  We complete the study of the supersingular locus in the fiber at $p$ of a
Shimura 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 $n =2,3$. The
supersingular locus is uniformized by a formal scheme $\Ncal$ which is a moduli
space of so-called unitary $p$-divisible groups. It depends on the choice of a
unitary isocrystal $\Nbf$. We define a stratification of $\Ncal$ indexed by
vertices of the Bruhat-Tits building attached to the reductive group of
automorphisms of $\Nbf$. 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
$\Ncal$) 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.
