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

Abstract

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 λ of P. Given a principal G-bundle E_G over X, let E_G(λ) be the line bundle over E_G/P associated to the principal P-bundle E_G→E_G/P for the character λ. We prove that E_G is strongly semistable if and only if the line bundle E_G(λ) is numerically effective. For any connected reductive algebraic group H over k, a similar criterion is proved for strongly semistable H-bundles.