On the classification of irreducible representations of affine Hecke algebras with unequal parameters

Let $R$ be a root datum with affine Weyl group $W^e$, and let $H=H(R,q)$ be an affine Hecke algebra with positive, possibly unequal, parameters $q$. Then $H$ is a deformation of the group algebra $\mathbb{C}[W^e]$, so it is natural to compare the representation theory of $H$ and of $W^e$. We define a map from irreducible $H$-representations to $W^e$-representations and we show that, when extended to the Grothendieck groups of finite dimensional representations, this map becomes an isomorphism, modulo torsion. The map can be adjusted to a (nonnatural) continuous bijection from the dual space of $H$ to that of $W^e$. We use this to prove the affine Hecke algebra version of a conjecture of Aubert, Baum and Plymen, which predicts a strong and explicit geometric similarity between the dual spaces of $H$ and $W^e$. An important role is played by the Schwartz completion $S=S(R,q)$ of $H$, an algebra whose representations are precisely the tempered $H$-representations. We construct isomorphisms $\zeta_\epsilon : S (R,q^\epsilon) \to S (R,q)$ $(ϵ>0)$ and injection $\zeta_0 : S (W^e) = S (R,q^0) \to S (R,q)$,dependingcontinuouslyon$\epsilon$. Although ${\zeta }_0$ is not surjective, it behaves like an algebra isomorphism in many ways. Not only does ${\zeta }_0$ extend to a bijection on Grothendieck groups of finite dimensional representations, it also induces isomorphisms on topological $K$-theory and on periodic cyclic homology (the first two modulo torsion). This proves a conjecture of Higson and Plymen, which says that the $K$-theory of the $C^{\ast }$-completion of an affine Hecke algebra $H(R,q)$ does not depend on the parameter(s) $q$.