 – m37 )} ′± 710 = {(m1 , m34 , m5 , m3 , m46 , m7 )± ; ±(m1 – m35 – m4 )} ′± 711 = {(m1 , m34 , m56 , m3 , m45 , m67 )± ; ±(m1 – m36 – m4 )} ′′± 712 = {(m1 , m35 , m6 , m3 , m4 , m57 )± ; ±(m1 – m36 – m45 )} ′± 730 = {(m3 , m4 , m5 , m1 , m36 , m7 )± ; ±(m1 – m1,35 – m34 )} ′± 712 = {(m1 , m34 , m57 , m3 , m45 , m6 )± ; ±(m1 – m37 – m4 )} ′± 731 = {(m3 , m4 , m56 , m1 , m35 , m67 )± ; ±(m1 – m1,36 – m34 )}

(8.79)

– , 7′– contain a m1 = 21 (m1 + 3m3 + 4m4 + 3m5 + 2m6 + m7 ). When all mk = 1 the ERs 720 30 minimal irrep of conformal weight d = 3, 7, respectively. Finally R41 : ± 720 = {(m2 , m3 , m4 , m25 , m6 , m7 )± ; ±(m1 – m24 )} ± 721 = {(m2 , m3 , m45 , m24 , m56 , m7 )± ; ±(m1 – m25 )} ± 722 = {(m2 , m3 , m46 , m24 , m5 , m67 )± ; ±(m1 – m26 )} ± 723 = {(m2 , m3 , m47 , m24 , m5 , m6 )± ; ±(m1 – m27 )} ′± 710 = {(m2 , m34 , m5 , m23 , m46 , m7 )± ; ±(m1 – m25 – m4 )} ′± 711 = {(m2 , m34 , m56 , m23 , m45 , m67 )± ; ±(m1 – m26 – m4 )} ′± 730 = {(m23 , m4 , m5 , m2 , m36 , m7 )± ; ±(m1 – m25 – m34 )} ′± 712 = {(m2 , m34 , m57 , m23 , m45 , m6 )± ; ±(m1 – m27 – m4 )} ′± 731 = {(m23 , m4 , m56 , m2 , m35 , m67 )± ; ±(m1 – m26 – m34 )} ′′± 722 = {(m23 , m4 , m57 , m2 , m35 , m6 )± ; ±(m1 – m27 – m34 )}

(8.80)

– contains a m1 = 21 (2m2 + 3m3 + 4m4 + 3m5 + 2m6 + m7 ). When all mk = 1 the ER 720 minimal irrep of conformal weight d = 7/2.

8.3.2.3 Further Reduction of Multiplets There are nine physically relevant and essentially different further reductions of multiplets denoted by R4ab , (a, b) = (13), (14), (15), (16), (17), (24), (25), (26), and (35). They contain 36 ERs/GVMs each (cf. [151]) of which only 6 are physically relevant. We present the latter starting with R413 :

302

8 IDOs and Parabolic Relations

′± 710 = {(m2 , m4 , m5 , m2 , m46 , m7 )± ; ±(m1 – m2,45 – m4 )} ′± 711 = {(m2 , m4 , m56 , m2 , m45 , m67 )± ; ±(m1 – m2,46 – m4 )} ′± 712 = {(m2 , m4 , m57 , m2 , m45 , m6 )± ; ±(m1 – m2,47 – m4 )}

(8.81)

′– contains a minimal irrep m1 = 21 (2m2 +4m4 +3m5 +2m6 +m7 ). When all mk = 1 the ER 710 of conformal weight d = 6. These ERs are connected by invariant differential operators as shown below:

747 ′– χ11

′– χ10

′– χ12

536

646 536

′+ χ12

′+ χ10

′+ χ11

•

637 747

Next is type R414 : ± 721 = {(m2 , m3 , m5 , m23 , m56 , m7 )± ; ±(m1 – m23,5 )} ± 722 = {(m2 , m3 , m56 , m23 , m5 , m67 )± ; ±(m1 – m23,56 )} ± 723 = {(m2 , m3 , m57 , m23 , m5 , m6 )± ; ∓(m1 – m23,57 )}

(8.82)

– contains a minimal m1 = 21 (2m2 + 3m3 + 3m5 + 2m6 + m7 ). When all mk = 1 the ER 721 irrep of conformal weight d = 11/2. – – – 721 → 722 → 723 646

747

●

+ + + 723 → 722 → 721 747

637

(8.83)

Next is type R415 : ± 721 = {(m2 , m3 , m4 , m24 , m6 , m7 )± ; ±(m1 – m24 )} ′± 711 = {(m2 , m34 , m6 , m23 , m4 , m67 )± ; ±(m1 – m24,6 – m4 )} ′± 731 = {(m23 , m4 , m6 , m2 , m34 , m67 )± ; ±(m1 – m24,6 – m34 )}

(8.84)

– , 7′– contain a minimal m1 = 21 (2m2 + 3m3 + 4m4 + 2m6 + m7 ). When all mk = 1 the ERs 721 11 ± form a KS doublet, irrep of conformal weight d = 5, 7, respectively. From these ERs 721 the rest are related as follows:

325

′– χ11

747 ′+ χ11

′– χ11

747

′+ χ31

325

8.3 The Lie Algebra su(n,n) and Parabolically Related

303

Next is type R416 : ± 722 = {(m2 , m3 , m45 , m24 , m5 , m7 )± ; ±(m1 – m25 )} ′± 711 = {(m2 , m34 , m5 , m23 , m45 , m7 )± ; ±(m1 – m25 – m4 )} ′± 731 = {(m23 , m4 , m5 , m2 , m35 , m7 )± ; ±(m1 – m25 – m34 )}

(8.85)

– contains a minimal m1 = 21 (2m2 + 3m3 + 4m4 + 3m5 + m7 ). When all mk = 1 the ER 722 irrep of conformal weight d = 11/2.

– ′– ′– 722 → 711 → 731

′+ ′+ + 731 → 711 → 722

(8.86)

± 722 = {(m2 , m3 , m46 , m24 , m5 , m6 )± ; ±(m1 – m26 )}

(8.87)

435

325

●

325

426

Next is type R417 :

′± 711 = {(m2 , m34 , m56 , m23 , m45 , m6 )± ; ±(m1 – m26 – m4 )} ′± 731 = {(m23 , m4 , m56 , m2 , m35 , m6 )± ; ±(m1 – m26 – m34 ) = ± 21 (m5 – m3 )} – contains a minimal m1 = 21 (2m2 + 3m3 + 4m4 + 3m5 + 2m6 ). When all mk = 1 the ER 722 irrep of conformal weight d = 6.

325

′– χ31

536

′– χ11

– χ22

+ χ22

′+ χ11

435

426 536

′+ χ31

325

Next is type R424 : ± 721 = {(m1 , m3 , m5 , m3 , m56 , m7 )± ; ±(m1 – m3,5 )}

(8.88)

± 722 = {(m1 , m3 , m56 , m3 , m5 , m67 )± ; ±(m1 – m3,56 )} ± 723 = {(m1 , m3 , m57 , m3 , m5 , m6 )± ; ± 21 (m1 – m3,57 )} – contains a minimal irrep m1 = 21 (m1 + 3m3 + 3m5 + 2m6 + m7 ). When all mk = 1 the ER 721 of conformal weight d = 2.

– – – 721 → 722 → 723 646

747

●

+ + + 723 → 722 → 721 747

637

(8.89)

304

8 IDOs and Parabolic Relations

Next is type R425 : ± = {(m1 , m3 , m4 , m34 , m6 , m7 )± ; ±(m1 – m34 )} 721 ′± 711 = {(m1 , m34 , m6 , m3 , m4 , m67 )± ; ′± 731 = {(m3 , m4 , m6 , m1 , m34 , m67 )± ; = ± 21 (m7 – m1,3 )}

(8.90)

±(m1 – m34,6 – m4 )} ±(m1 – m1,34,6 – m34 )

– , 7′– contain a minimal m1 = 21 (m1 + 3m3 + 4m4 + 2m6 + m7 ). When all mk = 1 the ERs 721 11 ± form a KS irrep of conformal weight d = 9/2, 11/2, respectively. From these ERs 721 doublet, the rest are related as follows: ′– ′+ 711 → 731 747

●

′– ′+ 731 → 711

(8.91)

747

Next is type R426 : ± = {(m1 , m3 , m45 , m34 , m5 , m7 )± ; ±(m1 – m35 )} 722

(8.92)

′± 711 = {(m1 , m34 , m5 , m3 , m45 , m7 )± ; ±(m1 – m35 – m4 )} ′± 731 = {(m3 , m4 , m5 , m1 , m35 , m7 )± ; ±(m1 – m1,35 – m34 ) = ± 21 (m5,7 – m1,3 )} – contains a minimal irrep m1 = 21 (m1 + 3m3 + 4m4 + 3m5 + m7 ). When all mk = 1 the ER 722 ′± of conformal weight d = 5. From these ERs 731 form a KS doublet, the rest are related as follows: – ′– 722 → 711 435

●

′+ + 711 → 722

(8.93)

426

Finally, type R435 : ± = {(m1 , m2 , m4 , m4 , m6 , m7 )± ; ±(m1 – m4 )} 711 ′± 711 = {(m12 , m4 , m6 , m2 , m4 , m67 )± ; ′± 721 = {(m2 , m4 , m6 , m12 , m4 , m67 )± ; = ± 21 (m7 – m1 )}

±(m1 – m2,4,6 – m4 ) = ±(m1 – m12,4,6 – m4 )

(8.94) ± 21 m1,7 }

– , 7′– contain a minimal m1 = 21 (m1 + 2m2 + 4m4 + 2m6 + m7 ). When all mk = 1 the ERs 711 11 ± form a KS doublet, irrep of conformal weight d = 4, 7, respectively. From these ERs 711 the rest are related as follows:

114

′– χ21

747 ′+ χ11

′– χ11

747

′+ χ21

114

8.3 The Lie Algebra su(n,n) and Parabolically Related

305

8.3.2.4 Yet Further Reduction of Multiplets There are six physically relevant and essentially different further reductions of multiplets denoted by R4abc , (a, b, c) = (1, 3, 5), (1, 3, 6), (1, 3, 7), (1, 4, 6), (1, 4, 7), and (2, 4, 6). They contain 26 ERs/GVMs each (cf. [151]) of which only 2 are physically relevant. We present the latter doublets: ± 7135 = {(m2 , m4 , m6 , m2 , m4 , m67 )± ; ± 21 m7 },

R4135

± 7136 = {(m2 , m4 , m5 , m2 , m45 , m7 )± ; ± 21 m5,7 },

R4136

± 7137 = {(m2 , m4 , m56 , m2 , m45 , m6 )± ; ± 21 m5 },

R4137

± 7146 = {(m2 , m3 , m5 , m23 , m5 , m7 )± ; ± 21 m3,5,7 },

R4146

± 7147 = {(m2 , m3 , m56 , m23 , m5 , m6 )± ; ± 21 m3,5 },

R4147

± 7246 = {(m1 , m3 , m5 , m3 , m5 , m7 )± ; ± 21 m1,3,5,7 },

R4246

(8.95)

– contain a minimal irrep of conformal weight d = When all mk = 1 the ERs 7... 15/2, 7, 15/2, 13/2, 7, 6, respectively. In each doublet the KS operator from 7– to 7+ has degenerated to a differential operator.

8.3.2.5 Last Reduction of Multiplets There are further reductions of the multiplets - quadruple, etc., but only one quadruple reduction contains representations of physical interest. Namely, this is the multiplet R41357 , which may be obtained from the main multiplet by setting formally m1 = m3 = m5 = m7 = 0. These multiplets contain 19 ERs/GVMs but only one is physically relevant - a singlet with c = 0 (d = 8) and being M self conjugated: 7min = {(m2 , m4 , m6 , m2 , m4 , m6 ); 0}.

(8.96)

These ERs contain minimal irreps (in the more general understanding) characterized by three positive integers which are denoted in this context as m2 , m4 , m6 . Each such irrep is the kernel of the three invariant differential operators Dm2 ,!15 , Dm4 ,!26 , Dm6 ,!37 , as given on the ﬁgure below: χmin 215

637 426

306

8 IDOs and Parabolic Relations

8.4 Multiplets and Representations for sp(n, R) and sp(r, r) 8.4.1 Preliminaries This section is based mostly on [146, 147, 149]. Let G = sp(n, R), the split real form of sp(n, C) = G C . The maximal compact subgroup of G is K ≅ u(1) ⊕ su(n). We choose a maximal parabolic P = MAN such that A ≅ so(1, 1), while the factor M = sl(n, R) has the same ﬁnite-dimensional (nonunitary) representations as the ﬁnite-dimensional (unitary) representations of the semisimple subalgebra su(n) of K. Finally, note that dimR N = n(n + 1)/2. We label the signature of the ERs of G as follows: 7 = {n1 , . . . , nn–1 ; c },

nj ∈ N,

c = d – (n + 1)/2,

(8.97)

where the last entry of 7 labels the characters of A, and the ﬁrst n – 1 entries are labels of the ﬁnite-dimensional nonunitary irreps of M, (or of the ﬁnite-dimensional unitary irreps of su(n)). Below we shall use the following conjugation on the ﬁnite-dimensional entries of the signature: (n1 , . . . , nn–1 )∗ ≐ (nn–1 , . . . , n1 ).

(8.98)

The ERs in the multiplet are related also by the intertwining KS integral operators which are deﬁned for any ER, though we present them as acting in doublets (which are important building blocks): G± : C7∓ → C7± , 7– = {n1 , . . . , nn–1 ; –c },

7+ = {(n1 , . . . , nn–1 )∗ ; c}.

(8.99)

Further, we need more explicitly the root system of the algebra sp(n, F), F = C, R. In terms of the orthonormal basis :i , i = 1, . . . , n, the positive roots are given by B+ = {:i ± :j , 1 ≤ i < j ≤ n; 2:i , 1 ≤ i ≤ n},

(8.100)

while the simple roots are 0 = {!i = :i – :i+1 , 1 ≤ i ≤ n – 1; !n = 2:n }.

(8.101)

8.4 Multiplets and Representations for sp(n, R) and sp(r, r)

307

From these the compact roots are those that form (by restriction) the root system of the semisimple part of KC , the rest are noncompact, i.e., compact: !ij ≡ :i – :j , 1 ≤ i < j ≤ n, noncompact: "ij ≡ :i + :j , 1 ≤ i ≤ j ≤ n.

