Chapter 1. Introduction 5

Chapter 2. Basics about Lie Groups and Homogeneous Spaces 7 2.1. Definitions, Invariant Vector Fields and Forms 7 2.2. Invariant Volume Forms and the Modular Function 10 2.3. Homogeneous Spaces 12

Chapter 3. Representations, Submodules, Characters and the Convolution Algebra of a Homogeneous Space 23

3.1. Representations and Characters 23 3.2. Definitions and Basic Properties of the Convolution Algebra 28 3.3. Isotropic Functions and Approximations to the Identity 33 3.4. Symmetric and Weakly Symmetric Spaces 36

Chapter 4. Compact Groups and Homogeneous Spaces 39 4.1. Complete Reducibility of Representations 39 4.2. The L^{2} Convolution Algebra of a Compact Space 48

Chapter 5. Compact Symmetric and Weakly Symmetric Spaces 51 5.1. The Decomposition of L^{2}(G/K) for Weakly Symmetric Spaces 51 5.2. Diagonalization of Invariant Linear Operators on Compact Weakly Symmetric spaces 56 5.3. Abelian Groups and Spaces with Commutative Convolution Algebra 57

Appendix A. Some Results form Analysis 61 A.1. Bounded Integral Operators 61 A.2. Spectral Theorem for Commuting Compact Selfadjoint and Normal Operators on a Hilbert Space 63 A.3. Miscellaneous analytic facts. 64

Appendix B. Radon Transforms and Spherical Functions on Finite Homogeneous Spaces 67

B.1. Introduction 67 B.2. Finite Homogeneous Spaces 67 B.3. Injectivity Results for Radon Transforms 68 B.4. The Convolution Algebra of a Finite G-Space 71 B.5. Finite Symmetric Spaces 73 B.6. Invariant Linear Operators on Finite Symmetric Spaces 76 B.7. Radon Transforms for Doubly Transitive Actions 78

Appendix C. Fiber Integral and the Coarea Formula 81 C.1. The basic geometry of the fibers of a smooth map 81 C.2. Fiber Integrals and the Coarea Formula 84 C.3. The Lemma on Fiber Integration 86 C.4. Remarks on the coarea formula and fiber integration 89

Appendix D. Isoperimetric Constants and Sobolev Inequalities 91 D.1. Relating Integrals to Volume and Surface Area 91 D.2. Sobolev Inequalities 92 D.3. McKean’s and Cheeger’s lower bounds on the first eigenvalue 94 D.4. Hölder Continuity 96 Problems 98

Bibliography 101

Index 103