Differential Operators and Infinitesimally Equivariant Bundles

We study AV -modules, as in the work of Billig and collaborators, from a more geometric perspective.We show that if the underlying sheaf is a vector bundle, then the covariant derivative by a vector field depends almost O -linearly on the vector field.More precisely, we will show that a certain Lie map is a differential operator.This strengthens a theorem of the author and Rocha, in the sense that the bound on the order of a certain differential operator is improved upon quadratically.