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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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