Topological K-theory of affine Hecke algebras

Abstract

Let H(R,q) be an affine Hecke algebra with a positive parameter function q. We are interested in the topological K-theory of H(R,q), that is, the K-theory of its C*-completion C*_r (R,q). We will prove that $K_{\ast }(C{\ast }_r(R,q))$ does not depend on the parameter q. 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 not 1, just crossed products of commutative algebras with finite Weyl groups. We explicitly determine $K_{\ast }(C{\ast }_r(R,q))$ for all classical root data R, and for some others as well. This will be useful to analyse the K-theory of the reduced C*-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 Γacting on a smooth manifold M. We develop a method to calculate the K-theory of the crossed product $C(M)⋊\Gamma$. In contrast to the equivariant Chern character of Baum and Connes, our method can also detect torsion elements in these K-groups.

In the second version, paragraph 1.2 was moved to an appendix. Apart from that, only a few minor corrections

References in corpus (2)

• Formal degrees of unipotent discrete series representations and the exotic Fourier transform
• A uniform classification of discrete series representations of affine Hecke algebras

Cited by in corpus (2)

• Affine Hecke algebras and their representations
• Hochschild homology of twisted crossed products and twisted graded Hecke algebras