Analysis on Lie Groups: An Introduction

This self-contained text concentrates on the perspective of analysis, assuming only elementary knowledge of linear algebra and basic differential calculus. The author describes, in detail, many interesting examples, including formulas which have not previously appeared in book form. Topics covered include the Haar measure and invariant integration, spherical harmonics, Fourier analysis and the heat equation, Poisson kernel, the Laplace equation and harmonic functions. Perfect for advanced undergraduates and graduates in geometric analysis, harmonic analysis and representation theory, the tools developed will also be useful for specialists in stochastic calculation and the statisticians. With numerous exercises and worked examples, the text is ideal for a graduate course on analysis on Lie groups.

Preface	p. ix
The linear group	p. 1
Topological groups	p. 1
The group GL (n, R)	p. 2
Examples of subgroups of GL (n, R)	p. 5
Polar decomposition in GL (n, R)	p. 7
The orthogonal group	p. 11
Gram decomposition	p. 13
Exercises	p. 14
The exponential map	p. 18
Exponential of a matrix	p. 18
Logarithm of a matrix	p. 25
Exercises	p. 29
Linear Lie groups	p. 36
One parameter subgroups	p. 36
Lie algebra of a linear Lie group	p. 38
Linear Lie groups are submanifolds	p. 41
Campbell-Hausdorff formula	p. 44
Exercises	p. 47
Lie algebras	p. 50
Definitions and examples	p. 50
Nilpotent and solvable Lie algebras	p. 56
Semi-simple Lie algebras	p. 62
Exercises	p. 69
Haar measure	p. 74
Haar measure	p. 74
Case of a group which is an open set in R[superscript n]	p. 76
Haar measure on a product	p. 78
Some facts about differential calculus	p. 81
Invariant vector fields and Haar measure on a linear Lie group	p. 86
Exercises	p. 90
Representations of compact groups	p. 95
Unitary representations	p. 95
Compact self-adjoint operators	p. 98
Schur orthogonality relations	p. 103
Peter-Weyl Theorem	p. 107
Characters and central functions	p. 115
Absolute convergence of Fourier series	p. 117
Casimir operator	p. 119
Exercises	p. 123
The groups SU (2) and SO(3), Haar measures and irreducible representations	p. 127
Adjoint representation of SU(2)	p. 127
Haar measure on SU(2)	p. 130
The group SO(3)	p. 133
Euler angles	p. 134
Irreducible representations of SU(2)	p. 136
Irreducible representations of SO(3)	p. 142
Exercises	p. 149
Analysis on the group SU(2)	p. 158
Fourier series on SO(2)	p. 158
Functions of class C[superscript k]	p. 160
Laplace operator on the group SU(2)	p. 163
Uniform convergence of Fourier series on the group SU(2)	p. 167
Heat equation on SO(2)	p. 172
Heat equation on SU(2)	p. 176
Exercises	p. 182
Analysis on the sphere and the Euclidean space	p. 186
Integration formulae	p. 186
Laplace operator	p. 191
Spherical harmonics	p. 194
Spherical polynomials	p. 200
Funk-Hecke Theorem	p. 204
Fourier transform and Bochner-Hecke relations	p. 208
Dirichlet problem and Poisson kernel	p. 212
An integral transform	p. 220
Heat equation	p. 225
Exercises	p. 227
Analysis on the spaces of symmetric and Hermitian matrices	p. 231
Integration formulae	p. 231
Radial part of the Laplace operator	p. 238
Heat equation and orbital integrals	p. 242
Fourier transforms of invariant functions	p. 245
Exercises	p. 246
Irreducible representations of the unitary group	p. 249
Highest weight theorem	p. 249
Weyl formulae	p. 253
Holomorphic representations	p. 260
Polynomial representations	p. 264
Exercises	p. 269
Analysis on the unitary group	p. 274
Laplace operator	p. 274
Uniform convergence of Fourier series on the unitary group	p. 276
Series expansions of central functions	p. 278
Generalised Taylor series	p. 284
Radial part of the Laplace operator on the unitary group	p. 288
Heat equation on the unitary group	p. 292
Exercises	p. 297
Bibliography	p. 299
Index	p. 301
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