Harder–Narasimhan Filtrations which are not split by the Frobenius maps

Abstract

We will produce a smooth projective scheme X over ℤ, a rank 2 vector bundle V on X with a line subbundle L having the following property. For a prime p, let Fp be the absolute Fobenius of Xp, and let Lp ⊂ Vp be the restriction of L ⊂ V. Then for almost all primes p, and for all t ≥ 0, \((F_p^*)^t L_P \subset (F_p^*)^t V_p\) is a non-split Harder-Narasimhan filtration. In particular, \((F_p^*)^t V_p\) is not a direct sum of strongly semistable bundles for any t. This construction works for any full flag veriety G/B, with semisimple rank of G ≥ 2. For the construction, we will use Borel–Weil–Bott theorem in characteristic 0, and Frobenius splitting in characteristic p.