Torus equivariant D-modules and hypergeometric systems

TL;DR

This paper develops a D-module framework to interpret A-hypergeometric systems as torus-equivariant versions of classical hypergeometric equations, revealing new structural insights.

Contribution

It constructs a functor on torus-equivariant D-modules that preserves key properties and relates classical hypergeometric systems to binomial D-modules through saturation.

Findings

Functor preserves holonomicity, regularity, and monodromy reducibility.

Application to binomial D-modules yields saturations of classical hypergeometric equations.

Provides new D-module perspective on hypergeometric systems.

Abstract

We formalize, at the level of D-modules, the notion that A-hypergeometric systems are equivariant versions of the classical hypergeometric equations. For this purpose, we construct a functor on a suitable category of torus equivariant D-modules and show that it preserves key properties, such as holonomicity, regularity, and reducibility of monodromy representation. We also examine its effect on solutions, characteristic varieties, and singular loci. When applied to certain binomial D-modules, our functor produces saturations of the classical hypergeometric differential equations, a fact that sheds new light on the D-module theoretic properties of these classical systems.