Representation Theory of Semisimple Groups

"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

Auteur

Anthony W. Knapp With a new preface by the author

Texte du rabat

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.

Résumé
Offers a survey of representation theory of semisimple Lie groups. Suitable for both graduate students and researchers, this book states the theorems precisely, and gives many illustrative examples or classes of examples. It includes for the reader a useful 300-item bibliography and an extensive section of notes.

Contenu
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
8. Irreducible Finite-Dimensional Complex-Linear Representations of 91(2,C) 30
9. Finite-Dimensional Representations of 91(2,C) 31
10. Irreducible Unitary Representations of SL(2,C) 33
11. Irreducible Unitary Representations of SL(2,08) 35
12. Use of SU(1,1) 39
13. Plancherel Formula 41
14. Problems 42 CHAPTER III. C VECTORS AND THE UNIVERSAL ENVELOPING ALGEBRA
    l. Universal Enveloping Algebra 46
15. Actions on Universal Enveloping Algebra 50
16. C Vectors 55
17. Garding Subspace. Problems 57 CHAPTER IV. REPRESENTATIONS OF COMPACT LIE GROUPS
18. Examples of Root Space Decompositions 60
19. Roots 65
20. Abstract Root Systems and Positivity 72
21. Weyl Group, Algebraically 78
22. Weights and Integral Forms 81
23. Centalizers of Tori 86
24. Theorem of the Highest Weight 89
25. Verma Modules 93
26. Weyl Group, Analytically 100
27. Weyl Character Formula 104
28. Problems 109 CHAPTER V. STRUCTURE THEORY FOR NONCOMPACT GROUPS
    l. Cartan Decomposition and the Unitary Trick 113
29. Iwasawa Decomposition 116
30. Regular Elements, Weyl Chambers, and the Weyl Group 121
31. Other Decompositions 126
32. Parabolic Subgroups 132
33. Integral Formulas 137
34. Borel-Weil Theorem 142
35. Problems 147 CHAPTER VI. HOLOMORPHIC DISCRETE SERIES
36. Holomorphic Discrete Series for SU(1,1) 150
37. Classical Bounded Symmetric Domains 152
38. Harish-Chandra Decomposition 153
39. Holomorphic Discrete Series 158
40. Finiteness of an Integral 161
41. Problems 164 CHAPTER VII. INDUCED REPRESENTATIONS
42. Three Pictures 167
43. Elementary Properties 169
44. Bruhat Theory 172
45. Formal Intertwining Operators 174
46. Gindikin-Karpelevic Formula 177
47. Estimates on Intertwining Operators, Part I 181
48. Analytic Continuation of Intertwining Operators, Part I 183
49. Spherical Functions 185
50. Finite-Dimensional Representations and the H function 191
51. Estimates on Intertwining Operators, Part II 196
52. Tempered Representations and Langlands Quotients 198
53. Problems 201 CHAPTER VIII. ADMISSIBLE REPRESENTATIONS
    l. Motivation 203
54. Admissible Representations 205
55. Invariant Subspaces 209
56. Framework for Studying Matrix Coefficients 215
57. Harish-Chandra Homomorphism 218
58. Infinitesimal Character 223
59. Differential Equations Satisfied by Matrix Coefficients 226
60. Asymptotic Expansions and Leading Exponents 234
61. First Application: Subrepresentation Theorem 238
62. Second Application: Analytic Continuation of Interwining Operators, Part II 239
63. Third Application: Control of K-Finite Z(gc)-Finite Functions 242
64. Asymptotic Expansions near the Walls 247
65. Fourth Application: Asymptotic Size of Matrix Coefficients 253
66. Fifth Application: Identification of Irreducible Tempered Representations 258
67. Sixth Application: Langlands Classification of Irreducible Admissible Representations 266
68. Problems 276 CHAPTER IX. CONSTRUCTION OF DISCR