3 research outputs found
Restrictions of generalized Verma modules to symmetric pairs
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 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.Comment: 31 pages, To appear in Transformation Group
Über das Synthese-Problem für nilpotente Liesche Gruppen
Poguntke D. Über das Synthese-Problem für nilpotente Liesche Gruppen. Mathematische Annalen. 1984;269(4):431-467