Simplicity of Rings of Differential Operators in Prime Characteristic

2002/09/20 by Karen E. Smith, Michel Van den Bergh · 2 citations
Mathematics · #math.RT #math.AC #math.RA #msc:16S32 #msc:16G60 #msc:13A35

paper · pdf

published as Proc. London Math. Soc. (3) 75 (1997), no. 1, 32--62 · 30 pages; Latex file; One minor difference between this version and published version: Incorrect justification for one easy statement in proof of Proposition 3.1.6 corrected

arxiv created 2002/09/20 · arxiv updated 2009/11/30

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 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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• Hilbert-Kunz series, F-signature series, and weak p-fractals
• On the uniform positivity of F-signature under reduction modulo p

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