2016/10/31 by Maarten Solleveld
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Affine transformation #Commutative property #Computation #Conjecture #Diffeomorphism #Equivariant map #Finite group #Homotopy and Cohomology in Algebraic Topology #Manifold (fluid mechanics) #Torsion (gastropod) #math.KT #math.RT #msc:19L47 #msc:20C08 #msc:46E80
paper · pdf · doi:10.2140/akt.2018.3.395
published as Ann. K-Th. 3 (2018) 395-460 · In the second version, paragraph 1.2 was moved to an appendix. Apart from that, only a few minor corrections
openalex created_date 2016/11/04 · arxiv created 2018/06/29 · openalex publication_date 2018/07/16 · arxiv updated 2018/07/25 · openalex updated_date 2026/08/05
Let [math] be an affine Hecke algebra with a positive parameter function [math] . We are interested in the topological K-theory of its [math] -completion [math] . We prove that [math] does not depend on the parameter [math] , 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.\n¶ Thus, for the computation of these K-groups it suffices to work out the case [math] . These algebras are considerably simpler than for [math] , just crossed products of commutative algebras with finite Weyl groups. We explicitly determine [math] for all classical root data [math] . This will be useful for analyzing the K-theory of the reduced [math] -algebra of any classical [math] -adic group.\n¶ For the computations in the case [math] , we study the more general situation of a finite group [math] acting on a smooth manifold [math] . We develop a method to calculate the K-theory of the crossed product [math] . In contrast to the equivariant Chern character of Baum and Connes, our method can also detect torsion elements in these K-groups.