ON THE FIRST RESTRICTED COHOMOLOGY OF A REDUCTIVE LIE ALGEBRA AND ITS BOREL SUBALGEBRAS

Let k be an algebraically closed field of characteristic p > 0 and let G be a connected reductive group over k. Let B be a Borel subgroup of G and let g and b be the Lie algebras of G and B. Denote the first Frobenius kernels of G and B by G(1) and B-1. Furthermore, denote the algebras of regular functions on G and g by k[G] and k[g], and similarly for B and b. The group G acts on k[G] via the conjugation action and on k[g] via the adjoint action. Similarly, B acts on k[B] via the conjugation action and on k[b] via the adjoint action. We show that, under certain mild assumptions, the cohomology groups H-1 (G(1), k[g]), H-1 (B-1, k[b]), H-1(G(1), k[G]) and H-1 (B-1, k[B]) are zero. We also extend all our results to the cohomology for the higher Frobenius kernels.