Reducibility of Scalar Generalized Verma Modules of Minimal Parabolic Type II
Jing Jiang, Siying Wu
TL;DR
The paper tackles the reducibility problem for scalar generalized Verma modules induced from minimal parabolic subalgebras in exceptional Lie algebras. It leverages the Gelfand–Kirillov dimension, together with Lusztig's a-function, the RS algorithm, and PyCox, to determine the first diagonal-reducible point by computing GK-dim of simple quotients. The main results are explicit lists of reducible points for types $G_2$, $F_4$, and the exceptional $E_6$, $E_7$, and $E_8$, with comprehensive data organized in an Appendix. These findings provide concrete reducibility criteria across all minimal parabolic cases in the exceptional realm, enabling precise classification and further study of parabolic inductions in these algebras.
Abstract
Let g be a exceptional complex simple Lie algebra and q be a parabolic subalgebra. A generalized Verma module M is called a scalar generalized Verma module if it is induced from a one-dimensional representation of q. In this paper, we will determine the first diagonal-reducible point of scalar generalized Verma modules associated to minimal parabolic subalgebras of exceptional Lie algebras by computing explicitly the Gelfand-Kirillov dimension of the corresponding highest weight modules.