Invariant algebraic D-modules over affine algebraic groups

TL;DR

This paper investigates invariant algebraic D-modules on affine varieties under algebraic group actions, providing classifications and isomorphism criteria for specific groups like unipotent groups, tori, and Borel subgroups.

Contribution

It offers new classification results for invariant D-modules, especially for unipotent groups, and links their isomorphism classes to GIT quotients, advancing understanding of their structure.

Findings

Invariant D-modules correspond to Lie algebra representations for linear algebraic groups.

Two invariant D-modules are isomorphic iff they are in the same GIT fiber for unipotent groups.

Complete classification of invariant D-modules over algebraic tori and Borel subgroups.

Abstract

We study the invariant algebraic D-modules on an affine variety under the action of an algebraic group.For linear algebraic groups with the multiplication action by themselves, such D-modules correspond to representations of their Lie algebra. For unipotent algebraic groups, we show that two invariant D-modules are isomorphic if and only if they lie in the same fiber of the GIT (Geometric Invariant Theory) quotient of the space of representations under the action of conjugation. Additionally, we classify invariant D-modules over the algebraic torus and the Borel subgroup of the general linear group.