Let R \mathcal R be a root datum with affine Weyl group W W , and let H = H ( R , q ) \mathcal H = \mathcal H (\mathcal R,q) be an affine Hecke algebra with positive, possibly unequal, parameters q q . Then H \mathcal H is a deformation of the group algebra C [ W ] \mathbb {C} [W] , so it is natural to compare the representation theory of H \mathcal H and of W W . We define a map from irreducible H \mathcal H -representations to W W -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 \mathcal H to that of W W . 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 \mathcal H and W W . An important role is played by the Schwartz completion S = S ( R , q ) \mathcal S = \mathcal S (\mathcal R,q) of H \mathcal H , an algebra whose representations are precisely the tempered H \mathcal H -representations. We construct isomorphisms ζ ϵ : S ( R , q ϵ ) → S ( R , q ) ( ϵ > 0 ) \zeta _\epsilon : \mathcal S (\mathcal R,q^\epsilon ) \to \mathcal S (\mathcal R,q) \; (\epsilon >0) and injection ζ 0 : S ( W ) = S ( R , q 0 ) → S ( R , q ) \zeta _0 : \mathcal S (W) = \mathcal S (\mathcal R,q^0) \to \mathcal S (\mathcal R,q) , depending continuously on ϵ \epsilon . Although ζ 0 \zeta _0 is not surjective, it behaves like an algebra isomorphism in many ways. Not only does ζ 0 \zeta _0 extend to a bijection on Grothendieck groups of finite dimensional representations, it also induces isomorphisms on topological K 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 K -theory of the C ∗ C^* -completion of an affine Hecke algebra H ( R , q ) \mathcal H (\mathcal R,q) does not depend on the parameter(s) q q .