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

TL;DR

This paper investigates the irreducibility of certain modules associated with reductive Lie algebras, constructing new finite-dimensional modules and providing criteria for irreducibility in specific representation cases.

Contribution

It introduces a method to construct finite-dimensional irreducible modules and establishes irreducibility criteria for various classes of modules in the context of Lie algebra restrictions.

Findings

Criteria for irreducibility of modules in generalized Verma modules and discrete series cases

Construction of finite-dimensional irreducible modules using Zuckerman functors

Branching laws for cohomologically induced modules

Abstract

In this paper, we study the irreducibility of $\mathcal{U}(\mathfrak{g})^{G'}$-modules on the spaces of intertwining operators in the branching problem of reductive Lie algebras, and construct a family of finite-dimensional irreducible $\mathcal{U}(\mathfrak{g})^{G'}$-modules using the Zuckerman derived functors. We provide criteria for the irreducibility of $\mathcal{U}(\mathfrak{g})^{G'}$-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 $\mathcal{U}(\mathfrak{g})^{G'}$-modules, we give branching laws of cohomologically induced modules using ones of generalized Verma modules when $K'$ acts on $K/L_K$ transitively.