Restrictions of generalized Verma modules to symmetric pairs

Abstract

We initiate a new line of investigation on branching problems for generalized Verma modules with respect to reductive symmetric pairs $\left( {\mathfrak{g},\mathfrak{g}'} \right)$. In general, Verma modules may not contain any simple module when restricted to a reductive subalgebra. In this article we give a necessary and sufficient condition on the triple $\left( {\mathfrak{g},\mathfrak{g}',\mathfrak{p}} \right)$ such that the restriction ${\left. X \right|_{\mathfrak{g}'}}$ always contains simple $\mathfrak{g}'$-modules for any $\mathfrak{g}$-module X lying in the parabolic BGG category ${\mathcal{O}^\mathfrak{p}}$ attached to a parabolic subalgebra $\mathfrak{p}$ of $\mathfrak{g}$. Formulas are derived for the Gelfand–Kirillov dimension of any simple module occurring in a simple generalized Verma module. We then prove that the restriction ${\left. X \right|_{\mathfrak{g}'}}$ is generically multiplicity-free for any $\mathfrak{p}$ and any $X \in {\mathcal{O}^\mathfrak{p}}$ if and only if $\left( {\mathfrak{g},\mathfrak{g}'} \right)$ is isomorphic to (A _n , A _n-1), (B _n , D _n ), or (D _n+1, B _n ). Explicit branching laws are also presented.