Representation Theory of Semisimple Groups

by Knapp, Anthony W.

ISBN13: 9780691084015

ISBN10: 0691084017

Format: Hardcover

Pub. Date: 1986-08-01

Publisher(s): Princeton Univ Pr

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Summary

The intention of this book is to give a survey of the representation theory of semisimple Lie groups, including results and techniques, in a way that reflects the spirit of the subject, corresponds more to a person's natural learning process, and stops at the end of a single volume.

Author Biography

Anthony W. Knapp is Emeritus Professor of Mathematics, State University of New York at Stony Brook.

Preface to the Princeton Landmarks in Mathematics Edition	p. xiii
Preface	p. xv
Acknowledgments	p. xix
Scope of the Theory
The Classical Groups	p. 3
Cartan Decomposition	p. 7
Representations	p. 10
Concrete Problems in Representation Theory	p. 14
Abstract Theory for Compact Groups	p. 14
Application of the Abstract Theory to Lie Groups	p. 23
Problems	p. 24
Representations of SU(2), SL(2, R), and SL(2, C)
The Unitary Trick	p. 28
Irreducible Finite-Dimensional Complex-Linear Representations of sl(2, C)	p. 30
Finite-Dimensional Representations of sl(2, C)	p. 31
Irreducible Unitary Representations of SL(2, C)	p. 33
Irreducible Unitary Representations of SL(2, R)	p. 35
Use of SU(1, 1)	p. 39
Plancherel Formula	p. 41
Problems	p. 42
C[superscript [infinity] Vectors and the Universal Enveloping Algebra
Universal Enveloping Algebra	p. 46
Actions on Universal Enveloping Algebra	p. 50
C[superscript [infinity] Vectors	p. 51
Garding Subspace	p. 55
Problems	p. 57
Representations of Compact Lie Groups
Examples of Root Space Decompositions	p. 60
Roots	p. 65
Abstract Root Systems and Positivity	p. 72
Weyl Group, Algebraically	p. 78
Weights and Integral Forms	p. 81
Centalizers of Tori	p. 86
Theorem of the Highest Weight	p. 89
Verma Modules	p. 93
Weyl Group, Analytically	p. 100
Weyl Character Formula	p. 104
Problems	p. 109
Structure Theory for Noncompact Groups
Cartan Decomposition and the Unitary Trick	p. 113
Iwasawa Decomposition	p. 116
Regular Elements, Weyl Chambers, and the Weyl Group	p. 121
Other Decompositions	p. 126
Parabolic Subgroups	p. 132
Integral Formulas	p. 137
Borel-Weil Theorem	p. 142
Problems	p. 147
Holomorphic Discrete Series
Holomorphic Discrete Series for SU(1, 1)	p. 150
Classical Bounded Symmetric Domains	p. 152
Harish-Chandra Decomposition	p. 153
Holomorphic Discrete Series	p. 158
Finiteness of an Integral	p. 161
Problems	p. 164
Induced Representations
Three Pictures	p. 167
Elementary Properties	p. 169
Bruhat Theory	p. 172
Formal Intertwining Operators	p. 174
Gindikin-Karpelevic Formula	p. 177
Estimates on Intertwining Operators, Part I	p. 181
Analytic Continuation of Intertwining Operators, Part I	p. 183
Spherical Functions	p. 185
Finite-Dimensional Representations and the H function	p. 191
Estimates on Intertwining Operators, Part II	p. 196
Tempered Representations and Langlands Quotients	p. 198
Problems	p. 201
Admissible Representations
Motivation	p. 203
Admissible Representations	p. 205
Invariant Subspaces	p. 209
Framework for Studying Matrix Coefficients	p. 215
Harish-Chandra Homomorphism	p. 218
Infinitesimal Character	p. 223
Differential Equations Satisfied by Matrix Coefficients	p. 226
Asymptotic Expansions and Leading Exponents	p. 234
First Application: Subrepresentation Theorem	p. 238
Second Application: Analytic Continuation of Interwining Operators, Part II	p. 239
Third Application: Control of K-Finite Z(g[superscript C])-Finite Functions	p. 242
Asymptotic Expansions near the Walls	p. 247
Fourth Application: Asymptotic Size of Matrix Coefficients	p. 253
Fifth Application: Identification of Irreducible Tempered Representations	p. 258
