On the cohomology of the ramified PEL unitary Rapoport-Zink space of signature $(1,n-1)$

TL;DR

This paper investigates the cohomology of ramified PEL unitary Rapoport-Zink spaces of signature (1,n-1), extending previous unramified case methods, and provides explicit descriptions of cohomology in specific low-dimensional cases.

Contribution

It develops a method to analyze the cohomology of ramified Rapoport-Zink spaces using Bruhat-Tits stratification, generalizing prior unramified case techniques, and describes cohomology of supersingular loci for small n.

Findings

Cohomology of certain strata is isomorphic to generalized Deligne-Lusztig varieties.

Unipotent representations contribute to only two cuspidal series.

Complete cohomology descriptions obtained for small n in specific cases.

Abstract

In this paper, we study the cohomology of the ramified PEL unitary Rapoport-Zink space of signature $(1,n-1)$ by using the Bruhat-Tits stratification on its special fiber. As such, we apply the same method that we developped for the unramified case in two previous papers. More precisely, we first investigate the cohomology of a given closed Bruhat-Tits stratum. It is isomorphic to a generalized Deligne-Lusztig variety which is in general not smooth, and is associated to a finite group of symplectic similitudes. We determine the weights of the Frobenius and most of the unipotent representations occuring in its cohomology. This computation involves the spectral sequence associated to a stratification by classical Deligne-Lusztig varieties, which are parabolically induced from Coxeter varieties of smaller symplectic groups. In particular, all the unipotent representations contribute to only two cuspidal series. Then, we introduce the analytical tubes of the closed Bruhat-Tits strata, which give an open cover of the generic fiber of the Rapoport-Zink space. Using the associated \v{C}ech spectral sequence, we prove that certain cohomology groups of the Rapoport-Zink space at hyperspecial level fail to be admissible if $n$ is large enough. Eventually, when $n=2$ in the split case, when $n=3$ and when $n=4$ in the non-split case, we give a complete description of the cohomology of the supersingular locus of the associated Shimura variety at hyperspecial level, in terms of automorphic representations. In particular, certain automorphic representations occur with a multiplicity depending on $p$. Thus, our computations in the ramified case recover all the main features of the unramified case, despite new technical difficulties due to the closed Bruhat-Tits strata not being smooth.