گروه‌های کاهشی واقعی II، جلد 132-II (ریاضیات خالص و کاربردی) (شماره 2)

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. 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 Sequel to "Real Reductive Groups I", this book emphasizes more analytical aspects of representation theory, but retains its focus on the interaction between algebra, analysis and geometry, like the first volume. The two volumes comprise a complete introduction to representation theory.