Table Of Content

Pure and Applied 11at.tetatit s,: 19 ,Harmonic Analysis on Homogeneous Spaces Nolan K Wallac h lt 11 Harmonic Analysis on Homogeneous Spaces Harmonic Analysis on Homogeneous Spaces NOLAN R. WALLACH Department of Mathematics Rutgers, The State University of New Jersey New Brunswick, New Jersey 1973 MARCEL DEKKER, INC., New York Contents Preface xi Suggestions to the Reader xv Chapter 1. Vector Bundles 1 1.1 Introduction 1.2 Preliminary Concepts 1.3 Operations on Vector Bundles 4 1.4 Cross Sections 7 1.5 Unitary Structures 10 1.6 K(X) 11 1.7 Differential Operators 12 1.8 The Complex Laplacian 17 1.9 Exercises 19 1.10 Notes 21 Chapter 2. Elementary Representation Theory 22 2.1 Introduction 22 2.2 Representations 23 2.3 Finite Dimensional Representations 24 2.4 Induced Representations 27 2.5 Invariant Measures on Lie Groups 30 2.6 The Regular Representation 32 2.7 Completely Continuous Representations 34 2.8 The Peter-Weyl Theorem 36 2.9 Characters and Orthogonality Relations 39 2.10 Exercises 41 2.11 Notes 43 V Vi CONTENTS Chapter 3. Basic Structure Theory of Compact Lie Groups and Semisimple Lie Algebras 45 3.1 Introduction 45 3.2 Some Linear Algebra 46 3.3 Nilpotent Lie Algebras 51 3.4 Semisimple Lie Algebras 51 3.5 Cartan Subalgebras 54 3.6 Compact Lie Groups 59 3.7 Real Forms 64 3.8 The Euler Characteristic of a Compact Homogeneous Space 67 3.9 Automorphisms of Compact Lie Algebras 70 3.10 The Weyl Group 74 3.11 Exercises 78 3.12 Notes 81 Chapter 4. The Topology and Representation Theory of Compact Lie Groups 83 4.1 Introduction 83 4.2 The Universal Enveloping Algebra 84 4.3 Representations of Lie Algebras 87 4.4 P-Extreme Representations 90 4.5 The Theorem of the Highest Weight 92 4.6 Representations and Topology of Compact Lie Groups 94 4.7 Holomorphic Representations 99 4.8 The Weyl Integral Formula 101 4.9 The Weyl Character Formula 103 4.10 The Ring of Virtual Representations 107 4.11 Exercises 110 4.12 Notes 113 Chapter 5. Harmonic Analysis on a Homogeneous Vector Bundle 114 5.1 Introduction 114 5.2 Homogeneous Vector Bundles 114 5.3 Frobenius Reciprocity 116 5.4 Homogeneous Differential Operators 119 5.5 The Symbol and Formal Adjoint of a Homogeneous Differential Operator 122 5.6 The Laplacian 123 5.7 The Sobolev Spaces 126 5.8 Globally Hypoelliptic Differential Operators 133 5.9 Bott's Index Theorem 137 5.A Appendix: The Fourier Integral Theorem 138 5.10 Exercises 141 5.11 Notes 143 CONTENTS vii Chapter 6. Holomorphic Vector Bundles over Flag Manifolds 144 6.1 Introduction 144 6.2 Generalized Flag Manifolds 144 6.3 Holomorphic Vector Bundles over Generalized Flag Manifolds 153 6.4 An Alternating Sum Formula 157 6.5 Exercises 158 6.6 Notes 159 Chapter 7. Analysis on Semisimple Lie Groups 160 7.1 Introduction 160 7.2 The Cartan Decomposition of a Semisimple Lie Group 160 7.3 The Iwasawa Decomposition of a Semisimple Lie Algebra 163 7.4 The Iwasawa Decomposition of a Semisimple Lie Group 164 7.5 The Fine Structure of Semisimple Lie Groups 168 7.6 The Integral Formula for the Isawasa Decomposition 176 7.7 Integral Formulas for the Adjoint Action 179 7.8 Integral Formulas for the Adjoint Representation 183 7.9 Semisimple Lie Groups with One Conjugacy Class of Cartan Subalgebra 186 7.10 Differential Operators on a Reductive Lie Algebra 190 7.11 A Formula for Semisimple Lie Groups with One Conjugacy Class of Cartan Subalgebra 201 7.12 The Fourier Expansion of Ff 206 7.13 Exercises 211 7.14 Notes 213 Chapter 8. Representations of Semisimple Lie Groups 215 8.1 Introduction 215 8.2 Finite Dimensional Unitary Representations 216 8.3 The Principal Series 218 8.4 Other Realizations of the Principal Series 220 8.5 Finite Dimensional Subrepresentations of the Nonunitary Principal Series 224 8.6 The Character of K-Finite Representation 232 8.7 Characters of Admissible Representations 241 8.8 The Character of a Principal Series Representation 244 8.9 The Weyl Group Revisited 247 8.10 The Intertwining Operators 252 8.11 The Analytic Continuation of the Intertwining Operators 265 8.12 The Asymptotics of the Principal Series for Semisimple Lie Groups of Split Rank 1 277 8.13 The Composition Series of the Principal Series 285 8.14 The Normalization of the Intertwining Operators 291 8.15 The Plancherel Measure 293 Viii CONTENTS 8.16 Exercises 301 8.17 Notes 303 Appendix 1. Review of Differential Geometry 307 A.1.1 Manifolds 307 A.1.2 Tangent Vectors 309 A.1.3 Vector Fields 311 A.1.4 Partitions of Unity 312 Appendix 2. Lie Groups 313 A.2.1 Basic Notions 313 A.2.2 The Exponential Map 315 A.2.3 Lie Subalgebras and Lie Subgroups 317 A.2.4 Homogeneous Spaces 318 A.2.5 Simply Connected Lie Groups 319 Appendix 3. A Review of Multilinear Algebra 321 A.3.1 The Tensor Product 321 Appendix 4. Integration on Manifolds 325 A.4.1 k-forms 325 A.4.2 Integration on Manifolds 327 Appendix 5. Complex Manifolds 330 A.5.1 Basic Concepts 330 A.5.2 The Holomorphic and Antiholomorphic Tangent Spaces 331 A.5.3 Complex Lie Groups 332 Appendix 6. Elementary Functional Analysis 334 A.6.1 Banach Spaces 334 A.6.2 Hilbert Spaces 337 Appendix 7. Integral Operators 343 A.7.1 Measures on Locally Compact Spaces 343 A.7.2 Integral Operators 345 CONTENTS ix Appendix 8. The Asymptotics for Certain Sturm- Liouville Systems 347 A.8.1 The Systems 347 A.8.2 The Asymptotics 347 Bibliography 353 Index 3 59