Mackey analogy as deformation of $\mathcal{D}$-modules

TL;DR

This paper demonstrates that the Mackey analogy for real reductive groups can be understood as a deformation of twisted alculus modules over the flag variety, revealing a geometric origin of the bijection.

Contribution

It introduces a geometric framework using families of twisted alculus modules to explain the Mackey analogy as a deformation process.

Findings

The Mackey analogy arises from natural deformations of alculus modules.

Provides a geometric interpretation linking representation theory and alculus.

Establishes a new perspective on the relationship between real reductive groups and their Cartan motion groups.

Abstract

Given a real reductive group Lie group $G_\mathbb{R}$, the Mackey analogy is a bijection between the set of irreducible tempered representations of $G_\mathbb{R}$ and the set of irreducible unitary representations of its Cartan motion group. We show that this bijection arises naturally from families of twisted $\mathcal{D}$-modules over the flag variety of $G_\mathbb{R}$.