Third cohomology for Frobenius kernels and related structures

Abstract

Let $G$ be a simple simply connected group scheme defined over ${\mathbb{F}}_p$ and $k$ be an algebraically closed field of characteristic $p>0$. Moreover, let $B$ be a Borel subgroup of $G$ and $U$ be the unipotent radical of $B$. In this paper the authors compute the third cohomology group for $B$ and its Frobenius kernels, $B_r$, with coefficients in a one-dimensional representation. These computations hold with relatively mild restrictions on the characteristic of the field. As a consequence of our calculations, the third ordinary Lie algebra cohomology group for $\mathfrak{u}=\text{Lie }U$ with coefficients in $k$ is determined, as well as the third $G_r$-cohomology with coefficients in the induced modules $H^0(\lambda )$.