Real Reductive Groups I (eBook)

Front Cover 1
Real Reductive Groups I 4
Copyright Page 5
Content 8
Preface 14
Introduction 16
Chapter 0. Background Material 22
Introduction 22
0.1. Invariant measures on homogeneous spaces 22
0.2. The structure of reductive Lie algebras 25
0.3. The structure of compact Lie groups 27
0.4. The universal enveloping algebra of a Lie algebra 29
0.5. Some basic representation theory 31
0.6. Modules over the universal enveloping algebra 34
Chapter 1. Elementary Representation Theory 38
Introduction 38
1.1. General properties of representations 39
1.2. Schur's lemma 41
1.3. Square integrable representations 43
1.4. Basic representation theory of compact groups. 45
1.5. A class of induced representations 50
1.6. C? Vectors and Analytic Vectors 52
1.7. Representations of compact lie groups 56
1.8. Further results and comments 60
Chapter 2. Real Reductive Groups 62
Introduction 62
2.1. The definition of a real reductive group 63
2.2. Parabolic pairs 69
2.3. Cartan subgroups 77
2.4. Integration formulas 81
2.5. The Weyl character formula 86
2.A. Appendices to Chapter 2 89
2.A.1. Some linear algebra 89
2.A.2. Norms on real reductive groups 91
Chapter 3. The Basic Theory of (g, K)-Modules 94
Introduction 94
3.1. The Chevalley restriction theorem 95
3.2. The Harish-Chandra isomorphism of the center of the universal enveloping algebra 98
3.3. (g, K)-modules 101
3.4. A basic theorem of Harish-Chandra 103
3.5. The subquotient theorem 107
3.6. The spherical principal series 113
3.7. A Lemma of Osborne 116
3.8. The subrepresentation theorem 118
3.9. Notes and further results 121
3.A. Appendices to Chapter 3 124
3.A.1. Some associative algebra 124
3.A.2. A Lemma of Harish-Chandra 125
Chapter 4. The Asymptotic Behavior of Matrix Coefficients 128
Introduction 128
4.1. The Jacquet module of an admissible (g, K)-module 129
4.2. Three applications of the Jacquet module 133
4.3. Asymptotic behavior of matrix coefficients 135
4.4. Asymptotic expansions of matrix coefficients 139
4.5. Harish-Chandra's ?-function 146
4.6. Notes and further results 151
4.A. Appendices to Chapter 4 152
4.A.1. Asymptotic expansions 152
4.A.2. Some inequalities 154
Chapter 5. The Langlands Classification 158
Introduction 158
5.1. Tempered (g, K)-modules 159
5.2. The principal series 161
5.3. The intertwining integrals 165
5.4. The Langlands classification 170
5.5. Some applications of the classification 173
5.6. SL(2, R) 177
5.7. SL(2, C) 180
5.8. Notes and further results 184
5.A. Appendices to Chapter 5 185
5.A.1. A Lemma of Langlands 185
5.A.2. An a priori estimate 187
5.A.3. Square integrability and the polar decomposition 189
Chapter 6. A Construction of the Fundamental Series 194
Introduction 194
6.1. Relative Lie algebra cohomology 195
6.2. A construction of ( f, K)-modules 197
6.3. The Zuckerman functors 200
6.4. Some vanishing theorems 205
6.5. Blattner type formulas 209
6.6. Irreducibility 214
6.7. Unitarizability 217
6.8. Temperedness and square integrability 222
6.9. The case of disconnected G 224
6.10. Notes and further results 227
6.A. Appendices to Chapter 6 228
6.A.1. Some homological algebra 228
6.A.2. Partition functions 232
6.A.3. Tensor products with finite dimensional representations 233
6.A.4. Infinitesimally unitary modules 241
Chapter 7. Cusp Forms on G 246
Introduction 246
7.1. Some Fréchet spaces of functions on G 247
7.2. The Harish-Chandra transform 251
7.3. Orbital integrals on a reductive Lie algebra 255
7.4. Orbital integrals on a reductive Lie group 264
7.5. The orbital integrals of cusp forms 271
7.6. Harmonic analysis on the space of cusp forms 275
7.7. Square integrable representations revisited 280
7.8. Notes and further results 285
7.A. Appendices to Chapter 7 286
7.A.1. Some linear algebra 286
7.A.2. Radial components on the Lie algebra 289
7.A.3. Radial components on the Lie group 294
7.A.4. Some harmonic analysis on Tori 298
7.A.5. Fundamental solutions of certain differential operators 303
Chapter 8. Character Theory 310
Introduction 310
8.1. The Character of an admissible representation 311
8.2. The K-character of a (g, K)-module 315
8.3. Harish-Chandra's regularity theorem on the Lie algebra 317
8.4. Harish-Chandra's regularity theorem on the Lie group 332
8.5. Tempered invariant Z(g)-finite distributions on G 334
8.6. Harish-Chandra's basic inequality 341
8.7. The completeness of the ?? 344
8.A. Appendices to Chapter 8 347
8.A.1. Trace class operators 347
8.A.2. Some operations on distributions 352
8.A.3. The radial component revisited 358
8.A.4. The orbit structure on a real reductive Lie algebia 363
8.A.5. Some technical results for Harish-Chandra's regularity theorem 370
Chapter 9. Unitary Representations and (g, K)-Cohomology 374
Introduction 374
9.1. Tensor products of finite dimensional representations 375
9.2. Spinors 380
9.3. The Dirac operator 386
9.4. (g, K)-cohomology 389
9.5. Some results of Kumaresan, Parthasarathy, Vogan, Zuckerman 394
9.6. u-cohomology 402
9.7. A theorem of Vogan-Zuckerman 409
9.8. Further results 415
9.A. Appendices to Chapter 9 417
9.A.1. Weyl groups 417
9.A.2. Spectral sequences 419
Bibliography 424
Index 432
Pure and Applied Mathematics 434