(8.102)

Thus, the only noncompact simple root is !n = "nn . Finally, we give the correspondence between the signatures 7 and the highest weight D. The explicit connection is ni = mi ,

c = – 21 (m!˜ + mn ) = – 21 (m1 + ⋅ ⋅ ⋅ + mn–1 + 2mn ),

(8.103)

where !˜ = "11 is the highest root. For n = 2r the algebra sp(2r, R) with M-factor sl(2r, R) is parabolically related to G = sp(r, r) with M-factor su∗ (2r), noting that (su∗ (2r))C = sl(2r, C). The algebra sp(r, r) has maximal compact subalgebra K = sp(r) ⊕ sp(r) and has discrete series representations but no highest/lowest weight representations. There are several types of multiplets: the main type, (which contains maximal number of ERs/GVMs, the ﬁnite-dimensional and the discrete series representations), and various reduced types of multiplets. The multiplets of the main type are in 1-to-1 correspondence with the ﬁnite-dimensional irreps of sp(n, R), i.e., they will be labeled by the n positive Dynkin labels mi ∈ N. The number of ERs in the main multiplets is according to (8.12): |W(G C , HC )| |W(sp(n, C))| 2n (n)! n = =2 = C )| |W(sl(n, C))| ((n)!) |W(MC , Hm

(8.104)

It is difﬁcult to give explicitly the multiplets for general n. Thus, we present the cases 3 ≤ n ≤ 6. Note that the cases n = 1, 2 were considered in Section 4.7 (recalling that sp(1, R) ≅ sl(2, R)), in Chapter 5 (recalling that sp(2, R) ≅ so(3, 2)), respectively. Note also that the case sp(1, 1) was considered in Section 8.2 (recalling that sp(1, 1) ≅ so(4, 1)). 8.4.2 The Case sp(3, R) The main multiplets R3m of sp(3, R) contain 8(= 23 ) ERs/GVMs whose signatures are given in the following pair-wise manner: 0 1 70± = (m1 , m2 )± ; ± 21 (m12 + 2m3 ) 1 0 7a± = (m1 , m2 + 2m3 )± ; ± 21 m12 1 0 7b± = (m12 , m2 + 2m3 )± ; ± 21 m1 1 0 7c± = (m2 , m12 + 2m3 )± ; ∓ 21 m1 and the notation (. . .)± employs the conjugation (8.98) as per (8.99).

(8.105)

308

8 IDOs and Parabolic Relations

The multiplets are given explicitly in the ﬁgure below: χ0– 333 χa– 223 113 χc–

322

322

χb–

•

113

χb+

χc+

212 χa+ 311 χ0+

(8.106)

8.4.2.1 Reduced Multiplets for sp(3, R) The reduced multiplets of type R31 contain 6 ERs/GVMs, however, only two of them are physically relevant: ± 7b1 = {(m2 , m2 + 2m3 )± ; 0}.

(8.107)

They have the same conformal weight, i.e., they are related by KS operators G± that have degenerated to the M-conjugation. Here belongs also a singlet: 7b1 = {(m2 , m2 ); 0}.

(8.108)

The reduced multiplets of type R32 also contain 6 ERs/GVMs with only two physically relevant. They are given a standard KS doublet: ± 7b2 = {(m1 , 2m3 )± ; ± 21 m1 }.

(8.109)

The reduced multiplets of type R33 contain 4 ERs/GVMs whose signatures can be given in the following pair-wise manner: 70± = {(m1 , m2 )± ; ± 21 m12 }, ± 7b3

±

= {(m12 , m2 ) ;

± 21 m1 }.

(8.110)

8.4 Multiplets and Representations for sp(n, R) and sp(r, r)

309

The ERs 70+ from (8.110) contain the limits of holomorphic discrete series positive energy d = 2 + 21 m12 . Here we shall use the relation of (8.110) to the main multiplet of the lower sp-case sp(2, R) ≅ so(3, 2) considered in Chapter 5. Namely, if we drop the 2nd M-entry in (8.110), and then replace m12 ↦ m1 + 2m2 , then we obtain exactly the main multiplet of so(3, 2). This occurrence is typical for all sp-cases. 8.4.2.2 Special Multiplets for sp(3, R) The special multiplets R3s also contain 8 ERs/GVMs with signatures: 70± 7a± 7b± 7c±

= = = =

{(m1 , m2 )± ; ± 21 (m12 + ,)}, {(m1 , m2 + ,)± ; ± 21 m12 }, {(m12 , m2 + ,)± ; ± 21 m1 }, {(m2 , m12 + ,)± ; ∓ 21 m1 },

(8.111)

where , ∈ 2N – 1. They have two components - one is a KS-doublet 70± , the other component is given in the ﬁgure below: 7a– → 7b– → 7c– 223

113

●

7c+ → 7b+ → 7a+ . 113

212

(8.112)

There are three minimal UIRs situated in 70– , 7a– and 7c+ with m1 = m2 = , = 1: – The one in 70– has trivial su(3) irrep and d = 21 . –

The one in 7a– has three-dimensional su(3) irrep and d = 1.

–

The one in 7c+ has six-dimensional su(3) irrep and d = 32 .

Finally, there are special reduced multiplets R3ks , k = 1, 2, in parallel to the reduced multiplets of type R3k . These are given by formulae (8.107) and (8.109), for k = 1, 2, respectively, with the replacement 2m3 ↦ , ∈ 2N – 1. They form KS doublets inheriting the original properties. 8.4.3 The Case sp(4, R) and sp(2, 2) The main multiplets R4m of sp(4, R) contain 16(= 24 ) ERs/GVMs whose signatures are given in the following pair-wise manner: 70± 7a± 7b± 7c± 7d± 7e± 7f± 7g±

= = = = = = = =

{(m1 , m2 , m3 )± ; ± 21 (m13 + 2m4 )} {(m1 , m2 , m3 + 2m4 )± ; ± 21 m13 } {(m1 , m23 , m3 + 2m4 )± ; ± 21 m12 } {(m12 , m3 , m23 + 2m4 )± ; ± 21 m1 } {(m1 , m23 + 2m4 , m3 )± ; ± 21 m12 } {(m2 , m3 , m13 + 2m4 )± ; ∓ 21 m1 } {(m12 , m3 + 2m4 , m23 )± ; ± 21 m1 } {(m2 , m3 + 2m4 , m13 )± ; ∓ 21 m1 }

(8.113)

310

8 IDOs and Parabolic Relations

The main multiplets are given explicitly in the ﬁgure below: χ0– 444 χa– 334

χc– χe–

114 433

224 433

χg–

433

χb–

χd–

χf–

224

114

323

• 323

χf+

χg+

114

422

422 χd+

χe+

114

213

χc+

422 213

χb+ 312 χa+ 411 χ0+

(8.114)

8.4.3.1 Reduced Multiplets for sp(4, R) and sp(2, 2) The reduced multiplets of type R41 contain 12 ERs/GVMs from which four are physically relevant: 7c± = {(m2 , m3 , m23 + 2m4 )± ; 0}, 7f± = {(m2 , m3 + 2m4 , m23 )± ; 0}.

(8.115)

They form two KS doublets with KS operators G± that have degenerated to the M-conjugation. Here belongs also one more KS degenerate doublet: 7f±′ = {(m2 , m3 , m23 )± ; 0}.

(8.116)

8.4 Multiplets and Representations for sp(n, R) and sp(r, r)

311

The reduced multiplets of type R42 contain 12 ERs/GVMs from which four are physically relevant: 7b± = {(m1 , m3 , m3 + 2m4 )± ; ± 21 m1 }, 7d± = {(m1 , m3 + 2m4 , m3 )± ; ± 21 m1 },

(8.117)

7b– → 7d– ● 7d+ → 7b+ .

(8.118)

433

422

Here belongs also one more KS doublet: 7d± = {(m1 , m3 , m3 )± ; ± 21 m1 }

(8.119)

The reduced multiplets of type R43 contain 12 ERs/GVMs from which four are physically relevant: 7b± = {(m1 , m2 , 2m4 )± ; ± 21 m12 }, 7f± = {(m12 , 2m4 , m2 )± ; ± 21 m1 },

(8.120)

7b– → 7f– → ● 7f+ → 7b+ .

(8.121)

433

114

422

Here belongs also a family of singlets (m ∈ N): 7d′ = {(m2 , m, m2 ); 0}

(8.122)

The reduced multiplets of type R44 contain 8 ERs/GVMs whose signatures can be given in the following pair-wise manner: 70± = {(m1 , m2 , m3 )± ; ± 21 m13 }, 7b± = {(m1 , m23 , m3 )± ; ± 21 m12 }, 7c± = {(m12 , m3 , m23 )± ; ± 21 m1 }, 7e± = {(m2 , m3 , m13 )± ; ∓ 21 m1 }.

(8.123)

Similarly to the observation at the end of Subsection 8.4.2.1 we notice that the above reduced sp(4, R) multiplet is 1-to-1 with the main sp(3, R) multiplet (8.105), thus we can use diagram (8.106).

312

8 IDOs and Parabolic Relations

8.4.3.2 Special Multiplets for sp(4, R) and sp(2, 2) The special multiplets R4s contain 16 ERs/GVMs whose signatures are given by the main multiplet signatures (8.113) by the replacement 2m4 ↦ , ∈ 2N – 1: 70± 7a± 7b± 7c± 7d± 7e± 7f± 7g±

= = = = = = = =

{(m1 , m2 , m3 )± ; 52 ± 21 (m13 + ,)} {(m1 , m2 , m3 + ,)± ; 52 ± 21 m13 } {(m1 , m23 , m3 + ,)± ; 52 ± 21 m12 } {(m12 , m3 , m23 + ,)± ; 52 ± 21 m1 } {(m2 , m3 , m13 + ,)± ; 52 ∓ 21 m1 } {(m2 , m3 + ,, m13 )± ; 52 ∓ 21 m1 } {(m12 , m3 + ,, m23 )± ; 52 ± 21 m1 } {(m1 , m23 + ,, m3 )± ; 52 ± 21 m12 }

(8.124)

They consist of a KS doublet 70± , two conjugated quadruplets: 7a– → 7b– → 7c– → 7e– 334

224

114

●

7e+ → 7c+ → 7b+ → 7a+ 114

213

312

(8.125)

and a sextet: χg– 323

114 χf–

χd–

χf+

χd+ 213

224 323

114 χg+

There are four minimal UIRs situated in 70– , 7a– , 7d– , and 7e+ with m1 = m2 = m3 = , = 1: – The one in 70– has trivial su(4) irrep and d = 21 . –

The one in 7a– has fundamental su(4) irrep and d = 1.

–

The one in 7d– has 20-dimensional su(4) irrep and d = 32 .

–

The one in 7e+ has 45-dimensional su(4) irrep and d = 2.

Finally, there are special reduced multiplets R4ks , k = 1, 2, 3, in parallel to the reduced multiplets of type R4k , with the replacement 2m4 ↦ , ∈ 2N – 1. First we give R41s : 7c± = {(m2 , m3 , m23 + ,)± ; 0} 7f± = {(m2 , m3 + ,, m23 )± ; 0}

(8.126)

8.4 Multiplets and Representations for sp(n, R) and sp(r, r)

313

Here we have the KS doublets 7c± and 7f± , both degenerated to M-conjugation. Next is R42s containing the KS doublets 7b± and 7d± : 7b± = {(m1 , m3 , m3 + ,)± ; ± 21 m1 } 7d± = {(m1 , m3 + ,, m3 )± ; ± 21 m1 }

(8.127)

Next in R42s we have the KS doublets 7b± and 7f– → ● 7f+ : 114

7b± = {(m1 , m2 , ,)± ; ± 21 m12 } 7f± = {(m12 , ,, m2 )± ; ± 21 m1 }

(8.128)

8.4.4 The Case sp(5, R) The main multiplets R5m of sp(5, R) contain 32(= 25 ) ERs/GVMs with signatures: 70± = {(m1 , m2 , m3 , m4 )± ; ± 21 (m14 + 2m5 )} 7a± = {(m1 , m2 , m3 , m4 + 2m5 )± ; ± 21 m14 } 7b± = {(m1 , m2 , m34 , m4 + 2m5 )± ; ± 21 m13 } 7c± = {(m1 , m23 , m4 , m34 + 2m5 )± ; ± 21 m12 } 7c±′ = {(m1 , m2 , m34 + 2m5 , m4 )± ; ± 21 m13 } 7d± = {(m12 , m3 , m4 , m24 + 2m5 )± ; ± 21 m1 } 7d±′ = {(m1 , m23 , m4 + 2m5 , m34 )± ; ± 21 m12 } 7e± = {(m2 , m3 , m4 , m14 + 2m5 )± ; ∓ 21 m1 } 7e±′ = {(m12 , m3 , m4 + 2m5 , m24 )± ; ± 21 m1 } 7e±′′ = {(m1 , m24 , m4 + 2m5 , m3 )± ; ± 21 m12 } 7f± = {(m2 , m3 , m4 + 2m5 , m14 )± ; ∓ 21 m1 } 7f±′ = {(m12 , m34 , m4 + 2m5 , m23 )± ; ± 21 m1 } 7f±′′ = {(m1 , m24 + 2m5 , m4 , m3 )± ; ± 21 m12 } 7g± = {(m2 , m34 , m4 + 2m5 , m13 )± ; ∓ 21 m1 } 7g±′ = {(m12 , m34 + 2m5 , m4 , m23 )± ; ± 21 m1 } 7h± = {(m2 , m34 + 2m5 , m4 , m13 )± ; ∓ 21 m1 }

(8.129)

314

8 IDOs and Parabolic Relations

The multiplets are given explicitly in the ﬁgure below:

χ0– 555 χa– 445 χb– 544

335

χc′–

χc– 544

225 115 544

225

544

χe–

434

χe′–

115

434

χf– 434

χf–″ 225

533 χh+

533

533

χh–

533 214

324 χg+

115

324

214

χg′–

115

∙

115

+ χe″

533

324 533

χg′+

– χe″

225

χf–′

115

χg– 324

χf+″

335

– χd′

χd–

423

423

χf+′

115

522

522

214

+ χd′

522

313

522

χe′+

423

χc′+

χf+

χb+

χd+ χc+

χe+

115

214

313

412

χa+ 511 χ0+

(8.130)

8.4 Multiplets and Representations for sp(n, R) and sp(r, r)

315

8.4.4.1 Reduced Multiplets for sp(5, R) The main reduced multiplets of types R5k , k = 1, 2, 3, 4, contain 24 ERs/GVMs from which only eight are physically relevant. We present the latter ﬁrst for R51 : 7e± = {(m2 , m3 , m4 , m24 + 2m5 )± ; 0}, 7f± = {(m2 , m3 , m4 + 2m5 , m24 )± ; 0}, 7g± = {(m2 , m34 , m4 + 2m5 , m23 )± ; 0}, 7h± = {(m2 , m34 + 2m5 , m4 , m23 )± ; 0};

(8.131)

then for R52 : 7d± = {(m1 , m3 , m4 , m34 + 2m5 )± ; ± 21 m1 }, 7e±′ = {(m1 , m3 , m4 + 2m5 , m34 )± ; ± 21 m1 }, 7f±′ = {(m1 , m34 , m4 + 2m5 , m3 )± ; ± 21 m1 }, 7g±′ = {(m1 , m34 + 2m5 , m4 , m3 )± ; ± 21 m1 };

(8.132)

then for R53 : 7c± = {(m1 , m2 , m4 , m4 + 2m5 )± ; ± 21 m12 }, 7d±′ = {(m1 , m2 , m4 + 2m5 , m4 )± ; ± 21 m12 }, 7f±′ = {(m12 , m4 , m4 + 2m5 , m2 )± ; ± 21 m1 }, 7g±′ = {(m12 , m4 + 2m5 , m4 , m2 )± ; ± 21 m1 };

(8.133)

for R54 : 7b± = {(m1 , m2 , m3 , 2m5 )± ; ± 21 m13 }, 7d±′ = {(m1 , m23 , 2m5 , m3 )± ; ± 21 m12 }, 7e±′ = {(m12 , m3 , 2m5 , m23 )± ; ± 21 m1 }, 7f± = {(m2 , m3 , 2m5 , m13 )± ; ∓ 21 m1 }.

