Let
$\mathcal{\mathscr{H}}\left(\mathcal{\mathcal{R}},q\right)$
be an affine Hecke algebra with a positive parameter function
$q$.
We are interested in the topological Ktheory of its
${C}^{\ast}$completion
${C}_{r}^{\ast}\left(\mathcal{\mathcal{R}},q\right)$. We prove that
${K}_{\ast}\left({C}_{r}^{\ast}\left(\mathcal{\mathcal{R}},q\right)\right)$ does not depend
on the parameter
$q$,
solving a longstanding conjecture of Higson and Plymen. For this we use
representationtheoretic methods, in particular elliptic representations of Weyl groups
and Hecke algebras.
Thus, for the computation of these Kgroups it suffices to work out the case
$q=1$.
These algebras are considerably simpler than for
$q\ne 1$, just
crossed products of commutative algebras with finite Weyl groups. We explicitly determine
${K}_{\ast}\left({C}_{r}^{\ast}\left(\mathcal{\mathcal{R}},q\right)\right)$ for all classical
root data
$\mathcal{\mathcal{R}}$.
This will be useful for analyzing the Ktheory of the reduced
${C}^{\ast}$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
$\Gamma $ acting on a
smooth manifold
$M$.
We develop a method to calculate the Ktheory of the crossed product
$C\left(M\right)\u22ca\Gamma $. In
contrast to the equivariant Chern character of Baum and Connes, our method can
also detect torsion elements in these Kgroups.
