Mackey analogy via $\mathcal{D}$-modules in the example of $SL(2,\mathbb{R})$

TL;DR

This paper explores the Mackey analogy for $SL(2,\mathbb{R})$ using $\mathcal{D}$-modules, proposing a geometric conjecture to deepen understanding of the relationship between representations of real reductive groups and their Cartan motion groups.

Contribution

It introduces a geometric approach via $\mathcal{D}$-modules to understand the Mackey-Higson conjecture for $SL(2,\mathbb{R})$, proposing a new conjecture for the general case.

Findings

Study of admissible $\mathcal{D}$-modules over flag varieties of $SL(2,\mathbb{R})$

Formulation of a conjecture linking geometric $\mathcal{D}$-module structures to the Mackey-Higson correspondence

Provides a framework for understanding the conjecture in the general case

Abstract

A conjecture by Mackey and Higson claims that there is close relationship between irreducible representations of a real reductive group and those of its Cartan motion group. The case of irreducible tempered unitary representations has been verified recently by Afgoustidis. We study the admissible representations of $SL(2,\mathbb{R})$ by considering families of $\D$-modules over its flag varieties. We make a conjecture which gives a geometric understanding of the Makcey-Higson bijection in the general case.