{
  "id": "1009.5279",
  "version": "v2",
  "published": "2010-09-27T15:00:08.000Z",
  "updated": "2010-10-28T05:23:29.000Z",
  "title": "Double flag varieties for a symmetric pair and finiteness of orbits",
  "authors": [
    "Kyo Nishiyama",
    "Hiroyuki Ochiai"
  ],
  "comment": "20 pages",
  "journal": "Journal of Lie Theory, volume 21 (2011), 79--99",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let G be a reductive algebraic group over the complex number filed, and K = G^{\\theta} be the fixed points of an involutive automorphism \\theta of G so that (G, K) is a symmetric pair. We take parabolic subgroups P and Q of G and K respectively and consider a product of partial flag varieties G/P and K/Q with diagonal K-action. The double flag variety G/P \\times K/Q thus obtained is said to be of finite type if there are finitely many K-orbits on it. A triple flag variety G/P^1 \\times G/P^2 \\times G/P^3 is a special case of our double flag varieties, and there are many interesting works on the triple flag varieties. In this paper, we study double flag varieties G/P \\times K/Q of finite type. We give efficient criterion under which the double flag variety is of finite type. The finiteness of orbits is strongly related to spherical actions of G or K. For example, we show a partial flag variety G/P is K-spherical if a product of partial flag varieties G/P \\times G/\\theta(P) is G-spherical. We also give many examples of the double flag varieties of finite type, and for type AIII, we give a classification when P = B is a Borel subgroup of G.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-10-28T05:23:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M15",
      "53C35",
      "14M17"
    ],
    "keywords": [
      "double flag variety",
      "symmetric pair",
      "partial flag variety",
      "finite type",
      "partial flag varieties g/p"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.5279N"
    }
  }
}