{
  "id": "1104.5472",
  "version": "v1",
  "published": "2011-04-28T19:06:46.000Z",
  "updated": "2011-04-28T19:06:46.000Z",
  "title": "Commuting involutions and degenerations of isotropy representations",
  "authors": [
    "Dmitri I. Panyushev"
  ],
  "comment": "31 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $\\sigma_1$ and $\\sigma_2$ be commuting involutions of a semisimple algebraic group $G$. This yields a $Z_2\\times Z_2$-grading of $\\g=\\Lie(G)$, $\\g=\\bigoplus_{i,j=0,1}\\g_{ij}$, and we study invariant-theoretic aspects of this decomposition. Let $\\g<\\sigma_1>$ be the $Z_2$-contraction of $\\g$ determined by $\\sigma_1$. Then both $\\sigma_2$ and $\\sigma_3:=\\sigma_1\\sigma_2$ remain involutions of the non-reductive Lie algebra $\\g<\\sigma_1>$. The isotropy representations related to $(\\g<\\sigma_1>, \\sigma_2)$ and $(\\g<\\sigma_1>, \\sigma_3)$ are degenerations of the isotropy representations related to $(\\g, {\\sigma_2})$ and $(\\g, {\\sigma_3})$, respectively. We show that these degenerated isotropy representations retain many good properties. For instance, they always have a generic stabiliser and their algebras of invariants are often polynomial. We also develop some theory on Cartan subspaces for various $Z_2$-gradings associated with the $Z_2\\times Z_2$-grading of $\\g$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-04-28T19:06:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13A50",
      "14L30",
      "17B40",
      "22E46"
    ],
    "keywords": [
      "commuting involutions",
      "degenerations",
      "degenerated isotropy representations retain",
      "study invariant-theoretic aspects"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1104.5472P"
    }
  }
}