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 polynomial functions on G and g by k[G] and k[g], and similar 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.