(8.134)

The relevant intertwining operators in these four cases are inherited from diagram (8.130). There are further reductions, from which again we present the physically relevant ERs:

316

8 IDOs and Parabolic Relations

7g± = {(m2 , m4 , m4 + 2m5 , m2 )± ; 0}, 7f± = {(m2 , m3 , 2m5 , m23 )± ; 0},

R514

7e± = {(m2 , m3 , m4 , m24 )± ; 0},

R515a

7h± = {(m2 , m34 , m4 , m23 )± ; 0},

R513

R515b

7e±′ = {(m1 , m3 , 2m5 , m3 )± ; ± 21 m1 },

R524

7d± = {(m1 , m3 , m4 , m34 )± ; ± 21 m1 },

R525a

7f±′ = {(m1 , m34 , m4 , m3 )± ; ± 21 m1 },

R525b

7c± = {(m1 , m2 , m4 , m4 )± ; ± 21 m12 },

R535a

7g±′ = {(m12 , m4 , m4 , m2 )± ; ± 21 m1 },

R535b

7s = {(m2 , m4 , m4 , m2 ); 0},

R5135

(8.135)

the last being a M-self-conjugate singlet. Finally, the main reduced multiplets of type R55 contain 16 ERs/GVMs whose signatures can be given in the following pair-wise manner: 70± = {(m1 , m2 , m3 , m4 )± ; ± 21 m14 } 7b± = {(m1 , m2 , m34 , m4 )± ; ± 21 m13 } 7c± = {(m1 , m23 , m4 , m34 )± ; ± 21 m12 } 7d± = {(m12 , m3 , m4 , m24 )± ; ± 21 m1 } 7e± = {(m2 , m3 , m4 , m14 )± ; ∓ 21 m1 } 7e±′′ = {(m1 , m24 , m4 , m3 )± ; ± 21 m12 } 7f±′ = {(m12 , m34 , m4 , m23 )± ; ± 21 m1 } 7g± = {(m2 , m34 , m4 , m13 )± ; ∓ 21 m1 }

(8.136)

The above reduced sp(5, R) multiplets are 1-to-1 with the main sp(4, R) multiplets, thus we can use diagram (8.114).

8.4.4.2 Special Multiplets for sp(5, R) The special multiplets R5s also contain 32 ERs/GVMs whose signatures are given by the formulae of the main multiplets R5m (8.129) with the change: 2m5 ↦ , ∈ 2N – 1. They contain a KS doublet 70± , then: 7a– → 7b– → 7c– → 7d– → 7e– 445

335

225

115

●

7e+ → 7d+ → 7c+ → 7b+ → 7a+ 115

214

313

412

(8.137)

8.4 Multiplets and Representations for sp(n, R) and sp(r, r)

317

and the third component is χc′– 335 – χd′

χe′–

225

434

115

434

434

115

χf– χg– 324 χg′+

225

χf–′

χf–″ 225

324

115

324

χh– 324

214 + χe″

χg–′

115

χh+

214

χf+″

– χe″

χf+′

χg+

115

423

423

χf+

115 χe′+

423

χc′+

+ χd′

214

313

There are ﬁve minimal UIRs situated in 70– , 7a– , 7e+ , 7c–′ and 7f–′′ with m1 = m2 = m3 = m4 = , = 1: – The one in 70– has have trivial su(5) irrep and d = 21 . –

The one in 7a– has fundamental su(5) irrep and d = 1.

–

The one in 7c–′ has 50-dimensional su(5) irrep and d = 32 .

–

The one in 7f–′′ has 175-dimensional su(5) irrep and d = 2.

–

The one in 7e+ has 70-dimensional su(5) irrep and d = 52 .

Finally, there are special reduced multiplets R5ks , in parallel to the reduced multiplets of type R5k , etc, with the replacement 2m4 ↦ , ∈ 2N – 1.

8.4.5 The Case sp(6, R) and sp(3, 3) The main multiplets R6m of sp(6, R) contain 64(= 26 ) ERs/GVMs whose signatures and diagram are given below: 70± = {(m1 , m2 , m3 , m4 , m5 )± ; ± 21 (m15 + 2m6 )} 7a±

±

= {(m1 , m2 , m3 , m4 , m5 + 2m6 ) ;

± 21 m15 }

(8.138)

318

8 IDOs and Parabolic Relations

7b± = {(m1 , m2 , m3 , m45 , m5 + 2m6 )± ; ± 21 m14 } 7c± = {(m1 , m2 , m34 , m5 , m45 + 2m6 )± ; ± 21 m13 } 7c±′ = {(m1 , m2 , m3 , m45 + 2m6 , m5 )± ; ± 21 m14 } 7d± = {(m1 , m23 , m4 , m5 , m35 + 2m6 )± ; ± 21 m12 } 7d±′ = {(m1 , m2 , m34 , m5 + 2m6 , m45 )± ; ± 21 m13 } 7e± = {(m12 , m3 , m4 , m5 , m25 + 2m6 )± ; ± 21 m1 } 7e±′ = {(m1 , m23 , m4 , m5 + 2m6 , m35 )± ; ± 21 m12 } 7e±′′ = {(m1 , m2 , m35 , m5 + 2m6 , m4 )± ; ± 21 m13 } 7f± = {(m2 , m3 , m4 , m5 , m15 + 2m6 )± ; ∓ 21 m1 } 7f±′ = {(m12 , m3 , m4 , m5 + 2m6 , m25 )± ; ± 21 m1 } 7f±′′ = {(m1 , m23 , m45 , m5 + 2m6 , m34 )± ; ± 21 m12 } 7f±′′′ = {(m1 , m2 , m35 + 2m6 , m5 , m4 )± ; ± 21 m13 } 7g± = {(m2 , m3 , m4 , m5 + 2m6 , m15 )± ; ∓ 21 m1 } 7g±′ = {(m12 , m3 , m45 , m5 + 2m6 , m24 )± ; ± 21 m1 } 7g±′′ = {(m1 , m23 , m45 + 2m6 , m5 , m34 )± ; ± 21 m12 } 7h± = {(m2 , m3 , m45 , m5 + 2m6 , m14 )± ; ∓ 21 m1 } 7h±′ = {(m12 , m3 , m45 + 2m6 , m5 , m24 )± ; ± 21 m1 } 7h±′′ = {(m2 , m3 , m45 + 2m6 , m5 , m14 )± ; ∓ 21 m1 } 7j± = {(m2 , m34 , m5 , m45 + 2m6 , m13 )± ; ∓ 21 m1 } 7j±′ = {(m12 , m34 , m5 , m45 + 2m6 , m23 )± ; ± 21 m1 } 7j±′′ = {(m1 , m24 , m5 , m45 + 2m6 , m3 )± ; ± 21 m12 } 7k± = {(m2 , m34 , m5 + 2m6 , m45 , m13 )± ; ∓ 21 m1 } 7k±′ = {(m12 , m34 , m5 + 2m6 , m45 , m23 )± ; ± 21 m1 } 7k±′′ = {(m1 , m24 , m5 + 2m6 , m45 , m3 )± ; ± 21 m12 } 7± = {(m2 , m35 , m5 + 2m6 , m4 , m13 )± ; ∓ 21 m1 } 7±′ = {(m12 , m35 , m5 + 2m6 , m4 , m23 )± ; ± 21 m1 } 7±′′ = {(m1 , m25 , m5 + 2m6 , m4 , m3 )± ; ± 21 m12 } ± 7m = {(m2 , m35 + 2m6 , m5 , m4 , m13 )± ; ∓ 21 m1 } ± ± 1 7m ′ = {(m12 , m35 + 2m6 , m5 , m4 , m23 ) ; ± 2 m1 } ± ± 1 7m ′′ = {(m1 , m25 + 2m6 , m5 , m4 , m3 ) ; ± 2 m12 }

8.4 Multiplets and Representations for sp(n, R) and sp(r, r)

319

χ0– 666 χa– 556 χb–

446

655

χc–

655

336

χd–′

χd–

226

655

– χe′

χe– χf–

116

655

655

116

χg–

χf′–

644

435

116 215

644 χk–

325

116

215

644 χl+″

424 314

+ χg″

633 215

χk′+

633 + χe″

424 215

χf+″

+ χd′

413

622 χb+

χg′+

– χm″

325 χj+

χe′+

χh+

523 116

523 215

χf+′

χg+

622 116

622 215

622 314

622

– χm′

226

424 χj+′ 633 116

523 314

633

116

116

633

+ χj″ 424

523 413

χc′+

424 + χh′

633

424

χh+″ 633 116 2 15

633 314

χk+ 3 25

116

534 215

534 + χk″

χf+′″

325

χl′+

χl′–

– χm

325

– χl″

226 633

633

χl+ •

534

– χk″

534

534

– χf′″

435

116

χl–

325

644

+ χm″

534

644

+ χm′

435 226

435 116

χm+

644

– χk′

336

– χg″

226 χh–′

644

χh–″

325

325

226 116

116

χj–

644

χj–″

644

χj–′

644

– χf″

435

435

χe–″

336

226

– χg′

116

χh–

545

545

545

545

336

226

χc–′

446

χf+

χe+

χd+

χc+

512

χa+

611 χ0+

(8.139)

320

8 IDOs and Parabolic Relations

8.4.5.1 Reduced Multiplets for sp(6, R) and sp(3, 3) The reduced multiplets of types R6k , k = 1, 2, 3, 4, 5, contain each 48 ERs/GVMs from which only 16 are physically relevant. For R61 : 7e± = {(m2 , m3 , m4 , m5 , m25 + 2m6 )± ; 0}, 7f±′ = {(m2 , m3 , m4 , m5 + 2m6 , m25 )± ; 0}, 7g±′ = {(m2 , m3 , m45 , m5 + 2m6 , m24 )± ; 0}, 7h±′ = {(m2 , m3 , m45 + 2m6 , m5 , m24 )± ; 0}, 7j±′ = {(m2 , m34 , m5 , m45 + 2m6 , m23 )± ; 0}, 7k±′ = {(m2 , m34 , m5 + 2m6 , m45 , m23 )± ; 0}, 7±′ = {(m2 , m35 , m5 + 2m6 , m4 , m23 )± ; 0}, ± ± 7m ′ = {(m2 , m35 + 2m6 , m5 , m4 , m23 ) ; 0}.

(8.140)

For R62 : 7d± = {(m1 , m3 , m4 , m5 , m35 + 2m6 )± ; ± 21 m1 }, 7e±′ = {(m1 , m3 , m4 , m5 + 2m6 , m35 )± ; ± 21 m1 }, 7f±′′ = {(m1 , m3 , m45 , m5 + 2m6 , m34 )± ; ± 21 m1 }, 7g±′′ = {(m1 , m3 , m45 + 2m6 , m5 , m34 )± ; ± 21 m1 }, 7j±′′ = {(m1 , m34 , m5 , m45 + 2m6 , m3 )± ; ± 21 m1 }, 7k±′′ = {(m1 , m34 , m5 + 2m6 , m45 , m3 )± ; ± 21 m1 }, 7±′′ = {(m1 , m35 , m5 + 2m6 , m4 , m3 )± ; ± 21 m1 }, ± ± 1 7m ′′ = {(m1 , m35 + 2m6 , m5 , m4 , m3 ) ; ± 2 m1 }.

For R63 : 7d± = {(m1 , m2 , m4 , m5 , m45 + 2m6 )± ; ± 21 m12 }, 7e±′ = {(m1 , m2 , m4 , m5 + 2m6 , m45 )± ; ± 21 m12 }, 7f±′′ = {(m1 , m2 , m45 , m5 + 2m6 , m4 )± ; ± 21 m12 }, 7g±′′ = {(m1 , m2 , m45 + 2m6 , m5 , m4 )± ; ± 21 m12 }, 7j± = {(m2 , m4 , m5 , m45 + 2m6 , m12 )± ; ∓ 21 m1 }, 7j±′ = {(m12 , m4 , m5 , m45 + 2m6 , m2 )± ; ± 21 m1 },

(8.141)

8.4 Multiplets and Representations for sp(n, R) and sp(r, r)

321

7k± = {(m2 , m4 , m5 + 2m6 , m45 , m12 )± ; ∓ 21 m1 }, 7k±′ = {(m12 , m4 , m5 + 2m6 , m45 , m2 )± ; ± 21 m1 }.

(8.142)

For R64 : 7c± = {(m1 , m2 , m3 , m5 , m5 + 2m6 )± ; ± 21 m13 }, 7d±′ = {(m1 , m2 , m3 , m5 + 2m6 , m5 )± ; ± 21 m13 }, 7j± = {(m2 , m3 , m5 , m5 + 2m6 , m13 )± ; ∓ 21 m1 }, 7j±′ = {(m12 , m3 , m5 , m5 + 2m6 , m23 )± ; ± 21 m1 }, 7j±′′ = {(m1 , m23 , m5 , m5 + 2m6 , m3 )± ; ± 21 m12 }, 7k± = {(m2 , m3 , m5 + 2m6 , m5 , m13 )± ; ∓ 21 m1 }, 7k±′ = {(m12 , m3 , m5 + 2m6 , m5 , m23 )± ; ± 21 m1 }, 7k±′′ = {(m1 , m23 , m5 + 2m6 , m5 , m3 )± ; ± 21 m12 }.

(8.143)

For R65 : 7b± = {(m1 , m2 , m3 , m4 , 2m6 )± ; ± 21 m14 }, 7e±′′ = {(m1 , m2 , m34 , 2m6 , m4 )± ; ± 21 m13 }, 7f±′′ = {(m1 , m23 , m4 , 2m6 , m34 )± ; ± 21 m12 }, 7g±′ = {(m12 , m3 , m4 , 2m6 , m24 )± ; ± 21 m1 }, 7h± = {(m2 , m3 , m4 , 2m6 , m14 )± ; ∓ 21 m1 }, 7k± = {(m2 , m34 , 2m6 , m4 , m13 )± ; ∓ 21 m1 }, 7k±′ = {(m12 , m34 , 2m6 , m4 , m23 )± ; ± 21 m1 }, 7k±′′ = {(m1 , m24 , 2m6 , m4 , m3 )± ; ± 21 m12 }.

