The second cohomology of small irreducible modules for simple algebraic groups

Let G be a simple, simply connected and connected algebraic group over an algebraically closed field of characteristic p>0, and let V be a rational G-module such that dim V <= p. According to a result of Jantzen, V is completely reducible, and H^1(G,V)=0. In this paper we show that H^2(G,V) = 0 unless some composition factor of V is a non-trivial Frobenius twist of the adjoint representation of G.