{
  "id": "1008.4544",
  "version": "v3",
  "published": "2010-08-26T16:58:22.000Z",
  "updated": "2012-03-29T13:18:21.000Z",
  "title": "Restrictions of generalized Verma modules to symmetric pairs",
  "authors": [
    "Toshiyuki Kobayashi"
  ],
  "comment": "31 pages, To appear in Transformation Groups",
  "journal": "Transformation Groups 17 (2012), pp. 523-546",
  "doi": "10.1007/s00031-012-9180-y",
  "categories": [
    "math.RT",
    "math-ph",
    "math.CO",
    "math.MP"
  ],
  "abstract": "We initiate a new line of investigation on branching problems for generalized Verma modules with respect to complex reductive symmetric pairs (g,k). Here we note that Verma modules of g may not contain any simple module when restricted to a reductive subalgebra k in general. In this article, using the geometry of K_C orbits on the generalized flag variety G_C/P_C, we give a necessary and sufficient condition on the triple (g,k, p) such that the restriction X|_k always contains simple k-modules for any g-module $X$ lying in the parabolic BGG category O^p attached to a parabolic subalgebra p of g. Formulas are derived for the Gelfand-Kirillov dimension of any simple k-module occurring in a simple generalized Verma module of g. We then prove that the restriction X|_k is multiplicity-free for any generic g-module X \\in O if and only if (g,k) is isomorphic to a direct sum of (A_n,A_{n-1}), (B_n,D_n), or (D_{n+1},B_n). We also see that the restriction X|_k is multiplicity-free for any symmetric pair (g, k) and any parabolic subalgebra p with abelian nilradical and for any generic g-module X \\in O^p. Explicit branching laws are also presented.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-03-29T13:18:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E47",
      "22F30",
      "53C35"
    ],
    "keywords": [
      "restriction",
      "generic g-module",
      "parabolic subalgebra",
      "complex reductive symmetric pairs",
      "parabolic bgg category"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.4544K"
    }
  }
}