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Representation Theory of Semisimple Groups : An Overview Based on Examples (PMS-36) - Anthony W. Knapp

Representation Theory of Semisimple Groups

An Overview Based on Examples (PMS-36)


Published: 1st December 2001
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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.

Winner of the 1997 Leroy P. Steele Prize, American Mathematics Society "Anthony Knapp has written a marvelous book... Written with accuracy, style, and a genuine desire to communicate the materials... This is one of the finest books I have ever had the pleasure to read, and I recommend it in the strongest possible terms to anyone wishing to appreciate the intricate beauty and technical difficulty of representation theory of semisimple Lie groups."--R. J. Plymen, Bulletin of the London Mathematical Society "Each [theme] is developed carefully and thoroughly, with beautifully worked examples and proofs that reflect long experience in teaching and research... This result is delightful: a readable text that loses almost none of its value as a reference work."--David A. Vogan Jr., Bulletin of the American Mathematical Society

Preface to the Princeton Landmarks in Mathematics Editionp. xiii
Prefacep. xv
Acknowledgmentsp. xix
Scope of the Theory
The Classical Groupsp. 3
Cartan Decompositionp. 7
Representationsp. 10
Concrete Problems in Representation Theoryp. 14
Abstract Theory for Compact Groupsp. 14
Application of the Abstract Theory to Lie Groupsp. 23
Problemsp. 24
Representations of SU(2), SL(2, R), and SL(2, C)
The Unitary Trickp. 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 Formulap. 41
Problemsp. 42
C[superscript [infinity] Vectors and the Universal Enveloping Algebra
Universal Enveloping Algebrap. 46
Actions on Universal Enveloping Algebrap. 50
C[superscript [infinity] Vectorsp. 51
Garding Subspacep. 55
Problemsp. 57
Representations of Compact Lie Groups
Examples of Root Space Decompositionsp. 60
Rootsp. 65
Abstract Root Systems and Positivityp. 72
Weyl Group, Algebraicallyp. 78
Weights and Integral Formsp. 81
Centalizers of Torip. 86
Theorem of the Highest Weightp. 89
Verma Modulesp. 93
Weyl Group, Analyticallyp. 100
Weyl Character Formulap. 104
Problemsp. 109
Structure Theory for Noncompact Groups
Cartan Decomposition and the Unitary Trickp. 113
Iwasawa Decompositionp. 116
Regular Elements, Weyl Chambers, and the Weyl Groupp. 121
Other Decompositionsp. 126
Parabolic Subgroupsp. 132
Integral Formulasp. 137
Borel-Weil Theoremp. 142
Problemsp. 147
Holomorphic Discrete Series
Holomorphic Discrete Series for SU(1, 1)p. 150
Classical Bounded Symmetric Domainsp. 152
Harish-Chandra Decompositionp. 153
Holomorphic Discrete Seriesp. 158
Finiteness of an Integralp. 161
Problemsp. 164
Induced Representations
Three Picturesp. 167
Elementary Propertiesp. 169
Bruhat Theoryp. 172
Formal Intertwining Operatorsp. 174
Gindikin-Karpelevic Formulap. 177
Estimates on Intertwining Operators, Part Ip. 181
Analytic Continuation of Intertwining Operators, Part Ip. 183
Spherical Functionsp. 185
Finite-Dimensional Representations and the H functionp. 191
Estimates on Intertwining Operators, Part IIp. 196
Tempered Representations and Langlands Quotientsp. 198
Problemsp. 201
Admissible Representations
Motivationp. 203
Admissible Representationsp. 205
Invariant Subspacesp. 209
Framework for Studying Matrix Coefficientsp. 215
Harish-Chandra Homomorphismp. 218
Infinitesimal Characterp. 223
Differential Equations Satisfied by Matrix Coefficientsp. 226
Asymptotic Expansions and Leading Exponentsp. 234
First Application: Subrepresentation Theoremp. 238
Second Application: Analytic Continuation of Interwining Operators, Part IIp. 239
Third Application: Control of K-Finite Z(g[superscript C])-Finite Functionsp. 242
Asymptotic Expansions near the Wallsp. 247
Fourth Application: Asymptotic Size of Matrix Coefficientsp. 253
