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

Abstract

Given a real reductive linear 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, established by Higson and Afgoustidis. We show that this bijection arises naturally from families of twisted \({\mathcal {D}}\)-modules over the ag variety of \(G_{\mathbb {R}}\).