A criterion for the strongly semistable principal bundles over a curve in positive characteristic

Let X be an irreducible smooth projective curve over an algebraically closed field k of positive characteristic and G a simple linear algebraic group over k. Fix a proper parabolic subgroup P of G and a nontrivial anti-dominant character &#955; of P. Given a principal G-bundle E<SUB>G</SUB> over X, let E<SUB>G</SUB>(&#955;) be the line bundle over E<SUB>G</SUB>/P associated to the principal P-bundle E<SUB>G</SUB>&#8594;E<SUB>G</SUB>/P for the character &#955;. We prove that E<SUB>G</SUB> is strongly semistable if and only if the line bundle E<SUB>G</SUB>(&#955;) is numerically effective. For any connected reductive algebraic group H over k, a similar criterion is proved for strongly semistable H-bundles