Paperback
Published: 7th October 2001
ISBN: 9780691090894
Number Of Pages: 800
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 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 |
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: 7th October 2001
Country of Publication: US
Dimensions (cm): 23.39 x 15.6
x 4.01
Weight (kg): 1.09
Edition Type: Revised