{
  "id": "1204.1118",
  "version": "v1",
  "published": "2012-04-05T04:26:53.000Z",
  "updated": "2012-04-05T04:26:53.000Z",
  "title": "Closed orbits on partial flag varieties and double flag variety of finite type",
  "authors": [
    "Kensuke Kondo",
    "Kyo Nishiyama",
    "Hiroyuki Ochiai",
    "Kenji Taniguchi"
  ],
  "comment": "7 pages",
  "categories": [
    "math.RT",
    "math.AG"
  ],
  "abstract": "Let $ G $ be a connected reductive algebraic group over $ \\C $. We denote by $ K = (G^{\\theta})_{0} $ the identity component of the fixed points of an involutive automorphism $ \\theta $ of $ G $. The pair $ (G, K) $ is called a symmetric pair. Let $Q$ be a parabolic subgroup of $K$. We want to find a pair of parabolic subgroups $P_{1}$, $P_{2}$ of $G$ such that (i) $P_{1} \\cap P_{2} = Q$ and (ii) $P_{1} P_{2}$ is dense in $G$. The main result of this article states that, for a simple group $G$, we can find such a pair if and only if $(G, K)$ is a Hermitian symmetric pair. The conditions (i) and (ii) yield to conclude that the $K$-orbit through the origin $(e P_{1}, e P_{2})$ of $G/P_{1} \\times G/P_{2}$ is closed and it generates an open dense $G$-orbit on the product of partial flag variety. From this point of view, we also give a complete classification of closed $K$-orbits on $G/P_{1} \\times G/P_{2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-04-05T04:26:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M15",
      "14M17",
      "14M27"
    ],
    "keywords": [
      "partial flag variety",
      "double flag variety",
      "finite type",
      "closed orbits",
      "parabolic subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1204.1118K"
    }
  }
}