{
  "id": "1907.05890",
  "version": "v1",
  "published": "2019-07-14T13:28:52.000Z",
  "updated": "2019-07-14T13:28:52.000Z",
  "title": "Distinguished representations of SO(n+1,1) x SO(n,1), periods and branching laws",
  "authors": [
    "Toshiyuki Kobayashi",
    "Birgit Speh"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1801.00158",
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "Given irreducible representations $\\Pi$ and $\\pi$ of the rank one special orthogonal groups $G=SO(n+1,1)$ and $G'=SO(n,1)$ with nonsingular integral infinitesimal character, we state in terms of $\\theta$-stable parameter necessary and sufficient conditions so that \\[ \\operatorname{Hom}_{G'}(\\Pi|_{G'}, \\pi )\\not = \\{0\\}. \\] In the special case that both $\\Pi$ and $\\pi$ are tempered, this implies the Gross--Prasad conjectures for tempered representations of $SO(n+1,1) \\times SO(n,1)$ which are nontrivial on the center. We apply these results to construct nonzero periods and distinguished representations. If both $\\Pi$ and $ \\pi$ have the trivial infinitesimal character $\\rho$ then we use a theorem that the periods are nonzero on the minimal $K$-type to obtain a nontrivial bilinear form on the $({\\mathfrak g},K)$-cohomology of the representations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-14T13:28:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E30",
      "11F40",
      "22E45",
      "22E46",
      "53A30",
      "58J70"
    ],
    "keywords": [
      "distinguished representations",
      "branching laws",
      "nonsingular integral infinitesimal character",
      "nontrivial bilinear form",
      "special orthogonal groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}