Simplicity of Rings of Differential Operators in Prime Characteristic

Details Simplicity of Rings of Differential Operators in Prime Characteristic Simplicity of Rings of Differential Operators in Prime Characteristic

2002-09-20

arxiv.org/abs/math/0209275v1

Proc. London Math. Soc. (3) 75 (1997), no. 1, 32--62

Mathematics

Representation Theory

30 pages; Latex file; One minor difference between this version and published version: Incorrect justification for one easy st

Scientific paper

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.

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