Chimney retractions in affine buildings encode orbits in affine flag varieties

TL;DR

This paper explores the connection between the geometry of retractions in affine buildings and the combinatorics of folded galleries, providing a unified approach to study orbits in affine flag varieties, with applications to groups over local fields.

Contribution

It introduces labeled folded galleries and relates them to chimney retractions, generalizing previous work to a broader class of affine buildings and groups.

Findings

Established a correspondence between folded galleries and orbit structures in affine flag varieties.

Generalized previous results to arbitrary affine buildings and groups with affine Tits systems.

Provided a framework for analyzing double coset intersections using geometric and combinatorial tools.

Abstract

This paper determines the relationship between the geometry of retractions and the combinatorics of folded galleries for arbitrary affine buildings, and so provides a unified framework to study orbits in affine flag varieties. We introduce the notion of labeled folded galleries for any affine building X and use these to describe the preimages of chimney retractions. When X is the building for a group with an affine Tits system, such as the Bruhat-Tits building for a group over a local field, we can then relate labeled folded galleries and shadows to double coset intersections in affine flag varieties. This result generalizes the authors' previous joint work with Naqvi on groups over function fields.