Preface

Summary

Locally compact groups arise in many diverse areas of mathematics, the physical sciences, and engineering and the presence of the group is usually felt through unitary representations of the group. This observation underlies the importance of understanding such representations and how they may be constructed, combined, or decomposed. Of particular importance are the irreducible unitary representations. In the middle of the last century, G.W. Mackey initiated a program to develop a systematic method for identifying all the irreducible unitary representations of a given locally compact group G. We denote the set of all unitary equivalence classes of irreducible unitary representations of G by Ĝ. Mackey's methods are only effective when G has certain restrictive structural characteristics; nevertheless, time has shown that many of the groups that arise in important problems are appropriate for Mackey's approach. The program Mackey initiated received contributions from many researchers with some of the most substantial advances made by R. J. Blattner and J. M. G. Fell. Fell's work is particularly important in studying Ĝ as a topological space. At the core of this program is the inducing construction, which is a method of building a unitary representation of a group from a representation of a subgroup.

The main goal of this book is to make the theory of induced representations accessible to a wider audience. As the book progresses, we provide a large number of examples to illustrate the theory. A few particular groups reappear at various stages in the development of the material as more and more can be said about them.