Tannakian classification of equivariant principal bundles on toric varieties

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.