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.
Author(s): Jacques Faraut
Series: Cambridge Studies in Advanced Mathematics 110
Edition: 1
Publisher: Cambridge University Press
Year: 2008
Language: English
Pages: 314
Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
1.1 Topological groups......Page 13
1.2 The group GL(n,R)......Page 14
1.3 Examples of subgroups of GL(n,R)......Page 17
1.4 Polar decomposition in GL(n,R)......Page 19
1.5 The orthogonal group......Page 23
1.6 Gram decomposition......Page 25
1.7 Exercises......Page 26
2.1 Exponential of a matrix......Page 30
2.2 Logarithm of a matrix......Page 37
2.3 Exercises......Page 41
3.1 One parameter subgroups......Page 48
3.2 Lie algebra of a linear Lie group......Page 50
3.3 Linear Lie groups are submanifolds......Page 53
3.4 Campbell–Hausdorff formula......Page 56
3.5 Exercises......Page 59
4.1 Definitions and examples......Page 62
4.2 Nilpotent and solvable Lie algebras......Page 68
4.3 Semi-simple Lie algebras......Page 74
4.4 Exercises......Page 81
5.1 Haar measure......Page 86
5.2 Case of a group which is an open set in Rn......Page 88
5.3 Haar measure on a product......Page 90
5.4 Some facts about differential calculus......Page 93
5.5 Invariant vector fields and the Haar measure on a linear Lie group......Page 98
5.6 Exercises......Page 102
6.1 Unitary representations......Page 107
6.2 Compact self-adjoint operators......Page 110
6.3 Schur orthogonality relations......Page 115
6.4 Peter–Weyl Theorem......Page 119
6.5 Characters and central functions......Page 127
6.6 Absolute convergence of Fourier series......Page 129
6.7 Casimir operator......Page 131
6.8 Exercises......Page 135
7.1 Adjoint representation of SU(2)......Page 139
7.2 Haar measure on SU(2)......Page 142
7.3 The group SO(3)......Page 145
7.4 Euler angles......Page 146
7.5 Irreducible representations of SU(2)......Page 148
7.6 Irreducible representations of SO(3)......Page 154
7.7 Exercises......Page 161
8.1 Fourier series on SO(2)......Page 170
8.2 Functions of class Ck......Page 172
8.3 Laplace operator on the group SU(2)......Page 175
8.4 Uniform convergence of Fourier series on the group SU(2)......Page 179
8.5 Heat equation on SO(2)......Page 184
8.6 Heat equation on SU(2)......Page 188
8.7 Exercises......Page 194
9.1 Integration formulae......Page 198
9.2 Laplace operator......Page 203
9.3 Spherical harmonics......Page 206
9.4 Spherical polynomials......Page 212
9.5 Funk–Hecke Theorem......Page 216
9.6 Fourier transform and Bochner–Hecke relations......Page 220
9.7 Dirichlet problem and Poisson kernel......Page 224
9.8 An integral transform......Page 232
9.9 Heat equation......Page 237
9.10 Exercises......Page 239
10.1 Integration formulae......Page 243
10.2 Radial part of the Laplace operator......Page 250
10.3 Heat equation and orbital integrals......Page 254
10.4 Fourier transforms of invariant functions......Page 257
10.5 Exercises......Page 258
11.1 Highest weight theorem......Page 261
11.2 Weyl formulae......Page 265
11.3 Holomorphic representations......Page 272
11.4 Polynomial representations......Page 276
11.5 Exercises......Page 281
12.1 Laplace operator......Page 286
12.2 Uniform convergence of Fourier series on the unitary group......Page 288
12.3 Series expansions of central functions......Page 290
12.4 Generalised Taylor series......Page 296
12.5 Radial part of the Laplace operator on the unitary group......Page 300
12.6 Heat equation on the unitary group......Page 304
12.7 Exercises......Page 309
Bibliography......Page 311
Index......Page 313
