Differential Operators on Hyperplane Arrangements

Abstract

Let 𝒜 be an arrangement of hyperplanes in an ℓ-dimensional vector space over a field k of characteristic zero. Let S = k[x ₁,…, x _ℓ], and let 𝒟(𝒜) be the subring of the ℓth Weyl algebra of linear differential operators preserving the defining ideal for 𝒜. We show that 𝒟(𝒜) = ⊕𝒟^(m)(𝒜), as an S-module, where 𝒟^(m)(𝒜) is the submodule of 𝒟(𝒜) of operators that are of homogeneous degree m as polynomials in the partial derivatives (with coefficients to the left). We also generalize some well known results for Der(𝒜) = 𝒟⁽¹⁾(𝒜) to higher order operators. When 𝒜 is a generic arrangement we find explicit S-module generators for 𝒟(𝒜), and use this to show that 𝒟(𝒜) is finitely generated as a k-algebra. From this we deduce the corresponding results for 𝒟(R), where R is the coordinate ring of the variety of the hyperplane arrangement.

Key Words:

• Ring of differential operator
• Derivations
• Hyperplane arrangement

Mathematics Subject Classification:

• Primary 13N10
• Secondary 14J70

Acknowledgments

The author would like to express his appreciation to his advisors. To Gísli Másson for introducing him to the subject of rings of differential operators, and for proposing the study of operators on central hyperplane arrangements. To Ralf Fröberg and Clas Löfwall for their guidance and support during the work on this paper.

Notes

^#Communicated by S. Wiegand.