Newton-Hodge filtration for F-crystals with additional structure

We study F-crystals with self-dual structure over base schemes of characteristic p. We generalize Katz's Newton-Hodge Decomposition Theorem and Newton-Hodge Filtration Theorem to F-crystals with self-dual structure. These results can be thought of as a group theoretic interpretation of Katz's theorems for the group of symplectic similitudes and the group of orthogonal similitudes. Then we make preliminary investigations into how these results might be generalized further to the case of any connected reductive group G..