The Ekedahl-Oort Stratification and the Semi-Module Stratification

In this paper we compare the $\mathbb{J}$-stratification (or the semi-module stratification) and the Ekedahl-Oort stratification of affine Deligne-Lusztig varieties in the superbasic case. In particular, we classify the cases where the $\mathbb{J}$-stratification gives a refinement of the Ekedahl-Oort stratification, which include many interesting cases such that the affine Deligne-Lusztig variety admits a simple geometric structure.