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We study Lie algebroids in positive characteristic and moduli spaces of theirmodules. In particular, we show a Langton's type theorem for the correspondingmoduli spaces. We relate Langton's construction to Simpson's construction ofgr-semistable Griffiths transverse filtration. We use it to prove a recentconjecture of Lan-Sheng-Zuo that semistable systems of Hodge sheaves onliftable varieties in positive characteristic are strongly semistable.

Semistable modules over Lie algebroids in positive characteristic