Principal bundles over finite fields

Abstract: Let M be an irreducible smooth projective variety defined over \bar{{\mathbb F}_p}. Let \pi(M, x_0) be the fundamental group scheme of M with respect to a base point x_0. Let G be a connected semisimple linear algebraic group over \bar{{\mathbb F}_p}. Fix a parabolic subgroup P \subsetneq G, and also fix a strictly anti-dominant character \chi of P. Let E_G \to M be a principal G-bundle such that the associated line bundle E_G(\chi) \to E_G/P is numerically effective. We prove that E_G is given by a homomorphism \pi(M, x_0)\to G. As a consequence, there is no principal G-bundle E_G \to M such that degree(\phi^*E_G(\chi)) > 0 for every pair (Y ,\phi), where Y is an irreducible smooth projective curve, and \phi: Y\to E_G/P is a nonconstant morphism.