Weyl character formula in KK-theory

TL;DR

This paper explores the connection between the Baum-Connes conjecture and geometric representation theory, specifically recasting the Weyl character formula within Kasparov's KK-theory and extending it to noncompact groups.

Contribution

It reformulates the Weyl character formula using KK-theory and demonstrates its potential extension to noncompact groups under the Baum-Connes conjecture.

Findings

Reformulation of the Weyl character formula in KK-theory

Extension of the formula to noncompact groups contingent on Baum-Connes conjecture

Linking topological K-theory approaches with operator K-theory in representation theory

Abstract

The purpose of this paper is to begin an exploration of connections between the Baum-Connes conjecture in operator $\K$-theory and the geometric representation theory of reductive Lie groups. Our initial goal is very modest, and we shall not stray far from the realm of compact groups, where geometric representation theory amounts to elaborations of the Weyl character formula such as the Borel-Weil-Bott theorem. We shall recast the topological $\K$-theory approach to the Weyl character formula, due basically to Atiyah and Bott, in the language of Kasparov's $\K\K$-theory. Then we shall show how, contingent on the Baum-Connes conjecture, our $\K\K$-theoretic Weyl character formula can be carried over to noncompact groups.