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

Abstract: 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".