{
  "id": "1503.06385",
  "version": "v1",
  "published": "2015-03-22T05:05:51.000Z",
  "updated": "2015-03-22T05:05:51.000Z",
  "title": "Differential equations and singular vectors in Verma modules",
  "authors": [
    "Wei Xiao"
  ],
  "comment": "11 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Xu introduced a system of partial differential equations to investigate singular vectors in the Verma modules of highest weight $\\lambda$ over $\\frsl(n,\\bbC)$. He proved that the solution space of this system in the space of truncated power series is spanned by $\\{\\sigma(1)\\ |\\ \\sigma\\in S_n\\}$. We present an explicit formula of the solution $s_\\alpha(1)$ for every positive root $\\alpha$ and showed directly that $s_\\alpha(1)$ is a polynomial if and only if $\\langle\\lambda+\\rho,\\alpha\\rangle$ is a nonnegative integer. From this, we can recover a formula of singular vectors given by Malikov et al.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-22T05:05:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B20",
      "22E47"
    ],
    "keywords": [
      "singular vectors",
      "verma modules",
      "partial differential equations",
      "explicit formula",
      "highest weight"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}