{
  "id": "1809.08743",
  "version": "v1",
  "published": "2018-09-24T03:57:22.000Z",
  "updated": "2018-09-24T03:57:22.000Z",
  "title": "A multiplicity one theorem for $\\mathrm{GL}_n$ and $\\mathrm{SL}_n$ over complete discrete valuation rings",
  "authors": [
    "Shiv Prakash Patel",
    "Pooja Singla"
  ],
  "comment": "13 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathfrak{o}$ be the ring of integers of a non-archimedian local field with maximal ideal $\\mathfrak{p}$ and finite residue field of characteristic $p.$ For $\\ell \\geq 1,$ let $\\mathfrak{o}_\\ell = \\mathfrak{o}/\\mathfrak{p}^\\ell,$ $\\mathbf{G}(\\mathfrak{o}_\\ell) = {\\rm GL}_{n}(\\mathfrak{o}_\\ell)$ or ${\\rm SL}_{n}(\\mathfrak{o}_{\\ell})$ and $U(\\mathfrak{o}_{\\ell}) \\subset \\mathbf{G}(\\mathfrak{o}_{\\ell})$ be the subgroup of upper triangular unipotent matrices. We prove that the induced representation $\\mathrm{Ind}^{\\mathbf{G}(\\mathfrak{o}_\\ell)}_{U(\\mathfrak{o}_\\ell)}(\\theta)$ of $\\mathbf{G}(\\mathfrak{o}_{\\ell})$ obtained from a non-degenerate character $\\theta$ of $U(\\mathfrak{o}_\\ell)$ is multiplicity free. This is analogous to the multiplicity one theorem in Gelfand-Graev representation for Chevalley groups. We also prove that for all $\\mathbf{G}(\\mathfrak{o}_\\ell),$ except for ${\\rm SL}_{n}(\\mathfrak{o}_{\\ell})$ with even $p$ or $p \\mid n,$ an irreducible representation of $\\mathbf{G}(\\mathfrak{o}_\\ell)$ is a constituent of $\\mathrm{Ind}^{\\mathbf{G}(\\mathfrak{o}_\\ell)}_{U(\\mathfrak{o}_\\ell)}(\\theta)$ if and only if it is regular.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-24T03:57:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G05",
      "20G25"
    ],
    "keywords": [
      "complete discrete valuation rings",
      "multiplicity",
      "upper triangular unipotent matrices",
      "non-archimedian local field",
      "finite residue field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}