Topological K-theory of affine Hecke algebras

Abstract

Let $\mathcal{\mathscr{H}}\left(\mathcal{ℛ},q\right)$ be an affine Hecke algebra with a positive parameter function $q$. We are interested in the topological K-theory of its $C^{\ast }$-completion $C_r^{\ast }\left(\mathcal{ℛ},q\right)$. We prove that $K_{\ast }\left(C_r^{\ast }\left(\mathcal{ℛ},q\right)\right)$ does not depend on the parameter $q$, solving a long-standing conjecture of Higson and Plymen. For this we use representation-theoretic methods, in particular elliptic representations of Weyl groups and Hecke algebras.

Thus, for the computation of these K-groups it suffices to work out the case $q=1$. These algebras are considerably simpler than for $q\ne 1$, just crossed products of commutative algebras with finite Weyl groups. We explicitly determine $K_{\ast }\left(C_r^{\ast }\left(\mathcal{ℛ},q\right)\right)$ for all classical root data $\mathcal{ℛ}$. This will be useful for analyzing the K-theory of the reduced $C^{\ast }$-algebra of any classical $p$-adic group.

For the computations in the case $q=1$, we study the more general situation of a finite group $\mathrm{\Gamma }$ acting on a smooth manifold $M$. We develop a method to calculate the K-theory of the crossed product $C\left(M\right)⋊\mathrm{\Gamma }$. In contrast to the equivariant Chern character of Baum and Connes, our method can also detect torsion elements in these K-groups.