Dimensional Criteria for Semisimplicity of Representations
GJ McNinch · Proceedings of the London Mathematical Society · 1998
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 ℓ. It is shown that a rational representation of G is semisimple provided that its dimension does not exceed ℓp. Furthermore, this result is improved by introducing a certain quantity C which is a quadratic function of ℓ. 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 L1 and L2 are irreducible representations of G. The essential problem is to study the possible extensions between L1 and L2 provided dim L1 + dim L2 is smaller than Cp. In this paper, all relevant simple modules Li are characterized, the restricted Lie algebra cohomology with coefficients in Li 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.