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 ...