Introduction to the Representation Theory of Compact and Locally Compact Groups

Alain Robert

doi:10.1017/CBO9780511661891

Because of their significance in physics and chemistry, representation of Lie groups has been an area of intensive study by physicists and chemists, as well as mathematicians. This introduction is designed for graduate students who have some knowledge of finite groups and general topology, but is otherwise self-contained. The author gives direct and concise proofs of all results yet avoids the heavy machinery of functional analysis. Moreover, representative examples are treated in some detail.

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Introduction to the Representation Theory of Compact and Locally Compact Groups

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

Foreword
pp vii-viii

Conventional notations and terminology
pp ix-x

PART I - REPRESENTATIONS OF COMPACT GROUPS
pp 1-2

1 - Compact groups and Haar measures
pp 3-12

2 - Representations, general constructions
pp 13-20

3 - A geometrical application
pp 21-28

4 - Finite-dimensional representations of compact groups (Peter-Weyl theorem)
pp 29-39

5 - Decomposition of the regular representation
pp 40-52

6 - Convolution, Plancherel formula & Fourier inversion
pp 53-62

7 - Characters and group algebras
pp 63-77

8 - Induced representations and Frobenius-Weil reciprocity
pp 78-89

9 - Tannaka duality
pp 90-94

10 - Representations of the rotation group
pp 95-108

PART II - REPRESENTATIONS OF LOCALLY COMPACT GROUPS
pp 109-110

11 - Groups with few finite-dimensional representations
pp 111-116

12 - Invariant measures on locally compact groups and homogeneous spaces
pp 117-127

13 - Continuity properties of representations
pp 128-132

14 - Representations of G and of L1(G)
pp 133-144

15 - Schur's lemma: unbounded version
pp 145-150

16 - Discrete series of locally compact groups
pp 151-163

17 - The discrete series of S12(ℝ)
pp 164-171

18 - The principal series of S12(ℝ)
pp 172-178

19 - Decomposition along a commutative subgroup
pp 179-186

20 - Type I groups
pp 187-193

21 - Getting near an abstract Plancherel formula
pp 194-200

Epilogue
pp 201-201

References
pp 202-203

Index
pp 204-205

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Introduction to the Representation Theory of Compact and Locally Compact Groups

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