Algebra publication
Title
Simplicity of rings of differential operators in
prime characteristic
Author(s)
K.E Smith, M. Van den Bergh.
Year
1995
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-> 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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