## Annals of K-Theory

### Topological K-theory of affine Hecke algebras

Maarten Solleveld

#### 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 its $C ∗$-completion $C r ∗ ( ℛ , q )$. We prove that $K ∗ ( C r ∗ ( ℛ , q ) )$ 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 $K ∗ ( C r ∗ ( ℛ , q ) )$ for all classical root data $ℛ$. 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
Revised: 18 September 2017
Accepted: 19 October 2017
First available in Project Euclid: 24 July 2018

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

#### 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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