Basic loci in Shimura varieties of Coxeter type

TL;DR

This paper classifies cases where the basic locus in certain Shimura varieties can be described as unions of Deligne-Lusztig sets linked to Coxeter elements, revealing structural properties of Newton and Ekedahl-Oort strata.

Contribution

It provides a classification of basic loci in Shimura varieties of Coxeter type where these loci are unions of classical Deligne-Lusztig sets, extending prior work.

Findings

Basic loci correspond to unions of Deligne-Lusztig sets in classified cases

Newton and Ekedahl-Oort strata exhibit favorable properties in these cases

The classification clarifies the structure of reductions of Shimura varieties

Abstract

This paper is a contribution to the general problem of giving an explicit description of the basic locus in the reduction modulo $p$ of Shimura varieties. Motivated by \cite{Vollaard-Wedhorn} and \cite{Rapoport-Terstiege-Wilson}, we classify the cases where the basic locus is (in a natural way) the union of classical Deligne-Lusztig sets associated to Coxeter elements. We show that if this is satisfied, then the Newton strata and Ekedahl-Oort strata have many nice properties.