This chapter provides the definitions and many of the basic properties of the objects of abstract harmonic analysis, including locally compact groups, their representations, and various algebras associated with both the groups and their representations. Besides providing the notational conventions used throughout the book, the necessary concepts are organized in a manner useful for the development of the theory of induced representations. For most of the propositions and theorems, we do not provide proofs as these are generally known and accessible in existing monographs. In Section 1.8, we provide a brief guide to the existing literature for the reader who seeks a more comprehensive treatment of a topic. However, we do provide full proofs in Section 1.3, which contains the tools for analysis on coset spaces that may not be so well known but are essential in defining induced representations and proving many of the key theorems.

Locally compact groups

A topological group G is a set with the structure of both a group and a topological space such that the group product is a continuous map from G × G into G and the group inverse is continuous on G. The group product of x and y in G will be denoted multiplicatively as xy and the inverse of x is x−1 except in a few specific cases such as the group of integers or the real numbers.

In this final chapter, we present applications of the theory of induced representations where knowing an explicit expression for an induced representation of a specific group is used.

In Section 7.1, we apply Mackey's theory, and one realization of the induced representations giving the irreducible representations to study the asymptotic behavior of coefficient functions of those representations. The main theorem is that those coefficient functions of infinite-dimensional irreducible representations of motion groups vanish at infinity. As a consequence, one can conclude that the image of a motion group under any irreducible representation is closed in the unitary group with the weak operator topology.

Section 7.2 is concerned with introducing methods for constructing self-adjoint idempotents, or projections, in L1(G) for certain kinds of groups G. A key observation is that the support in Ĝ of a projection must be a compact open set. After reviewing how projections arise for compact and abelian G, we turn to the noncompact, nonabelian situation. Drawing upon the theory developed in Chapters 4 and 5, we identify groups with nontrivial compact open sets in their duals. For appropriate groups G and explicit induced representations of those groups, coefficient functions of those representations can be modified to produce nontrivial projections in L1(G).

Finally, certain identities arising in the construction of projections can be exploited to produce generalizations of the continuous wavelet transform. This is explored in Section 7.3.

The inducing construction in Chapter 2 gives one the power to create unitary representations of a group G when representations of a closed subgroup H are given. One can also find conditions under which the induced representation is irreducible. However, a key question is often the converse: Is every irreducible representation of G equivalent to one induced from a proper subgroup? The imprimitivity theorem presented in this chapter is an invaluable tool in answering this question for many groups.

The full definition and some basic properties of systems of imprimitivity are introduced in Section 3.1. An induced representation is part of what is called, in Section 3.2, an induced system of imprimitivity.

In Section 3.2, we show that if two representations are induced from some subgroup, then the intertwining space of these representations can be identified with the intertwining space for the corresponding systems of imprimitivity. In Section 3.3, we state the imprimitivity theorem and provide a proof in the special case when the system of imprimitivity is living over a discrete coset space (that is, the corresponding subgroup is open). This prepares the way to understanding the general proof, which is presented in Section 3.4.

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.