Restrictions of generalized Verma modules to symmetric pairs

TL;DR

This paper investigates the restrictions of generalized Verma modules to symmetric pairs, establishing conditions for simple module containment, deriving formulas for module dimensions, and identifying cases of multiplicity-free restrictions with explicit branching laws.

Contribution

It provides a necessary and sufficient geometric condition for simple module containment in restrictions and characterizes when restrictions are multiplicity-free, including explicit branching laws.

Findings

Restriction contains simple modules under specific geometric conditions.

Restriction is multiplicity-free for certain symmetric pairs and parabolic subalgebras.

Formulas for Gelfand-Kirillov dimensions of simple modules are derived.

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.