A gallery model for affine flag varieties via chimney retractions

Elizabeth Milićević, Yusra Naqvi, Petra Schwer and Anne Thomas

Abstract

This paper provides a unified combinatorial framework to study orbits in 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, we relate these retractions and their effect on certain minimal galleries to double coset intersections in the corresponding affine flag variety.