Rational and line bundle cohomology for reductive algebraic groups

Abstract

In general little is known about the structure of Hⁱ(λ), the higher right-derived functors of the induction from a Borel subgroup of a simple reductive algebraic group (not of type A₁) defined over a field k of positive characteristic relative to an integral weight. In this work, we prove several results which help elucidate it. In particular we demonstrate how these higher right-derived functors are closely related to Weyl modules defined over certain parabolic subgroups. Then this relationship is exploited to recover Andersen's well-known computation of the socle of the first derived functor and to give vanishing results for some of these derived functors. Furthermore, parabolic subgroups are used to show that certain second derived functors are isomorphic to the base field and further vanishing results on H²(λ). Finally, we explore the relationship between the cohomologies for the algebraic group and its Borel subgroup. In so doing, it is proven that the first cohomology of the algebraic group with coefficients in Hⁱ(λ) vanishes.