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Abstract
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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
and use these to describe the preimages of chimney retractions. When
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.
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Keywords
retraction, chimney retraction, Bruhat–Tits buildings,
affine buildings, Coxeter group, affine flag variety,
double coset intersections
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Mathematical Subject Classification
Primary: 20E42
Secondary: 05E45, 14M15, 20G25, 51E24
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Milestones
Received: 1 August 2022
Revised: 16 January 2023
Accepted: 15 February 2023
Published: 13 September 2023
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