(8.144)

The relevant invariant differential operators in these ﬁve cases are inherited from diagram (8.139). There are further reductions, from which again present the physically relevant ERs: 7j±′ = {(m2 , m4 , m5 , m45 + 2m6 , m2 )± ; 0},

R613a

7k±′ = {(m2 , m4 , m5 + 2m6 , m45 , m2 )± ; 0},

R613b

7h±′ = {(m2 , m3 , m5 + 2m6 , m5 , m23 )± ; 0},

R614a

7ˆ±′ = {(m2 , m3 , m5 , m5 + 2m6 , m23 )± ; 0},

R614b

j 7f±′ 7±′

= {(m2 , m3 , m4 , 2m6 , m24 )± ; 0}, = {(m2 , m34 , 2m6 , m4 , m23 )± ; 0},

R615a R615b

322

8 IDOs and Parabolic Relations

7e± = {(m2 , m3 , m4 , m5 , m25 )± ; 0},

R616a

7g±′ = {(m2 , m3 , m45 , m5 , m24 )± ; 0},

R616b

± ± 7m ′ = {(m2 , m35 , m5 , m4 , m23 ) ; 0},

R616c

7˜j±′ = {(m2 , m34 , m5 , m45 , m23 )± ; 0},

R616d

(8.145)

7f±′′ = {(m1 , m3 , m5 , m5 + 2m6 , m3 )± ; ± 21 m1 },

R624a

7k±′′ = {(m1 , m3 , m5 + 2m6 , m5 , m3 )± ; ± 21 m1 },

R624b

7e±′ = {(m1 , m3 , m4 , 2m6 , m34 )± ; ± 21 m1 },

R625a

7±′′ = {(m1 , m34 , 2m6 , m4 , m3 )± ; ± 21 m1 },

R625b

7d± = {(m1 , m3 , m4 , m5 , m35 )± ; ± 21 m1 },

R626a

7g±′′ = {(m1 , m3 , m45 , m5 , m34 )± ; ± 21 m1 },

R626b

7j±′′ = {(m1 , m34 , m5 , m45 , m3 )± ; ± 21 m1 },

R626c

± ± 1 7m ′′ = {(m1 , m35 , m5 , m4 , m3 ) ; ± 2 m1 },

R626d

7e±′ = {(m1 , m2 , m4 , 2m6 , m4 )± ; ± 21 m12 },

R635a

7k± = {(m2 , m4 , 2m6 , m4 , m12 )± ; ∓ 21 m1 },

R635b

7d± = {(m1 , m2 , m4 , m5 , m45 )± ; ± 21 m12 },

R636a

7f±′′ = {(m1 , m2 , m45 , m5 , m4 )± ; ± 21 m12 },

R636b

7j± = {(m2 , m4 , m5 , m45 , m12 )± ; ∓ 21 m1 },

R636c

7j±′ = {(m12 , m4 , m5 , m45 , m2 )± ; ± 21 m1 },

R636d

7c± = {(m1 , m2 , m3 , m5 , m5 )± ; ± 21 m13 },

R646a

7j± = {(m2 , m3 , m5 , m5 , m13 )± ; ∓ 21 m1 },

R646b

7j±′ = {(m12 , m3 , m5 , m5 , m23 )± ; ± 21 m1 },

R646c

7j±′′ = {(m1 , m23 , m5 , m5 , m3 )± ; ± 21 m12 },

R646d

(8.146)

(8.147)

(8.148)

Finally, there are triple reductions with physically relevant ERs: 7j±′ = {(m2 , m4 , m5 , m45 , m2 )± ; 0},

R6136

7h±′ = {(m2 , m3 , m5 , m5 , m23 )± ; 0},

R6146

7f±′′ = {(m1 , m3 , m5 , m5 , m3 )± ; ± 21 m1 }, 7k′ = {(m2 , m4 , 2m6 , m4 , m2 ); 0}, the last one being a singlet.

R6135

R6246 (8.149)

8.4 Multiplets and Representations for sp(n, R) and sp(r, r)

323

The reduced multiplets of type R66 contain 32 ERs/GVMs whose signatures can be given in the following pair-wise manner: 70± = {(m1 , m2 , m3 , m4 , m5 )± ; ± 21 m15 }, 7c±′ = {(m1 , m2 , m3 , m45 , m5 )± ; ± 21 m14 }, 7d±′ = {(m1 , m2 , m34 , m5 , m45 )± ; ± 21 m13 }, 7e±′ = {(m1 , m23 , m4 , m5 , m35 )± ; ± 21 m12 }, 7f±′ = {(m12 , m3 , m4 , m5 , m25 )± ; ± 21 m1 }, 7f±′′′ = {(m1 , m2 , m35 , m5 , m4 )± ; ± 21 m13 }, 7g± = {(m2 , m3 , m4 , m5 , m15 )± ; ∓ 21 m1 }, 7g±′′ = {(m1 , m23 , m45 , m5 , m34 )± ; ± 21 m12 }, 7h±′ = {(m12 , m3 , m45 , m5 , m24 )± ; ± 21 m1 }, 7h±′′ = {(m2 , m3 , m45 , m5 , m14 )± ; ∓ 21 m1 }, 7k± = {(m2 , m34 , m5 , m45 , m13 )± ; ∓ 21 m1 }, 7k±′ = {(m12 , m34 , m5 , m45 , m23 )± ; ± 21 m1 }, 7k±′′ = {(m1 , m24 , m5 , m45 , m3 )± ; ± 21 m12 }, 7± = {(m2 , m35 , m5 , m4 , m13 )± ; ∓ 21 m1 }, 7±′ = {(m12 , m35 , m5 , m4 , m23 )± ; ± 21 m1 }, 7±′′ = {(m1 , m25 , m5 , m4 , m3 )± ; ± 21 m12 }.

(8.150)

The above reduced sp(6, R) multiplets are 1-to-1 with the main sp(5, R) multiplets, thus we can use diagram (8.130).

8.4.5.2 Special Multiplets for sp(6, R) and sp(3, 3) The special multiplets R5s also contain 64 ERs/GVMs whose signatures are given by the formulae of the main multiplets (8.138) with the change: 2m6 ↦ , ∈ 2N – 1. They have four components - one is a KS-doublet 70± , the second component is given as follows: 7a– → 7b– → 7c– → 7d– → 7e– → 7f– 556

546

336

●

226

116

7f+ → 7e+ → 7d+ → 7c+ → 7b+ → 7a+ 116

215

314

413

512

(8.151)

324

8 IDOs and Parabolic Relations

the third component (containing 7c±′ ) and fourth component (containing 7f±′′′ ) are given in the ﬁgure below:

446

– χd′

χf–′

226

545

545

226

1 16

χg–

5 45

545

336

χe′–

χg–′

226

226 116

116

χk′–

435 116

116

χk–

+ χm

534

+ χm′

325

534

χl″+

+ χk″

+ χg″

116

215

215

424

314

f ′″ + χe″

215

χd+′

325 χ+ j

424 χj′+

χg′+

116

χe+′

χ+ h

523 116

523 215

523 314

523 413

χc+′

χf+″

116

116

χj″+

314 χ+

χk+ 325

424

424

– χm′

m

+ χh″

k′

424 + χh′

– χm″

424

χ+

χl″–

226

χl′–

χ–

116

534 215

534

534

325

325

χl′+

534

χl+ •

116

+ χm″

215

– χk″

116

χl–

325

215

435

– χh′ 435 226

– χh″

325

325

– χg″

435

χj–′

χj–

χf+′″

336 χj–″

435

435

– χe″

336

χf–″

116

χh–

χc′–

χf+′

χg+

226

8.5 SO∗ (4n) Case

325

There are six minimal UIRs with m1 = m2 = m3 = m4 = m5 = , = 1: – The one in 70– has have trivial su(6) irrep and d = 21 . –

The one in 7a– has fundamental su(6) irrep and d = 1.

–

The one in 7c–′ has 105-dimensional su(6) irrep and d = 32 .

–

The one in 7f–′′′ has 980-dimensional su(6) irrep and d = 2.

–

– has 1764-dimensional su(6) irrep and d = 5 . The one in 7m ′′ 2

–

The one in 7f+ has 252-dimensional su(6) irrep and d = 3.

8.4.6 Summary for sp(n, R) We summarize some facts that are explicit for n ≤ 6 and conjectural for n > 6. In each main multiplet the ERs 70– contain the ﬁnite-dimensional irreps of sp(n, R). In each main multiplet the ERs 70+ contain the (anti)holomorphic discrete series irreps of sp(n, R). The holomorphic discrete series have positive energy d = 21 (n + 1 + m1 n–1 ) + -, - ∈ N. Each special multiplet of sp(n, R) contains [ n2 ] + 1 connected components, or submultiplets, although if we do not take into account the Knapp-Stein operators then there would be n + 1 connected components. In each special multiplet the ERs 70+ contain positive energy irreps with d = 21 (n + 1 + m1 n–1 ) + 21 ,, , ∈ 2N – 1. In the reduced multiplets Rnn the ERs 70+ contain the limits of (anti) holomorphic discrete series irreps of sp(n, R). The limits of holomorphic discrete series have positive energy d = 21 (n + 1 + m1 n–1 ). There are n minimal UIRs for sp(n, R) with conformal weights d = 1 n 2 , 1, . . . , 2 whose corresponding ERs are denoted in the corresponding ﬁgures as – – 70 , 7a , . . .. Note that in each case the KS operator acting from the ER containing a minimal UIR degenerates to a differential operator of degree n, n – 1, . . . , 1, (respectively to the above enumeration). Note further that there is no differential operator with image – that is, an ER containing a minimal UIR – that can be used as an equivalent deﬁnition of a minimal UIR. The only operator that acts to such an ER is the conjugate nondegenerate KS integral operator. When n = 2r the above summary is valid for sp(r, r) dropping the term “(anti)holomorphic,” e.g., replacing the words “holomorphic discrete series” by “discrete series.”

8.5 SO∗ (4n) Case This section is based mostly on [147, 154]. Let G = so∗ (4n). We choose a maximal parabolic P = MAN such that A ≅ so(1, 1), M = su∗ (2n). Since the algebras so∗ (4n) belong to the class we called “conformal Lie algebras,” we have KC ≅ u(1)C ⊕ sl(2n, C) ≅ AC ⊕ MC .

(8.152)

326

8 IDOs and Parabolic Relations

Here we have the series of algebras: so∗ (4), so∗ (8), so∗ (12), . . . However, the ﬁrst two cases are reduced to well-known conformal algebras due to the coincidences: so∗ (4) ≅ so(3) ⊕ so(2, 1), and so∗ (8) ≅ so(6, 2). Thus, we shall study the algebra G6 ≡ so∗ (12). We label the signature of the ERs of G6 as follows: 7 = {n1 , n2 , n3 , n4 , n5 ; c },

nj ∈ Z+ ,

c=d–

15 2,

(8.153)

where the last entry of 7 labels the characters of A, and the ﬁrst ﬁve entries are labels of the ﬁnite-dimensional nonunitary irreps of M6 = su∗ (6). Below we shall use the following conjugation on the ﬁnite-dimensional entries of the signature: (n1 , n2 , n3 , n4 , n5 )∗ ≐ (n5 , n4 , n3 , n2 , n1 ).

(8.154)

The ERs in the multiplets are related also by the KS operators deﬁned for any ER, the general action being GKS : C7 → C7′ , 7 = {n1 , . . . , n5 ; c },

7′ = {(n1 , . . . , n5 )∗ ; –c }.

(8.155)

Further, we need the root system of the complexiﬁcation G6C = so(12, C). The positive roots are given standardly as !ij = :i – !j , "ij = :i + :j ,

1 ≤ i < j ≤ 6, 1 ≤ i < j ≤ 6.

(8.156) (8.157)

From these the compact roots are those that form (by restriction) the root system of the semisimple part of KC , i.e., !ij in (8.156), while the roots "ij in (8.157) are noncompact. Further, we give the correspondence between the signatures 7 and the highest weight D. The connection is through the Dynkin labels mi , i = 1, . . . , 5. The explicit connection is n i = mi ,

c = – 21 (m1 + 2m2 + 3m3 + 4m4 + 2m5 + 3m6 ).

(8.158)

Finally, we remind that the above considerations are applicable also for the parabolically related algebra so(6, 6) with parabolic M-factor sl(6, R). It has discrete series representations but no highest/lowest weight representations.

8.5 SO∗ (4n) Case

327

8.5.1 Main Multiplets The multiplets of the main type are in 1-to-1 correspondence with the ﬁnitedimensional irreps of so∗ (12), i.e., they are labeled by the six positive Dynkin labels mi ∈ N. The number of ERs/GVMs in the main multiplets is (8.12): C |W(G6C , HC )| / |W(MC 6 , Hm )| = |W(so(12, C))| / |W(sl(6, C))| = 32

(8.159)

C are Cartan subalgebras of G C , MC , resp. where HC , Hm 6 6 The signatures of the 32 ERs/GVMs of a main multiplet R12 can be given as follows:

70± = {(m1 , m2 , m3 , m4 , m5 )± ; ± 21 m16,26,34,4,6 }, 7a± = {(m1 , m2 , m3 , m4,6 , m5 )± ; ± 21 m16,25,34,4 }, 7b± = {(m1 , m2 , m34 , m6 , m45 )± ; ± 21 m16,25,3 }, 7c± = {(m1 , m2 , m35 , m6 , m4 )± ; ± 21 m14,24,3,6 }, 7c±′ = {(m1 , m23 , m4 , m6 , m35 )± ; ± 21 m16,2,45 }, 7d± = {(m1 , m23 , m45 , m6 , m34 )± ; ± 21 m14,2,4,6 }, 7d±′ = {(m12 , m3 , m4 , m6 , m25 )± ; ± 21 m1,36,45 }, 7e± = {(m1 , m24 , m5 , m4,6 , m3 )± ; ± 21 m13,2,6 }, 7e±′ = {(m12 , m3 , m45 , m6 , m24 )± ; ± 21 m1,34,4,6 }, 7e±′′ = {(m2 , m3 , m4 , m6 , m15 )± ; ± 21 (–m1 + m3 + 2m4 + 2m5 + m6 )}, 7f± = {(m12 , m34 , m5 , m4,6 , m23 )± ; ± 21 (m1 + m3 + m6 )}, 7f±′ = {(m2 , m3 , m45 , m6 , m14 )± ; ± 21 (–m1 + m3 + 2m4 + m6 )}, 7f±′′ = {(m1 , m24,6 , m5 , m4 , m3 )± ; ± 21 (m1 + 2m2 + m3 – m6 )}, 7g± = {(m13 , m4 , m5 , m34,6 , m2 )± ; ± 21 (m1 – m3 + m6 )}, 7g±′ = {(m12 , m34,6 , m5 , m4 , m23 )± ; ± 21 (m1 + m3 – m6 )}, 7g±′′ = {(m2 , m34 , m5 , m4,6 , m13 )± ; ± 21 (–m1 + m3 + m6 )},

(8.160)

328

8 IDOs and Parabolic Relations

where (k1 , k2 , k3 , k4 , k5 )– = (k1 , k2 , k3 , k4 , k5 ), (k1 , k2 , k3 , k4 , k5 )+ = (k1 , k2 , k3 , k4 , k5 )∗ . The multiplets are given explicitly in the ﬁgure below: χ0– 656

χa–

446

χb– 545

336

χc– 336

χe–

χf″–

634

χg–

424

χf′+

+ χg″

116

226

+ χe″

χe′+

116

325

116 634 χg′– •

226

325 χ +

g′

116

χf+

116 435 – χg″

116 215

634 χ+

325

χf″+

215

634

g

χe+

215 424

215

523

χc′+

χd+

314

χc+

314 523

χb+ 413

χa+ 612

χ0+

where as before we use the notation: D± = D(7± ).

116

χe′–

325

634

523 + χd′

545

435

χf–

424

523

226 – χd′

435

116 634

545

χd–

226

χc′–

545

χf′–

– χe″

8.5 SO∗ (4n) Case

329

As before matters are arranged so that in every multiplet the ERs with signature 70± have the standard properties in a main multiplet. In particular, the ER 70+ contains a unitary discrete series subrepresentation in an inﬁnite-dimensional subspace D. It is annihilated by the operator G– , and is the image of the operator G+ . All the above is valid also for the algebra so(6, 6). Furthermore, since so∗ (12) has highest/lowest weight series representations then for so∗ (12) the ER 70+ contains both a holomorphic discrete series representation and a conjugate antiholomorphic discrete series representation. The conformal weight of the positive energy irrep is restricted by d ≥ 15.

8.5.2 Reduced Multiplets and Minimal Irreps The main reduced multiplet R12 6 contains 24 ERs/GVMs of which only eight are physically relevant: 70± = {(m1 , m2 , m3 , m4 , m5 )± ; ± 21 (m1 + 2m2 + 3m3 + 4m4 + 2m5 )}, 7e± = {(m1 , m24 , m5 , m4 , m3 )± ; ± 21 (m1 + 2m2 + m3 )}, 7f± = {(m12 , m34 , m5 , m4 , m23 )± ; ± 21 (m1 + m3 )}, 7g± = {(m2 , m34 , m5 , m4 , m13 )± ; ± 21 (–m1 + m3 )}.

