Representation Theory of Semisimple Groups: An Overview Based on ExamplesPrinceton University Press, 2001 - 773 pages In this classic work, Anthony W. Knapp offers a survey of representation theory of semisimple Lie groups in a way that reflects the spirit of the subject and corresponds to the natural learning process. This book is a model of exposition and an invaluable resource for both graduate students and researchers. Although theorems are always stated precisely, many illustrative examples or classes of examples are given. To support this unique approach, the author includes for the reader a useful 300-item bibliography and an extensive section of notes. |
Table des matières
Scope of thE THEORY 1 The Classical Groups 2 Cartan Decomposition 3 Representations | 3 |
4 Concrete Problems in Representation Theory | 4 |
5 Abstract Theory for Compact Groups | 5 |
6 Application of the Abstract Theory to Lie Groups | 6 |
7 Problems | 22 |
REPRESENTATIONS OF SU2 SL2 R AND SL2 | 28 |
2 Irreducible FiniteDimensional ComplexLinear Representations of sl2 | 30 |
3 FiniteDimensional Representations of sl2 | 32 |
7 Construction of Discrete Series | 309 |
8 Limitations on K Types | 320 |
9 Lemma on Linear Independence | 328 |
10 Problems | 330 |
GLOBAL CHARACTERS 1 Existence | 333 |
2 Character Formulas for SL2 R | 338 |
5 Problems | 347 |
4 Differential Equations | 354 |
4 Irreducible Unitary Representations of SL2 | 35 |
6 Use of SU1 | 39 |
7 Plancherel Formula | 42 |
REPRESENTATIONS OF COMPACT LIE GROUPS 1 Examples of Root Space Decompositions | 60 |
2 Roots | 65 |
3 Abstract Root Systems and Positivity | 72 |
4 Weyl Group Algebraically | 78 |
5 Weights and Integral Forms | 81 |
6 Centalizers of Tori | 86 |
7 Theorem of the Highest Weight | 89 |
8 Verma Modules | 93 |
9 Weyl Group Analytically | 100 |
10 Weyl Character Formula | 104 |
8 Problems | 109 |
STRUCTURE THEORY FOR NONCOMPACT GROUPS 1 Cartan Decomposition and the Unitary Trick | 113 |
2 Iwasawa Decomposition | 116 |
3 Regular Elements Weyl Chambers and the Weyl Group | 121 |
4 Other Decompositions | 126 |
5 Parabolic Subgroups | 132 |
6 Integral Formulas | 137 |
7 BorelWeil Theorem | 142 |
8 Problems | 147 |
HOLOMORPHIC Discrete SERIES 1 Holomorphic Discrete Series for SU1 1 | 150 |
2 Classical Bounded Symmetric Domains | 152 |
3 HarishChandra Decomposition | 153 |
4 Holomorphic Discrete Series | 158 |
5 Finiteness of an Integral | 161 |
6 Problems | 164 |
INDUCED REPRESENTATIONS 1 Three Pictures | 167 |
2 Elementary Properties | 169 |
3 Bruhat Theory | 172 |
4 Formal Intertwining Operators | 174 |
5 GindikinKarpelevič Formula | 177 |
6 Estimates on Intertwining Operators Part I | 181 |
7 Analytic Continuation of Intertwining Operators Part I | 183 |
8 Spherical Functions | 185 |
9 FiniteDimensional Representations and the H function | 191 |
10 Estimates on Intertwining Operators Part II | 196 |
11 Tempered Representations and Langlands Quotients | 198 |
12 Problems | 201 |
ADMISSIBLE REPRESENTATIONS 1 Motivation | 203 |
C VECTORS and the Universal ENVELOPING ALGEBRA 1 Universal Enveloping Algebra | 204 |
2 Admissible Representations | 205 |
3 Invariant Subspaces | 209 |
2 Actions on Universal Enveloping Algebra 3 C Vectors | 211 |
4 Framework for Studying Matrix Coefficients | 215 |
5 HarishChandra Homomorphism | 218 |
6 Infinitesimal Character | 223 |
7 Differential Equations Satisfied by Matrix Coefficients | 226 |
8 Asymptotic Expansions and Leading Exponents | 234 |
Subrepresentation Theorem | 238 |
Analytic Continuation of Interwining Operators Part II | 239 |
Control of KFinite ZgFinite Functions | 242 |
4 Gårding Subspace | 244 |
