The basic locus of the unitary Shimura variety with parahoric level structure, and special cycles

TL;DR

This paper investigates the structure of the basic locus in a unitary Shimura variety with parahoric level, revealing its uniformization by Rapoport-Zink spaces, the nature of its irreducible components, and properties of special cycles.

Contribution

It characterizes the irreducible components of the basic locus as Deligne-Lusztig varieties and analyzes the intersection behavior of special cycles within the Rapoport-Zink space.

Findings

Irreducible components are Deligne-Lusztig varieties.

Intersection behavior is governed by Bruhat-Tits building.

Defined and studied intersection multiplicities of special cycles.

Abstract

In this paper, we study the basic locus in the fiber at $p$ of a certain unitary Shimura variety with a certain parahoric level structure. The basic locus $\widehat{\mathcal{M}^{ss}}$ is uniformized by a formal scheme $\mathcal{N}$ which is called Rapoport-Zink space. We show that the irreducible components of the induced reduced subscheme $\mathcal{N}_{red}$ of $\mathcal{N}$ are Deligne-Lusztig varieties and their intersection behavior is controlled by a certain Bruhat-Tits building. Also, we define special cycles in $\mathcal{N}$ and study their intersection multiplicities.