Equivariant D-modules on binary cubic forms

TL;DR

This paper classifies and constructs equivariant D-modules on the space of binary cubic forms, revealing their quiver representation structure and providing explicit character formulas and cohomology calculations.

Contribution

It explicitly describes the category of GL_2-equivariant D-modules on binary cubic forms as a quiver representation category, classifies indecomposables, and constructs simple modules.

Findings

The category is of tame quiver representation type.

There are exactly 14 simple equivariant D-modules.

Provides formulas for characters and computes local cohomology groups.

Abstract

We consider the space X = Sym^3(C^2) of binary cubic forms, equipped with the natural action of the group GL_2 of invertible linear transformations of C^2. We describe explicitly the category of GL_2-equivariant coherent D_X-modules as the category of representations of a quiver with relations. We show moreover that this quiver is of tame representation type and we classify its indecomposable representations. We also give a construction of the simple equivariant D_X-modules (of which there are 14), and give formulas for the characters of their underlying GL_2-representations. We conclude the article with an explicit calculation of (iterated) local cohomology groups with supports given by orbit closures.