{
  "id": "1406.3154",
  "version": "v2",
  "published": "2014-06-12T08:57:14.000Z",
  "updated": "2014-07-22T16:29:29.000Z",
  "title": "A Distributional Treatment of Relative Mirabolic Multiplicity One",
  "authors": [
    "Maxim Gurevich"
  ],
  "comment": "15 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "We study the role of the mirabolic subgroup $P$ of $G=\\mathbf{GL}_n(F)$ ($F$ a $p$-adic field) in smooth irreducible representations of $G$ that possess a non-zero invariant functional relative to a subgroup of the form $H_{k} = \\mathbf{GL}_k(F)\\times \\mathbf{GL}_{n-k}(F)$. We show that if a non-zero $H_1$-invariant functional exists on a representation, then every $P\\cap H_1$-invariant functional must equal to a scalar multiple of it. When $k>1$, we give a reduction of the same problem to a question about invariant distributions on the nilpotent cone of the tangent space of the symmetric space $G/H_k$. Some new distributional methods, which are suitable for a setting of non-reductive groups, are developed.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-07-22T16:29:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G25",
      "22E50"
    ],
    "keywords": [
      "relative mirabolic multiplicity",
      "distributional treatment",
      "adic field",
      "tangent space",
      "nilpotent cone"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.3154G"
    }
  }
}