Annals of K-Theory

Topological K-theory of affine Hecke algebras

Maarten Solleveld

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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 C r ( , q ) . We prove that K ( C r ( , 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 q 1 , just crossed products of commutative algebras with finite Weyl groups. We explicitly determine K ( C r ( , 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.

Article information

Source
Ann. K-Theory, Volume 3, Number 3 (2018), 395-460.

Dates
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
https://projecteuclid.org/euclid.akt/1532397764

Digital Object Identifier
doi:10.2140/akt.2018.3.395

Mathematical Reviews number (MathSciNet)
MR3830198

Zentralblatt MATH identifier
06911673

Subjects
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]

Keywords
topological K-theory affine Hecke algebra Weyl group crossed product algebra

Citation

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


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