Cohomology of the Bruhat-Tits strata in the unramified unitary Rapoport-Zink space of signature $(1,n-1)$

TL;DR

This paper computes the $\,ar{ ext{Q}}_ ext{ ext{ell}}$-adic cohomology of certain Deligne-Lusztig varieties associated with Bruhat-Tits strata in an unramified unitary Rapoport-Zink space, revealing their representation-theoretic structure.

Contribution

It describes the $\,ar{ ext{Q}}_ ext{ ext{ell}}$-adic cohomology of non-classical Deligne-Lusztig varieties linked to Bruhat-Tits strata, connecting geometric stratifications with representation theory.

Findings

Cohomology groups are computed using spectral sequences from Ekedahl-Oort stratification.

Irreducible representations appear in only two unipotent Harish-Chandra series.

One of the series is part of the principal series.

Abstract

In [Inventiones mathematicae, 184 (2011)], Vollaard and Wedhorn defined a stratification on the special fiber of the unitary unramified PEL Rapoport-Zink space with signature $(1,n-1)$. They constructed an isomorphism between the closure of a stratum, called a closed Bruhat-Tits stratum, and a Deligne-Lusztig variety which is not of classical type. In this paper, we describe the $\ell$-adic cohomology groups over $\overline{\mathbb Q_{\ell}}$ of these Deligne-Lusztig varieties, where $\ell \not = p$. The computations involve the spectral sequence associated to the Ekedahl-Oort stratification of a closed Bruhat-Tits stratum, which translates into a stratification by Coxeter varieties whose cohomology is known. Eventually, we find out that the irreducible representations of the finite unitary group which appear inside the cohomology contribute to only two different unipotent Harish-Chandra series, one of them belonging to the principal series.