On the Reducibility of Scalar Generalized Verma Modules of Abelian Type

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.Comment: 23 pages, 2 figures, 4 tables, To appear in Algebras and Representation Theor

Gelfand-Kirillov dimensions and Reducibility of scalar type generalized Verma modules for classical Lie algebras

“…So the scalar generalized Verma module M I (λ) (with highest weight λ = zξ 1 ) is reducible if and only if z ∈ 1 − min{1, 2} + Z ≥0 = Z ≥0 . This result is the same with [10] and [16].…”

“…Kubo [16] found some practical reducibility criteria for all the scalar generalized Verma modules associated with maximal parabolic subalgebras. By using Kubo's result, He [10] gave the reducibily for all scalar-type generalized Verma modules of Hermitian symmetric pairs. Then He-Kubo-Zierau [11] found the reducibily for all scalar-type generalized Verma modules associated with maximal parabolic subalgebras.…”

On the space of 𝐾-finite solutions to intertwining differential operators

“…We further consider a variant of Problem A for g-homomorphisms between generalized Verma modules in Theorem 5.21. We also classify in Corollary 5.22 the reducibility of the generalized Verma module in consideration; this recovers a result of [1,14,15] for the pair (g, p) = (sl(n, C), p 1,n−1 ).…”

Polology of Superconformal Blocks

“…There are mathematical papers explicitly working out Jantzen criterion for scalar (spin zero) parabolic Verma modules for semisimple Lie algebras. See e.g [39]. for recent discussion.…”