$p$-kernels occurring in an isogeny class of $p$-divisible groups

TL;DR

This paper provides a combinatorial criterion to identify p-kernels in isogeny classes of p-divisible groups and applies it to determine the non-emptiness of certain affine Deligne-Lusztig varieties for the general linear group.

Contribution

It introduces a new combinatorial criterion based on root systems to analyze p-kernels in isogeny classes of p-divisible groups and applies this to affine Deligne-Lusztig varieties.

Findings

Criterion for p-kernels in isogeny classes established

Non-emptiness conditions for affine Deligne-Lusztig varieties derived

Connection between root system combinatorics and geometric properties

Abstract

We give a criterion which allows to determine, in terms of the combinatorics of the root system of the general linear group, which p-kernels occur in an isogeny class of p-divisible groups over an algebraically closed field of positive characteristic. As an application we obtain a criterion for the non-emptiness of certain affine Deligne-Lusztig varieties associated to the general linear group.