Invariant algebraic D-modules over affine algebraic groups

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.