Foliations in deformation spaces of local G-shtukas

TL;DR

This paper extends the theory of local G-shtukas by showing their isogeny to slope divisible forms after base change, and establishes a product structure on Newton strata, impacting the understanding of affine Deligne-Lusztig varieties.

Contribution

It generalizes results from p-divisible groups to local G-shtukas, proving isogeny to slope divisible forms and describing the structure of Newton strata.

Findings

Local G-shtukas are isogenous to slope divisible ones after base change.

Established a product structure on Newton strata of deformations.

Proved bounds and equidimensionality of affine Deligne-Lusztig varieties.

Abstract

We study local G-shtukas with level structure over a base scheme whose Newton polygons are constant on the base. We show that after a finite base change and after passing to an \'etale covering, such a local G-shtuka is isogenous to a completely slope divisible one, generalizing corresponding results for p-divisible groups by Oort and Zink. As an application we establish a product structure up to finite morphism on the closed Newton stratum of the universal deformation of a local G-shtuka, similarly to Oort's foliations for p-divisible groups and abelian varieties. This also yields bounds on the dimensions of affine Deligne-Lusztig varieties and proves equidimensionality of affine Deligne-Lusztig varieties in the affine Grassmannian.