Invariant Algebraic $D$-Modules on Connected Reductive Groups

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.