The imprimitivity theorem
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 ...