{
  "id": "1008.0177",
  "version": "v3",
  "published": "2010-08-01T14:23:16.000Z",
  "updated": "2013-12-03T18:04:40.000Z",
  "title": "On the classification of irreducible representations of affine Hecke algebras with unequal parameters",
  "authors": [
    "Maarten Solleveld"
  ],
  "comment": "105 pages. The third version is nearly identical to the published one. Compared to the first two versions there are several minor changes",
  "journal": "Representation Theory 16 (2012), 1--87",
  "categories": [
    "math.RT"
  ],
  "abstract": "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) $q$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-12-03T18:04:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C08",
      "20G25"
    ],
    "keywords": [
      "unequal parameters",
      "irreducible representations",
      "finite dimensional representations",
      "classification",
      "affine hecke algebra version"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Represent. Theory"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 105,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.0177S"
    }
  }
}