Two problems in the representation theory of reduced enveloping algebras

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 $Z_{\chi }(\lambda )\otimes Z_{{\chi }^{\prime }}(\mu )$ 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 $\chi \in {\mathfrak{g}}^{\ast }$. 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 $\mathbb{C}$.

arXiv comment33 pages. Version 2: Minor corrections