On the stability of homogeneous vector bundles

Abstract

Let G be a connected semisimple linear algebraic group over an algebraically closed field k and P ⊂ G a parabolic sub-group without any simple factor. Let V be an irreducible P-module and E_P (V ) = (G× V)/P the associated vector bundle over G/P. We prove that E_P (V ) is stable with respect to any polarization on G/P. In [Um] this was proved under the assumption that the characteristic of k is zero and the question was asked whether it remains valid when the characteristic is positive.