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Dimensional criteria for semisimplicity of representations. (English) Zbl 0891.20032
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 $$C$$ which is a quadratic function of $$\ell$$. Roughly speaking, it is shown that any rational $$G$$ module of dimension less than $$Cp$$ 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 extension between $$L_1$$ and $$L_2$$ provided $$\dim L_1+\dim L_2$$ is smaller than $$Cp$$. 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 analyzed. These data are then exploited to obtain the needed control of the extension theory.

##### MSC:
 20G05 Representation theory for linear algebraic groups
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