Invariant Algebraic Connections on Connected Reductive Groups

Abstract

Let $G$ be a connected complex reductive algebraic group. We study finite-rank flat algebraic connections on trivial vector bundles whose connection forms are left invariant, allowing arbitrary algebraic horizontal morphisms. We prove that a flat algebraic connection on $G$ is regular-singular if and only if its pullback to the canonical finite central cover is isomorphic to a left-invariant flat algebraic connection on a trivial vector bundle. Moreover, every regular-singular algebraic connection on $G$ is a direct summand of such a connection. We characterize the essential image of pullback from the abelianization by the vanishing of a derived-monodromy obstruction and show that pullback is an equivalence precisely when $G^{\mathrm{der}}$ is simply connected. For semisimple $G$, regular-singular connections are classified by finite-dimensional representations of the finite central kernel of the simply connected cover. We also classify regular-singular connections on $\mathrm{GL}_r$ by pullback along the determinant, give a $\mathrm{PGL}_2$ counterexample to the abelianization classification of left-invariant trivial-bundle algebraic connections, and compute de Rham and Betti cohomology in the semisimple case.