{
  "id": "math/0210111",
  "version": "v2",
  "published": "2002-10-07T22:17:25.000Z",
  "updated": "2006-01-10T17:25:33.000Z",
  "title": "Holomorphic extension of representations: (I) automorphic functions",
  "authors": [
    "Bernhard Kroetz",
    "Robert J. Stanton"
  ],
  "comment": "84 pages, published version",
  "journal": "Ann. of Math. (2), Vol. 159 (2004), no. 2, 641--724",
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "Let G be a connected, real, semisimple Lie group contained in its complexification G_C, and let K be a maximal compact subgroup of G. We construct a K_C-G double coset domain in G_C, and we show that the action of G on the K-finite vectors of any irreducible unitary representation of G has a holomorphic extension to this domain. For the resultant holomorphic extension of K-finite matrix coefficients we obtain estimates of the singularities at the boundary, as well as majorant/minorant estimates along the boundary. We obtain L^\\infty bounds on holomorphically extended automorphic functions on G/K in terms of Sobolev norms, and we use these to estimate the Fourier coefficients of combinations of automorphic functions in a number of cases, e.g. of triple products of Maass forms.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-01-10T17:25:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E45",
      "11F70"
    ],
    "keywords": [
      "representation",
      "semisimple lie group",
      "maximal compact subgroup",
      "k-finite matrix coefficients",
      "resultant holomorphic extension"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Princeton University and the Institute for Advanced Study",
      "journal": "Ann. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 84,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math.....10111K"
    }
  }
}