(8.161)

The ERs 70± form a doublet, while the rest are connected by invariant differential operators as shown below:

116

χf–

χe–

χg–

325

χf+

•

226 325

χg+

χe+ 215

116

The ER 70+ has the standard properties of the main reduced multiplets. It contains the unitary irrep called limits of discrete series in an inﬁnite-dimensional subspace D ′ . It is annihilated by the operator G– , and is the image of the operator G+ . This is valid for the algebra so(6, 6), while for so∗ (12) (as before) the structure is more reﬁned, i.e., the ER with signature 70+ contains both a limit of holomorphic discrete series representation and a conjugate limit of antiholomorphic discrete series representation. The positive energy conformal weight is restricted by d ≥ 17/2. The direct sum of these limit irreps forms the invariant subspace D ′ mentioned above. The ER 7e– contains a minimal irrep {(1, 3, 1, 1, 1); c = –2} of conformal weight d = 11/2.

330

8 IDOs and Parabolic Relations

The next reduced multiplet R12 5 also has eight physically relevant ERs/GVMs: 7c± = {(m1 , m2 , m34 , m6 , m4 )± ; ± 21 (m1 + 2m2 + 3m3 + 2m4 + m6 )}, 7d± = {(m1 , m23 , m4 , m6 , m34 )± ; ± 21 (m1 + 2m2 + m3 + 2m4 + m6 )}, 7e± = {(m12 , m3 , m4 , m6 , m24 )± ; ± 21 (m1 + m3 + 2m4 + m6 )}, 7f± = {(m2 , m3 , m4 , m6 , m14 )± ; ± 21 (–m1 + m3 + 2m4 + m6 )},

7c– → 7d– → 7e– → 7f– 336

226

116

●

7f+ → 7e+ → 7d+ → 7c+ . 116

215

314

(8.162)

(8.163)

The ER 7c– contains a minimal irrep {(1, 1, 2, 1, 1); c = –9/2} of conformal weight d = 3. The next reduced multiplet R12 4 also has eight physically relevant ERs/GVMs: 7b± = {(m1 , m2 , m3 , m6 , m5 )± ; ± 21 (m1 + 2m2 + 3m3 + 2m5 + m6 )}, 7e± = {(m1 , m23 , m5 , m6 , m3 )± ; ± 21 (m1 + 2m2 + m3 + m6 )}, 7f± = {(m12 , m3 , m5 , m6 , m23 )± ; ± 21 (m1 + m3 + m6 )}, 7g± = {(m2 , m3 , m5 , m6 , m13 )± ; ± 21 (–m1 + m3 + m6 )}.

(8.164)

The ERs 7b± form a doublet, while the rest are related as follows: 7e– → 7f– → 7g– 226

116

●

7g+ → 7f+ → 7e+ . 116

215

(8.165)

The ER 7e– contains a minimal irrep {(1, 2, 1, 1, 1); c = –5/2} of conformal weight d = 5. It is the kernel for the operator D2,26 and for an additional operator D6,34 (intertwining with unphysical representation). The next multiplet R12 3 also has eight physically relevant ERs/GVMs: 7b± = {(m1 , m2 , m4 , m6 , m45 )± ; ± 21 (m1 + 2m2 + 2m4 + 2m5 + m6 )}, 7c± = {(m1 , m2 , m45 , m6 , m4 )± ; ± 21 (m1 + 2m2 + 2m4 + m6 )}, 7f± = {(m12 , m4 , m5 , m4,6 , m2 )± ; ± 21 (m1 + m6 )}, 7g± = {(m2 , m4 , m5 , m4,6 , m12 )± ; ± 21 (–m1 + m6 )}.

(8.166)

They are related as follows: 7b– → 7c– 545

●

7c+ → 7b+ 523

(8.167)

8.5 SO∗ (4n) Case

331

χg– 634

116

χf–

χf+

• 634

116

χg+

The ER 7b– contains a minimal irrep {(1, 1, 1, 1, 2); c = –4} of conformal weight d = 7/2. It is the kernel for the operator D5,45 and for an additional operator D2,26 . The ER 7f– contains a minimal irrep {(2, 1, 1, 2, 1); c = –1} of conformal weight d = 13/2. It is the kernel for the operators D1,16 and D6,34 The next multiplet R12 2 also has eight physically relevant ERs/GVMs: 7d± = {(m1 , m3 , m4 , m6 , m35 )± ; ± 21 (m1 + m3 + 2m4 + 2m5 + m6 )} 7e± = {(m1 , m3 , m45 , m6 , m34 )± ; ± 21 (m1 + m3 + 2m4 + m6 )} 7f± = {(m1 , m34 , m5 , m4,6 , m3 )± ; ± 21 (m1 + m3 + m6 )} 7g± = {(m1 , m34,6 , m5 , m4 , m3 )± ; ± 21 (m1 + m3 – m6 )}

(8.168)

They are related as follows: 7d– → 7e– → 7f– → 7g– 545

435

634

●

7g+ → 7f+ → 7e+ → 7d+ 424

634

(8.169)

523

The ER 7d– contains a minimal irrep {(1, 1, 1, 1, 3); c = –4} of conformal weight d = 7/2. It is the kernel for the operator D5,45 and for an additional operator D1,16 . The next multiplet R12 1 also has eight physically relevant ERs/GVMs: 7d± = {(m2 , m3 , m4 , m6 , m25 )± ; ± 21 (m3 + 2m4 + 2m5 + m6 )} 7e± = {(m2 , m3 , m45 , m6 , m24 )± ; ± 21 (m3 + 2m4 + m6 )} 7f± = {(m2 , m34 , m5 , m4,6 , m23 )± ; ± 21 (m3 + m6 )} 7g± = {(m23 , m4 , m5 , m34,6 , m2 )± ; ± 21 (–m3 + m6 )}

(8.170)

related as follows: χg– 634

325

χf–

χe–

χd– 545

χf+

•

435 634

χd+

χe+ 424

523

325

χg+

The ER 7d– contains a minimal irrep {(1, 1, 1, 1, 4); c = –9/2} of conformal weight d = 3.

332

8 IDOs and Parabolic Relations

There are many further reductions of multiplets from which the physically relevant representations appear in nine doublets: 7a± = {(m1 , m34 , m5 , m4 , m3 )± ; ± 21 (m1 + m3 )},

R12 26

7b± = {(m2 , m3 , m4 , m6 , m24 )± ; ± 21 (m3 + 2m4 + m6 )},

R12 15

7c± = {(m1 , m3 , m4 , m6 , m34 )± ; ± 21 (m1 + m3 + 2m4 + m6 )},

R12 25

7d± = {(m1 , m2 , m4 , m6 , m4 )± ; ± 21 (m1 + 2m2 + 2m4 + m6 )},

R12 35

7e± = {(m2 , m3 , m5 , m6 , m23 )± ; ± 21 (m3 + m6 )},

R12 14

7f± = {(m1 , m3 , m5 , m6 , m3 )± ; ± 21 (m1 + m3 + m6 )}, 7g± = {(m2 , m34 , m5 , m4 , m23 )± ; ± 21 m3 }, 7h± = {(m12 , m4 , m5 , m4 , m2 )± ; ± 21 m1 }, 7j± = {(m2 , m4 , m5 , m4,6 , m2 )± ; ± 21 m6 },

R12 24

R12 16 R12 36 R12 13 .

(8.171)

These doublets are standardly related by the KS operators acting in doublets, yet in the last three cases we have the phenomenon of degeneration of KS operator from 7– to 7+ to differential operator: 7g– → ● 7g+ , 325

7h– → ● 7h+ , 116

7j– → ● 7j+ . 634

(8.172)

These last three contain minimal irreps with conformal weight d = 7 and M-irreps: (1, 2, 1, 1, 2), (2, 1, 1, 1, 1), and (1, 1, 1, 2, 1), respectively, which are conserved currents w.r.t. the invariant differential operators shown on (8.172). Finally, we have a triple reduction – the last three doublets from (8.171) and (8.172) reducing to a self-conjugate singlet: 7s = {(m2 , m4 , m5 , m4 , m2 ); 0},

R12 136 .

(8.173)

It contains a family of subrepresentations characterized by three positive integers which are denoted in our context as m2 , m4 , m5 , with c = 0, thus, conformal weight d = 152 . The subrepresentations are kernels of the invariant differential operators D2,15 , D4,24 (which intertwine with two unphysical representations).

8.6 The Lie Algebras E7(–25) and E7(7) This section is based mostly on [144, 147]. Let G = E7(–25) . The maximal compact subgroup is K ≅ e6 ⊕ so(2). We work with maximal parabolic P = M ⊕ A ⊕ N with M ≅ E6(–26) . We label the signature of the ERs of G as follows: 7 = {n1 , . . . , n6 ; c },

nj ∈ N,

c = d – 9,

(8.174)

8.6 The Lie Algebras E7(–25) and E7(7)

333

where the last entry of 7 labels the characters of A, and the ﬁrst six entries are labels of the ﬁnite-dimensional nonunitary irreps of M, (or of the ﬁnite-dimensional unitary irreps of the compact e6 ). The signatures expressed through the Dynkin labels: c = – 21 (m!˜ + m7 )

n i = mi , = –

1 2 (2m1

(8.175)

+ 2m2 + 3m3 + 4m4 + 3m5 + 2m6 + 2m7 ).

The same holds for the parabolically related exceptional Lie algebra E7(7) (with M-factor E6(6) ). Its maximal compact subgroup is K ≅ su(8). This algebra has discrete series representations (as rankG = rankK), but no highest/lowest weight representations. The M-noncompact roots of the complex algebra E7 are !7 , !17 , . . . , !67 , !1,37 , !2,47 , !17,4 , !27,4 ,

(8.176)

!17,34 , !17,35 , !17,36 , !17,45 , !17,46 , !27,45 , !27,46 , !17,25,4 , !17,26,4 , !17,35,4 , !17,36,4 , !17,26,45 , !17,36,45 , !17,26,35,4 , !17,26,45,4 , ˜ !17,16,35,4 = !, using notation !ij = !i + !i+1 + ⋅ ⋅ ⋅ + !j , i < j, !ij,k = !k,ij = !i + !i+1 + ⋅ ⋅ ⋅ + !j + !k ,

(8.177) i ≐ { k ,

= (,1 , . . . , ,k ) ∈ k A+ | ,1 ≥ . . . ≥ ,k },

(9.13)

where some ordering of H∗ (e.g., the lexicographical one) is implemented. Let , ∈ A+ and let Pk> (,) = # of elements k - = (-1 , . . . , -k ) ∈ k A+> such that 3(k -) = ,, each such ele( ment being taken with multiplicity kj=1 P(-j ). Clearly, Pk (0) = 1, Pk> (") = 1, ∀" ∈ BS . Now one can prove the following: Proposition 1: Let D ∈ H∗ and let D k V,

≐ {v ∈ k V D | H v = (kD(H) – ,(H)) v}.

(9.14)

Then we have kV

D

= ⊕

, ∈ A+

D k V, ,

dim k V D, = Pk> (,),

D k V, = + k - ∈k A> 3(k -) = ,

...

"1 ∈ B+ i G"1 = -1 i "11 ≤"12 ≤...≤"1n 1

"k ∈ B+ i G"k = -k i "k1 ≤"k2 ≤...≤"kn k

× D k V0 kV

D

X"–1 1

. . . X"–1 n1

⊗ ...

⊗ X"–k 1

ˆ F v0 , ⊗

. . . X"–k n

k

ˆ F v0 , = {1u ⊗ . . . ⊗ 1u } ⊗ = Sk (G – ) k V D0 ,

G + k V D0 = 0,

(9.15)

9.3 Singular Vectors of k-Verma Modules

357

where the ordering of the root system in the sums is inherited from the ordering of H∗ which is implemented. Proof. Completely analogous to the classical case k = 1 [120].

∎

9.3 Singular Vectors of k-Verma Modules 9.3.1 Definition In contrast to the ordinary Verma modules (k = 1), the k-Verma modules for k ≥ 2 are reducible independently of the highest weight, which is natural taking into account their tensor product character. This we show by exhibiting singular vectors for arbitrary highest weights. We call a singular vector of a k-Verma module k V D a vector vs ∈ k V D , such that vs ∉ k V D0 and X vs = 0,

∀ X ∈ G+,

H vs = (kD(H) – ,(H)) vs ,

, ∈ A+ ,

, ≠ 0, ∀ H ∈ H,

(9.16)

i.e., vs is homogeneous; vs ∈ k V D, for some , ∈ A+ . The space Sk (G – ) vs is a submodule of k V D isomorphic to the Verma module kD–, = Sk (G – ) ⊗ v0′ where v0′ is the highest weight vector of k V kD–, ; the isomorphism kV ˆ 0′ . is realized by vs ↦ {1u ⊗ . . . 1u }⊗v In the next subsections we show some explicit examples for the cases k = 2, 3. 9.3.2 k = 2 We consider now the case k = 2 i.e., bi-Verma modules (= 2-Verma modules). We take a weight , = n!, where n ∈ N and ! ∈ BS is any simple root. We have dim 2 V Dn! = [n/2] + 1, where [x] is the biggest integer not exceeding x. The possible singular vectors have the following form: n! 2 vs =

[n/2]

D #nj

X!–

n–j

j ˆ v0 . ⊗ X!– ⊗

(9.17)

j=0 D are determined from the ﬁrst condition in (9.16) with X = X + – all The coefﬁcients #nj ! other cases of (9.16) are fulﬁlled automatically. Thus we have

Proposition 2: The singular vectors of the bi-Verma (= 2-Verma) module 2 V D of weight , = n!, where n ∈ N and ! ∈ BS is any simple root, are given by formula (9.17) with the coefﬁcients #nj given explicitly (up to multiplicative renormalization) by

358

9 Multilinear Invariant Differential Operators from New Generalized Verma Modules

n A(Dˆ + 1 – n + j) A(Dˆ + 1 – j) j A(Dˆ + 1 – n) A(Dˆ + 1)

D = # #(n, D)(–1) j (2 – $ ˆ #nj 0 j,n/2 )

Dˆ ≡ D(H! ), ⎧ ⎪ ⎪ ⎨1, for ˆ #(n, D) = 1, for ⎪ ⎪ ⎩0, for

H! = [X!+ , X!– ],

n

ˆ even and arbitrary D; odd, Dˆ = n – 1, n – 2, . . . , (n – 1)/2;

n

odd, Dˆ ≠ n – 1, n – 2, . . . , (n – 1)/2;

n

and #0 is an arbitrary nonzero constant. ∎

Proof. By direct veriﬁcation.

