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

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