G-connections on principal bundles over complete G-varieties

Let X be a complete variety over an algebraically closed field k of characteristic zero, equipped with an action of an algebraic group G. Let H be a reductive group. We study the notion of G-connection on a principal H-bundle. We give necessary and sufficient criteria for the existence of G-connections extending the Atiyah-Weil type criterion for holomorphic connections obtained by Azad and Biswas. We also establish a relationship between the existence of G-connection and equivariant structure on a principal H-bundle, under the assumption that G is semisimple and simply connected. These results have been obtained by Biswas et al. when the underlying variety is smooth.