{
  "id": "1907.09093",
  "version": "v1",
  "published": "2019-07-22T02:49:46.000Z",
  "updated": "2019-07-22T02:49:46.000Z",
  "title": "Dual pairs in the Pin-group and duality for the corresponding spinorial representation",
  "authors": [
    "Clément Guérin",
    "Gang Liu",
    "Allan Merino"
  ],
  "comment": "22 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "In this paper, we give a complete picture of Howe correspondence for the setting ($O(E, b), Pin(E, b), \\Pi$), where $O(E, b)$ is an orthogonal group (real or complex), $Pin(E, b)$ is the two-fold Pin-covering of $O(E, b)$, and $\\Pi$ is the spinorial representation of $Pin(E, b)$. More precisely, for a dual pair ($G, G'$) in $O(E, b)$, we determine explicitly the nature of its preimages $(\\tilde{G}, \\tilde{G'})$ in $Pin(E, b)$, and prove that apart from some exceptions, $(\\tilde{G}, \\tilde{G'})$ is always a dual pair in $Pin(E, b)$; then we establish the Howe correspondence for $\\Pi$ with respect to $(\\tilde{G}, \\tilde{G'})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-22T02:49:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E46",
      "20G05"
    ],
    "keywords": [
      "dual pair",
      "corresponding spinorial representation",
      "howe correspondence",
      "complete picture",
      "orthogonal group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}