Construction of Irreducible $\mathcal{U}(\mathfrak{g}{)}^{G^{\prime }}$-Modules and Discretely Decomposable Restrictions

Abstract

In this paper, we study the irreducibility of $\mathcal{U}(\mathfrak{g}{)}^{G^{\prime }}$-modules on the spaces of intertwining operators in the branching problem of reductive Lie algebras, and construct a family of finite-dimensional irreducible $\mathcal{U}(\mathfrak{g}{)}^{G^{\prime }}$-modules using the Zuckerman derived functors. We provide criteria for the irreducibility of $\mathcal{U}(\mathfrak{g}{)}^{G^{\prime }}$-modules in the cases of generalized Verma modules, cohomologically induced modules, and discrete series representations. We treat only discrete decomposable restrictions with certain dominance conditions (quasi-abelian and in the good range). To describe the $\mathcal{U}(\mathfrak{g}{)}^{G^{\prime }}$-modules, we give branching laws of cohomologically induced modules using ones of generalized Verma modules when $K^{\prime }$ acts on $K/L_K$ transitively.