On the Connes-Kasparov isomorphism, II: The Vogan classification of essential components in the tempered dual

TL;DR

This paper computes the Connes-Kasparov morphism for real reductive groups using Vogan's classification, confirming the conjecture for these groups and advancing understanding of their C*-algebraic structure.

Contribution

It provides the explicit computation of the Connes-Kasparov morphism leveraging Vogan's classification, completing the verification of the conjecture for these groups.

Findings

Confirmed the Connes-Kasparov conjecture for connected, linear, real reductive groups.

Computed the reduced C*-algebra up to Morita equivalence using Vogan's classification.

Established the Morita equivalence and the Connes-Kasparov morphism explicitly.

Abstract

This is the second of two papers dedicated to the computation of the reduced C*-algebra of a connected, linear, real reductive group up to C*-algebraic Morita equivalence, and the verification of the Connes-Kasparov conjecture for these groups. These results were originally announced by Antony Wassermann in 1987. In Part I we presented the Morita equivalence and the Connes-Kasparov morphism. In this part we shall compute the morphism using David Vogan's description of the tempered dual.