A gallery model for affine flag varieties via chimney retractions. (English) Zbl 07903040
Summary: 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.
MSC:
| 20E42 | Groups with a \(BN\)-pair; buildings |
| 05E10 | Combinatorial aspects of representation theory |
| 05E45 | Combinatorial aspects of simplicial complexes |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 20G25 | Linear algebraic groups over local fields and their integers |
| 51E24 | Buildings and the geometry of diagrams |
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