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 $\mathrm{\Sigma }$–filtered algebra associated to $X$, generalizing the notion of a compatible $\mathrm{\Sigma }$–filtered vector space due to Klyachko, where $\mathrm{\Sigma }$ 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 $\mathrm{\Sigma }$–filtered algebras associated to $X$, when the characteristic of $K$ is $0$.