Equivariant reduction to torus of a principal bundle

Abstract

Let M be an irreducible projective variety, over an algebraically closed field k of characteristic zero, equipped with an action of a connected algebraic group S over k . Let E_G be a principal G-bundle over M equipped with a lift of the action of S on M, where G is a connected reductive linear algebraic group. Assume that E_G admits a reduction of structure group to a maximal torus T ⊂G. We give a necessary and sufficient condition for the existence of a T-reduction of E_G which is left invariant by the action of S on E_G.