{
  "id": "1912.00341",
  "version": "v1",
  "published": "2019-12-01T07:06:38.000Z",
  "updated": "2019-12-01T07:06:38.000Z",
  "title": "Casimir elements associated with Levi subalgebras of simple Lie algebras and their applications",
  "authors": [
    "Dmitri I. Panyushev"
  ],
  "comment": "30 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathfrak g$ be a simple Lie algebra, $\\mathfrak h$ a Levi subalgebra, and $C_{\\mathfrak h}\\in U(\\mathfrak h)$ the Casimir element defined via the restriction of the Killing form on $\\mathfrak g$ to $\\mathfrak h$. We study $C_{\\mathfrak h}$-eigenvalues in $\\mathfrak g/\\mathfrak h$ and related $\\mathfrak h$-modules. Without loss of generality, one may assume that $\\mathfrak h$ is a maximal Levi. Then $\\mathfrak g$ is equipped with the natural $\\mathbb Z$-grading $\\mathfrak g=\\bigoplus_{i\\in\\mathbb Z}\\mathfrak g(i)$ such that $\\mathfrak g(0)=\\mathfrak h$ and $\\mathfrak g(i)$ is a simple $\\mathfrak h$-module for $i\\ne 0$. We give explicit formulae for the $C_\\mathfrak h$-eigenvalues in each $\\mathfrak g(i)$, $i\\ne 0$, and relate eigenvalues of $C_\\mathfrak h$ in $\\bigwedge^\\bullet\\mathfrak g(1)$ to the dimensions of abelian subspaces of $\\mathfrak g(1)$. We also prove that if $\\mathfrak a\\subset\\mathfrak g(1)$ is abelian, whereas $\\mathfrak g(1)$ is not, then $\\dim\\mathfrak a\\le \\dim\\mathfrak g(1)/2$. Moreover, if $\\dim\\mathfrak a=(\\dim\\mathfrak g(1))/2$, then $\\mathfrak a$ has an abelian complement. The $\\mathbb Z$-gradings of height $\\le 2$ are closely related to involutions of $\\mathfrak g$, and we provide a connection of our theory to (an extension of) the \"strange formula\" of Freudenthal-de Vries.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-01T07:06:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B20",
      "17B22",
      "15A75"
    ],
    "keywords": [
      "simple lie algebra",
      "levi subalgebra",
      "casimir element",
      "applications",
      "explicit formulae"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}