On the equivariant reduction of structure group of a principal bundle to a Levi subgroup

Abstract

Let M be an irreducible projective variety, defined over an algebraically closed field k of characteristic zero, equipped with an action of a group Γ. Let E_G be a principal G-bundle over M, where G is a connected reductive linear algebraic group defined over k, equipped with a lift of the action of Γ on M. We give conditions for E_G to admit a Γ-equivariant reduction of structure group to H, where H⊂G is a Levi subgroup. We show that for any principal G-bundle E_G, there is a naturally associated conjugacy class of Levi subgroups of G. Given a Levi subgroup H in this conjugacy class, the principal G-bundle E_G admits a Γ-equivariant reduction of structure group to H, and furthermore, such a reduction is unique up to an automorphism of E_G that commutes with the action of ? on E_G.