On the Existence of Gr-semistable Filtrations of Orthogonal/Symplectic $\lambda$-connections

In this paper, we study the existence of gr-semistable filtrations of orthogonal/symplectic $\lambda$-connections. It is known that gr-semistable filtrations always exist for flat bundles in arbitrary characteristic. However, we found a counterexample of orthogonal flat bundles of rank 5 in positive characteristic. The central new idea in this example is the notion of quasi gr-semistability for orthogonal/symplectic $\lambda$-connections. We establish the equivalence between gr-semistability and quasi gr-semistablity for an orthogonal/symplectic $\lambda$-connection. This provides a way to determine whether an orthogonal/symplectic $\lambda$-connection is gr-semistable. As an application, we obtain a characterization of gr-semistable orthogonal $\lambda$-connections of rank $\leq 6$.