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

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 $K_T^{\ast }(X)$ 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 $K_G^{\ast }(X)$ to be isomorphic to the subgroup of Weyl invariants of $K_T^{\ast }(X)$.

22 pages, minor errors corrected, some examples added

Cited by in corpus (1)

• Divided difference operators in equivariant $KK$-theory