The intent of this book is to give students of mathematics and mathematicians in diverse fields an entry into the subject of harmonic analysis on homogeneous spaces. It is hoped that the book could be used as a supplement to a standard one-year course in Lie groups and Lie algebras or as the main text in a more unorthodox course on the subject.
Pains have been taken to build the subject matter gradually from fairly easy material to the more advanced material. Exercises have been included at the end of each chapter to help the reader's comprehension of the material.
We have also adopted a decimal system of labeling statements, lemmas,theorems, etc. (For example, 3.7.2(1) means the first displayed formula or assertion in the second paragraph of the seventh section of Chapter 3.) We felt after much soul-searching that this system of presentation would in the long-run make the book more accessible since it simplifies the cross-referencing.
However, once one decides to consistently use a system of labeling such as ours, one is forced to write the material of the book in a "quantized" form.
The writing also becomes dry and discussions of material are hard to fit into the text. It is hoped that the underlying mathematics will motivate the reader to tolerate the exposition. To somewhat temper the dryness of exposition,each chapter has an introduction which will, hopefully, help the reader to organize and motivate the material of the chapter.
Author(s): Nolan R Wallach
Series: Pure and applied mathematics, 19
Publisher: M. Dekker
Year: 1973
Language: English
Commentary: Copyright Page & page 195 are missing
Pages: C, XV+ 361
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
