Topological K-theory of affine Hecke algebras

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 will 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 to analyse 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.

On Completions of Hecke Algebras

“…Theorem 5.2 was one of the motivations for the author to write a paper about the K-theory of C * -completions of (extended) affine Hecke algebras [Sol5]. It enables us to show that the K-groups of the reduced C * -algebras of the above groups are torsion-free.…”

“…By Morita invariance and Theorem 5.2, it suffices to show that the K-theory of the C * -completion of an extended affine Hecke algebra as in Theorem 5.2 is a finitely generated free abelian group. It was checked in [Sol5,(62)] that the Künneth theorem for topological K-theory [Sch] applies to such algebras. Thus we only need to prove the result when the underlying root datum is of type GL n , Sp 2n or SO 2n+1 and Γ is trivial, and when the root datum is of type (SO 2n ) e and Γ is as in (91).…”

“…Thus we only need to prove the result when the underlying root datum is of type GL n , Sp 2n or SO 2n+1 and Γ is trivial, and when the root datum is of type (SO 2n ) e and Γ is as in (91). These K-groups were computed in [Sol5], see respectively Theorem 3.2, Theorem 3.3, (128), and Proposition 3.5. They are free abelian and have finite rank.…”

“…This can, for instance, be used to compute the topological K-theory of these algebras. Indeed, in [Sol5] the author determined the K-theory of C * r (R, q) for many root data R (it does not depend on q). These calculations, together with Theorem 1 for classical groups, lead to a result which is useful in relation with the Baum-Connes conjecture.…”

Stratified Langlands duality in the $A_n$ tower

“…In [5] we gave a new proof of the Baum-Connes conjecture for the extended affine Weyl groups associated to compact connected semi-simple Lie groups, and showed that Langlands duality for the Lie groups induces an isomorphism of K-theory for the corresponding extended affine Weyl groups. In his consideration of the K-theory of Hecke algebras, Solleveld [7,8] determined the right hand side of the Baum-Connes conjecture for the affine Weyl groups associated to a number of Lie Groups including SL n (C). In order to do so he computed the extended quotients T n //W up to homotopy by constructing the quotients S n //W .…”