{
  "id": "1801.10298",
  "version": "v1",
  "published": "2018-01-31T04:43:05.000Z",
  "updated": "2018-01-31T04:43:05.000Z",
  "title": "$(\\mathfrak{g},K)$-module of $\\mathrm{O}(p,q)$ associated with the finite-dimensional representation of $\\mathfrak{sl}_2$",
  "authors": [
    "Takashi Hashimoto"
  ],
  "comment": "20 pages, 1 figure",
  "categories": [
    "math.RT"
  ],
  "abstract": "The main aim of this paper is to construct irreducible $(\\mathfrak{g},K)$-modules of $\\mathrm{O}(p,q)$ corresponding to the finite-dimensional representation of $\\mathfrak{sl}_2$ of dimension $m+1$ under the Howe duality, to find the $K$-type formula, the Gelfand-Kirillov dimension and the Bernstein degree of them, where $m$ is a non-negative integer. The $K$-type formula for $m=0$ shows that it is nothing but the $(\\mathfrak{g},K)$-module of the minimal representation of $\\mathrm{O}(p,q)$. One finds that the Gelfand-Kirillov dimension is equal to $p+q-3$ not only for $m=0$ but for any $m$ satisfying $m+3 \\leq (p+q)/2$ when $p, q \\geq 3$ and $p+q$ is even, and that the Bernstein degree for $m$ is equal to $(m+1)$ times that for $m=0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-31T04:43:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E46",
      "17B20"
    ],
    "keywords": [
      "finite-dimensional representation",
      "bernstein degree",
      "gelfand-kirillov dimension",
      "type formula",
      "main aim"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}