{
  "id": "1805.01327",
  "version": "v1",
  "published": "2018-05-03T14:33:56.000Z",
  "updated": "2018-05-03T14:33:56.000Z",
  "title": "Minimal dimensional representations of reduced enveloping algebras for $\\mathfrak{gl}_n$",
  "authors": [
    "Simon M. Goodwin",
    "Lewis Topley"
  ],
  "comment": "24 pages, comments welcome",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "Let $\\mathfrak g = \\mathfrak{gl}_N(k)$, where $k$ is an algebraically closed field of characteristic $p > 0$, and $N \\in \\mathbb Z_{\\ge 1}$. Let $\\chi \\in \\mathfrak g^*$ and denote by $U_\\chi(\\mathfrak g)$ the corresponding reduced enveloping algebra. The Kac--Weisfeiler conjecture, which was proved by Premet, asserts that any finite dimensional $U_\\chi(\\mathfrak g)$-module has dimension divisible by $p^{d_\\chi}$, where $d_\\chi$ is half the dimension of the coadjoint orbit of $\\chi$. Our main theorem gives a classification of $U_\\chi(\\mathfrak g)$-modules of dimension $p^{d_\\chi}$. As a consequence, we deduce that they are all parabolically induced from a 1-dimensional module for $U_0(\\mathfrak h)$ for a certain Levi subalgebra $\\mathfrak h$ of $\\mathfrak g$; we view this as a modular analogue of M{\\oe}glin's theorem on completely primitive ideals in $U(\\mathfrak{gl}_N(\\mathbb C))$. To obtain these results, we reduce to the case $\\chi$ is nilpotent, and then classify the 1-dimensional modules for the corresponding restricted $W$-algebra.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-03T14:33:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "minimal dimensional representations",
      "kac-weisfeiler conjecture",
      "finite dimensional",
      "coadjoint orbit",
      "main theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}