On the Reducibility of Scalar Generalized Verma Modules of Abelian Type

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.