Harmonic analysis on homogeneous spaces

Front Matter
Cover
S Title
Harmonic Analysis on Homogeneous Spaces
Contents
Preface
Suggestions to the Reader

CHAPTER 1   Vector Bundles
1.1 Introduction
1.2 Preliminary Concepts
1.3 Operations on Vector Bundles
1.4 Cross Sections
1.5 Unitary Structures
1.6 K(X)
1.7 Differential Operators
1.8 The Complex Laplacian
1.9 Exercises
1.10 Notes

CHAPTER 2   Elementary Representation Theory
2.1 Introduction
2.2 Representations
2.3 Finite Dimensional Representations
2.4 Induced Representations
2.5 Invariant Measures on Lie Groups
2.6 The Regular Representation
2.7 Completely Continuous Representations
2.8 The Peter-Weyl Theorem
2.9 Characters and Orthogonality Relations
2.10 Exercises
2.11 Notes

CHAPTER 3   Basic Structure Theory of Compact Lie Groups and Semisimple Lie Algebras
3.1 Introduction
3.2 Some Linear Algebra
3.3 Nilpotent Lie Algebras
3.4 Semisimple Lie Algebras
3.5 Cartan Subalgebras
3.6 Compact Lie Groups
3.7 Real Forms
3.8 The Euler Characteristic of a Compact Homogeneous Space
3.9 Automorphisms of Compact Lie Algebras
3.10 The Weyl Group
3.11 Exercises
3.12 Notes

CHAPTER 4   The Topology and Representation Theory of Compact Lie Groups
4.1 Introduction
4.2 The Universal Enveloping Algebra
4.3 Representations of Lie Algebras
4.4 P-Extreme Representations
4.5 The Theorem of the Highest Weight
4.6 Representations and Topology of Compact Lie Groups
4.7 Holomorphic Representations
4.8 The Weyl Integral Formula
4.9 The Weyl Character Formula
4.10 The Ring of Virtual Representations
4.11 Exercises
4.12 Notes

CHAPTER 5   Harmonic Analysis on a Homogeneous Vector Bundle
5.1 Introduction
5.2 Homogeneous Vector Bundles
5.3 Frobenius Reciprocity
5.4 Homogeneous Differential Operators
5.5 The Symbol and Formal Adjoint of a Homogeneous Differential Operator
5.6 The Laplacian
5.7 The Sobolev Spaces
5.8 Globally Hypoelliptic Differential Operators
5.9 Bott's Index Theorem
5.A Appendix : The Fourier Integral Theorem
5.10 Exercises
5.11 Notes

CHAPTER 6   Holomorphic Vector Bundles over Flag Manifolds
6.1 Introduction
6.2 Generalized Flag Manifolds
6.3 Holomorphic Vector Bundles over Generalized Flag Manifolds
6.4 An Alternating Sum Formula
6.5 Exercises
6.6 Notes

CHAPTER 7   Analysis on Semisimple Lie Groups
7.1 Introduction
7.2 The Cartan Decomposition of a Semisimple Lie Group
7.3 The Iwasawa Decomposition of a Semisimple Lie Algebra
7.6 The Integral Formula for the Iwasawa Decomposition
7.7 Integral Formulas for the Adjoint Action
7.8 Integral Formulas for the Adjoint Representation
7.9 Semisimple Lie Groups with One Conjugacy Class of Cartan Subalgebra
7.10 Differential Operators on a Reductive Lie Algebra
7.11 A Formula for Semisimple Lie Groups with One Conjugacy Class of Cartan Subalgebra
7.12 The Fourier Expansion of F_f
7.13 Exercises
7.14 Notes

CHAPTER 8   Representations of Semisimple Lie Groups
8.1 Introduction
8.2 Finite Dimensional Unitary Representations
8.3 The Principal Series
8.4 Other Realizations of the Principal Series
8.5 Finite Dimensional Subrepresentations of the Nonunitary Principal Series
8.6 The Character of K-Finite Representation
8.7 Characters of Admissible Representations
8.8 The Character of a Principal Series Representation
8.9 The Weyl Group Revisited
8.10 The Intertwining Operators
8.11 The Analytic Continuation of the Intertwining Operators
8.12 The Asymptotics of the Principal Series for Semisimple Lie Groups of Split Rank 1
8.13 The Composition Series of the Principal Series
8.14 The Normalization of the Intertwining Operators
8.15 The Plancherel Measure
8.16 Exercises
8.17 Notes

Back Matter

APPENDIX 1 Review of Differential Geometry
A.1.1 Manifolds
A.1.2 Tangent Vectors
A.1.3 Vector Fields
A.1.4 Partitions of Unity

APPENDIX 2 Lie Groups
A.2.1 Basic Notions
A.2.3 Lie Subalgebras and Lie Subgroups
A.2.4 Homogeneous Spaces
A.2.5 Simply Connected Lie Groups

APPENDIX 3 A Review of Multilinear Algebra
A.3.1 The Tensor Product

APPENDIX 4 Integration on Manifolds
A.4.1 k-forms
A.4.2 Integration on Manifolds

APPENDIX 5 Complex Manifolds
A.5.1 Basic Concepts
A.5.2 The Holomorphic and Antiholomorphic Tangent Spaces
A.5.3 Complex Lie Groups

APPENDIX 6 Elementary Functional Analysis
A.6.1 Banach Spaces
A.6.2 Hilbert Spaces

APPENDIX 7 Integral Operators
A.7.1 Measures on Locally Compact SpacesA.
A.7.2 Integral Operators

APPENDIX 8 The Asymptotics for Certain Sturm-Liouville Systems
A.8.1 The Systems
A.8.2 The Asymptotics

Bibliography

Index