Ekedahl–Oort strata and Kottwitz–Rapoport strata

We study Ekedahl–Oort strata on the moduli space Ag of g-dimensional principally polarized abelian varieties in positive characteristic, and Kottwitz–Rapoport strata on its variants AJ with parahoric level structure. First, we show that every Ekedahl–Oort stratum is isomorphic to a parahoric Kottwitz–Rapoport stratum. Second, both supersingular Ekedahl–Oort strata and supersingular Kottwitz–Rapoport strata are isomorphic to disjoint unions of Deligne–Lusztig varieties (see Hoeve (2010) [10], and Görtz and Yu (2010) [5], resp.), and here we compare these isomorphisms. Finally we give an explicit description of Kottwitz–Rapoport strata contained in the supersingular locus in the general parahoric case.