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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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Unramified Galois Involutions

Unramified Galois Involutions

Chapter:
(p.275) Chapter Thirty Two Unramified Galois Involutions
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/9780691166902.003.0032

This chapter describes the fixed point building of an automorphism of a Bruhat-Tits building Ξ‎ which induces an unramified Galois involution on the building at infinity Ξ‎. An element of G (for example, a Galois involution of Δ‎) is unramified if the subgroup of G it generates is unramified. Before presenting the main result, the chapter presents the notation stating that Δ‎ = Ξ‎ is the building at infinity of Ξ‎ with respect to its complete system of apartments and G = Aut(Δ‎), followed by definitions. The central theorem shows how an unramified Galois involution of Δ‎ is obtained. Here Γ‎ := τ‎ is a descent group of both Δ‎ and Ξ‎, there is a canonical isomorphism from Δ‎Γ‎ to (Ξ‎Γ‎), where Ξ‎Γ‎ and Ξ‎Γ‎ are the fixed point buildings.

Keywords:   fixed point building, automorphism, Bruhat-Tits building, unramified Galois involution, apartments, descent group, canonical isomorphism

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