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

Author: Maarten Solleveld
Journal: Represent. Theory 16 (2012), 1-87
MSC (2010): Primary 20C08; Secondary 20G25
Published electronically: January 11, 2012
MathSciNet review: 2869018
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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$.

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Additional Information

Maarten Solleveld
Affiliation: Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstraße 3-5, 37073 Göttingen, Germany

Received by editor(s): September 27, 2010
Received by editor(s) in revised form: May 31, 2011
Published electronically: January 11, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.