A gallery model for affine flag varieties via chimney retractions

TL;DR

This paper introduces a new combinatorial framework using chimney retractions to analyze orbits in affine flag varieties via Bruhat-Tits buildings, unifying existing retraction concepts and providing recursive computational tools.

Contribution

It formulates the notion of chimney retractions for arbitrary affine buildings, generalizing known retractions, and links these to double coset intersections in affine flag varieties.

Findings

Defined chimney retractions for affine buildings

Developed recursive formulas for minimal galleries

Connected retractions to double coset intersections

Abstract

This paper provides a unified combinatorial framework to study orbits in certain affine flag varieties via the associated Bruhat-Tits buildings. We first formulate, for arbitrary affine buildings, the notion of a chimney retraction. This simultaneously generalizes the two well-known notions of retractions in affine buildings: retractions from chambers at infinity and retractions from alcoves. We then present a recursive formula for computing the images of certain minimal galleries in the building under chimney retractions, using purely combinatorial tools associated to the underlying affine Weyl group. Finally, for Bruhat-Tits buildings in the function field case, we relate these retractions and their effect on minimal galleries to double coset intersections in the corresponding affine flag variety.