Graded cofinite rings of differential operators

In this paper we study subalgebras A of the algebra D(X) of differential operators on a smooth variety X which are big in the following sense: using the order of a differential operator, the ring D(X) is equipped with a filtration. Its associated graded algebra D(X) is commutative and can be regarded as the set of regular functions on the cotangent bundle ofX . The subalgebra A inherits a filtration from D(X) and its associated graded algebraA is a subalgebra of D(X). We call A graded cofinite in D(X) if D(X) is a finitely generated A-module. Our guiding example of a graded cofinite subalgebra is the algebra of invariants D(X) where W is a finite group acting on X . Other examples can be constructed as follows. Let φ : X → Y be a finite dominant morphism onto a normal variety Y . Then we put