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Let $ S $ be the special fibre of the good reduction of a Shimura variety ofHodge type. By constructing adapted deformations for the associated$p$-divisible groups of $ S $, we manage to construct a morphism from $S$ tosome quotient sheaf of the loop group associated with $S$. We show that thegeometric fibres of this morphism give back the Ekedahl-Oort strata of $S$. Forany geometric point $ x $ of $ S $, we give a deformation over $ W(k(x)) $ ofthe $ p $-divisible group associated with $ x $ by (non-canonically)constructing a Breuil-Kisin window (which corresponds to a $ p $-divisiblegroup over $ W(k(x)) $ by the work of Kisin). This map in a sense gives aconceptual interpretation of Viehmann's new invariants "truncations of levelone of elements in the loop group".

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