On the reducibility of scalar generalized Verma modules associated to two-step nilpotent parabolic subalgebras

TL;DR

This paper investigates when scalar generalized Verma modules, induced from certain parabolic subalgebras of simple Lie algebras, are reducible, focusing on two-step nilpotent cases and visualizing reducibility points.

Contribution

It characterizes reducibility of scalar generalized Verma modules for specific Lie algebras associated with two-step nilpotent parabolic subalgebras using Gelfand-Kirillov dimension.

Findings

Reducibility points form a two-dimensional complex plane

Modules exist only for specific Lie algebras: sl(n,C), so(2n,C), E6

Provides a geometric visualization of reducibility

Abstract

Let $\mathfrak{g}$ be a simple complex Lie algebra.A generalized Verma module induced from a one-dimensional representation of a parabolic subalgebra of $\mathfrak{g}$ is called a scalar generalized Verma module of $\mathfrak{g}$. In this article, we use Gelfand-Kirillov dimension to determine the reducibility of scalar generalized Verma modules of $\mathfrak{g}$ associated to a two-step nilpotent parabolic subalgebra of non-maximal type. Such a module exists only when $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$, $\mathfrak{so}(2n,\mathbb{C})$ or $E_6$. We find that the reducible points of these modules can be drawn in a two-dimensional complex plane.