Pseudodifferential operators and the Connes-Kasparov isomorphism

TL;DR

This paper computes the K-theory of a category of pseudodifferential operators on symmetric spaces of real reductive Lie groups, leveraging the Connes-Kasparov isomorphism and relating to representation theory.

Contribution

It establishes a K-theoretic computation of pseudodifferential operators' categories and connects it to the Connes-Kasparov isomorphism and Vogan's parametrization, including a Fourier isomorphism for rank-one groups.

Findings

K-theory of pseudodifferential operator categories computed

Relation to Connes-Kasparov isomorphism established

K-theoretic version of Vogan's theorem proved for rank-one groups

Abstract

We compute the K-theory of the C*-category generated by order zero, equivariant, properly supported, classical pseudodifferential operators acting on sections of homogeneous bundles over the symmetric space of a real reductive Lie group G. Our result uses the Connes-Kasparov isomorphism for G, and in fact is equivalent to the Connes-Kasparov isomorphism. We relate our computation to David Vogan's well-known parametrization of the tempered irreducible representations of G with real infinitesimal character. When the reductive group G has real rank one, we formulate and prove a Fourier isomorphism theorem for equivariant order zero pseudodifferential operators on the symmetric space, and use it to prove a K-theoretic version of Vogan's theorem.