Buch, Englisch, 794 Seiten, Format (B × H): 156 mm x 234 mm, Gewicht: 1182 g
An Overview Based on Examples (PMS-36)
Buch, Englisch, 794 Seiten, Format (B × H): 156 mm x 234 mm, Gewicht: 1182 g
Reihe: Princeton Mathematical Series
ISBN: 978-0-691-09089-4
Verlag: Princeton University Press
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.
Autoren/Hrsg.
Weitere Infos & Material
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. A
