Representation Theory

Abstract:

Let $\mathcal R$ be a root datum with affine Weyl group $W$, and let $\mathcal H = \mathcal H (\mathcal R,q)$ be an affine Hecke algebra with positive, possibly unequal, parameters $q$. Then $\mathcal H$ is a deformation of the group algebra $\mathbb {C} [W]$, so it is natural to compare the representation theory of $\mathcal H$ and of $W$.

We define a map from irreducible $\mathcal H$-representations to $W$-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 $\mathcal H$ to that of $W$. 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 $\mathcal H$ and $W$.

An important role is played by the Schwartz completion $\mathcal S = \mathcal S (\mathcal R,q)$ of $\mathcal H$, an algebra whose representations are precisely the tempered $\mathcal H$-representations. We construct isomorphisms $\zeta _\epsilon : \mathcal S (\mathcal R,q^\epsilon ) \to \mathcal S (\mathcal R,q) \; (\epsilon >0)$ and injection $\zeta _0 : \mathcal S (W) = \mathcal S (\mathcal R,q^0) \to \mathcal S (\mathcal R,q)$, 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 $\mathcal H (\mathcal R,q)$ does not depend on the parameter(s) $q$.