Topological K-theory of affine Hecke algebras

TL;DR

This paper proves that the topological K-theory of affine Hecke algebra C*-completions is independent of the parameter q, and explicitly computes these K-groups for classical root data, aiding analysis of p-adic groups.

Contribution

It demonstrates parameter independence of K-theory for affine Hecke algebras and provides explicit calculations for classical cases using representation theoretic methods.

Findings

K-theory of C*_r (R,q) does not depend on q

Explicit K-theory computations for classical root data

Develops a method to compute K-theory of crossed products with torsion detection

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 $K_* (C*_r (R,q))$ 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 $K_* (C*_r (R,q))$ 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 \Gamma acting on a smooth manifold M. We develop a method to calculate the K-theory of the crossed product $C(M) \rtimes \Gamma$. In contrast to the equivariant Chern character of Baum and Connes, our method can also detect torsion elements in these K-groups.