Representation Theory of Semisimple Groups

Preface to the Princeton Landmarks in Mathematics Edition xiii Preface xv Acknowledgments xix CHAPTER I. SCOPE OF THE THEORY 1. The Classical Groups 3 2. Cartan Decomposition 7 3. Representations 10 4. Concrete Problems in Representation Theory 14 5. Abstract Theory for Compact Groups 14 6. Application of the Abstract Theory to Lie Groups 23 7. Problems 24 CHAPTER II. REPRESENTATIONS OF SU(2), SL(2,R), AND SL(2,C) l. The Unitary Trick 28 2. Irreducible Finite-Dimensional Complex-Linear Representations of 91(2,C) 30 3. Finite-Dimensional Representations of 91(2,C) 31 4. Irreducible Unitary Representations of SL(2,C) 33 5. Irreducible Unitary Representations of SL(2,08) 35 6. Use of SU(1,1) 39 7. Plancherel Formula 41 8. Problems 42 CHAPTER III. C VECTORS AND THE UNIVERSAL ENVELOPING ALGEBRA l. Universal Enveloping Algebra 46 2. Actions on Universal Enveloping Algebra 50 3. C Vectors 55 4. Garding Subspace. Problems 57 CHAPTER IV. 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 11. Problems 109 CHAPTER V. STRUCTURE THEORY FOR NONCOMPACT GROUPS l. 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. Borel-Weil Theorem 142 8. Problems 147 CHAPTER VI. HOLOMORPHIC DISCRETE SERIES 1. Holomorphic Discrete Series for SU(1,1) 150 2. Classical Bounded Symmetric Domains 152 3. Harish-Chandra Decomposition 153 4. Holomorphic Discrete Series 158 5. Finiteness of an Integral 161 6. Problems 164 CHAPTER VII. INDUCED REPRESENTATIONS 1. Three Pictures 167 2. Elementary Properties 169 3. Bruhat Theory 172 4. Formal Intertwining Operators 174 5. Gindikin-Karpelevic Formula 177 6. Estimates on Intertwining Operators, Part I 181 7. Analytic Continuation of Intertwining Operators, Part I 183 8. Spherical Functions 185 9. Finite-Dimensional Representations and the H function 191 10. Estimates on Intertwining Operators, Part II 196 11. Tempered Representations and Langlands Quotients 198 12. Problems 201 CHAPTER VIII. ADMISSIBLE REPRESENTATIONS l. Motivation 203 2. Admissible Representations 205 3. Invariant Subspaces 209 4. Framework for Studying Matrix Coefficients 215 5. Harish-Chandra Homomorphism 218 6. Infinitesimal Character 223 7. Differential Equations Satisfied by Matrix Coefficients 226 8. Asymptotic Expansions and Leading Exponents 234 9. First Application: Subrepresentation Theorem 238 10. Second Application: Analytic Continuation of Interwining Operators, Part II 239 11. Third Application: Control of K-Finite Z(gc)-Finite Functions 242 12. Asymptotic Expansions near the Walls 247 13. Fourth Application: Asymptotic Size of Matrix Coefficients 253 14. Fifth Application: Identification of Irreducible Tempered Representations 258 15. Sixth Application: Langlands Classification of Irreducible Admissible Representations 266 16. Problems 276 CHAPTER IX. 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 SU(1,1) 287 5. Finite-Dimensional Spherical Representations 300 6. Duality in the General Case 303 7. Construction of Discrete Series 309 8. Limitations on K Types 320 9. Lemma on Linear Independence 328 10. Problems 330 CHAPTER X. GLOBAL CHARACTERS l. Existence 333 2. Character Formulas for SL(2,R) 338 3. Induced Characters 347 4. Differential Equations 354 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 CHAPTER XI. INTRODUCTION TO PLANCHEREL FORMULA l. Constructive Proof for SU(2) 385 2. Constructive Proof for SL(2,C) 387 3. Constructive Proof for SL(2,R) 394 4. Ingredients of Proof for General Case 401 5. Scheme of Proof for General Case 404 6. Properties of F f 407 7. Hirai's Patching Conditions 421 8. Problems 425 CHAPTER XII. 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. Schmid's Identity 473 10. Problems 476 CHAPTER XIII. PLANCHEREL FORMULA 1. Ideas and Ingredients 482 2. Real-Rank-One Groups, Part I 482 3. Real-Rank-One Groups, Part II 485 4. Averaged Discrete Series 494 5. Sp (2,R) 502 6. General Case 511 7. Problems 512 CHAPTER XIV. IRREDUCIBLE TEMPERED REPRESENTATIONS l. SL(2,R) from a More General Point of View 515 2. Eisenstein Integrals 520 3. Asymptotics of Eisenstein Integrals 526 4. The il Functions for Intertwining Operators 535 5. First Irreducibility Results 540 6. Normalization of Intertwining Operators and Reducibility 543 7. Connection with Plancherel Formula when dim A = 1 547 8. Harish-Chandra's Completeness Theorem 553 9. R Group 560 10. Action by Weyl Group on Representations of M 568 11. Multiplicity One Theorem 577 12. Zuckerman Tensoring of Induced Representations 584 13. Generalized Schmid Identities 587 14. Inversion of Generalized Schmid Identities 595 15. Complete Reduction of Induced Representations 599 16. Classification 606 17. Revised Langlands Classification 614 18. Problems 621 CHAPTER XV. MINIMAL K TYPES l. Definition and Formula 626 2. Inversion Problem 635 3. Connection with Intertwining Operators 641 4. Problems 647 CHAPTER XVI. UNITARY REPRESENTATIONS 1. SL(2,U8) and SL(2,C) 650 2. Continuity Arguments and Complementary Series 653 3. Criterion for Unitary Representations 655 4. Reduction to Real Infinitesimal Character 660 5. Problems 665 APPENDIX A: ELEMENTARY THEORY OF LIE GROUPS l. Lie Algebras 667 2. Structure Theory of Lie Algebras 668 3. Fundamental Group and Covering Spaces 670 4. Topological Groups 673 5. Vector Fields and Submanifolds 674 6. Lie Groups 679 APPENDIX B: REGULAR SINGULAR POINTS OF PARTIAL DIFFERENTIAL EQUATIONS 1. Summary of Classical One-Variable Theory 685 2. Uniqueness and Analytic Continuation of Solutions in Several Variables 690 3. Analog of Fundamental Matrix 693 4. Regular Singularities 697 5. Systems of Higher Order 700 6. Leading Exponents and the Analog of the Indicial Equation 703 7. Uniqueness of Representation 710 APPENDIX C: ROOTS AND RESTRICTED ROOTS FOR CLASSICAL GROUPS 1. Complex Groups 713 2. Noncompact Real Groups 713 3. Roots vs. Restricted Roots in Noncompact Real Groups 715 NOTES 719 REFERENCES 747 INDEX OF NOTATION 763 INDEX 767