Invariant Algebraic $D$-Modules on Connected Reductive Groups

TL;DR

This paper classifies finite-rank invariant algebraic $D$-modules on complex reductive groups, linking them to group representations and simplifying their structure through pullback maps, with applications to cohomology and local systems.

Contribution

It provides an explicit classification of invariant $D$-modules on connected reductive groups, extending known results for semisimple and general linear groups, and relates these modules to group representations.

Findings

Invariant $D$-modules on semisimple groups correspond to representations of the finite central kernel.

For general linear groups, invariant $D$-modules are obtained via pullback from the determinant map.

Applications include insights into cohomology and local systems for semisimple groups.

Abstract

We study finite-rank 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. We prove our main results for semisimple groups, for general linear groups, and more generally for connected reductive groups. For a connected semisimple complex algebraic group, invariant $D$-modules are classified by representations of the finite central kernel of the simply connected cover. For a general linear group, every invariant $D$-module is obtained by pullback along the determinant map, reducing the classification to the one-dimensional torus case. For a connected reductive group, we relate invariant $D$-modules via pullback along the abelianization map. We also derive applications concerning cohomology and the associated local systems for semisimple groups.