The algebraic Mackey-Higson bijections

TL;DR

This paper constructs explicit algebraic correspondences between the admissible duals of a semisimple Lie group and its Cartan motion group, providing evidence for a conjectured algebraic isomorphism.

Contribution

It introduces a new method using algebraic families of Harish-Chandra modules to relate the admissible duals of these groups, and proves the conjecture for SL(2,R).

Findings

Conjecture holds for SL(2,R).

Explicit algebraic correspondences are constructed.

Provides an equivalent characterization for the bijections in the SL(2,R) case.

Abstract

For a connected semisimple Lie group $G$ we describe an explicit collection of correspondences between the admissible dual of $G$ and the admissible dual of the Cartan motion group associated with $G$. We conjecture that each of these correspondences induces an algebraic isomorphism between the admissible duals. The constructed correspondences are defined in terms of algebraic families of Harish-Chandra modules. We prove that the conjecture holds in the case of $SL_2(\mathbb{R})$, and in that case we give an equivalent characterization for the bijections.