  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 $\zeta_\epsilon : S
(R,q^\epsilon) \to S (R,q)$ $(\epsilon >0)$ and injection $\zeta_0 : S (W^e) =
S (R,q^0) \to S (R,q)$, depending continuously on $\epsilon$.
  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 $C^*$-completion of an affine Hecke algebra $H (R,q)$ does
not depend on the parameter(s) $q$.
