The Bernstein-Gelfand-Gelfand complex for rank one semi simple Lie groups as a Kasparov module

TL;DR

This paper constructs a Fredholm module representing the Kasparov gamma element for rank one semisimple Lie groups, advancing the proof of the Baum-Connes conjecture for these groups.

Contribution

It introduces a new Kasparov module construction for rank one semisimple Lie groups, crucial for proving the Baum-Connes conjecture in this setting.

Findings

Constructed a Fredholm module for the gamma element

Proved the Baum-Connes conjecture for rank one semisimple Lie groups

Enhanced understanding of G-equivariant KK-theory

Abstract

We construct a Fredhom module representing the Kasparov gamma element in G-equivariant KK-theory for G a semisimple Lie group of real rank one. This is the main step of our proof of the Baum-Connes conjecture for such groups.