Fifth Application: Identification of Irreducible Tempered Representationsp. 258
Sixth Application: Langlands Classification of Irreducible Admissible Representationsp. 266
Problemsp. 276
Construction of Discrete Series
Infinitesimally Unitary Representationsp. 281
A Third Way of Treating Admissible Representationsp. 282
Equivalent Definitions of Discrete Seriesp. 284
Motivation in General and the Construction in SU(1, 1)p. 287
Finite-Dimensional Spherical Representationsp. 300
Duality in the General Casep. 303
Construction of Discrete Seriesp. 309
Limitations on K Typesp. 320
Lemma on Linear Independencep. 328
Problemsp. 330
Global Characters
Existencep. 333
Character Formulas for SL(2, R)p. 338
Induced Charactersp. 347
Differential Equationsp. 354
Analyticity on the Regular Set, Overview and Examplep. 355
Analyticity on the Regular Set, General Casep. 360
Formula on the Regular Setp. 368
Behavior on the Singular Setp. 371
Families of Admissible Representationsp. 374
Problemsp. 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 Casep. 401
Scheme of Proof for General Casep. 404
Properties of F[subscript f]p. 407
Hirai's Patching Conditionsp. 421
Problemsp. 425
Exhaustion of Discrete Series
Boundedness of Numerators of Charactersp. 426
Use of Patching Conditionsp. 432
Formula for Discrete Series Charactersp. 436
Schwartz Spacep. 447
Exhaustion of Discrete Seriesp. 452
Tempered Distributionsp. 456
Limits of Discrete Seriesp. 460
Discrete Series of Mp. 467
Schmid's Identityp. 473
Problemsp. 476
Plancherel Formula
Ideas and Ingredientsp. 482
Real-Rank-One Groupsp. 482
Real-Rank-One Groups, Part IIp. 485
Averaged Discrete Seriesp. 494
Sp (2, R)p. 502
General Casep. 511
Problemsp. 512
Irreducible Tempered Representations
SL(2, R) from a More General Point of Viewp. 515
Eisenstein Integralsp. 520
Asymptotics of Eisenstein Integralsp. 526
The [eta] Functions for Intertwining Operatorsp. 535
First Irreducibility Resultsp. 540
Normalization of Intertwining Operators and Reducibilityp. 543
Connection with Plancherel Formula when dim A = 1p. 547
Harish-Chandra's Completeness Theoremp. 553
R Groupp. 560
Action by Weyl Group on Representations of Mp. 568
Multiplicity One Theoremp. 577
Zuckerman Tensoring of Induced Representationsp. 584
Generalized Schmid Identitiesp. 587
Inversion of Generalized Schmid Identitiesp. 595
Complete Reduction of Induced Representationsp. 599
Classificationp. 606
Revised Langlands Classificationp. 614
Problemsp. 621
Minimal K Types
Definition and Formulap. 626
Inversion Problemp. 635
Connection with Intertwining Operatorsp. 641
Problemsp. 647
Unitary Representations
SL(2, R) and SL(2, C)p. 650
Continuity Arguments and Complementary Seriesp. 653
Criterion for Unitary Representationsp. 655
Reduction to Real Infinitesimal Characterp. 660
Problemsp. 665
Elementary Theory of Lie Groups
Lie Algebrasp. 667
Structure Theory of Lie Algebrasp. 668
Fundamental Group and Covering Spacesp. 670
Topological Groupsp. 673
Vector Fields and Submanifoldsp. 674
Lie Groupsp. 679
Regular Singular Points of Partial Differential Equations
Summary of Classical One-Variable Theoryp. 685
Uniqueness and Analytic Continuation of Solutions in Several Variablesp. 690
Analog of Fundamental Matrixp. 693
Regular Singularitiesp. 697
Systems of Higher Orderp. 700
Leading Exponents and the Analog of the Indicial Equationp. 703
Uniqueness of Representationp. 710
Roots and Restricted Roots for Classical Groups
Complex Groupsp. 713
Noncompact Real Groupsp. 713
Roots vs. Restricted Roots in Noncompact Real Groupsp. 715
Notesp. 719
Referencesp. 747
Index of Notationp. 763
Indexp. 767
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780691090894
ISBN-10: 0691090890
Series: Princeton Landmarks in Mathematics & Physics (Paperback)
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 800
Published: 1st December 2001
Country of Publication: US
Dimensions (cm): 23.39 x 15.6  x 4.01
Weight (kg): 1.09
Edition Type: Revised