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

TL;DR

This paper investigates the reducibility of scalar type generalized Verma modules for classical Lie algebras by calculating their Gelfand-Kirillov dimensions, providing criteria for their simplicity or reducibility.

Contribution

It offers a method to determine reducibility of scalar type generalized Verma modules via Gelfand-Kirillov dimension calculations for classical Lie algebras.

Findings

Criteria for reducibility based on Gelfand-Kirillov dimension

Explicit calculations for classical Lie algebras

Characterization of simple quotients of generalized Verma modules

Abstract

Let $\mathfrak{g}$ be a classial Lie algebra and $\mathfrak{p}$ be a maximal parabolic subalgebra. Let $M$ be a generalized Verma module induced from a one dimensional representation of $\mathfrak{p}$. Such $M$ is called a scalar type generalized Verma module. Its simple quotient $L$ is a highest weight moudle. In this paper, we will determine the reducibility of such scalar type generalized Verma modules by computing the Gelfand-Kirillov dimension of $L$.