{
  "id": "0905.4244",
  "version": "v3",
  "published": "2009-05-26T16:56:59.000Z",
  "updated": "2013-08-04T10:20:31.000Z",
  "title": "Spherical functions on spherical varieties",
  "authors": [
    "Yiannis Sakellaridis"
  ],
  "comment": "78 pages, to appear in the American Journal of Mathematics",
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "Let X=H\\G be a homogeneous spherical variety for a split reductive group G over the integers o of a p-adic field k, and K=G(o) a hyperspecial maximal compact subgroup of G=G(k). We compute eigenfunctions (\"spherical functions\") on X=X(k) under the action of the unramified (or spherical) Hecke algebra of G, generalizing many classical results of \"Casselman-Shalika\" type. Under some additional assumptions on X we also prove a variant of the formula which involves a certain quotient of L-values, and we present several applications such as: (1) a statement on \"good test vectors\" in the multiplicity-free case (namely, that an H-invariant functional on an irreducible unramified representation \\pi is non-zero on \\pi^K), (2) the unramified Plancherel formula for X, including a formula for the \"Tamagawa measure\" of X(o), and (3) a computation of the most continuous part of H-period integrals of principal Eisenstein series.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-08-04T10:20:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50"
    ],
    "keywords": [
      "spherical variety",
      "spherical functions",
      "hyperspecial maximal compact subgroup",
      "principal eisenstein series",
      "p-adic field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 78,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0905.4244S"
    }
  }
}