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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.

20G05 Representation theory for linear algebraic groups
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