Tannakian Classification of Equivariant Principal Bundles on Toric Varieties

Abstract

Let X be a complete toric variety equipped with the action of a torus T, and G a reductive algebraic group, defined over an algebraically closed field K. We introduce the notion of a compatible ∑-filtered algebra associated to X, generalizing the notion of a compatible ∑-filtered vector space due to Klyachko, where ∑ denotes the fan of X. We combine Klyachko's classification of T-equivariant vector bundles on X with Nori's Tannakian approach to principal G-bundles, to give an equivalence of categories between T-equivariant principal G-bundles on X and certain compatible ∑-filtered algebras associated to X, when the characteristic of K is 0.