Sixth Application: Langlands Classification of Irreducible Admissible Representations	p. 266
Problems	p. 276
Construction of Discrete Series
Infinitesimally Unitary Representations	p. 281
A Third Way of Treating Admissible Representations	p. 282
Equivalent Definitions of Discrete Series	p. 284
Motivation in General and the Construction in SU(1, 1)	p. 287
Finite-Dimensional Spherical Representations	p. 300
Duality in the General Case	p. 303
Construction of Discrete Series	p. 309
Limitations on K Types	p. 320
Lemma on Linear Independence	p. 328
Problems	p. 330
Global Characters
Existence	p. 333
Character Formulas for SL(2, R)	p. 338
Induced Characters	p. 347
Differential Equations	p. 354
Analyticity on the Regular Set, Overview and Example	p. 355
Analyticity on the Regular Set, General Case	p. 360
Formula on the Regular Set	p. 368
Behavior on the Singular Set	p. 371
Families of Admissible Representations	p. 374
Problems	p. 383
Introduction to Plancherel Formula
Constructive Proof for SU(2)	p. 385
Constructive Proof for SL(2, C)	p. 387
Constructive Proof for SL(2, R)	p. 394
Ingredients of Proof for General Case	p. 401
Scheme of Proof for General Case	p. 404
Properties of F[subscript f]	p. 407
Hirai's Patching Conditions	p. 421
Problems	p. 425
Exhaustion of Discrete Series
Boundedness of Numerators of Characters	p. 426
Use of Patching Conditions	p. 432
Formula for Discrete Series Characters	p. 436
Schwartz Space	p. 447
Exhaustion of Discrete Series	p. 452
Tempered Distributions	p. 456
Limits of Discrete Series	p. 460
Discrete Series of M	p. 467
Schmid's Identity	p. 473
Problems	p. 476
Plancherel Formula
Ideas and Ingredients	p. 482
Real-Rank-One Groups	p. 482
Real-Rank-One Groups, Part II	p. 485
Averaged Discrete Series	p. 494
Sp (2, R)	p. 502
General Case	p. 511
Problems	p. 512
Irreducible Tempered Representations
SL(2, R) from a More General Point of View	p. 515
Eisenstein Integrals	p. 520
Asymptotics of Eisenstein Integrals	p. 526
The [eta] Functions for Intertwining Operators	p. 535
First Irreducibility Results	p. 540
Normalization of Intertwining Operators and Reducibility	p. 543
Connection with Plancherel Formula when dim A = 1	p. 547
Harish-Chandra's Completeness Theorem	p. 553
R Group	p. 560
Action by Weyl Group on Representations of M	p. 568
Multiplicity One Theorem	p. 577
Zuckerman Tensoring of Induced Representations	p. 584
Generalized Schmid Identities	p. 587
Inversion of Generalized Schmid Identities	p. 595
Complete Reduction of Induced Representations	p. 599
Classification	p. 606
Revised Langlands Classification	p. 614
Problems	p. 621
Minimal K Types
Definition and Formula	p. 626
Inversion Problem	p. 635
Connection with Intertwining Operators	p. 641
Problems	p. 647
Unitary Representations
SL(2, R) and SL(2, C)	p. 650
Continuity Arguments and Complementary Series	p. 653
Criterion for Unitary Representations	p. 655
Reduction to Real Infinitesimal Character	p. 660
Problems	p. 665
Elementary Theory of Lie Groups
Lie Algebras	p. 667
Structure Theory of Lie Algebras	p. 668
Fundamental Group and Covering Spaces	p. 670
Topological Groups	p. 673
Vector Fields and Submanifolds	p. 674
Lie Groups	p. 679
Regular Singular Points of Partial Differential Equations
Summary of Classical One-Variable Theory	p. 685
Uniqueness and Analytic Continuation of Solutions in Several Variables	p. 690
Analog of Fundamental Matrix	p. 693
Regular Singularities	p. 697
Systems of Higher Order	p. 700
Leading Exponents and the Analog of the Indicial Equation	p. 703
Uniqueness of Representation	p. 710
Roots and Restricted Roots for Classical Groups
Complex Groups	p. 713
Noncompact Real Groups	p. 713
Roots vs. Restricted Roots in Noncompact Real Groups	p. 715
Notes	p. 719
References	p. 747
Index of Notation	p. 763
Index	p. 767
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