We give the lowest cases of the above general formula for illustration ( ﬁxing the overall constant #0 appropriately): ! 2 vs 2! 2 vs 3! 2 vs

4! 2 vs

1 0 ˆ v0 , Dˆ = 0 = X!– ⊗ 1u ⊗ 2 ˆ v0 , = Dˆ X!– ⊗ 1u – (Dˆ – 1)X!– ⊗ X!– ⊗ 0 3 = Dˆ X!– ⊗ 1u 1 2 ˆ v0 , – 3(Dˆ – 2) X!– ⊗ X!– ⊗

(9.18) ∀Dˆ

Dˆ = 1, 2

4 = Dˆ (Dˆ – 1) X!– ⊗ 1u 3 – 4(Dˆ – 1)(Dˆ – 3) X!– ⊗ X!–

2 2 ˆ v0 , ⊗ + 3 (Dˆ – 2)(Dˆ – 3) X!– ⊗ X!–

∀Dˆ

Proposition 2 conﬁrms that bi-Verma modules are always reducible since they possess singular vectors independently of D. In fact, they have an inﬁnite number of singular vectors of weights n!i , for any even positive integer n and any simple root !i . Moreover, they possess singular vectors of other weights, also independent of D. For example we consider weights ,n = n" = n(!1 + !2 ), where " is a positive root, and !1 , and !2 are two simple roots, e.g., of equal minimal length ( for simplicity). Then there exist singular vectors of these weights given by, e.g., " 2 vs

= D1 X1– X2– ⊗ 1u – D2 X2– X1– ⊗ 1u ˆ v0 , – (D1 + D2 + 1) X1– ⊗ X2– ⊗ Di ≡ D(H!i ), i = 1, 2

∀D (9.19)

9.3 Singular Vectors of k-Verma Modules

2" 2 vs

=

359

a1 (X3– )2 ⊗ 1u + a2 X2– X3– X1– ⊗ 1u

+ a3 (X2– )2 (X1– )2 ⊗ 1u + b1 X3– X1– ⊗ X2– + b2 X2– X3– ⊗ X1–

(9.20)

+ c1 X2– (X1– )2 ⊗ X2– + c2 (X2– )2 X1– ⊗ X1– + d1 X3– ⊗ X3– + d2 X3– ⊗ X2– X1–

ˆ v0 , + d3 X2– X1– ⊗ X2– X1– + d4 (X2– )2 ⊗ (X1– )2 ⊗

∀D

where for the two solutions (present in this case) the coefﬁcients are a1 = D21 D2 (D1 + D2 + 1)(D2 + 1), a2 = –D1 D2 (D1 + D2 + 1)(D1 – D2 – 2), a3 = –D1 D2 (D1 + D2 + 1), b1 = –D1 (D1 + D2 + 1)(D1 + D2 )(D2 + 2), b2 = D1 D2 (D1 + D2 + 1)(D1 + D2 ), c1 = D1 (D1 + D2 + 1)(D1 + D2 ), c2 = D2 (D1 + D2 + 1)(D1 + D2 ), d1 = –D21 (D1 + D2 )(D22 + D2 + 1), d2 = D1 (D1 + D2 ), (D1 D2 + 2D1 – D22 + D2 ), d3 = –(D1 + D2 )(D21 + D1 D2 + D22 ), d4 = 0,

(9.21)

a1 = 2D21 D2 (D1 + D2 + 1)(D22 – D2 – 1 + D1 D2 ), a2 = –2D1 (D1 + D2 + 1)(D1 + D2 – 1)(D1 – D2 )(D2 + 1), a3 = D1 (D1 + D2 + 1)(D1 + D2 – 1)(D1 – D2 ), b1 = –2D1 (D1 + D2 + 1)2 (D1 + D2 )(D2 – 1), b2 = 2D1 (D1 + D2 + 1)(D1 + D2 )(D22 – 2D2 – 1 + D1 D2 + D1 ), c1 = 0, c2 = –2(D1 + D2 + 1)(D1 + D2 – 1)(D1 + D2 )(D1 – D2 ), d1 = –2D21 (D1 + D2 )(D2 – 1)(D1 D2 + D1 + D22 ), d2 = –2D1 (D1 + D2 )(D2 – 1)(D22 – 2D2 – 1 – D21 ), d3 = –2(D1 + D2 )(D2 – 1)(D1 D2 + D1 + D22 ), d4 = D1 (D1 + D2 )(D1 + D2 + 1)2 .

(9.22)

360

9 Multilinear Invariant Differential Operators from New Generalized Verma Modules

Furthermore, the structure of the Verma modules is additionally complicated since there is no factorization of the singular vectors as happened for k = 1. For the latter consider the weight ,0 = 2!1 + !2 , with !1 and !2 as in the previous example. Then there exists a singular vector of this weight given by – 2 – ,0 X2 ⊗ 1 u 2 vs = (2D1 D2 + D1 – D2 – 1) X1 2 – 2D1 D2 X1– X2– X1– ⊗ 1u + (D1 + D2 + 1) X1– ⊗ X2– – 2 (D1 – 1) (D2 + 1) X1– X2– ⊗ X1– ˆ 0, + 2D2 (D1 – 1) X2– X1– ⊗ X1– ⊗v

∀ D1 , D2 .

(9.23)

9.3.3 k = 3 Here we consider the case k = 3, i.e., tri-Verma modules over arbitrary G. Consider a weight , = n!, where n ∈ N and ! ∈ BS is any simple root. We ﬁrst note the dimension of the weight space. ! " ! " dim3 V Dn! = n – 3 n6 1 + n6 + $ n , n . (9.24) 6

6

The possible singular vectors have the following form:

j k n–j–k n! D ˆ v0 . X!– ⊗ #njk ⊗ X!– ⊗ X!– 3 vs =

(9.25)

j,k∈Z+ n–j–k≥j≥k

D are determined from the ﬁrst condition (9.16) with X = X + ; all other The coefﬁcients #njk ! cases in (9.16) are fulﬁlled automatically. We give now the singular vectors for n ≤ 6 (again using Dˆ = D(H! )): 1 0 – ! ˆ v0 , Dˆ = 0, (9.26) 3 vs = X! ⊗ 1u ⊗ 1u ⊗ 0 – 2 2! ˆ ⊗ 1u ⊗ 1u 3 vs = D X! 1 ˆ ˆ v0 , – (Dˆ – 1)X!– ⊗ X!– ⊗ 1u ⊗ ∀D, 2 3! – 3 ˆ ⊗ 1u ⊗ 1u 3 v s = D X!

2 – 3 Dˆ (Dˆ – 2) X!– ⊗ X!– ⊗ 1u

ˆ v0 , + 2 (Dˆ – 1)(Dˆ – 2)X!– ⊗ X!– ⊗ X!– ⊗ 4! 3 vs

ˆ ∀D,

4 = Dˆ (Dˆ – 1) X!– ⊗ 1u ⊗ 1u 3 – 4 (Dˆ – 1)(Dˆ – 3) X!– ⊗ X!– ⊗ 1u

2 2 ˆ v0 , + 3 (Dˆ – 2)(Dˆ – 3) X!– ⊗ X!– ⊗ 1u ⊗

ˆ ∀D,

9.3 Singular Vectors of k-Verma Modules

′4! 3 vs

=

X!–

4

3 ⊗ 1u ⊗ 1u + 8 X!– ⊗ X!– ⊗ 1u

2 ˆ v0 , + 12 X!– ⊗ X!– ⊗ X!– ⊗ 5! 3 vs

361

Dˆ = 1,

2 5 = Dˆ (Dˆ – 1) X!– ⊗ 1u ⊗ 1u ˆ Dˆ – 1)(Dˆ – 4) X – 4 ⊗ X – ⊗ 1u – 5 D( ! ! – 3 – 2 + 2 Dˆ (Dˆ – 3) (Dˆ – 4) X! ⊗ X! ⊗ 1u 3 + 8 (Dˆ – 1)(Dˆ – 3)(Dˆ – 4) X!– ⊗ X!– ⊗ X!–

2 2 ˆ v0 , –6 (Dˆ – 2)(Dˆ – 3)(Dˆ – 4) X!– ⊗ X!– ⊗ X!– ⊗

′5! 3 vs

2 3 = 2 X!– ⊗ X!– u ⊗ 1u 3 ˆ v0 , – X!– ⊗ X!– ⊗ X!– ⊗

6! 3 vs

Dˆ = 2,

4 2 = Dˆ (Dˆ – 1) (Dˆ – 2) X!– ⊗ X!– ⊗ 1u 4 – (Dˆ – 1)2 (Dˆ – 2) X!– ⊗ X!– ⊗ X!– 3 3 – Dˆ (Dˆ – 1)(Dˆ – 3) X!– ⊗ X!– ⊗ 1u 3 2 + 2 (Dˆ – 1)(Dˆ – 2)(Dˆ – 3) X!– ⊗ X!– ⊗ X!– 2 2 2 ˆ v0 , ⊗ – (Dˆ – 2)2 (Dˆ – 3) X!– ⊗ X!– ⊗ X!–

′6! 3 vs

ˆ ∀D,

ˆ ∀D,

6 = Dˆ (Dˆ – 1) (Dˆ – 2) X!– ⊗ 1u ⊗ 1u 5 – 6 (Dˆ – 1)(Dˆ – 2)(Dˆ – 5) X!– ⊗ X!– ⊗ 1u 4 2 + 5 (Dˆ – 2)(Dˆ – 4)(Dˆ – 5) X!– ⊗ X!– ⊗ 1u

3 3 ˆ v0 , – 10 (Dˆ – 3)(Dˆ – 4)(Dˆ – 5) X!– ⊗ X!– ⊗ 1u ⊗

ˆ ∀D.

We give those examples in order to point out some new features appearing for triVerma modules in comparison with the bi-Verma modules. – Independently of the value of D(H! ) there exists a singular vector at any level n!, except the lowest n = 1, while for bi-Verma modules singular vectors at odd levels exist only for special values of D(H! ); –

There exists more than one singular vector at any ﬁxed level n! for n ≥ 6 and arbitrary D(H! ). For special values of D(H! ) there exists a second singular vector for n = 4, 5.

362

9 Multilinear Invariant Differential Operators from New Generalized Verma Modules

Similar facts hold for k-Verma modules for k > 3. These questions are left for further study. Here we would like to demonstrate on examples the usefulness of these modules, which we do in the following sections.

9.4 Multilinear Invariant Differential Operators Here we generalize our construction of invariant differential operators to multilinear invariant differential operators. We have the following. Proposition 3: Let the signature 7 of an ER be such that the k-Verma module k V D with , highest weight D = D7 , has a singular vector k vs ∈ k V D, , i.e., (9.16) is satisﬁed for some + , ∈ A . Let us denote , , k vs = k P

ˆ v0 , ⊗

(9.27)

where k P , ∈ Sk (G – ) is some concrete polynomial as in (9.15c). Then there exists a multilinear invariant differential operator which we denote by k I,D such that D k I,

: > ⊗ ⋅ ⋅ ⋅ ⊗ > ↦ 8, # $% &

> ∈ C 7 , 8 ∈ C 7′ ;

(9.28)

k D k I,

○

k

j=1 1 #u

7 (X) ⊗ ⋅ ⋅ ⋅ ⊗ 1u = L 7′ (X) ○ k I D, , ⊗ ⋅⋅⋅ ⊗ L $% &

∀X ∈ G,

k

where 7′ is uniquely determined (up to the representation parameters of the discrete subgroup M d ) so that D′ = D7′ = kD – ,. The operator is given explicitly by the same polynomial as in (9.27), i.e.: D k I, (>

#

⊗ ⋅ ⋅ ⋅ ⊗ >) = k4 P , (> ⊗ ⋅ ⋅ ⋅ ⊗ >), # $% & $% & k

(9.29)

k

where the hat on k P , symbolizes the right action (deﬁned as for k = 1), the explicit action of a typical term of k P , being (cf. (9.15)). 4 X"–1 . . . X"–1 ⊗ ⋅ ⋅ ⋅ ⊗ X –k . . . X –k > ⊗ ⋅ ⋅ ⋅ ⊗ > "1 "n # $% & n1 1 k

= Xˆ "–1 . . . Xˆ "–1 1

n1

k

– – ˆ ˆ > ... X k ...X k > .

Proof. Completely analogous to the case k = 1.

"1

"n

(9.30)

k

∎

9.5 Bilinear Operators for SL(n, R) and SL(n, C)

363

Remark 2: The analog of the intertwining property (9.28) on the group level, i.e., D k I,

′

˜ ˜ ○ h(g) ⊗ ⋅ ⋅ ⋅ ⊗ h(g) = T 7 (g) ○ k I D, , # $% &

∀g ∈ G

(9.31)

k

will hold, in general, for less values of D than (9.28). This in sharp contrast with the k = 1 case, where there is no difference in this respect. An additional feature on the group level common for all k ≥ 1 is that some discrete representation parameters of 7, not represented in D, get ﬁxed. ♢ Remark 3: Let us stress that since we have realized arbitrary representations in the spaces of scalar-valued functions > then also the invariant differential operators are scalar operators in all cases; geometrically speaking, these operators intertwine (tensor products of) line bundles. This simplicity may be contrasted with the proliferation of tensor indices in the approaches relying on Weyl’s SO(n) polynomial invariant theory. ♢ Finally, we should mention that the simplest formulae are obtained of one restricts the ˜ functions to the conjugate to N subgroup N: ˜ ˜ ≐ >(n), ˜ n˜ ∈ N}. C˜ 7 ≐ {6 = R> | > ∈ C 7 , (R>)(n)

(9.32)

Clearly, the elements of C 7 and consequently C˜ 7 are almost determined by their values on N˜ because of right covariance and because N˜ is an open dense submanifold of G/P, P = MAN. The ER T 7 acts in this space by

˜ = e-(H) D, (m–1 )6(n˜ ′ ), T 7 (g) 6 (n)

˜ = D, (m) >(n) ˜ = v0 , D, (m) F (n), ˜ D, (m) 6(n)

(9.33)

˜ m ∈ M, a = exp(H), and H ∈ A, and we have used the Bruhat de˜ n˜ ′ ∈ N, where g ∈ G, n, –1 composition g n˜ = n˜ ′ man, (n ∈ N). One may easily check that the restriction operator R intertwines the two representations, i.e.: T 7 (g) R = RT 7 (g),

∀g ∈ G.

(9.34)

9.5 Bilinear Operators for SL(n, R) and SL(n, C) 9.5.1 Setting In the present section we restrict ourselves to the case G = SL(n, R), mentioning also which results are extendable to SL(n, C). We use the following matrix representations for G, its Lie algebra G and some subgroups and subalgebras:

364

9 Multilinear Invariant Differential Operators from New Generalized Verma Modules

G = SL(n, R) = {g ∈ GL(n, R) | det g = 1},

(9.35)

G = sl(n, R) = {X ∈ gl(n, R) | tr X = 0}, K = SO(n) = {g ∈ SL(n, R) | g g t = g t g = 1n }, A0 = {X ∈ sl(n, R) | X diagonal}, A0 = exp(A0 ) = {g ∈ SL(n, R) | g diagonal}, 1 0 M0 = m ∈ K | ma = am, ∀a ∈ A0 = {m = diag($1 , $2 , . . . , $n ) | $k = ±, N0 = {X = (aij ) ∈ gl(n, R) | aij = 0,

$1 $2 . . . $n = 1} = M0d ,

i ≥ j},

N0 = exp(N0 ) = {n = (nij ) ∈ GL(n, R) | nii = 1, 0 = {X = (aij ) ∈ gl(n, R) | aij = 0, N

nij = 0,

i > j},

n˜ ij = 0,

i < j}.

i ≤ j},

0 ) = {n˜ = (n˜ ij ) ∈ GL(n, R) | n˜ ii = 1, N˜ 0 = exp(N

Since the algebra sl(n, R) is maximally split then the Bruhat decomposition with the minimal parabolic 0 G = sl(n, R) = N0 ⊕ A0 ⊕ N

(9.36)

may be viewed as a restriction from C to R of the triangular decomposition of its complexiﬁcation G C = sl(n, C) = G+C ⊕ H ⊕ G–C .

