The second cohomology of small irreducible modules for simple algebraic groups

Let G be a connected, simply connected, quasisimple alge-braic group over an algebraically closed field of characteristic p> 0, and let V be a rational G-module such that dimV ≤ p. According to a result of Jantzen, V is completely reducible, and H1(G,V) = 0. In this paper we show that H2(G,V) = 0 unless some composition factor of V is a nontrivial Frobenius twist of the adjoint representation of G. 1. Introduction. Let G be a quasisimple, connected, simply connected algebraic group over the algebraically closed field k of characteristic p> 0. By a G-module V, we always understand a rational G-module (one given by a morphism of algebraic groups G → GL(V)). In this paper, we study the cohomology of