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
.