{
  "id": "0806.2917",
  "version": "v1",
  "published": "2008-06-18T08:11:49.000Z",
  "updated": "2008-06-18T08:11:49.000Z",
  "title": "Kostant's problem and parabolic subgroups",
  "authors": [
    "Johan Kåhrström"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\frak g$ be a finite dimensional complex semi-simple Lie algebra with Weyl group $W$ and simple reflections $S$. For $I\\subseteq S$ let $\\frak g_I$ be the corresponding semi-simple subalgebra of $\\frak g$. Denote by $W_I$ the Weyl group of $\\frak g_I$ and let $w_o$ and $w^I_o$ be the longest elements of $W$ and $W_I$, respectively. In this paper we show that the answer to Kostant's problem, i.e. whether the universal enveloping algebra surjects onto the space of all ad-finite linear transformations of a given module, is the same for the simple highest weight $\\frak g_I$-module $L_I(x)$ of highest weight $x\\cdot 0$, $x\\in W_I$, as the answer for the simple highest weight $\\frak g$-module $L(x w^I_o w_o)$ of highest weight $(x w^I_o w_o)\\cdot 0$. We also give a new description of the unique quasi-simple quotient of the Verma module $\\Delta(e)$ with the same annihilator as $L(y)$, $y\\in W$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-06-18T08:11:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "kostants problem",
      "parabolic subgroups",
      "simple highest weight",
      "finite dimensional complex semi-simple lie",
      "dimensional complex semi-simple lie algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.2917K"
    }
  }
}