Vol. 3, No. 3, 2018

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ISSN: 2379-1691 (e-only)
ISSN: 2379-1683 (print)
Topological K-theory of affine Hecke algebras

Maarten Solleveld

Vol. 3 (2018), No. 3, 395–460
Abstract

Let (,q) be an affine Hecke algebra with a positive parameter function q. We are interested in the topological K-theory of its C-completion Cr(,q). We prove that K(Cr(,q)) 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 q1, just crossed products of commutative algebras with finite Weyl groups. We explicitly determine K(Cr(,q)) for all classical root data . This will be useful for analyzing 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 Γ acting on a smooth manifold M. We develop a method to calculate the K-theory of the crossed product C(M) Γ. In contrast to the equivariant Chern character of Baum and Connes, our method can also detect torsion elements in these K-groups.

Keywords
topological K-theory, affine Hecke algebra, Weyl group, crossed product algebra
Mathematical Subject Classification 2010
Primary: 20C08, 46L80
Secondary: 19L47
Milestones
Received: 6 November 2016
Revised: 18 September 2017
Accepted: 19 October 2017
Published: 16 July 2018
Authors
Maarten Solleveld
IMAPP
Radboud Universiteit Nijmegen
Nijmegen
Netherlands