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

Let M be an irreducible projective variety, defined over an algebraically closed field k of characteristic zero, equipped with an action of a group Gamma. 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 Gamma on M. We give conditions for E-G to admit a Gamma-equivariant reduction of structure group to H, where H subset of 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 Gamma-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 Gamma on E-G. (c) 2005 Elsevier SAS. All rights reserved.