Semistability of Frobenius Direct Image of Representations of Cotangent Bundles

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 . : GLn(k) . GLm(k) a rational GLn(k)-representation of degree at most d such that . maps the radical RGLn(k)) of GLn(k) into the radical R(GLm(k)) of GLm(k). We show that if FN.X(E) is semistable for some integer N.max0<r<m(mr).logp(dr), then the induced rational GLm(k)-bundle E(GLm(k)) is semistable. As an application, if dimX = n, we get a sufficient condition for the semistability of Frobenius direct image FX.(..(.1X)), where ..(.1X) is the vector bundle obtained from .1X via the rational representation ..