  We give a simple and natural proof for the compatibility of the Hasse
invariant with duality. We then study a $p$-divisible group with an action of
the ring of integers of a finite ramified extension of $\mathbb{Q}_p$. We
suppose that it satisfies the Pappas-Rapoport condition ; in that case the
Hasse invariant is a product of partial Hasse invariants, each of which can be
expressed in terms of primitive Hasse invariants. We then show that the dual of
the $p$-divisible group naturally satisfies a Pappas-Rapoport condition, and
prove the compatibility with the duality for the partial and primitive Hasse
invariants.
