On the first restricted cohomology of a reductive Lie algebra and its Borel subalgebras

TL;DR

This paper investigates the first restricted cohomology groups of a reductive Lie algebra and its Borel subalgebras over an algebraically closed field of positive characteristic, establishing their vanishing under certain conditions.

Contribution

It proves the vanishing of specific first Frobenius kernel cohomology groups for reductive groups and Borel subgroups, extending results to higher Frobenius kernels.

Findings

H^1(G_1,k[g]) = 0 under mild assumptions

H^1(B_1,k[b]) = 0 under mild assumptions

Results extended to higher Frobenius kernels

Abstract

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.