The Higson-Mackey analogy for finite extensions of complex semisimple groups

TL;DR

This paper extends Higson's analogy between representations of complex semisimple groups and their extensions, providing new insights into the Baum-Connes conjecture for a broader class of Lie groups.

Contribution

It generalizes Higson's one-to-one correspondence to Lie groups with finitely many components whose identity component is complex semisimple.

Findings

Extended Higson's results to broader class of Lie groups.

Verified Baum-Connes conjecture for these groups.

Utilized Mackey's description and twisted crossed product $C^*$-algebras.

Abstract

In the 1970's, George Mackey pointed out an analogy that exists between tempered representations of semisimple Lie groups and unitary representations of associated semidirect product Lie groups. More recently, Nigel Higson refined Mackey's analogy into a one-to-one correspondence for connected complex semisimple groups, and in doing so obtained a novel verification of the Baum-Connes conjecture with trivial coefficients for such groups. Here we extend Higson's results to any Lie group with finitely many connected components whose connected component of the identity is complex semisimple. Our methods include Mackey's description of unitary representations of group extensions involving projective unitary representations, as well as the notion of twisted crossed product $C^*$-algebra introduced independently by Green and Dang Ngoc.