Duality, refined partial Hasse invariants and the canonical filtration

Let $G$ be a $p$-divisible group over the ring of integers of $\mathbb{C}_p$, and assume that it is endowed with an action of the ring of integers of a finite unramified extension $F$ of $\mathbb{Q}_p$. Let us fix the type $\mu$ of this action on the sheaf of differentials $\omega_G$. V. Hernandez, following a construction of Goldring and Nicole, defined partial Hasse invariants for $G$. The product of these invariants is the $\mu$-ordinary Hasse invariant, and it is non-zero if and only if the $p$-divisible group is $\mu$-ordinary (i.e. the Newton polygon is minimal given the type of the action). \\ We show that if the valuation of the $\mu$-ordinary Hasse invariant is small enough, then each of these partial Hasse invariants is a product of other sections, the refined partial Hasse invariants. We also give a condition for the construction of these invariants over an arbitrary scheme of characteristic $p$. We then give a simple, natural and elegant proof of the compatibility with duality for the classical Hasse invariant, and show how to adapt it to the case of the refined partial Hasse invariants. Finally, we show how these invariants allow us to compute the partial degrees of the canonical filtration (if it exists).