{
  "id": "1612.01776",
  "version": "v1",
  "published": "2016-12-06T12:29:16.000Z",
  "updated": "2016-12-06T12:29:16.000Z",
  "title": "On completions of Hecke algebras",
  "authors": [
    "Maarten Solleveld"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Let G be a reductive p-adic group and let H(G)^s be a Bernstein block of the Hecke algebra of G. We consider two important topological completions of H(G)^s: a direct summand S(G)^s of the Harish-Chandra--Schwartz algebra of G and a two-sided ideal C*_r (G)^s of the reduced C*-algebra of G. These are useful for the study of all tempered smooth G-representations. We suppose that H(G)^s is Morita equivalent to an affine Hecke algebra H(R,q) -- as is known in many cases. The latter algebra also has a Schwartz completion S(R,q) and a C*-completion C*_r (R,q), both defined in terms of the underlying root datum R and the parameters q. We prove that, under some mild conditions, a Morita equivalence between H(G)^s and H(R,q) extends to Morita equivalences between S(G)^s and S(R,q), and between C*_r (G)^s and C*_r (R,q). We also check that our conditions are fulfilled in all known cases of such Morita equivalences between Hecke algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-06T12:29:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C08",
      "22E50",
      "22E35"
    ],
    "keywords": [
      "morita equivalence",
      "affine hecke algebra",
      "mild conditions",
      "root datum",
      "important topological completions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}