On the analogy between real reductive groups and Cartan motion groups. I: The Mackey-Higson bijection

TL;DR

This paper establishes a natural bijection between the tempered duals of a noncompact reductive Lie group and its Cartan motion group, confirming Mackey's conjecture on the analogy of their irreducible representations.

Contribution

It constructs a simple, natural bijection between the tempered duals of the groups and extends it to their admissible duals, advancing the understanding of their representation theory.

Findings

Confirmed Mackey's conjecture on the bijection of irreducible representations.

Established a natural correspondence between the tempered duals of the groups.

Extended the bijection to admissible duals.

Abstract

George Mackey suggested in 1975 that there should be analogies between the irreducible unitary representations of a noncompact reductive Lie group $G$ and those of its Cartan motion group $G_0$ $-$ the semidirect product of a maximal compact subgroup of $G$ and a vector space. He conjectured the existence of a natural one-to-one correspondence between "most" irreducible (tempered) representations of $G$ and "most" irreducible (unitary) representations of $G_0$. We here describe a simple and natural bijection between the tempered duals of both groups, and an extension to a one-to-one correspondence between the admissible duals.