Homogeneous Principal Bundles and Stability

Let G/P be a rational homogeneous variety, where P is a parabolic subgroup of a simple and simply connected linear algebraic group G defined over an algebraically closed field of characteristic zero. A homogeneous principal bundle over G/P is semistable (respectively, polystable) if and only if it is equivariantly semistable (respectively, equivariantly polystable). A stable homogeneous principal H–bundle (EH , ρ) is equivariantly stable, but the converse is not true in general. If a homogeneous principal H–bundle (EH , ρ) is equivariantly stable, but EH is not stable, then the principal H–bundle EH admits an action ρ ′ of G such that the pair (EH , ρ ) is a homogeneous principal H–bundle which is not equivariantly stable.