First cohomology of Frobenius kernels and truncated invariants

$k$ be an algebraically closed field of characteristic $p > 0$ and let $G$ be a connected reductive group over $k$ with Lie algebra ${g}$. Consider the rings $k[G]$ and $k[{g}]$ of regular functions on $G$ and ${g}$ as $G$-modules via the conjugation action. They have been studied extensively, for example in Kostant’s 1963 paper. I will discuss the result that, under some mild assumptions, the first restricted cohomology of these modules is zero. After this I will discuss the problem of describing the invariants in a certain finite dimensional quotient of $k[{g}]$.