Foreword

Summary

These are notes from a graduate course given in Lausanne (Switzerland) during the winter term 1978–79 (Convention romande des enseignements de 3^e cycle en mathématiques).

This term was devoted to a self-contained approach to representation theory for locally compact groups, using only integral methods. The sole prerequisite was a basic familiarity with the theory for finite groups (e.g. as contained in the first chapter of Serre 1967). For didactic reasons, I spent the first half of the term discussing compact groups, trying to be more elementary and more complete in this part. In particular, I have given several proofs of the main results. For example, the Peter-Weyl theorem is proved first with the use of the Stone-Weierstrass approximation theorem (p.33) and then without it (p.36). The “finiteness theorem” (irreducible representations of compact groups are finite dimensional) is proved first for Banach (or barrelled) spaces ((5.8)p.46), then in the general case (quasi-complete spaces) ((7.9)p.69) and finally in a more elementary fashion for Hilbert spaces ((8.5)p.81). Thus I hope that readers with various backgrounds will be able to benefit from these notes.

My way of introducing the subject has forced me to repeat some definitions in the second part where I gradually assume more from my reader (this is particularly so as far as measure theory is concerned). This part in no way claims to be complete and only has an introductory purpose.