  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 inclusionof R into 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.
