Let $W$ be a finite dimensional representation of a linearlyreductive group $G$ over a field $k$. Motivated by their work on classicalrings of invariants, Levasseur and Stafford asked whether thering of invariants under $G$ of the symmetric algebra of $W$ has a simple ringof 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 fieldof 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 howone 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 ofLevasseur and Stafford remains open in characteristic zero.

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1991 Mathematics Subject Classification:16S32, 16G60, 13A35.