Ekedahl-Oort stratifications of Shimura varieties via Breuil-Kisin windows

Let $S$ be the special fibre of the good reduction of a Shimura variety of Hodge type. By constructing adapted deformations for the associated $p$-divisible groups of $S$, we manage to construct a morphism from $S$ to some quotient sheaf of the loop group associated with $S$. We show that the geometric fibres of this morphism give back the Ekedahl-Oort strata of $S$. For any geometric point $x$ of $S$, we give a deformation over $W(k(x))$ of the $p$-divisible group associated with $x$ by (non-canonically) constructing a Breuil-Kisin window (which corresponds to a $p$-divisible group over $W(k(x))$ by the work of Kisin). This map in a sense gives a conceptual interpretation of Viehmann’s new invariants “truncations of level one of elements in the loop group”.