The supersingular locus of the Shimura variety of GU(1,n−1) II

Abstract

We complete the study of the supersingular locus \(\mathcal{M}^{\mathrm{ss}}\) in the fiber at p of a Shimura variety attached to a unitary similitude group GU(1,n−1) over ℚ in the case that p is inert. This was started by the first author in Can. J. Math. 62, 668–720 (2010) where complete results were obtained for n=2,3. The supersingular locus \(\mathcal{M}^{\mathrm{ss}}\) is uniformized by a formal scheme \(\mathcal{N}\) which is a moduli space of so-called unitary p-divisible groups. It depends on the choice of a unitary isocrystal N. We define a stratification of \(\mathcal{N}\) indexed by vertices of the Bruhat-Tits building attached to the reductive group of automorphisms of N. We show that the combinatorial behavior of this stratification is given by the simplicial structure of the building. The closures of the strata (and in particular the irreducible components of \(\mathcal{N}_{\mathrm{red}}\)) are identified with (generalized) Deligne-Lusztig varieties. We show that the Bruhat-Tits stratification is a refinement of the Ekedahl-Oort stratification and also relate the Ekedahl-Oort strata to Deligne-Lusztig varieties. We deduce that \(\mathcal{M}^{\mathrm{ss}}\) is locally a complete intersection, that its irreducible components and each Ekedahl-Oort stratum in every irreducible component is isomorphic to a Deligne-Lusztig variety, and give formulas for the number of irreducible components of every Ekedahl-Oort stratum of \(\mathcal{M}^{\mathrm{ss}}\).