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

TL;DR

This paper demonstrates a deformation process that contracts irreducible tempered representations of a reductive Lie group onto those of its associated Cartan motion group, establishing a concrete link between their representation theories.

Contribution

It constructs a deformation framework that realizes the bijection between irreducible tempered representations of G and unitary irreducible representations of G_0 as a continuous contraction process.

Findings

Established a deformation linking G and G_0 representations

Constructed a family of subspaces and evolution operators for contraction

Provided a concrete realization of the Mackey correspondence

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 had been suggested by Mackey in the 1970s, has recently been described by the author. In the present notes, 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 Fr\'echet space, a family of subspaces and evolution operators that contract $\pi$ onto the corresponding representation of $G_0$.