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 G1 and B1. 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 H1(G1,k[g]), H1(B1,k[b]), H1(G1,k[G]) and H1(B1,k[B]) are zero. We also extend all our results to the cohomology for the higher Frobenius kernels.