On a lemma of Samokhin

Abstract

Let G be a reductive algebraic group over an algebraically closed field \(\mathbb{k}\) of positive characteristic p, B a Borel subgroup of G, P a minimal parabolic subgroup of G containing B, and π: G/B→G/P the natural morphism. Using Orlov’s semiorthogonal decomposition of the bounded derived category of coherent sheaves on G/B to those on G/P, Samokhin derived a short exact sequence relating the Frobenius direct image of the structure sheaf of G/B to that of G/P, which was a key to his proof of the D-affinity of G/B for the symplectic group G of degree 4 over \(\mathbb{k}\). In this note we obtain his exact sequence from a short exact sequence of G n B-modules, G n the n-th Frobenius kernel of G, using representation theory.