Closure relations of Newton strata in Iwahori double cosets

TL;DR

This paper investigates the structure of Newton strata within Iwahori double cosets, revealing that some closures are not unions of strata and linking this to non-equidimensional affine Deligne-Lusztig varieties, with an explicit example for type A4.

Contribution

It demonstrates the existence of Newton strata closures that are not unions of strata and connects this to the geometry of affine Deligne-Lusztig varieties, providing a concrete example.

Findings

Some Newton strata closures cannot be expressed as unions of strata

Non-equidimensional affine Deligne-Lusztig varieties imply complex stratification behavior

Explicit example provided for group of type A4

Abstract

We consider the Newton stratification on Iwahori double cosets for a connected reductive group. We prove the existence of Newton strata whose closures cannot be expressed as a union of strata, and show how this is implied by the existence of non-equidimensional affine Deligne-Lusztig varieties. We also give an explicit example for a group of type $A_4$.