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Chimney retractions in affine buildings encode orbits in affine flag varieties

Elizabeth Milićević, Petra Schwer and Anne Thomas

Vol. 20 (2023), No. 2-3, 395–430

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.

retraction, chimney retraction, Bruhat–Tits buildings, affine buildings, Coxeter group, affine flag variety, double coset intersections
Mathematical Subject Classification
Primary: 20E42
Secondary: 05E45, 14M15, 20G25, 51E24
Received: 1 August 2022
Revised: 16 January 2023
Accepted: 15 February 2023
Published: 13 September 2023
Elizabeth Milićević
Department of Mathematics and Statistics
Haverford College
Haverford, PA
United States
Petra Schwer
Fakultät für Mathematik
Otto-von-Guericke Universität Magdeburg
Anne Thomas
School of Mathematics and Statistics
University of Sydney