{
  "id": "1011.3267",
  "version": "v1",
  "published": "2010-11-14T23:52:46.000Z",
  "updated": "2010-11-14T23:52:46.000Z",
  "title": "On the algebraic set of singular elements in a complex simple Lie algebra",
  "authors": [
    "Bertram Kostant",
    "Nolan Wallach"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be a complex simple Lie group and let $\\g = \\hbox{\\rm Lie}\\,G$. Let $S(\\g)$ be the $G$-module of polynomial functions on $\\g$ and let $\\hbox{\\rm Sing}\\,\\g$ be the closed algebraic cone of singular elements in $\\g$. Let ${\\cal L}\\s S(\\g)$ be the (graded) ideal defining $\\hbox{\\rm Sing}\\,\\g$ and let $2r$ be the dimension of a $G$-orbit of a regular element in $\\g$. Then ${\\cal L}^k = 0$ for any $k<r$. On the other hand, there exists a remarkable $G$-module $M\\s {\\cal L}^r$ which already defines $\\hbox{\\rm Sing}\\,\\g$. The main results of this paper are a determination of the structure of $M$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-11-14T23:52:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G20",
      "20G05",
      "22E10",
      "22E60",
      "22E45"
    ],
    "keywords": [
      "complex simple lie algebra",
      "singular elements",
      "algebraic set",
      "complex simple lie group",
      "polynomial functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1011.3267K"
    }
  }
}