Two problems in the representation theory of reduced enveloping algebras

TL;DR

This paper investigates two key problems in the representation theory of Lie algebras over fields of positive characteristic, focusing on tensor products of baby Verma modules and minimal-dimension modules of reduced enveloping algebras.

Contribution

It demonstrates a filtration structure for tensor products of baby Verma modules and constructs minimal-dimensional modules as quotients of base-changed simple modules in type A.

Findings

Tensor product of baby Verma modules admits a specific filtration.

Minimal-dimensional modules can be realized as quotients of base-changed simple modules.

Results are established under certain assumptions in type A.

Abstract

In this paper we consider two problems relating to the representation theory of Lie algebras ${\mathfrak g}$ of reductive algebraic groups $G$ over algebraically closed fields ${\mathbb K}$ of positive characteristic $p>0$. First, we consider the tensor product of two baby Verma modules $Z_{\chi}(\lambda)\otimes Z_{\chi'}(\mu)$ and show that it has a filtration of baby Verma modules of a particular form. Secondly, we consider the minimal-dimension representations of a reduced enveloping algebra $U_\chi({\mathfrak g})$ for a nilpotent $\chi\in{\mathfrak g}^{*}$. We show that under certain assumptions in type $A$ we can obtain the minimal-dimensional modules as quotients of certain modules obtained by base change from simple highest weight modules over ${\mathbb C}$.