Table of Contents
1 Introduction 1
1.1 Symmetries 1
1.2 Invariant Differential Operators 7
1.3 Sketch of Procedure 8
1.4 Organization of the Book 10
2 Lie Algebras and Groups 11
2.1 Generalities on Lie Algebras 11
2.1.1 Lie Algebras 11
2.1.2 Subalgebras, Ideals, and Factor-Algebras 12
2.1.3 Representations 13
2.1.4 Solvable Lie Algebras 13
2.1.5 Nilpotent Lie Algebras 14
2.1.6 Semisimple Lie Algebras 14
2.1.7 Examples 15
2.2 Elements of Group Theory 18
2.2.1 Definition of a Group 18
2.2.2 Group Actions 18
2.2.3 Subgroups and Factor-Groups 19
2.2.4 Homomorphisms 19
2.2.5 Direct and Semidirect Products of Groups 20
2.3 Structure of Semisimple Lie Algebras 21
2.3.1 Cartan Subalgebra 21
2.3.2 Lemmas on Root Systems 22
2.3.3 Weyl Group 25
2.3.4 Cartan Matrix 26
2.4 Classification of Kac-Moody Algebras 27
2.5 Realization of Semisimple Lie Algebras 34
2.5.1 Special Linear Algebra 34
2.5.2 Odd Orthogonal Lie Algebra 36
2.5.3 Symplectic Lie Algebra 38
2.5.4 Even Orthogonal Lie Algebra 40
2.5.5 Exceptional Lie Algebra G2 41
2.5.6 Exceptional Lie Algebra F4 42
2.5.7 Exceptional Lie Algebras El 44
2.6 Realization of Affine Kac-Moody Algebras 47
2.6.1 Realization of Affine Type 1 Kac-Moody Algebras 47
2.6.2 Realization of Affine Type 2 and 3 Kac-Moody Algebras 52
2.6.3 Root System for the Algebras AFF 2 & 3 55
2.7 Chevalley Generators, Serre Relations, and Cartan-Weyl Basis 58
2.8 Highest Weight Representations of Kac-Moody Algebras 61
2.9 Verma Modules 63
2.10 Irreducible Representations 71
2.10.1 Al 72
2.10.2 Cl 73
2.10.3 Bl 73
2.10.4 Dl 75
2.10.5 El 77
2.10.6 F4 78
2.10.7 G2 78
2.11 Characters of Highest Weight Modules 78
2.11.1 Irreducible Quotients of Reducible Verma Modules 78
2.11.2 Embedding Patterns and Mulltiplets 79
2.11.3 Characters of Generic Highest Weight Modules 86
2.11.4 Characters for Nondominant Weights 88
2.11.5 Characters in the Affine Case 88
2.11.6 Example of A1(1) 90
3 Real Semlsimple Lie Algebras 93
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 102
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 7ge; 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 :Ei6 119
3.12.2 EII : Eii6 120
3.12.3 EIII: Eiii6 122
3.12.4 EIV : Eiv6 122
3.12.5 EV: Ei7 123
3.12.6 EVI :Eii7 124
3.12.7 EVII :Eiii7 126
3.12.8 EVIII: Ei8 127
3.12.9 EIX: Eii8 129
3.12.10 FI: F i4 131
3.12.11 FII: F ii4 132
3.12.12 GI: G Ei2 132
4 Invariant Differential Operators 133
4.1 Lie Groups 133
4.1.1 Preliminaries 133
4.1.2 Classical Groups 134
4.1.3 Types of Lie Groups 135
4.1.4 Cartan Subgroups 136
4.1.5 Cartan and Iwasawa Decompositions 136
4.1.6 Parabolic Subgroups 137
4.2 Preliminaries on Group Representation Theory 138
4.2.1 Representations and Modules 138
4.2.2 Reducibility and Irreducibility 138
4.2.3 Operations on Representations 139
4.2.4 Induced Representations 140
4.3 Elementary Representations 140
4.3.1 Compact Lie Groups 140
4.3.2 Noncompact Lie Groups 141
4.3.3 Knapp-Stein Integral Operators 142
4.3.4 ERs of Complex Lie Groups 144
4.4 Unitary Irreducible Representations 145
4.5 Associated Verma Modules 146
4.6 Invariant Differential Operators 149
4.6.1 Canonical Construction 149
4.6.2 Multiplets of GVMs and ERs 152
4.7 Example of SL(2,M) 152
4.7.1 Elementary Representations 152
4.7.2 Discrete Series and Limits Thereof 153
4.7.3 Positive Energy Representations 154
4.8 Explicit Formulae for Singular Vectors 154
4.8.1 Al 155
4.8.2 Dl 156
4.8.3 El 157
4.8.4 Bl 157
4.8.5 Cl 158
4.8.6 F4 158
4.8.7 G2 159
4.8.8 Nonstraight Roots 160
5 Case of the Anti-de Sitter Group 162
5.1 Preliminaries 162
5.1.1 Lie Algebra 162
5.1.2 Finite-Dimensional Realization 164
5.1.3 Structure Theory 165
5.1.4 Lie Groups 166
5.2 Representations and Invariant Operators 166
5.2.1 Elementary Representations 166
5.2.2 Elementary Representations Induced from P0 168
