## T. Kobayashi, *Restrictions of generalized Verma modules to symmetric
pairs*,
Transformation Groups **17** (2012), no. 2, 523-546, (published online first 5 April 2012).
DOI: 10.1007/s00031-012-9180-y.
arXiv:1008.4544 [math.RT]..

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.

[
DOI |
preprint version(pdf) | arXiv |
IHES-preprint ]

© Toshiyuki Kobayashi