LAUSR

Let k be an algebraically closed field of characteristic p > 0, X a smooth projective variety over k with a fixed ample divisor H, FX : X → X the absolute Frobenius morphism on X. Let E be a rational GLn(k)-bundle on X, and ρ : GLn(k) → GLm(k) a rational GLn(k)-representation of degree at most d such that ρ maps the radical RGLn(k)) of GLn(k) into the radical R(GLm(k)) of GLm(k). We show that if $$F_X^{N*}(E)$$FXN*(E) is semistable for some integer $$N \ge {\max {_{0 < r < m}}}(_r^m) \cdot {\log _p}(dr)$$N≥max0