2010/08/31 by Toshiyuki Kobayashi · 2 citations
Mathematics · Physics and Astronomy · #math-ph #math.CO #math.MP #math.RT #msc:22E47 #msc:22F30 #msc:53C35
paper · pdf · doi:10.1007/s00031-012-9180-y
published as Transformation Groups 17 (2012), pp. 523-546 · 31 pages, To appear in Transformation Groups
arxiv created 2012/03/29 · arxiv updated 2012/06/05
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 KC orbits on the generalized flag variety GC/PC, 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 Op 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 ∈ O if and only if (g,k) is isomorphic to a direct sum of (An,An-1), (Bn,Dn), or (Dn+1,Bn). 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 ∈ Op. Explicit branching laws are also presented.