The basic locus of ramified unitary Shimura varieties of signature $(n-1,1)$ at maximal vertex level

TL;DR

This paper constructs and analyzes the stratification of ramified unitary Rapoport--Zink spaces, proving their geometric properties and linking them explicitly to Deligne--Lusztig varieties.

Contribution

It develops the local model theory for Bruhat--Tits strata, proving normality and Cohen--Macaulayness, and establishes an explicit isomorphism with Deligne--Lusztig varieties.

Findings

Proved normality and Cohen--Macaulayness of Bruhat--Tits strata.

Derived explicit dimension formulas for the strata.

Established a scheme-theoretical isomorphism with Deligne--Lusztig varieties.

Abstract

We construct the Bruhat--Tits stratification of the reduced locus of the ramified unitary Rapoport--Zink space of signature $(n-1,1)$, with the level being the stabilizer of a vertex lattice. We develop the local model theory for Bruhat--Tits strata, proving their normality and Cohen--Macaulayness, and provide precise dimension formulas. Additionally, we establish an explicit scheme-theoretical isomorphism between Bruhat--Tits strata and Deligne--Lusztig varieties.