Simplicity of rings of differential operators in prime characteristic
Let $W$ be a finite dimensional representation of a linearly
 reductive group $G$ over a field $k$. Motivated by their work on classical
 rings of invariants, Levasseur and Stafford asked whether the
 ring of invariants under $G$ of the symmetric algebra of $W$ has a simple ring
 of differential operators. In this paper, we show that this is true in prime characteristic.
 Indeed, if $R$ is a graded subring of a polynomial ring over a perfect field
 of characteristic $p>0$ and if the inclusion $R\hookrightarrow S$ splits, then
 $D_k(R)$ is a simple ring. In the last section of the paper, we discuss how
 one might try to deduce the characteristic zero case from this result.
 As yet, however, this is a subtle problem and the answer to the question of
 Levasseur and Stafford remains open in characteristic zero. http://www.luc.ac.be/Research/Algebra 1991 Mathematics Subject Classification: 16S32, 16G60, 13A35.