Holonomic $A\mathcal{V}$-modules for the affine space

TL;DR

This paper investigates holonomic modules over the Lie algebra of vector fields on affine space, establishing foundational properties and classifications, including their structure, finite length, and relation to differential operators.

Contribution

It introduces the concept of holonomic modules in this context, proves their finite length, classifies simple modules, and connects their representation maps to differential operators.

Findings

Simple holonomic modules are tensor products of Weyl algebra modules and finite-dimensional rak{gl}_n-modules.

Holonomic modules have finite length.

Representation maps of holonomic modules are differential operators.

Abstract

We study the growth of representations of the Lie algebra of vector fields on the affine space that admit a compatible action of the polynomial algebra. We establish the Bernstein inequality for these representations, enabling us to focus on modules with minimal growth, known as holonomic modules. We show that simple holonomic modules are isomorphic to the tensor product of a holonomic module over the Weyl algebra and a finite-dimensional $\mathfrak{gl}_n$-module. We also prove that holonomic modules have a finite length and that the representation map associated with a holonomic module is a differential operator. Finally, we present examples illustrating our results.