{
  "id": "2501.04500",
  "version": "v1",
  "published": "2025-01-08T13:39:37.000Z",
  "updated": "2025-01-08T13:39:37.000Z",
  "title": "Distinguished Representations for $\\rm{SL}_n(D)$ where $D$ is a quaternion division algebra over a $p$-adic field",
  "authors": [
    "Kwangho Choiy",
    "Shiv Prakash Patel"
  ],
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "Let $D$ be a quaternion division algebra over a non-archimedean local field $F$ of characteristic zero. Let $E/F$ be a quadratic extension and $\\rm{SL}_{n}^{*}(E) = {\\rm{GL}}_{n}(E) \\cap \\rm{SL}_{n}(D)$. We study distinguished representations of $\\rm{SL}_{n}(D)$ by the subgroup $\\rm{SL}_{n}^{*}(E)$. Let $\\pi$ be an irreducible admissible representation of $\\rm{SL}_{n}(D)$ which is distinguished by $\\rm{SL}_{n}^{*}(E)$. We give a multiplicity formula, i.e. a formula for the dimension of the $\\mathbb{C}$-vector space ${\\rm{Hom}}_{\\rm{SL}_{n}^{*}(E)} (\\pi, \\mathbbm{1})$, where $\\mathbbm{1}$ denotes the trivial representation of $\\rm{SL}_{n}^{*}(E)$. This work is a non-split inner form analog of a work by Anandavardhanan-Prasad which gives a multiplicity formula for $\\rm{SL}_{n}(F)$-distinguished irreducible admissible representation of $\\rm{SL}_{n}(E)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-08T13:39:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50",
      "11S37",
      "20G25",
      "22E35"
    ],
    "keywords": [
      "quaternion division algebra",
      "distinguished representations",
      "adic field",
      "non-split inner form analog",
      "multiplicity formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}