A quotient stack related to the Weyl algebra

TL;DR

This paper establishes an equivalence between the category of graded modules over the Weyl algebra and quasi-coherent sheaves on a specific quotient stack, revealing a geometric perspective on differential operator modules.

Contribution

It introduces a geometric interpretation of the Weyl algebra modules via a quotient stack with isotropy groups, connecting algebraic and geometric frameworks.

Findings

Category of graded A-modules is equivalent to sheaves on the stack X

Identifies the stack X with a coarse moduli space with isotropy groups at integers

Provides a new geometric perspective on differential operator modules

Abstract

Let A denote the ring of differential operators on the affine line with its two usual generators t and d/dt given degrees +1 and -1 respectively. Let X be the stack having coarse moduli space the affine line Spec k[z] and isotropy groups Z/2 at each integer point. Then the category of graded A-modules is equivalent to the category of quasi-coherent sheaves on X. Version 2: corrected typos and deleted appendix at referee's suggestion.