Operator K-Theory and Tempiric Representations

TL;DR

This paper connects Vogan's classification of irreducible representations of real reductive groups with the Connes-Kasparov isomorphism in operator K-theory, establishing an equivalence between representation theory and K-theoretic isomorphisms.

Contribution

It proves that the Connes-Kasparov isomorphism in operator K-theory is equivalent to a K-theoretic reformulation of Vogan's classification result.

Findings

Established the equivalence between Connes-Kasparov isomorphism and Vogan's classification.

Provided a K-theoretic interpretation of minimal K-types in tempered representations.

Unified operator K-theory with representation theory of real reductive groups.

Abstract

David Vogan proved that if $G$ is a real reductive group, and if $K$ is a maximal compact subgroup of $G$, then every irreducible representation of $K$ is included as a minimal $K$-type in precisely one tempered, irreducible unitary representation of $G$ with real infinitesimal character, and that moreover it is included there with multiplicity one and is the unique minimal $K$-type in that representation. We shall prove that the Connes-Kasparov isomorphism in operator $K$-theory is equivalent to a $K$-theoretic version of Vogan's result.