(9.37)

Accordingly, we may use for both cases the same Chevalley basis consisting of the 3(n – 1) generators Xi+ , Xi– , and Hi given explicitly by Xi+ = Ei,i+1 ,

Xi– = Ei+1,i ,

Hi = Eii – Ei+1,i+1 ,

i = 1, . . . , n – 1,

(9.38)

where Eij are the standard matrices with 1 on the intersection of the ith row and jth 0 , column and zeroes everywhere else. Note that Xi+ , Xi– , and Hi , resp., generate N0 , N and A0 , resp., over R and G+C , G–C , and H, resp., over C. 9.5.2 Minimal Parabolic We consider induction from the minimal parabolic case, i.e., P = M0 A0 N0 . The characters of the discrete group M0 = M0d are labeled by the signature : = (:1 , :2 , . . . , :n–1 ), :k = 0, 1:

9.5 Bilinear Operators for SL(n, R) and SL(n, C)

ch: (m) = ch: ($1 , $2 , . . . , $n ) ≐

n–1 '

($k ):k ,

m ∈ M0 .

365

(9.39)

k=1

The (nonunitary) characters - ∈ A∗0 of A0 are labeled by ck ∈ C, k = 1, . . . , n – 1, which is the value of - on Hk = diag(0, . . . , 0, 1, –1, 0, . . . , 0) ∈ A0 (with the unity on kth place), k = 1, . . . , n – 1, i.e., ck = -(Hk ):

chc (a) = chc exp

= exp

tk Hk

k

= exp

'

tk ck =

k

a=

'

tk -(Hk )

(9.40)

k c

aˆ kk ,

k

ak ∈ A – 0,

ak = exp tk Hk ∈ A0 ,

k

tk , aˆ k = exp tk ∈ R. Thus, the right covariance property is F (gman) =

n–1 ' c ($k ):k aˆ kk F (g)

(9.41)

k=1

and in this case we have scalar functions, i.e., > = F because M0 is discrete. Thus, the ER acts on the restricted functions as (cf. (9.33)) n–1 ' c ˜ = T c,: (g) 6 (n) ($k ):k aˆ kk 6(n˜ ′ )

(9.42)

k=1

(note that $k = ($k )–1 ). The functions 6 depend on the n(n – 1)/2 nontrivial elements of the matrices of N˜ 0 . For further use those will be denoted by zji , i.e., for n˜ ∈ N˜ 0 we have n˜ = (n˜ ij ) with n˜ ij = zji for i > j (cf. (9.35)). The correspondence between the ER with signature 7 = [c, :] and the highest weight D, used in the general construction of the previous section, here is very simple. D = – -, so that D(H) = – -(H). Further, we recall that the root system of sl(n, C) is given by roots ±!ij , i < j, so that !ij = !i + !i+1 + ⋅ ⋅ ⋅ + !j–1 for i + 1 < j and !i,i+1 = !i , where !i , i = 1, . . . , n – 1 are the simple roots with non-zero scalar products (!i , !i ) = !i (Hi ) = 2, (!i , !i+1 ) = !i (!i+1 ) = – 1, then !∨i = !i . Then we use (-, !i ) = -(Hi ) = ci . We also need the inﬁnitesimal version of (9.42): c (Y)6(n) ˜ ≐ L

d c,: ˜ T (exp tY) 6 (n) dt

, | t=0

Y ∈ G,

(9.43)

366

9 Multilinear Invariant Differential Operators from New Generalized Verma Modules

which we give for the Chevalley generators (9.38) explicitly

n n

c + i+1 k k ci + Ni – Ni+1 L (Xi ) = zi k = i+1

–

i–1

n

zsi+1 Dis +

s=1

(9.44)

k = i+2

zik Dki+1 ,

k = i+2

c (X – ) = –Di+1 – L i i

i–1

zsi Di+1 s ,

s=1 n

c (Hi ) = ci + L

Nik –

k = i+1

–

i–1

Nsi +

s=1

where Dij ≡

∂ , ∂zji

Nji ≡ zji

∂ , ∂zji

n

k Ni+1

k=i+2 i

Nsi+1 ,

s=1

and we are using the convention that when the lower

summation limit is bigger than the higher summation limit then the sum is zero. We also need the right action for the lowering generators which on the restricted functions is given explicitly by

– ˜ = Xˆ i 6(n)

Di+1 i +

n

k zi+1 Dki

˜ 6(n).

(9.45)

k = i+2

Naturally, the signature : representing the discrete subgroup M = Md0 is not present in (9.44) and (9.45). Thus, formulae (9.44) and (9.45) are valid also for the holomorphic ERs of SL(n, C). Now to obtain explicit examples of multilinear invariant differential operators it remains to substitute formula (9.45) in the corresponding formulae for the singular vectors of the k-Verma modules. We note that often a singular vector will produce many invariant differential operators. For example each formula valid for any simple root will produce n – 1 formula, and each formula valid for roots as !1 + !2 will produce n – 2 formulae for each !i + !i+1 . To save space we shall not write these formulae except in a few examples in the cases n = 2 and n = 3. 9.5.3 SL(2, R) Now we restrict ourselves to the case G = SL(2, R). We denote x = z12 , c = c1 , : = :1 , X ± = X1± , H = H1 . We start with bilinear invariant differential operators, k = 2. We combine Propositions 2 and 3. For G = sl(2, R) Proposition 2 gives all singular vectors of

9.5 Bilinear Operators for SL(n, R) and SL(n, C)

367

bi-Verma modules since all weights in A+ are of the form , = n!, n ∈ N. Thus we have the following theorem. Theorem 1: All bilinear invariant differential operators for the case of G = sl(2, R) are given by the formula D 2 In! (6)

=

[n/2]

D (n–j) (j) #nj 6 6 ,

(9.46)

j=0 D are given in Proposition 2. where 6(p) ≐ (∂x )p 6(x), ∂x ≐ ∂/∂x, and the coefﬁcients #nj The intertwining property is D 2 In!

c c (X) = L c′ (X) ○ 2 I Dn! , (X) ⊗ 1u + 1u ⊗ L ○ L

∀X ∈ G

c′ = 2(c + n).

c = – D(H! ),

(9.47)

Proof. Elementary combination of Propositions 2 and 3 in the present setting. In particular, D′ = 2D – n!, c′ = – D′ (H! ) = – 2D(H! ) + n!(H! ) = 2(c + n). ∎ As we mentioned, the corresponding intertwining property on the group level is restricting the values of D and, as for k = 1, of some discrete parameters not represented in D. In the present case, we have the following theorem. Theorem 2: All bilinear invariant differential operators for the case of G = SL(2, R) are given by formulae (9.46) and (9.18) with “integer” highest weight D(H! ) = p ∈ Z. The intertwining property is D 2 In!

′ ′ ○ T c,: (g) ⊗ T c,: (g) = T c ,: (g) ○ 2 I Dn! , c = – D(H! ) = – p,

∀g ∈ G,

(9.48)

c′ = 2(c + n) = 2(n – p),

: = :′ = p (mod 2).

Proof. Follows by using Theorem 1 and checking (9.48) for g = x ≠ 0 into –1/x.

0 1 which sends –1 0 ∎

Thus, the signatures of the two intertwined spaces coincide and are determined by the parameter p. Next we consider the example of invariant functions 6, i.e., functions for which the transformation law (9.42) has no multipliers; this happens iff c = : = 0. For these functions the bilinear invariant differential operators are given by the special case of Theorem 2 when D(H! ) = p = 0, which by (9.18) further restricts n to be even or n = 1. Formula (9.46) with (9.18) substituted simpliﬁes to

368

9 Multilinear Invariant Differential Operators from New Generalized Verma Modules

0 2 In! (6) =

n/2

j=1

1 n – j n n – 1 (n–j) (j) (–1)j–1 1 – $j, n 6 , n ∈ 2N; 6 2 2 n(n – 1) j j–1

(9.49)

= 6∂x 6 = 66′

(9.50)

0 2 I! (6)

and in addition we have ﬁxed the constant #0 for later convenience. Let us write out the ﬁrst several cases of (9.49). 1 ′ 2 6 2

0 2 I2! (6)

=

0 2 I4! (6)

3 = 6′′′ 6′ – (6′′ )2 2

0 2 I6! (6)

= 6(5) 6′ – 106(4) 6′′ + 10(6′′′ )2

(9.51)

where (standardly) 6′ ≡ ∂x 6 = 6(1) , 6′′ ≡ ∂x2 6 = 6(2) , and 6′′′ ≡ ∂x3 6 = 6(3) . Note that the second case of (9.51) (i.e., (9.49) for n = 4) was already given in (9.3). We give now two important technical statements. Lemma 1: For n > 2 the ( formal) substitution 6(x) ↦ operators (9.49) gives zero. 0 2 In! (60 )

= 0,

60 (x) ≡

!x – # , $ – "x

!x–# $–"x

in the invariant differential

n ∈ 2 + 2N.

(9.52)

Proof. By direct substitution. In the calculations one uses the fact ∂xm

!x – # (–1)m m!"m–1 , = $ – "x ($ – "x)m+1

m ∈ N.

(9.53)

After the substitution of (9.53) in (9.49) the resulting expression is proportional to (1 – 1)n–2 which is zero for n > 2 (the latter making clear why the lemma is not valid for n = 2). ∎ Lemma 2: Let 6, 8 ∈ Diff0 S1 , the group of orientation preserving diffeomorphisms of the circle S1 = R/20Z. Then we have n ○ 8) = 8′ 2 I 0n! (6) , n = 1, 2; ′ n 0 ′ 2 0 0 2 I n! (8) 2 In! (6 ○ 8) = 8 2 In! (6) + 6 0 2 In! (6

+ Pn (6, 8),

(9.54)

n ∈ 2 + 2N;

P4 (6, 8) = 0; Pn (6, 60 ) = 0,

n ∈ 4 + 2N.

(9.55)

9.5 Bilinear Operators for SL(n, R) and SL(n, C)

369

Proof. For (9.54) this is just substitution. Further we note that 0 2 In! (6

n ○ 60 ) = 6′0 2 I 0n! (6),

60 (x) =

!x – # $ – "x

0 and then (9.55) follows because of is just the intertwining property of 2 In! Lemma 1.

(9.56)

∎

Remark 4: We give an example from the last lemma. 2 ′′′ ′ 2 0 P6 (6, 8) = 10 8′ 36 6 – 4 6′′ 2 I4! (8) 2 3 –56′′ 6′ 8(4) 8′ – 68′′′ 8′′ 8′ + 6 8′′ .

(9.57)

Using (9.53) it is straightforward to show P6 (6, 60 ) = 0. In fact, the ﬁrst term in (9.57) vanishes for 8 = 60 because of (9.52). The vanishing of the second term in (9.57) prompts us that the trilinear expression in 8 is also a invariant differential operator. This is indeed so (cf. the last section for some more examples for trilinear operators). ♢ We can introduce now a hierarchy of SL(2, R) invariant 0 6 Schn (6) ≐ 2 In! ( )

dx 6′

n/2

Schn (6 ○ 60 ) = Schn (6) ○ 60 ,

n 2

differentials for every n ∈ 2N.

n ∈ 2N;

,

!x–# 60 (x) = $–"x ,

(9.58) (9.59)

where the second property (9.58) is just a restatement of (9.56) for n > 2 and (9.54) for n = 2. The usual Schwarzian Sch4 is one of these objects (cf. (9.5)). It has an additional property Sch4 (6 ○ 8) = Sch4 (6) ○ 8 + Sch4 (8) ,

6, 8 ∈ Diff0 S1

(9.60)

showing that it is a 1-cocycle on Diff0 S1 [339]. Let us introduce the following notation: 2 n (6) ≐ Sch

1 0 2 I (6) . 6′2 n!

(9.61)

It is well known (cf., e.g., [611]) that the famous KdV equation may be rewritten in the Krichever–Novikov form 2 4 ( f ) f ′ = 0, ∂t f + Sch

f = f (t, x),

(9.62)

370

9 Multilinear Invariant Differential Operators from New Generalized Verma Modules

2 4 ( f ) is the Schwarz derivative S[f ] (cf. (9.5)). To pass to the standard noting that Sch KdV form ∂t u + u′′′ – 6uu′ = 0,

(9.63)

2 4 ( f ). Motivated by the above we make the conjecone uses the substitution u = – 21 Sch ture that the following equations 2 n ( f ) f ′ = 0, ∂t f + Sch

n ∈ 2N + 2

(9.64)

are integrable (true for n = 4). It may happen (if the conjecture is true) that this hierarchy of equations coincides with the KdV one. Then formulae (9.64), (9.61), and (9.49) would give an explicit expression for the whole KdV hierarchy in the Krichever–Novikov form. Remark 5: One may also consider half-differentials and using (9.58a) and (9.50) write Sch1 (6) ≐ 2 I!0 (6)

dx 6′

1/2

1/2 = 6 6′ dx = 6 (d6)1/2

(9.65) ♢

The second property (9.58) then follows from (9.54).

Finally, we just mention the case when the resulting functions are invariant c′ = 0 ⇒ c = –n. This is only possible when n is even (cf. (9.18)). Formula (9.46) with (9.18) substituted simpliﬁes considerably –n 2 In! (6)

=

n/2

1 (–1)j 1 – $j,n/2 6(n–j) 6(j) 2

(9.66)

j=0

= 6(n) 6 – 6(n–1) 6′ + 6(n–2) 6′′ –6(n–3) 6′′′ + ⋅ ⋅ ⋅ +

2 1 (–1)n/2 6(n/2) , 2

n ∈ 2N

and we have ﬁxed the constant #0 appropriately. 9.5.4 SL(3, R) Now we consider the case G = SL(3, R). We denote x = z12 , y = z23 , and z = z13 . The right action is (6 = 6(x, y, z)). – Xˆ 1 6 = (∂x + y∂z ) 6,

– Xˆ 2 6 = ∂y 6,

– Xˆ 3 6 = ∂z 6,

and we have given it also for the nonsimple root vector X3– = [X2– , X1– ].

(9.67)

9.5 Bilinear Operators for SL(n, R) and SL(n, C)

371

The bilinear operator corresponding to (9.19) is 3 D 2 I! (6)

= 6 (D1 – D2 ) 6xy + y6yz – D2 6z – (D1 + D2 + 1) 6y (6x + y6z ),

where ! = !3 = !1 + !2 , 6x ≡ are given by 3 D 2 I2! (6)

(9.68)

∂6 ∂x , etc. The two bilinear operators corresponding to (9.20)

= (a1 + a2 + 2a3 ) 6 6zz + (a2 + 2a3 ) 6 6xyz

(9.69)

+ (a2 + 4a3 ) y 6 6yzz + a3 6 6xxyy + y 6xyyz + y2 6yyzz + (b1 + 2c1 ) 6y (6xz + y 6zz ) + (b2 + 2c2 ) (6x + y 6z ) 6yz + c1 6y 6xxy + 2y 6xyz + y2 6yzz + c2 (6x + y 6z ) 6xyy + y 6yyz + (d1 + d2 + d3 ) 62z + (d2 + 2d3 ) 6z 6xy + y 6yz 2 + d3 6xy + y 6yz + d4 62y (6x + y 6z )2 with constants as given in (9.21) or (9.22). The intertwining property is 3 D 2 In!

c c (X) = L c′ (X) ○ 32 I D , (X) ⊗ 1u + 1u ⊗ L ○ L n!

