A criterion for the strongly semistable principal bundles over a curve in positive characteristic (Q707255)

Let \(G\) be a connected reductive linear algebraic group over an algebraically closed field \(k\) of positive characteristic. A principal \(G\)-bundle \(E_G\) on an irreducible smooth projective curve \(X\) over \(k\) is called strongly semistable if its pullbacks by all powers of the Frobenius morphism are semistable. Fix a parabolic subgroup \(P\subset G\) that projects to a proper parabolic subgroup \(P_i\) of each simple factor \(G_i\) of \(G/Z(G)\), \(Z(G)\) being the centre of \(G\). Fix a character \(\lambda\) of \(P\) which is trivial on \(Z(G)\) such that the induced character of \(P/Z(G)\) gives a nontrivial antidominant character of each \(P_i\). The main result of the paper is that a principal \(G\)-bundle \(E_G\) is strongly semistable if and only if the associated line bundle \(E_G(\lambda)\) over \(E_G/P\) is numerically effective. In characteristic zero, the result remains valid for semistable \(G\)-bundles. For a general stable \(G\)-bundle \(E_G\) over a compact Riemann surface, \(E_G(\lambda)\) is ample on any proper closed subvariety of \(E_G/P\), which was proved by the authors and \textit{S. Subramanian} [Bull. Sci. Math. 128, 23--29 (2004; Zbl 1046.14011)].