A proof of the Baum-Connes conjecture for real semisimple Lie groups with coefficients on flag varieties

TL;DR

This paper proves the Baum-Connes conjecture with coefficients for real semisimple Lie groups acting on flag varieties by constructing an isomorphism in equivariant K-theory using KK-theory techniques.

Contribution

It provides a proof of the Baum-Connes conjecture with coefficients for a class of real semisimple Lie groups via a novel KK-theoretic approach.

Findings

The assembly map is an isomorphism for these groups.

The orbit structure of the group action is crucial for the proof.

The result confirms the conjecture in this specific setting.

Abstract

We consider the equivariant K-theory of a real semisimple Lie group which acts on the (complex) flag variety of its complexification group. We construct an assemble map in the framework of KK-theory and then we prove that it is an isomorphism. The prove relies on a careful study of the orbits of the real group action on the flag variety and then piecing together different orbits. This result is a special case of the Baum-Connes conjecture with coefficients.