A criterion for the strongly sernistable 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 lambda of P. Given a principal G-bundle E-G over X, let E-G(lambda) be the line bundle over E-G/P associated to the principal P-bundle E-G --> E-G/P for the character lambda. We prove that E-G is strongly semistable if and only if the line bundle E-G(lambda) is numerically effective. For any connected reductive algebraic group H over k, a similar criterion is proved for strongly semistable H-bundles. (C) 2004 Elsevier SAS. All rights reserved.