Rapoport-Zink (RZ) spaces are moduli spaces which classify the deformations of a $p$-divisible group with additional structures. It is equipped with compatible actions of $p$-adic and Galois groups, and their cohomology is believed to play a role in the local Langlands program. So far, the cohomology of RZ spaces is entirely known only in the cases of the Lubin-Tate tower and of the Drinfeld space; in particular both of them are RZ spaces of EL type.

In this talk, we consider the basic unramified PEL unitary RZ space with signature $(1,n-1)$ at hyperspecial level. In 2011, Vollaard and Wedhorn proved that its special fiber is stratified by generalized Deligne-Lusztig varieties, whose incidence relations mimic the combinatorics of the Bruhat-Tits building of a unitary group. We compute the cohomology of these strata and we draw some consequences on the cohomology of the RZ space, such as its non admissibility. When $n = 3, 4$ we deduce an automorphic description of the cohomology of the basic locus in the corresponding Shimura variety at hyperspecial level via p-adic uniformization.

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