On homogeneous vector bundles

Abstract

Let G be a simply connected linear algebraic group, defined over the field of complex numbers, whose Lie algebra is simple. Let P be a proper parabolic subgroup of G. Let E be a holomorphic vector bundle over G/P such that E admits a homogeneous structure. Assume that E is not stable. Then E admits a homogeneous structure with the following property: There is a nonzero subbundle F⊆E left invariant by the action of G such that degree(F)/rank(F)≥ degree(E)/rank(E).