Divided difference operators in equivariant $KK$-theory

Abstract

Let $G$ be a compact connected Lie group with a maximal torus $T$. Let $A$, $B$ be $G$-${\mathrm{C}}^{\ast }$-algebras. We define certain divided difference operators on Kasparov's $T$-equivariant $KK$-group $K{K}_T(A,B)$ and show that $K{K}_G(A,B)$ is a direct summand of $K{K}_T(A,B)$. More precisely, a $T$-equivariant $KK$-class is $G$-equivariant if and only if it is annihilated by an ideal of divided difference operators. This result is a generalization of work done by Atiyah, Harada, Landweber and Sjamaar.

27 pages. arXiv admin note: text overlap with arXiv:0906.1629 by other authors

References in corpus (2)

• K-theory of weight varieties
• Divided differences and the Weyl character formula in equivariant K-theory