Introduction to the Representation Theory of Compact and Locally Compact Groups

CONTENTS......Page 5
Foreword......Page 7
Conventional notations and terminology......Page 9
PART I : REPRESENTATIONS OF COMPACT GROUPS......Page 11
1. Compact groups and Haar measures p.......Page 13
Exercises......Page 21
2. Representations, general constructions......Page 23
Exercises......Page 30
3. A geometrical application......Page 31
Exercises......Page 38
4. Finite-dimensional representations of compact groups......Page 39
(Peter-Weyl theorem)......Page 48
5. Decomposition of the regular representation......Page 50
Exercises......Page 61
6. Convolution, Plancherel formula & Fourier inversion......Page 63
Exercises......Page 71
7. Characters and group algebras......Page 73
Exercises......Page 86
8. Induced representations and Frobenius-1Veil reciprocity......Page 88
Exercises......Page 99
9. Tannaka duality......Page 100
10. Representations of the rotation group......Page 105
Exercises......Page 117
PART II : REPRESENTATIONS OF LOCALLY COMPACT GROUPS......Page 119
11. Groups with few finite-dimensional representations......Page 121
Exercises......Page 126
12. Invariant measures on locally compact groups and homogeneous spaces......Page 127
Exercises......Page 136
13. Continuity properties of representations......Page 138
14. Representations of G and of L'(G)......Page 143
Exercises......Page 154
15. Schur's lemma : unbounded version......Page 155
Exercises......Page 160
16. Discrete series of locally compact groups......Page 161
Exercises......Page 172
17. The discrete series of S 12 (JR)......Page 174
Exercises......Page 181
18. The principal series of S12OR)......Page 182
19. Decomposition along a commutative subgroup......Page 189
Appendix: Note on Hilbertian integrals......Page 195
20. Type I groups......Page 197
Exercises......Page 203
21. Getting near an abstract Plancherel formula......Page 204
Epilogue......Page 211
References......Page 212
Index......Page 214