On two mod p period maps: Ekedahl–Oort and fine Deligne–Lusztig stratifications

Consider a Shimura variety of Hodge type admitting a smooth integral model S at an odd prime p ≥ 5. Consider its perfectoid cover Sad( p∞) and the Hodge–Tate period map introduced by Caraiani and Scholze. We compare the pull-back to Sad( p∞) of the Ekedahl–Oort stratification on the mod p special fiber of a toroidal compactification of S and the pull back to Sad( p∞) of the fine Deligne–Lusztig stratification on the mod p special fiber of the flag variety which is the target of the Hodge–Tate period map. An application to the non-emptiness of Ekedhal–Oort strata is provided.