Hitchin map for the moduli space of $\Lambda$-modules in positive characteristic

Building on Simpson's original definition over the complex numbers, we introduce the notion of restricted sheaf $\Lambda$ of rings of differential operators on a variety defined over a field of positive characteristic. We define the notion of $p$-curvature for $\Lambda$-modules and the analogue of the Hitchin map on the moduli space of $\Lambda$-modules. We show that under certain conditions this Hitchin map descends under the Frobenius map of the underlying variety and we give examples.