p-adic Simpson correspondences for principal bundles in abelian settings

We explore generalizations of the p-adic Simpson correspondence on smooth proper rigid spaces to principal bundles under rigid group varieties G. For commutative G, we prove that such a correspondence exists if and only if the Lie group logarithm is surjective. Second, we treat the case of general G on ordinary abelian varieties, in which case we prove a generalization of Faltings' "small"correspondence to general rigid groups. On abeloid varieties, we also prove an analog of the classical Corlette-Simpson correspondence for principal bundles under linear algebraic groups.