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Kostant convexity for affine buildings

Petra Hitzelberger1

1Mathematisches Institut der Westfälischen Wilhelms-Universität Münster, Einsteinstrasse 62, 48149 Münster.

Citation Information: Forum Mathematicum. Volume 22, Issue 5, Pages 959–971, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: 10.1515/forum.2010.051, February 2010

Publication History

Received:
2008-03-12
Revised:
2009-03-02
Published Online:
2010-02-12

Abstract

We prove an analogue of Kostant's convexity theorem for thick affine buildings and give an application for groups with affine BN-pairs. Recall that there are two natural retractions of the affine building onto a fixed apartment A: The retraction r centered at an alcove in A and the retraction ρ centered at a chamber in the spherical building at infinity. We prove that for each special vertex xA the set ρ(r –1(W .x)) is a certain convex hull of W .x. The proof can be reduced to a statement about Coxeter complexes and heavily relies on a character formula for highest weight representations of algebraic groups.

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[1]
Petra Hitzelberger
Advances in Mathematics, 2011, Volume 227, Number 1, Page 210

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