Semistable modules over Lie algebroids in positive characteristic

We study Lie algebroids in positive characteristic and moduli spaces of their modules. In particular, we show a Langtons type theorem for the corresponding moduli spaces. We relate Langtons construction to Simpsons construction of gr-semistable Griffiths transverse filtration. We use it to prove a recent conjecture of Lan-Sheng-Zuo that semistable systems of Hodge sheaves on liftable varieties in positive characteristic are strongly semistable.