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Simplicity of rings of differential operators in prime characteristic

Published online by Cambridge University Press:  01 July 1997

KE Smith
Affiliation:
Mathematics Department 2-167, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA. Email: [email protected]
M van den Bergh
Affiliation:
Limburgs Universitair Centrum, Departement WNI, Universitaire Campus, 3590 Diepenbeek, Belgium. Email: [email protected]
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Abstract

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.

Type
Research Article
Copyright
London Mathematical Society 1997

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