Classification of Invariant Algebraic D-Modules on Semisimple and General Linear Groups
Rudrendra Kashyap, Ruoxi Li
TL;DR
The paper studies left-translation invariant algebraic D-modules on complex affine groups by reformulating them as left-invariant flat algebraic connections on the trivial bundle, modulo algebraic gauge. It proves a complete classification in two key settings: (i) for connected semisimple groups, invariant D-modules of rank $n$ are determined by representations of the finite central kernel $\Gamma$ of the simply connected cover, giving a bijection to $\text{Hom}(\Gamma,\text{GL}_n)/\text{GL}_n$; and (ii) for $G=\text{GL}_r$, all invariant D-modules of rank $n$ arise from pullback along the determinant to the torus, reducing the GL theory to the torus case and yielding $\text{Hom}(\mathbb{Z},\text{GL}_n)/\text{GL}_n$. The approach combines Maurer–Cartan theory, gauge transformations, and descent along central isogenies, with Wei’s results in basic cases (torus, unipotent, Borel, and simply connected semisimple) serving as the building blocks. The work clarifies when descent along a finite central kernel suffices and provides explicit constructions in the GL₂ and general GL_r cases via central covers and determinant pullbacks. Overall, the paper delivers explicit moduli descriptions for invariant D-modules, linking monodromy data on tori to the global classification on semisimple and general linear groups, and highlighting the central role of finite centers in the invariant D-module landscape.
Abstract
We study left-translation invariant algebraic $D$-modules on complex affine algebraic groups. Using the standard description of these objects as left-invariant flat algebraic connections on the trivial vector bundle, modulo algebraic gauge transformations, we recast the classification problem as an explicit moduli problem for constant connections. Our main results treat the semisimple case and the general linear case. For a connected semisimple complex algebraic group $G$, we show that invariant $D$-modules of rank $n$ are determined by the finite central kernel of the simply connected cover of $G$: equivalently, their isomorphism classes are classified by $n$-dimensional complex representations of that finite group, up to conjugacy. For a general linear group, we prove that every invariant $D$-module of rank $n$ is obtained by pullback along the determinant map from the one-dimensional torus, so the classification reduces completely to the torus case.