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For a smooth algebraic variety $X$, we study the category of finitelygenerated modules over the ring of function of $X$ that has a compatible actionof the Lie algebra $\mathcal{V}$ of polynomials vector fields on $X$. We showthat the associated representation of $\mathcal{V}$ is given by a differentialoperator of order depending on the rank of the module. The order of thedifferential operator provides a natural measure of the complexity of therepresentation, with the simplest case being that of $D$-modules.

Annihilators of $A\mathcal{V}$-modules and differential operators