Divided differences and the Weyl character formula in equivariant K-theory

TL;DR

This paper explores the structure of equivariant K-theory for spaces with Lie group actions, linking divided differences, the Weyl character formula, and invariants under Weyl group actions.

Contribution

It demonstrates that the G-equivariant K-group is a direct summand of the T-equivariant K-group characterized by divided difference operators, extending the Weyl character formula.

Findings

Identifies the G-equivariant K-group as a subgroup of T-equivariant K-group annihilated by divided differences.

Provides conditions under which K_G^*(X) equals the Weyl invariants of K_T^*(X).

Connects the algebraic operators with topological K-theory structures.

Abstract

Let $X$ be a topological space and $G$ a compact connected Lie group acting on $X$. Atiyah proved that the $G$-equivariant K-group of $X$ is a direct summand of the $T$-equivariant K-group of $X$, where $T$ is a maximal torus of $G$. We show that this direct summand is equal to the subgroup of $K_T^*(X)$ annihilated by certain divided difference operators. If $X$ consists of a single point, this assertion amounts to the Weyl character formula. We also give sufficient conditions on $X$ for $K_G^*(X)$ to be isomorphic to the subgroup of Weyl invariants of $K_T^*(X)$.