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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 afinite unramified extension $F$ of $\mathbb{Q}_p$. Let us fix the type $\mu$ ofthis action on the sheaf of differentials $\omega_G$. V. Hernandez, following aconstruction of Goldring and Nicole, defined partial Hasse invariants for $G$.The product of these invariants is the $\mu$-ordinary Hasse invariant, and itis non-zero if and only if the $p$-divisible group is $\mu$-ordinary (i.e. theNewton polygon is minimal given the type of the action). \\ We show that if thevaluation of the $\mu$-ordinary Hasse invariant is small enough, then each ofthese partial Hasse invariants is a product of other sections, the refinedpartial Hasse invariants. We also give a condition for the construction ofthese invariants over an arbitrary scheme of characteristic $p$. We then give asimple, natural and elegant proof of the compatibility with duality for theclassical Hasse invariant, and show how to adapt it to the case of the refinedpartial Hasse invariants. Finally, we show how these invariants allow us tocompute the partial degrees of the canonical filtration (if it exists).

Duality, refined partial Hasse invariants and the canonical filtration