12 Asymptotic Expansions near the Walls | 247 |
Asymptotic Size of Matrix Coefficients | 253 |
Identification of Irreducible Tempered Representations | 258 |
Langlands Classification of Irreducible Admissible Representations | 266 |
16 Problems | 276 |
CONSTRUCTION OF DISCrete Series 1 Infinitesimally Unitary Representations | 281 |
2 A Third Way of Treating Admissible Representations | 282 |
3 Equivalent Definitions of Discrete Series | 284 |
4 Motivation in General and the Construction in SU1 1 | 287 |
5 FiniteDimensional Spherical Representations | 300 |
6 Duality in the General Case | 303 |
5 Analyticity on the Regular Set Overview and Example | 355 |
6 Analyticity on the Regular Set General Case | 360 |
7 Formula on the Regular Set | 368 |
8 Behavior on the Singular Set | 371 |
9 Families of Admissible Representations | 374 |
10 Problems | 383 |
INTRODUCTION to Plancherel Formula 1 Constructive Proof for SU2 | 385 |
2 Constructive Proof for SL2 C | 387 |
3 Constructive Proof for SL2 R | 394 |
4 Ingredients of Proof for General Case | 401 |
5 Scheme of Proof for General Case | 404 |
xiii | 405 |
6 Properties of F | 407 |
7 Hirais Patching Conditions | 421 |
8 Problems | 425 |
EXHAUSTION OF DISCRETE Series 1 Boundedness of Numerators of Characters | 426 |
2 Use of Patching Conditions | 432 |
3 Formula for Discrete Series Characters | 436 |
4 Schwartz Space | 447 |
5 Exhaustion of Discrete Series | 452 |
6 Tempered Distributions | 456 |
7 Limits of Discrete Series | 460 |
8 Discrete Series of M | 467 |
9 Schmids Identity | 473 |
10 Problems | 476 |
PLANCHEREL FORMULA 1 Ideas and Ingredients | 482 |
3 RealRankOne Groups Part II | 485 |
4 Averaged Discrete Series | 494 |
5 Sp 2 R | 502 |
XV | 509 |
6 General Case | 511 |
7 | 512 |
14 | 513 |
IRREDUCIBLE TEMPERED RepresentaTIONS 1 SL2 R from a More General Point of View | 515 |
23 | 516 |
2 Eisenstein Integrals | 520 |
3 Asymptotics of Eisenstein Integrals | 526 |
24 | 528 |
28 | 531 |
30 | 534 |
4 Then Functions for Intertwining Operators | 535 |
31 | 536 |
5 First Irreducibility Results | 540 |
6 Normalization of Intertwining Operators and Reducibility | 543 |
7 Connection with Plancherel Formula when dim A 1 | 547 |
33 | 555 |
35 | 557 |
39 | 563 |
42 | 565 |
46 | 569 |
51 | 574 |
15 Complete Reduction of Induced Representations | 599 |
7779 | 621 |
UNITARY REPRESENTATIONS | 650 |
ELEMENTARY THEORY OF LIE GROUPS | 667 |
REGULAR SINGULAR POINTS OF PARTIAL | 685 |
3 Analog of Fundamental Matrix | 693 |
5 Systems of Higher Order | 700 |
7 Uniqueness of Representation | 710 |
ROOTS AND Restricted Roots for Classical | 713 |
NOTES | 719 |
REFERENCES | 747 |
| 763 | |
Expressions et termes fréquents
a₁ abelian admissible representation apply Cartan subalgebra Cartan subgroup Com(G completes the proof compute conjugate Corollary corresponding decomposition define denote diag differential discrete series discrete series representation e₁ eigenvalue element equivalent exp H finite follows formula function given global character H₁ Haar measure Harish-Chandra Hence highest weight holomorphic implies induced infinitesimal character invariant irreducible unitary isomorphism Iwasawa decomposition K-finite K-finite matrix coefficients K-finite vectors Lemma Let G Lie algebra Lie group linear connected linear connected reductive matrix coefficients multiple noncompact nonzero notation obtain orthogonal parabolic subgroup polynomials positive roots positive system proof of Theorem Proposition prove real analytic representation of G restricted root right side scalar semisimple shows simply connected sinh space subset subspace t₁ U₁ unitary representation V₁ v₂ Weyl chamber Weyl group write