Supersingular Kottwitz-Rapoport strata and Deligne-Lusztig varieties

TL;DR

This paper explores the structure of supersingular strata within Siegel modular varieties with Iwahori level structure, revealing their decomposition into Deligne-Lusztig varieties and providing explicit descriptions in terms of abelian varieties.

Contribution

It provides a clear description of supersingular Kottwitz-Rapoport strata as unions of Deligne-Lusztig varieties and offers an explicit numerical characterization related to abelian varieties.

Findings

KR strata in supersingular locus are unions of Deligne-Lusztig varieties

Explicit numerical description of KR stratification in terms of abelian varieties

Clarification of the relationship between Newton and Kottwitz-Rapoport stratifications

Abstract

We investigate Siegel modular varieties in positive characteristic with Iwahori level structure. On these spaces, we have the Newton stratification, and the Kottwitz-Rapoport stratification; one would like to understand how these stratifications are related to each other. We give a simple description of all KR strata which are entirely contained in the supersingular locus as disjoint unions of Deligne-Lusztig varieties. We also give an explicit numerical description of the KR stratification in terms of abelian varieties.