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

We complete the study of the supersingular locus in the fiber at $p$ of a Shimura variety attached to a unitary similitude group $\text{GU}(1,n-1)$ over $\text{QQ}$ in the case that $p$ is inert. This was started by the first author in \cite{Vo_Uni} where complete results were obtained for $n=2,3$. The supersingular locus is uniformized by a formal scheme $\text{Ncal}$ which is a moduli space of so-called unitary $p$-divisible groups. It depends on the choice of a unitary isocrystal $\text{Nbf}$. We define a stratification of $\text{Ncal}$ indexed by vertices of the Bruhat-Tits building attached to the reductive group of automorphisms of $\text{Nbf}$. We show that the combinatorial behaviour of this stratification is given by the simplicial structure of the building. The closures of the strata (and in particular the reduced irreducible components of $\text{Ncal}$) 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 the supersingular locus 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 the supersingular locus.