This book is the sequel to "Real Reductive Groups I", and emphasizes the more analytical aspects of representation theory, while still retaining its focus on the interaction between algebra, analysis and geometry, like the first volume. It provides a self-contained introduction to abstract representation theory, covering locally compact groups, C- algebras, Von Neuman algebras, direct integral decompositions. In addition, it contains a proof of Harish-Chandra's plancherel theorem. Together, the two volumes comprise a complete introduction to representation theory. Both volumes are based on courses and lectures given by the author over the last 20 years. They are intended for research mathematicians and graduate-level students taking courses in representation theory and mathematical physics.
Author(s): Nolan Wallach
Series: Pure and Applied Mathematics Academic Pr
Publisher: Academic Press
Year: 1992
Language: English
Pages: 475
Front Cover......Page 1
Real Reductive Groups II......Page 4
Copyright Page......Page 5
Contents......Page 6
Preface......Page 10
Introduction......Page 12
Introduction......Page 16
10.1. The intertwining operators......Page 17
10.2. The proof of Theorem 10.1.5......Page 32
10.3. Limit formulas......Page 43
10.4. A generalization of L. Cohn’s determinant formula......Page 47
10.5. The Harish-Chandra μ-function......Page 54
10.6. Notes and further results......Page 62
10.A.l. Some constructions related to finite dimensional representations......Page 64
10.A.2. Some results related to Sterling's formula......Page 70
10.A.3. Miscellaneous results......Page 71
Introduction......Page 74
11.1. Some results on Weyl group invariants......Page 75
11.2. A lemma of Kostant......Page 82
11.3. Representations with small K-types......Page 84
11.4. The automatic continuity theorem......Page 92
11.5. Completions of (g, K)-modules......Page 99
11.6. Analysis of completions of (g, K)-modules......Page 103
11.7. The proof of the main theorem......Page 111
11.8. The action of f(G) on admissible representations......Page 118
11.9. Poisson integral representations......Page 120
11.10. Notes and further results......Page 125
11.A.l. Some results on the action of a compact group on a symmetric algebra......Page 126
11.A.2. Small K-types......Page 128
11.A.3. Some results on Verma modules......Page 141
11.A.4. Some functional analysis......Page 143
Introduction......Page 148
12.1. Characters of principal series representations......Page 149
12.2. The modules VQ|P,σ,iv......Page 154
12.3. The leading term......Page 159
12.4. The dependence of the leading term on parameters......Page 164
12.5. The leading term and intertwining operators......Page 173
12.6. The main inequality......Page 182
12.7. Wave packets......Page 197
12.8. The Harish-Chandra transform of a wave packet......Page 206
12.9. Notes......Page 215
12.A.1. Traces of certain kernel operators......Page 216
12.A.2. Some inequalities......Page 220
12.A.3. The topology of induced representations......Page 228
Introduction......Page 230
13.1. The Eisenstein integral......Page 231
13.2. The leading terms of Eisenstein integrals......Page 243
13.3. Wave packets of Eisenstein integrals......Page 250
13.4. The Harish-Chandra Plancherel theorem......Page 254
13.5. The calculation of μ(ω, υ) for the fundamental series......Page 262
13.6. The intertwining algebra of Ip,σ,iv and the irreducibility of the fundamental series......Page 264
13.7. Groups with one conjugacy class of Cartan subgroup......Page 271
13.8. The Plancherel theorem for L2(G/K)......Page 273
13.9. Notes and further results......Page 275
Introduction......Page 278
14.1. The basic theory of C* algebras......Page 280
14.2. The C* algebra of a locally compact group......Page 288
14.3. Quotients of C* algebras......Page 290
14.4. Density theorems......Page 294
14.5. Representations of C* algebras and positive functionals......Page 298
14.6. The topology on the unitary dual of a C* algebra......Page 309
14.7. The topology on the unitary dual of a locally compact group......Page 321
14.8. Direct integrals and Von Neumann algebras......Page 327
14.9. Direct integrals of representations of C* algebras and locally compact groups......Page 341
14.10. Decompositions of representations of CCR C* algebras and locally compact groups......Page 344
14.11. The Plancherel formula for CCR locally compact, unimodular groups......Page 355
14.12. The Plancherel formula for real reductive groups......Page 364
14.13. Notes and further results......Page 369
14.A. Some functional analysis......Page 370
Introduction......Page 378
15.1. The support of certain induced representations......Page 379
15.2. Some asymptotic expansions and estimates......Page 383
15.3. The Schwartz space for L2(N \ G; χ)......Page 390
15.4. The holomorphic continuation of the Jacquet integral......Page 396
15.5. First steps for the holomorphic continuation......Page 398
15.6. The completion of the proof of the holomorphic continuation......Page 408
15.7. Cusp forms revisited......Page 420
15.8. The first steps for the Plancherel theorem for generic χ......Page 427
15.9. The Plancherel theorem for L2N0 \ G; χ)......Page 437
15.10. Some examples of the Plancherel theorem for generic χ......Page 441
15.11. Notes and further results......Page 445
15.A. Appendix to Chapter 15......Page 450
Bibliography......Page 454
Index......Page 466
Pure and Applied Mathematics......Page 470
