On the Principal Bundles over a Flag Manifold: II

Let G be a connected semisimple linear algebraic group defined over an algebraically closed field k and P G a reduced parabolic subgroup that does not contain any simple factor of G .Let : P - H be a homomorphism, where H is a connected reductive linear algebraic group defined over k , with the property that the image (P ) is not contained in any proper parabolic subgroup of H .We prove that the principal H -bundle G P H over G/P constructed using is stable with respect to any polarization on G/P .When the characteristic of k is positive, the principal H -bundle G P H is shown to be strongly stable with respect to any polarization on G/P .