Table of contents for Symmetries and Laplacians : introduction to harmonic analysis, group representations, and applications / David Gurarie.


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Chapter 1. Basics of representation theory
1.1. Groups and group actions ......................................... 13
1.2. Regular and induced representations; Haar measure and
convolution algebras ............................................ 24
1.3. Irreducibility  and  decom position  ...............................................................  37
1.4. Lie groups and algebras; the infinitesimal method ....................................46
Chapter 2. Commutative Harmonic analysis
2.1. Fourier transform: inversion and Plancherel formula ................................. 62
2.2* Fourier transform on function-spaces ............................... .........  71
2.3. Some applications of Fourier analysis ........................................ 83
2.4. Laplacian and related differential equations ........................................  92
2.5* The Radon transform................................................... 120
Chapter 3. Representations of compact and finite groups
3.1. The Peter-Weyl theory...................................                  125
3.2. Induced representations and Frobenius reciprocity .................................. 137
3.3* Semidirect products .........................          ........................ 147
Chapter 4. Lie groups SU(2) and S0(3)
4.1. Lie groups and SU(2) and SO(3) and their Lie algebras ................................ 161
4.2. Irreducible representations of SU(2)............ ........................................   164
4.3* Matrix entries and characters of irreducible representations:
Legendre and Jacobi polynomials .....................................     171
4.4. Representations of SO(3): angular momentum and spherical harmonics..... 174
4.5* Laplacian on the n-sphere ................................. ...................... 184
Chapter 5. Classical compact Lie groups and algebras
5.1. Simple and semisimple Lie algebras; Weyl "unitary trick"....................... 191
5.2. Cartan subalgebra, root system, Weyl group .....................................   197
5.3. Highest weight representations ........................................ 206
5.4* Tensors and  Young  tableaux................................................ ..   217
5.5. Haar measure on compact semisimple Lie groups ...................................... 227
5.6. The Weyl character formulae ........................................ 231
5.7* Laplacians on  symmetric spaces................................................................ 241
Chapter 6. The Heisenberg group and semidirect products
6.1. Induced representations and the Mackey's group extension theory........... 257
6.2. The Heisenberg group and the oscillator representation ........................... 274
6.3* The Kirillov orbit method.     ..................................... 290
Chapter 7. Representations of SL,
7.1. Principal, complementary and discrete series ........................................... 305
7.2. Characters of irreducible representations ................................................ 313
7.3. The  Plancherel formula  on  SL,(R) ........................................................... 317
7.4. Infinitesimal representations of SL2; spherical functions and characters....... 328
7.5* Selberg trace formula.     ......................... 334
7.6* Laplacians on hyperbolic surfaces H/F.................................................... 348
7.7* SL,(C) and the Lorentz group ........................................ 361
Chapter 8. Lie groups and hamiltonian mechanics
8.1. Minimal action principle; Euler-Lagrange equations; canonical formalism... 369
8.2. Noether Theorem, conservation laws and Marsden-Weinstein reduction ..... 378
8.3. Classical examples ............................................. 385
8.4* Integrable systems related to classical Lie algebras.............................   393
8.5* The Kepler Problem and the Hydrogen atom .......................................... 408
Appendices:
A: Spectral decomposition of selfadjoint operators ........................................ 423
B: Integral operators ........................................                 427
C: A primer on Riemannian geometry: geodesics, connection, curvature........ 430
R eferences  ........................................ ..................... .. .. ..................... 439
List of frequently used notations................................................   447



Library of Congress subject headings for this publication: Harmonic analysis, Representations of groups