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Descent in Buildings (AM-190)$
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Bernhard Mühlherr, Holger P. Petersson, and Richard M. Weiss

Print publication date: 2015

Print ISBN-13: 9780691166902

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691166902.001.0001

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Forms of Residually Pseudo-Split Buildings

Forms of Residually Pseudo-Split Buildings

Chapter:
(p.297) Chapter Thirty Four Forms of Residually Pseudo-Split Buildings
Source:
Descent in Buildings (AM-190)
Author(s):

Bernhard M¨uhlherr

Holger P. Petersson

Richard M. Weiss

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691166902.003.0034

This chapter deals with forms of residually pseudo-split buildings. The proof rests on the fact that in every case, there is a Galois action of Γ‎ := GalL/K on Δ‎L whose fixed point building is isomorphic to Δ‎. A Tits index = (Π‎, Θ‎, A) is displayed by drawing the Coxeter diagram, bending edges where necessary so that vertices in the same Θ‎-orbit are conspicuously near to each other, and putting a circle around the set of vertices in each orbit of Θ‎ disjoint from A. The chapter presents the main result showing that every exceptional Bruhat-Tits building of rank at least 3 but not of type G˜2 with Tilde₂ is the fixed point building of an unramified group of order 2 or 4 acting on a residually pseudo-split building.

Keywords:   residually pseudo-split building, fixed point building, Tits index, Coxeter diagram, Bruhat-Tits building

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