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.