On the analogy between real reductive groups and Cartan motion groups: Contraction of irreducible tempered representations

Abstract

Attached to any reductive Lie group $G$ is a “Cartan motion group” $G_0$—a Lie group with the same dimension as $G$, but a simpler group structure. A natural one-to-one correspondence between the irreducible tempered representations of $G$ and the unitary irreducible representations of $G_0$, whose existence was suggested by Mackey in the 1970s, has recently been described by the author. In the present article, we use the existence of a family of groups interpolating between $G$ and $G_0$ to realize the bijection as a deformation: for every irreducible tempered representation $\pi$ of G, we build, in an appropriate Fréchet space, a family of subspaces, and evolution operators that contract $\pi$ onto the corresponding representation of $G_0$.