This paper is concerned with rationalrepresentations of reductive algebraic groups over fields of positive characteristic $p$.Let $G$ be a simplealgebraic group of rank $\ell$.It is shown that a rational representation of $G$ is semisimple provided thatits 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 finitelist of exceptional modules.

Suppose that $L_1$ and $L_2$ are irreducible representations of $G$. Theessential 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, therestricted Lie algebra cohomology with coefficients in $L_i$ is determined, and the decomposition of thecorresponding Weyl modules is analysed. These data are then exploited to obtain the needed control of theextension theory.

1991 Mathematics Subject Classification: 20G05.