{
  "id": "1306.6118",
  "version": "v4",
  "published": "2013-06-26T02:08:07.000Z",
  "updated": "2016-06-15T15:06:40.000Z",
  "title": "On multiplicity in restriction for $p$-adic groups",
  "authors": [
    "Kwangho Choiy"
  ],
  "comment": "The previous article entitled \"On multiplicity in the restriction for SL(m,D) over a p-adic field\" has been substantially enhanced and replaced by the present article",
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "We study the multiplicity occurring when irreducible smooth representations of $\\widetilde{\\bold G}(F)$ are restricted to $\\bold G(F)$ in a general setting, where $\\widetilde{\\bold G}$ is a connected reductive algebraic group over a $p$-adic field $F$ of characteristic 0 and $\\bold G$ is its closed $F$-subgroup sharing the same derived group. We first illuminate various quantitative aspects of the multiplicity for discrete series representations in the case of $\\widetilde{\\bold G} =\\rm{GL}_m(D)$ and $\\bold G = \\rm{SL}_m(D),$ where $D$ is a central division algebra of dimension $d^2$ over $F.$ We then investigate parallel phenomena occurring in restrictions of representations between the connected reductive $F$-groups in the general setting and their component groups, so-called $\\mathcal{S}$-groups, under the assumptions of the local Langlands conjecture and internal structure of $L$-packets. We also obtain the equality of multiplicities in the both sides, and provide a general formula of the multiplicity in the restriction of irreducible smooth representations of $\\widetilde{\\bold G}(F)$ to $\\bold G(F).$ This formula is given in terms of dimensions of irreducible representations of their $\\mathcal{S}$-groups and generalizes Hiraga and Saito's result in 2012 for the case of $\\rm{GL}_m(D)$ and $\\rm{SL}_m(D).$",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-03-12T18:15:48.000Z",
      "title": "On multiplicity in the restriction for SL(m,D) over a p-adic field",
      "abstract": "We study the multiplicity for discrete series representations of GL(m,D) when restricted to SL(m,D), where D is a central division algebra of dimension d^2 over a p-adic field F of characteristic 0. We express the multiplicity in terms of the quotient of the cardinalities of elliptic L-packets of SL(md,F) and SL(m,D). Its bound and explicit descriptions for several cases are also given.",
      "comment": "The previous article entitled \"Formal Degrees and Multiplicities in Restriction\" has been replaced by the present article",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2016-06-15T15:06:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50",
      "22E35",
      "11F70"
    ],
    "keywords": [
      "p-adic field",
      "multiplicity",
      "restriction",
      "discrete series representations",
      "central division algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.6118C"
    }
  }
}