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

TL;DR

This paper investigates modules over the ring of functions on a smooth algebraic variety with compatible Lie algebra actions, revealing their structure via differential operators whose order reflects their complexity.

Contribution

It introduces a new perspective on $A ext{-} ext{V}$-modules by linking their representations to differential operators of variable order based on module rank.

Findings

Representation of $ ext{V}$ is given by a differential operator.

Order of the differential operator measures representation complexity.

Simplest case corresponds to $D$-modules.

Abstract

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