The decomposition of certain abstract-induced modules over reductive algebraic groups

Abstract

Let G be a connected reductive algebraic group defined over an algebraically closed field F, and B be a Borel subgroup of G. Let k be another field. We determine the composition factors of the abstract induced module M(θ) = kG ⊗ kB θ (here kH is the group algebra of H over the field k and θ is a character of B over k) when char F = p is poistive and char k = p for a general character θ. In particular, when θ is trivial, we give the composition factors of M(tr) for any F and k.