Dimensional criteria for semisimplicity ofrepresentations

Abstract

This paper is concerned with rational representations of reductive algebraic groups over fields of positive characteristic $p$. Let $G$ be a simple algebraic group of rank $\ell$. It is shown that a rational representation of $G$ is semisimple provided that its dimension does not exceed $\ell p$. Furthermore, this result is improved by introducing a certain quantity $\mathcal{C}$ which is a quadratic function of $\ell$. Roughly speaking, it is shown that any rational $G$ module of dimension less than $\mathcal{C} p$ is either semisimple or involves a subquotient from a finite list of exceptional modules.

Suppose that $L_1$ and $L_2$ are irreducible representations of $G$. The essential problem is to study the possible extensions between $L_1$ and $L_2$ provided $\dim L_1 + \dim L_2$ is smaller than $\mathcal{C} p$. In this paper, all relevant simple modules $L_i$ are characterized, the restricted Lie algebra cohomology with coefficients in $L_i$ is determined, and the decomposition of the corresponding Weyl modules is analysed. These data are then exploited to obtain the needed control of the extension theory.

1991 Mathematics Subject Classification: 20G05.