Differential operators on hyperplane arrangements

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) = circle timesD((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.