Reducibility of scalar generalized Verma modules of minimal parabolic type

TL;DR

The paper addresses the reducibility of scalar generalized Verma modules induced from minimal parabolic subalgebras of classical complex simple Lie algebras. It leverages the Gelfand-Kirillov (GK) dimension and a Robinson–Schensted–Knuth based algorithm to compute GK-dimensions of highest-weight modules and uses the criterion that reducibility occurs when GK-dim$\,L(\lambda) < \dim \mathfrak{u}$. The authors establish that the first diagonal-reducible point for $M_I(z\widehat{\omega_p})$ typically occurs at $z=-1$ across types $A_n,B_n,C_n,D_n$, with detailed subcase analyses for integral and half-integral $z$ values; non-integral $z$ outside $\frac{1}{2}+\mathbb{Z}$ yield irreducibility. This furnishes explicit reducibility thresholds for scalar generalized Verma modules in the minimal-parabolic setting, extending prior results on Hermitian symmetric and maximal parabolic cases and enabling precise structural understanding via GK-dimension techniques.

Abstract

Let $\mathfrak{g}$ be a classical complex simple Lie algebra and $\mathfrak{q}$ be a parabolic subalgebra. Generalized Verma module $M$ is called a scalar generalized Verma module if it is induced from a one-dimensional representation of $\mathfrak{q}$. In this paper, we will determine the first diagonal-reducible point of scalar generalized Verma modules associated to minimal parabolic subalgebras by computing explicitly the Gelfand-Kirillov dimension of the corresponding highest weight modules.

Theorems & Definitions (23)

• Definition 2.1
• Definition 2.2
• Lemma 2.3: BX
• Lemma 2.4: BX
• Example 2.5
• Theorem 2.7: BXX
• Definition 2.8
• Example 2.9
• Definition 2.10
• Theorem 2.11: BX1 and BXX