Real Reductive Groups II (eBook)

Front Cover 1
Real Reductive Groups II 4
Copyright Page 5
Contents 6
Preface 10
Introduction 12
Chapter 10. Intertwining Operators 16
Introduction 16
10.1. The intertwining operators 17
10.2. The proof of Theorem 10.1.5 32
10.3. Limit formulas 43
10.4. A generalization of L. Cohn’s determinant formula 47
10.5. The Harish-Chandra µ-function 54
10.6. Notes and further results 62
10.A. Appendices to Chapter 10 64
10.A.l. Some constructions related to finite dimensional representations 64
10.A.2. Some results related to Sterling's formula 70
10.A.3. Miscellaneous results 71
Chapter 11. Completions of Admissible (g, K)-Modules 74
Introduction 74
11.1. Some results on Weyl group invariants 75
11.2. A lemma of Kostant 82
11.3. Representations with small K-types 84
11.4. The automatic continuity theorem 92
11.5. Completions of (g, K)-modules 99
11.6. Analysis of completions of (g, K)-modules 103
11.7. The proof of the main theorem 111
11.8. The action of f(G) on admissible representations 118
11.9. Poisson integral representations 120
11.10. Notes and further results 125
1l.A. Appendices to Chapter 11 126
11.A.l. Some results on the action of a compact group on a symmetric algebra 126
11.A.2. Small K-types 128
11.A.3. Some results on Verma modules 141
11.A.4. Some functional analysis 143
Chapter 12. The Theory of the Leading Term 148
Introduction 148
12.1. Characters of principal series representations 149
12.2. The modules VQ|P,s,iv 154
12.3. The leading term 159
12.4. The dependence of the leading term on parameters 164
12.5. The leading term and intertwining operators 173
12.6. The main inequality 182
12.7. Wave packets 197
12.8. The Harish-Chandra transform of a wave packet 206
12.9. Notes 215
12.A. Appendices to Chapter 12 216
12.A.1. Traces of certain kernel operators 216
12.A.2. Some inequalities 220
12.A.3. The topology of induced representations 228
Chapter 13. The Harish-Chandra Plancherel Theorem 230
Introduction 230
13.1. The Eisenstein integral 231
13.2. The leading terms of Eisenstein integrals 243
13.3. Wave packets of Eisenstein integrals 250
13.4. The Harish-Chandra Plancherel theorem 254
13.5. The calculation of µ(., .) for the fundamental series 262
13.6. The intertwining algebra of Ip,s,iv and the irreducibility of the fundamental series 264
13.7. Groups with one conjugacy class of Cartan subgroup 271
13.8. The Plancherel theorem for L2(G/K) 273
13.9. Notes and further results 275
Chapter 14. Abstract Representation Theory 278
Introduction 278
14.1. The basic theory of C* algebras 280
14.2. The C* algebra of a locally compact group 288
14.3. Quotients of C* algebras 290
14.4. Density theorems 294
14.5. Representations of C* algebras and positive functionals 298
14.6. The topology on the unitary dual of a C* algebra 309
14.7. The topology on the unitary dual of a locally compact group 321
14.8. Direct integrals and Von Neumann algebras 327
14.9. Direct integrals of representations of C* algebras and locally compact groups 341
14.10. Decompositions of representations of CCR C* algebras and locally compact groups 344
14.11. The Plancherel formula for CCR locally compact, unimodular groups 355
14.12. The Plancherel formula for real reductive groups 364
14.13. Notes and further results 369
14.A. Some functional analysis 370
Chapter 15. The Whittaker Plancherel Theorem 378
Introduction 378
15.1. The support of certain induced representations 379
15.2. Some asymptotic expansions and estimates 383
15.3. The Schwartz space for L2(N / G .)
15.4. The holomorphic continuation of the Jacquet integral 396
15.5. First steps for the holomorphic continuation 398
15.6. The completion of the proof of the holomorphic continuation 408
15.7. Cusp forms revisited 420
15.8. The first steps for the Plancherel theorem for generic . 427
15.9. The Plancherel theorem for L2N0 / G .)
15.10. Some examples of the Plancherel theorem for generic . 441
15.11. Notes and further results 445
15.A. Appendix to Chapter 15 450
Bibliography 454
Index 466
Pure and Applied Mathematics 470