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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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On the classification of irreducible representations of affine Hecke algebras with unequal parameters
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by Maarten Solleveld PDF
Represent. Theory 16 (2012), 1-87 Request permission

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) $q$.

References
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Additional Information
  • Maarten Solleveld
  • Affiliation: Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstraße 3-5, 37073 Göttingen, Germany
  • Email: Maarten.Solleveld@mathematik.uni-goettingen.de
  • Received by editor(s): September 27, 2010
  • Received by editor(s) in revised form: May 31, 2011
  • Published electronically: January 11, 2012
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Represent. Theory 16 (2012), 1-87
  • MSC (2010): Primary 20C08; Secondary 20G25
  • DOI: https://doi.org/10.1090/S1088-4165-2012-00406-X
  • MathSciNet review: 2869018