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We complete the study of the supersingular locus in the fiber at $p$ of aShimura variety attached to a unitary similitude group $\GU(1,n-1)$ over $\QQ$in the case that $p$ is inert. This was started by the first author in\cite{Vo_Uni} where complete results were obtained for $n =2,3$. Thesupersingular locus is uniformized by a formal scheme $\Ncal$ which is a modulispace of so-called unitary $p$-divisible groups. It depends on the choice of aunitary isocrystal $\Nbf$. We define a stratification of $\Ncal$ indexed byvertices of the Bruhat-Tits building attached to the reductive group ofautomorphisms of $\Nbf$. We show that the combinatorial behaviour of thisstratification is given by the simplicial structure of the building. Theclosures of the strata (and in particular the reduced irreducible components of$\Ncal$) are identified with (generalized) Deligne-Lusztig varieties. We showthat the Bruhat-Tits stratification is a refinement of the Ekedahl-Oortstratification and also relate the Ekedahl-Oort strata to Deligne-Lusztigvarieties. We deduce that the supersingular locus is locally a completeintersection, that its irreducible components and each Ekedahl-Oort stratum inevery irreducible component is isomorphic to a Deligne-Lusztig variety, andgive formulas for the number of irreducible components of every Ekedahl-Oortstratum of the supersingular locus.

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