Let $k$ be an algebraically closed field and let $G$ be a reductive linear algebraic group over $k$ with Lie algebra $\mathfrak g$. If the characteristic of $k$ is $0$ then, by a classical result of Kostant, $k[\mathfrak g]$ is free over the subalgebra $k[\mathfrak g]^G$. Is there an analog of this result when the characteristic of $k$ is positive?