Gelfand–Kirillov Dimension and Reducibility of Scalar Generalized Verma Modules for Classical Lie Algebras

Let \(\mathfrak{g}\) be a classical complex simple Lie algebra and \(\mathfrak{q}\) be a parabolic subalgebra. Let M be a generalized Verma module induced from a one dimensional representation of \(\mathfrak{q}\). Such M is called a scalar generalized Verma module. In this paper, we will determine the reducibility of scalar generalized Verma modules associated to maximal parabolic subalgebras by computing explicitly the Gelfand–Kirillov dimension of the corresponding highest weight modules.