Annals of K-Theory
- Ann. K-Theory
- Volume 3, Number 3 (2018), 395-460.
Topological K-theory of affine Hecke algebras
Let be an affine Hecke algebra with a positive parameter function . We are interested in the topological K-theory of its -completion . We prove that does not depend on the parameter , 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 . These algebras are considerably simpler than for , just crossed products of commutative algebras with finite Weyl groups. We explicitly determine for all classical root data . This will be useful for analyzing the K-theory of the reduced -algebra of any classical -adic group.
For the computations in the case , we study the more general situation of a finite group acting on a smooth manifold . We develop a method to calculate the K-theory of the crossed product . In contrast to the equivariant Chern character of Baum and Connes, our method can also detect torsion elements in these K-groups.
Ann. K-Theory, Volume 3, Number 3 (2018), 395-460.
Received: 6 November 2016
Revised: 18 September 2017
Accepted: 19 October 2017
First available in Project Euclid: 24 July 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 20C08: Hecke algebras and their representations 46L80: $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
Secondary: 19L47: Equivariant $K$-theory [See also 55N91, 55P91, 55Q91, 55R91, 55S91]
Solleveld, Maarten. Topological K-theory of affine Hecke algebras. Ann. K-Theory 3 (2018), no. 3, 395--460. doi:10.2140/akt.2018.3.395. https://projecteuclid.org/euclid.akt/1532397764