CONTENTS ix

Chapter 6. Compact Lie Groups and Homogeneous Spaces 119

6.1. Some Generalities on Lie Groups 119

6.2. Reductive Lie Groups and Lie Algebras 122

6.3. Cartan's Highest Weight Theory 127

6.4. The Peter-Weyl Theorem and the Plancherel Formula 131

6.5. Complex Flag Manifolds and Holomorphic Vector Bundles 133

6.6. Invariant Function Algebras 136

Chapter 7. Discrete Co-Compact Subgroups 141

7.1. Basic Properties of Discrete Subgroups 141

7.2. Regular Representations on Compact Quotients 146

7.3. The First Trace Formula for Compact Quotients 147

7.4. The Lie Group Case 148

Part 3.

INTRODUCTION TO COMMUTATIVE SPACES

Chapter 8. Basic Theory of Commutative Spaces 153

8.1. Preliminaries 153

8.2. Spherical Measures and Spherical Functions 156

8.3. Alternate Formulation in the Differentiable Setting 160

8.4. Positive Definite Functions 165

8.5. Induced Spherical Functions 168

8.6. Example: Spherical Principal Series Representations 170

8.7. Example: Double Transitivity and Homogeneous Trees 174

8.7A. Doubly Transitive Groups 174

8.7B. Homogeneous Trees 175

8.7C. A Special Case 176

Chapter 9. Spherical Transforms and Plancherel Formulae 179

9.1. Commutative Banach Algebras 179

9.2. The Spherical Transform 184

9.3. Bochner's Theorem 187

9.4. The Inverse Spherical Transform 191

9.5. The Plancherel Formula for K\G/K 192

9.6. The Plancherel Formula for G/K 194

9.7. The Multiplicity Free Criterion 197

9.8. Characterizations of Commutative Spaces 198

9.9. The Uncertainty Principle 199

9.9A. Operator Norm Inequalities for K\G/K 199

9.9B. The Uncertainty Principle for K\G/K 202

9.9C. Operator Norm Inequalities for G/K 203

9.9D. The Uncertainty Principle for G/K 204

9.10. The Compact Case 204