{
  "id": "1208.2084",
  "version": "v2",
  "published": "2012-08-10T04:14:52.000Z",
  "updated": "2013-07-27T14:33:15.000Z",
  "title": "On orbits in double flag varieties for symmetric pairs",
  "authors": [
    "Xuhua He",
    "Kyo Nishiyama",
    "Hiroyuki Ochiai",
    "Yoshiki Oshima"
  ],
  "comment": "47 pages, 3 tables; add all the details of the classification",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $ G $ be a connected, simply connected semisimple algebraic group over the complex number field, and let $ K $ be the fixed point subgroup of an involutive automorphism of $ G $ so that $ (G, K) $ is a symmetric pair. We take parabolic subgroups $ P $ of $ G $ and $ Q $ of $ K $ respectively and consider the product of partial flag varieties $ G/P $ and $ K/Q $ with diagonal $ K $-action, which we call a \\emph{double flag variety for symmetric pair}. It is said to be \\emph{of finite type} if there are only finitely many $ K $-orbits on it. In this paper, we give a parametrization of $ K $-orbits on $ G/P \\times K/Q $ in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of $ P \\subset G $ or $ Q \\subset K $ is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of $ K $-spherical flag varieties $ G/P $ and $ G $-spherical homogeneous spaces $ G/Q $.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-07-27T14:33:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M15",
      "53C35",
      "14M17"
    ],
    "keywords": [
      "flag variety",
      "double flag varieties",
      "symmetric pair",
      "finite type",
      "partial flag varieties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.2084H"
    }
  }
}