Cohomology of the basic unramified PEL unitary Rapoport-Zink space of signature $(1,n-1)$

TL;DR

This paper investigates the cohomology of a specific unitary Rapoport-Zink space, using spectral sequences and Bruhat-Tits stratification, revealing properties like semisimplicity and connecting to Shimura varieties and automorphic representations.

Contribution

It introduces a spectral sequence approach to analyze the cohomology of unramified PEL Rapoport-Zink spaces of signature (1,n-1), establishing semisimplicity and describing cohomology in special cases.

Findings

Proves semisimplicity of Frobenius action on cohomology.

Shows non-admissibility of the cohomology in general.

Provides explicit cohomology descriptions for n=3,4.

Abstract

In this paper, we study the cohomology of the unitary unramified PEL Rapoport-Zink space of signature $(1,n-1)$ at maximal level. Our method revolves around the spectral sequence associated to the open cover by the analytical tubes of the closed Bruhat-Tits strata in the special fiber, which were constructed by Vollaard and Wedhorn. The cohomology of these strata, which are isomorphic to generalized Deligne-Lusztig varieties, has been computed in a previous paper. This spectral sequence allows us to prove the semisimplicity of the Frobenius action and the non-admissibility of the cohomology in general. Via $p$-adic uniformization, we relate the cohomology of the Rapoport-Zink space to the cohomology of the supersingular locus of a Shimura variety with no level at $p$. In the case $n=3$ or $4$, we give a complete description of the cohomology of the supersingular locus in terms of automorphic representations.