Supersingular Kottwitz-Rapoport strata and Deligne-Lusztig varieties

We investigate the special fibers of Siegel modular varieties with Iwahori level structure. On these spaces, we have the Newton stratification, and the Kottwitz-Rapoport stratification; one would like to understand how these stratifications are related to each other. We give a simple description of (conjecturally all) KR strata which are entirely contained in the supersingular locus as disjoint unions of Deligne-Lusztig varieties. We also give an explicit numerical description of the KR stratification in terms of abelian varieties.