∀X ∈ G,

(9.70)

where ci = –Di = –D(Hi ), D′ = 2D–n!, c′i = –D′ (Hi ) = –2D(Hi )+n!(Hi ) = –2Di +n = 2ci +n, i = 1, 2, since !(Hi ) = (!1 + !2 )(Hi ) = 1. The case of invariant functions, i.e., ck = :k = 0 gives a trivial (zero) operator n = 2, while for n = 1 we have (up to scalar multiple) 3 0 2 I! (6) = 6y

(6x + y6z ).

(9.71)

In the case of invariant resulting functions, i.e., c′k = :′k = 0, D = n!/2, Di = n/2, we have from (9.68). 3 !/2 2 I! (6) =

1 6z + 2 6y (6x + y6z ), 2

(9.72)

372

9 Multilinear Invariant Differential Operators from New Generalized Verma Modules

while from (9.69) we have two operators corresponding to the two solutions given by (9.21) and (9.22), resp., 3 ! 2 I2! (6)

= 2 6 6zz – 2 y 6 6yzz – 6 6xxyy + y 6xyyz + y2 6yyzz

(9.73)

– 2 6y (6xz + y 6zz ) + 6 (6x + y 6z ) 6yz + 2 6y 6xxy + 2y 6xyz + y2 6yzz + 2 (6x + y 6z ) 6xyy + y 6yyz 2 – 2 62z – 2 6z 6xy + y 6yz – 2 6xy + y 6yz , 3 ′! 2 I2! (6)

= 62y (6x + y 6z )2 .

(9.74)

9.6 Examples with k ≥ 3 We return now to the SL(2, R) setting to give examples of trilinear invariant differential operators using the singular vectors of tri-Verma modules above. The trilinear invariant differential operators for the case of G = sl(2, R) are given by the formula

D D #njk 6(n–j–k) 6(j) 6(k) , (9.75) 3 In! (6) = j,k∈Z+ n–j–k≥j≥k

D are given from the expressions for the corresponding where the coefﬁcients #nj singular vectors of tri-Verma modules, e.g., those given in the previous subsection. If we pass to the group level then the possible weights are restricted to be “integer” (as in Theorem 2): D(H! ) = p ∈ Z and the corresponding intertwining property is c,: D c,: c,: c′ ,:′ (g) ○ 3 I Dn! , ∀g ∈ G, (9.76) 3 In! ○ T (g) ⊗ T (g) ⊗ T (g) = T

c = – D(H! ) = – p,

c′ = 3(c + n) = 3(n – p),

: = :′ = p (mod 2). Next we restrict to the example of invariant functions 6, i.e., c = : = 0. The trilinear invariant differential operators obtained from the singular vectors in (9.26) are 0 3 I! (6) 0 3 I2! (6) 0 3 I3! (6) 0 3 I4! (6) 0 3 I5! (6) 0 3 I6! (6) ′0 3 I6! (6)

= (6)2 6′ , 2 = 6 6′ , 3 = 6′ , 3 = 6 6′′′ 6′ – (6′′ )2 , 2 3 = 6′ 6′′′ 6′ – (6′′ )2 , 2 ′ 2 (4) 6 – 66′′′ 6′′ 6′ + 6 (6′′ )3 , =6 = 6 6(5) 6′ – 106(4) 6′′ + 10(6′′′ )2 .

We recall that the last but one case of (9.77) has appeared in (9.57).

(9.77)

9.6 Examples with k ≥ 3

!x–# $–"x

373

Analogously to Lemma 1, we note that for n > 3 the ( formal) substitution 6(x) ↦ in the invariant differential operators (9.77) gives zero: 0 3 In! (60 )

= 0,

!x – # , $ – "x

60 (x) ≡

n>3

(9.78)

which because of the factorization follows from Lemma 1 except for the last but one case in (9.77). Analogously to Lemma 2 for 6, 8 ∈ Diff0 S1 , one can check for the examples in (9.77): 0 3 In! (6

n ○ 8) = 8′ 3 I 0n! (6) ,

0 3 I5! (6

5 0 3 ○ 8) = 8′ 3 I5! (6) + 6′ 3 I 05! (8) ,

0 3 I6! (6

6 0 3 ○ 8) = 8′ 3 I6! (6) + 6′ 3 I 06! (8)

n = 1, 2, 3,

(9.79)

2 0 – 26′′ 6′ 8′ 2 I4! (8) . Consider now the case of resulting invariant functions, and c′ = :′ = 0 and p = D(H! ) = Dˆ = n. There is no operator for n = 1, while for n > 1 we get from (9.26): Dˆ = 2,

2! 3 vs

= 26′′ 62 – 6′2 6,

3! 3 vs

= 96′′′ 62 – 96′′ 6′ 6 + 4 6′3 ,

4! 3 vs

= 26(4) 62 – 26′′′ 6′ 6 + (6′′ )2 6,

5! 3 vs

= 256(5) 62 – 25 6(4) 6′ 6 + 56′′′ 6′′ 6

Dˆ = 4,

= 606(4) 6′′ 6 – 506(4) 6′2 – 45 6′′′2 6 + 606′′′ 6′′ 6′ – 24 (6′′ )3 ,

′6! 3 vs

Dˆ = 3,

Dˆ = 5,

+ 166′′′ 6′2 – 9(6′′ )2 6′ , 6! 3 vs

(9.80)

Dˆ = 6,

= 26(6) 62 – 26(5) 6′ 6 + 26(4) 6′′ 6 – 6′′′2 6,

Dˆ = 6.

We see that at lower levels there occur many factorizations and trilinear operators are actually determined by bilinear ones. We illustrate this by two statements for arbitrary k-Verma modules and the corresponding multilinear invariant differential operators. Proposition 4: The singular vectors of the k-Verma modules k V D of level n! with n ∈ N, n ≤ k, and ! ∈ BS in the case D(H! ) = 0 are given by ⎧ ⎫ ⎨ ⎬ n! – – ˆ v0 , k v s = #0 X ! ⊗ ⋅ ⋅ ⋅ ⊗ X ! ⊗ 1 u ⊗ ⋅ ⋅ ⋅ ⊗ 1 u ⊗ $% &⎭ ⎩# $% & # n

1 ≤ n ≤ k,

k–n

D(H! ) = 0.

(9.81)

374

9 Multilinear Invariant Differential Operators from New Generalized Verma Modules

∎

Proof. By direct veriﬁcation.

Proposition 5: The SL(2, R) multilinear invariant differential operators with the property: 0 k In!

′ ′

○ T c,: (g) ⊗ ⋅ ⋅ ⋅ ⊗ T c,: (g) = T c ,: (g) ○ k I 0n! , # $% &

∀g ∈ G

k

c = – D(H! ) = 0,

c′ = kn,

: = :′ = 0

(9.82)

are given by 0 k In! (6)

n = 6n–k 6′ ,

1 ≤ n ≤ k,

D(H! ) = 0.

(9.83)

Proof. Follows from Propositions 3 and 4. We note that the operators in (9.50), and the ﬁrst case of (9.51), and the ﬁrst three cases of (9.77) are partial cases of (9.83). ∎

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Author Index Ado 16 Bargmann 145 Barut–Ra¸czka 11 Bernstein–Gel’fand–Gel’fand 67 Borel 65 Bourbaki 44 Chevalley 59 Dirac 162 Dixmier 65 Gel’fand 145 Gross–Wallach 146 Harish-Chandra 102, 145, 173 Helgason 102 Humphreys 251 Iwasawa 102 Jacobson 134

Kac 27–29, 32, 33 Kac–Kazhdan 67 Kac–Moody 11 Kazhdan–Lusztig 238 Knapp–Stein 142, 173 Knapp–Zuckerman 142 Kostant 151 Langlands 142 Satake 119 Serre 59 Shapovalov 68 Verma 65 Warner 101 Weyl 87 Zhelobenko 141

Subject Index AFF 27 Di 170 E9 33 E! 22 En 33 E10 33 FIN 27 G × H 20 G-module 138 G ⋉ H 20 H+ 23 LD 66 M-graded 49 N!" 24 SO∗ (2n) 134 SU ∗ (2n) 134 Sp(p, r) 134 A+ 86 (w) 26 A0 95 A0 -restricted root spaces 95 A0 -restricted roots 95 H-diagonalizable 63 H0 95 K 94 M0 95 M0 -compact roots 96 P0 100 Q 94 1c 63 k-Verma modules 353 an = su(n + 1) 94 bn = so(2n + 1) 94 cn = sp(n) 94 dn = so(2n) 94 e(,) 86 en 94 f4 94 g2 94 gl(V) 16 gl(n) 16 gl(n, F) 16 sl(n, F) 17 sl(n, R) 103 so(n, F) 17

sp(n, F) 17 (left) G-space 18 associated roots to the subsingular vector 241 restricted Weyl group 143 2-cocycle 48 Abelian algebra 11 acting (from the left) 18 adjoint representation 13 affine line algebra 15 affine type 27 affinization 50 algebra 11 antidominant 239 antiholomorphic discrete series representation 279 anti-semidominant 239 antihomomorphism 20 associated parabolic subalgebras 104 associative algebra 11 Aut(B) 52 automorphism 20 basic representation 72 bi-Verma modules 357 Borel subalgebra 64 Borel subgroup B 137 Bruhat order 88 Cartan automorphisms 94 Cartan decomposition 94 Cartan decomposition of G 136 Cartan subgroups 136 Cartan subspace 95 Cartan–Weyl basis 24 Casimir operator 62 central charge 80 central extension 14 center 14 character of G 138 character of V 86 Chevalley generators 59 classical compact Lie groups 135

406

Subject Index

classical complex Lie groups 135 classical Weyl character formula 87 commutative algebra 11 commutative group 18 commutator 12 commutator of ideals 13 compact real form 94 compact roots 96 compact symplectic group Sp(m) 134 complementary series of unitary representations 145 completely reducible 13, 139 complex roots 96 composite singular vector 241 conditionally invariant equation 238 conservation laws 288 conserved currents 288 contragredient representation 139 covariant tensors of rank k 61 Coxeter number 30 cuspidal 102 cyclic module 64, 139 cyclic submodule 139 cyclic vector 139 defining irreps 72 degree 49 derivation 13 descendant of the singular vector 241 direct product 20 direct sum of algebras 13 discrete series 140 dominant 239 dominant element 86 dual Coxeter number 30 Dynkin diagram 28 Dynkin labels 70 effective action 18 elementary representation equivalent representations Euclidean Lie algebra 15 exact linear representation exact representation 139 extended Dynkin diagrams factor-algebra 13 factor-group 19 factor-module 139 factor-representation 139

141 139 139 29

Fibonacci numbers 105 finite group 18 finite type 27 first reduction points 283 fundamental representations 72 Galilei Lie algebra 16 Gauss decomposition 137 general linear algebra 16 general linear group GL(n, F) 18 generalized Cartan matrix 27 generalized partition function P(,) 65 generalized principal series representations 141 generalized Verma modules (GVMs) 149 golden ratio 106 group 18 group homomorphism 19 group isomorphism 20 group of automorphisms 22 group of inner automorphisms 22 half-differentials 370 Harish-Chandra parameters 69 height 25 highest root 25 highest weight 64 highest weight module (HWM) 64 highest weight vector 64 highest weight GVM 279 holomorphic discrete series representation 145 homogeneous space 18 hull 24 hyperbolic type 32 imaginary roots 51 Ind 27 IndGH (4) 140 indecomposable representation 13, 139 indefinite type 27 induced G-module 64 induced representation 140 inner automorphisms 20 inner derivations 13 inner products in H and H∗ 23 integrable highest weight modules 79 integral element 86 intertwining differential operator 150 intertwining operator 139

Subject Index

invariant differential equation 151 invariant form 15 invariant functions 367 invariant operator 139 invariant subgroup 19 invariant subspace 138 irreducible representation 13, 139 irreps 72 Iwasawa decomposition 96, 136

407

multilinear invariant differential operators 353 noncompact real form 95 noncompact roots 96 nonsimply laced algebras 47 nonsingular Verma modules 80 normal divisor 19 normalizer of A0 in K 138 normalizer of K in G 21

Jacobi identity 11 K 296 k-Verma modules 354 Kac character formula 87 Kazhdan-Lusztig polynomials 88 Killing form 15 KS operators 142 Langlands decomposition 100 left factor-space 19 length of reflection 26 Levi–Malcev decomposition 15 lexicographical ordering 24 Lie algebra 11 Lie group 133 limits of discrete series 153 linear group 139 linear quaternionic group SL(n, IH) 134 loop algebra 47 lowest weight 64 lowest weight GVM 279 lowest weight module 64 lowest weight vector 64 main Cartan subgroup 136 main vector subgroup 136 massless representations 170 Matn F 16 maximal compact subalgebra 94 maximal parabolic subalgebras 101 minimal Bruhat decomposition 96 minimal irreps 297 minimal parabolic subalgebra 100 minimal parabolic subgroup 137 minimal representations 289 Minkowski length 172 multilinear intertwining differential operators 353

orbit 18 order of root 25 order of the finite group 18 ordered partitions 104 orthogonal group O(n, F) 134 orthogonal Lie algebra 17 parabolic subgroup 137 parabolically related algebras 273 partially equivalent 139 partition function 71 Pauli matrices 190 Poincaré–Birkhoff–Witt (PBW) theorem 62 principal series of unitary representations 145 pseudo-Iwasawa decomposition 100 pseudoorthogonal group O(p, r) 134 pseudoreductive algebra 14 pseudounitary group U(p, r) 134 quasi-root 67 quaternionic discrete series representation 146 radical 15 rank of G 22 rank of G 136 real form 93 real rank 95 real rank of G 136 real roots 51 reduced functions 141 reduced reflection expression 26 reducible module 66 reducible representation 13, 138 reductive algebra 14 reflection s! 25 regular element 21 representation of the group G 138

408

Subject Index

representation space 138 restricted G-modules 63 restricted root system 95, 142 restricted Weyl group 96 restricted Weyl reflections 95 right congruence classes 19 right covariance 141 root 22 root length 23 root space decomposition 22 root system 22 roots associated to subsingular vector 240 Satake diagram 119 second order Casimir element 62 semidirect product 20 semidominant 239 Serre relations 59 shadow fields 275 simple algebra 14 simple root 25 simply laced algebras 47 singular vector 67 singular vector of a k-Verma module 357 singular vectors associated to the subsingular vector 241 singular Verma modules 80 special linear algebra 17 special linear group SL(n, F) 134 special orthogonal group SO(n, F) 134 special pseudoorthogonal group SO(p, r) 134 special pseudounitary group SU(p, r) 134 special unitary group SU(n) 134 spinor representations 74 split rank 95 split real form 93

standard parabolic subalgebra 100 straight root 155 strictly hyperbolic type 32 structure constants 12 subgroup 19 submodule 66, 139 subrepresentation 138 subroot 155 subsingular vector 238, 240 symmetric (antisymmetric) tensor 61 symmetrizable matrix 28 symplectic group Sp(m, F) 134 symplectic Lie algebra 17 tensor algebra 61 tensor representations 74 torus subgroup 136 transitive action 18 tri-Verma modules 360 twisted loop algebra 57 unitarily equivalent 140 unitary group U(n) 134 unitary representation 140 universal enveloping algebra U(G) 61 unordered partitions 104 vector irreps 72 vector subgroup 136 Verma module 65 Virasoro algebra 17 weight 63 weight space 63 weight vector 63, 138 Witt algebra 17

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Invariant Differential Operators. Noncompact Semisimple Lie Algebras and Groups

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