Invariant algebraic D-modules over affine algebraic groups
Yunsong Wei
TL;DR
The paper develops a complete framework for invariant algebraic $D$-modules on affine varieties under linear algebraic group actions, showing that on the group $G$ with left action these modules correspond to Lie algebra representations. For unipotent groups $U$ with nilpotent Lie algebra $\\mathfrak{n}$, invariant $D$-modules up to isomorphism are governed by algebraic gauge equivalence of $\\mathfrak{n}$-representations, and the classification reduces to representations of the abelian quotient $\\mathfrak{n}/[\\mathfrak{n},\\mathfrak{n}]$. The categorical quotient result $Hom_{Lie alg}(\\mathfrak{n},\\mathfrak{gl}_n)/\\!/GL_n(\\mathbb{C}) \cong S^n\\mathbb{C}^l$ (with $l=\dim(\\mathfrak{n}/[\\mathfrak{n},\\mathfrak{n}])$) yields concrete parameter spaces for these D-modules. The paper further provides explicit classifications for algebraic tori, Borel subgroups, and simply-connected semisimple groups, connecting invariant D-modules to moduli of flat connections and highlighting gauge-equivalence invariants across these cases.
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.