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

Alexandre Afgoustidis

doi:10.1215/00127094-2019-0071

paper

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

arXiv:1808.09525 · doi:10.1215/00127094-2019-0071

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 $π$ of G, we build, in an appropriate Fréchet space, a family of subspaces and evolution operators that contract $π$ onto the corresponding representation of $G_{0}$ .

Final version, to appear in the Duke Math. Journal (42 pages, 3 figures)