On the Reducibility of Scalar Generalized Verma Modules of Abelian Type

TL;DR

This paper characterizes when scalar generalized Verma modules associated with abelian type parabolic subalgebras of complex semisimple Lie algebras are reducible, using Jantzen's criterion and classification of unitary modules.

Contribution

It provides a complete parameter characterization for reducibility of scalar generalized Verma modules of abelian type, advancing understanding of their structure.

Findings

Complete reducibility criteria for scalar generalized Verma modules

Application of Jantzen's simplicity criterion

Utilization of Enright-Howe-Wallach classification

Abstract

A parabolic subalgebra $\mathfrak{p}$ of a complex semisimple Lie algebra $\mathfrak{g}$ is called a parabolic subalgebra of abelian type if its nilpotent radical is abelian. In this paper, we provide a complete characterization of the parameters for scalar generalized Verma modules attached to parabolic subalgebras of abelian type such that the modules are reducible. The proofs use Jantzen's simplicity criterion, as well as the Enright-Howe-Wallach classification of unitary highest weight modules.