Semistable Non Abelian Hodge theorem in positive characteristic

In this paper, we show that for any reductive group $G$ the moduli space of semistable $G$-Higgs bundles on a curve in characteristic $p$ is a twisted form of the moduli space of semistable flat $G$-connections. This is the semistable version of a previous result of Chen-Zhu, and the $G$-bundle version of a previous result of de Cataldo-Groechenig-Zhang. As a consequence, we show that the Decomposition Theorem for the Hitchin morphism for $G$-Higgs bundles has the same shape as that for the de Rham-Hitchin morphism for flat $G$-connections.