On the analogy between real reductive groups and Cartan motion groups. III: A proof of the Connes-Kasparov isomorphism

TL;DR

This paper provides a new proof of the Connes-Kasparov isomorphism for a broad class of reductive groups by leveraging Mackey's analogy and recent advances in representation theory, extending prior results from complex semisimple groups.

Contribution

It introduces a novel proof of the Connes-Kasparov isomorphism for general reductive groups, building on Mackey's analogy and recent work on representation rigidity.

Findings

Established the Connes-Kasparov isomorphism for general reductive groups.

Extended Higson's proof from complex semisimple to more general reductive groups.

Connected cohomological conjectures with representation theory deformation.

Abstract

Alain Connes and Nigel Higson pointed out in the 1990s that the Connes-Kasparov "conjecture"' for the K-theory of reduced groupe $C^\ast$-algebras seemed, in the case of reductive Lie groups, to be a cohomological echo of a conjecture of George Mackey concerning the rigidity of representation theory along the deformation from a reductive Lie group to its Cartan motion group. For complex semisimple groups, Nigel Higson established in 2008 that Mackey's analogy is a real phenomenon and does lead to a simple proof of the Connes-Kasparov isomorphism. We here turn to more general reductive groups and use our recent work on Mackey's proposal, together with Higson's work, to obtain a new proof of the Connes-Kasparov isomorphism.