5.2.3 Singular Vectors 168
5.2.4 Invariant Differential Operators 171
5.2.5 Reducible ERs 172
5.2.6 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 179
5.4 Character Formulae 183
5.4.1 Character Formulae of AdS Irreps 183
5.4.2 Character Formulae of Positive Energy UIRs 186
6 Conformal Case in 4D 188
6.1 Preliminaries 188
6.1.1 Realizations of the Group SU(2,2) 188
6.1.2 Lie Algebra of SU(2,2) 189
6.1.3 Restricted Root System, Bruhat and Iwasawa Decompositions 191
6.1.4 Restricted Weyl Group W(G, A0)-193
6.1.5 Parabolic Subalgebras 194
6.1.6 Complexified Lie Algebra 195
6.1.7 Compact and Noncompact Roots 198
6.1.8 Important Subgroups of G 199
6.2 Elementary Representations of SU(2,2) 202
6.2.1 ERs from the Minimal Parabolic Subgroup P0 203
6.2.2 ERs from the Maximal Cuspidal Parabolic Subgroup P1 204
6.2.3 ERs from the Maximal Noncuspidal Parabolic Subgroup P2 205
6.2.4 Noncompact Picture of the ERs 207
6.2.5 Properties of ERs 209
6.2.6 Integral Invariant Operators 211
6.3 Invariant Differential Operators and Multiplet Classification of the Reducible ERs 215
6.3.1 Explicit Expressions for the Invariant Differential Operators -215
6.3.2 Multiplet Classification: Case P0 217
6.3.3 Multiplet Classification: Case P2 225
6.3.4 Holomorphic Discrete Series and Lowest Weight Representations 229
6.3.5 Multiplet Classification: Case P1 231
7 Kazhdan-Lusztig Polynomials, Subsingular Vectors, and Conditionally Invariant Equations 238
7.1 Subsingular Vectors 239
7.1.1 Preliminaries 239
7.1.2 Definition 240
7.1.3 Bernstem-Gel'fand-Gei'fand Example 242
7.1.4 The Other Archetypal sl(4, C) Example 243
7.2 Kazhdan-Lusztig Polynomials 248
7.3 Characters of LWM and Nontrivial KL Polynomials 251
7.3.1 Preliminaries on Characters of Lowest Weight Modules 251
7.3.2 Case sl(4,C) 253
7.3.3 Related Character Formulae 259
7.4 KL Polynomials and Subsingular Vectors: A Conjecture 260
7.5 Conditionally Invariant Differential Equations 262
7.5.1 Preliminaries 262
7.5.2 Conditionally Invariant Operators 263
7.6 Application to sl (4, C) 264
7.6.1 Equations Arising from the BGG Example 265
7.6.2 Equations Arising from the Other Archetypal sl(4, C) Example 265
8 Invariant Differential Operators for Noncompact Lie Algebras Parabolically Related to Conformal Lie Algebras 271
8.1 Generalities 271
8.2 The Pseudo-Orthogonal Algebras so(p,q) 274
8.2.1 Choice of Parabolic Subalgebra 274
8.2.2 Main Multiplets 275
8.2.3 Reduced Multiplets and Their Representations 280
8.2.4 Conservation Laws for so(p,q) 288
8.2.5 Remarks on Shadow Fields and History 290
8.2.6 Case so(3,3) ≅ sl(4. R) 291
8.3 The Lie Algebra su(n,n) and Parabolically Related 292
8.3.1 Multiplets of su(3,3) and sl(6, R) 293
8.3.2 Multiplets of su(4,4), sl(8, R), and su*(8) 298
8.4 Multiplets and Representations for sp(n, R) and sp(r, r) 306
8.4.1 Preliminaries 306
8.4.2 The Case sp(3,R) 307
8.4.3 The Case sp(4, R) and sp(2,2) 309
8.4.4 The Case sp(5,R) 313
8.4.5 The Case sp(6,R) and sp(3,3) 317
8.4.6 Summary for sp(n, R) 325
8.5 SO*(4n) Case 325
8.5.1 Main Multiplets 327
8.5.2 Reduced Multiplets and Minimal Irreps 329
8.6 The Lie Algebras E7(-25) and E7(7) 332
8.6.1 Main Type of Multiplets 333
8.6.2 Reduced Multiplets 336
8.7 The Lie Algebras E6(-14), E6(6), and E6(2) 341
8.7.1 Main Type of Multiplets 343
8.7.2 Reduced Multiplets 346
9 Multilinear Invariant Differential Operators from New Generalized Verma Modules 353
9.1 Preliminaries 353
9.2 k-Verma Modules 355
9.3 Singular Vectors of k-Verma Modules 357
9.3.1 Definition-357
9.3.2 k = 2 357
9.3.3 k = 3 360
9.4 Multilinear Invariant Differential Operators 362
9.5 Bilinear Operators for SL(n, R) and SL(n, C) 363
9.5.1 Setting 363
9.5.2 Minimal Parabolic 364
9.5.3 SL(2, R) 366
9.5.4 SL(3,R) 370
9.6 Examples with k ≥ 3 372
Bibliography 375
Author Index 403
Subject Index 405
