The structure of affine buildings

381 Pages·2008·2.05 MB·English

by  Richard M. Weiss

Annals of Mathematics Studies
Number 168
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The Structure of Affine Buildings
Richard M. Weiss
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
2009
Copyright (cid:1)c 2009 by Princeton University Press
PublishedbyPrincetonUniversityPress,41WilliamStreet,Princeton,New
Jersey 08540
IntheUnitedKingdom: PrincetonUniversityPress,6OxfordStreet,Wood-
stock, Oxfordshire 0X20 1TW
All Rights Reserved
Library of Congress Cataloging-in-PublicationData
Weiss, Richard M. (Richard Mark), 1946–
The structure of affine buildings / Richard M. Weiss.
p. cm.
Includes bibliographical references and index.
ISBN: 978-0-691-13659-2(cloth : acid-free paper)
ISBN: 978-0-691-13881-7(paper : acid-free paper)
1. Buildings (Group theory) 2. Moufang loops. 3. Automorphisms. 4.
Affine algebraic groups. I. Title.
QA174.2 .W454 2008
512(cid:1).2–dc22 2008062106
British Library Cataloging-in-PublicationData is available
This book has been composed in LATEX
The publisher would like to acknowledge the author of this volume for pro-
viding the camera-readycopy from which this book was printed.
Printed on acid-free paper. ∞
press.princeton.edu
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
Contents
Preface vii
Chapter1. Affine Coxeter Diagrams 1
Chapter2. Root Systems 13
Chapter3. Root Data with Valuation 25
Chapter4. Sectors 39
Chapter5. Faces 45
Chapter6. Gems 53
Chapter7. Affine Buildings 59
Chapter8. The Building at Infinity 67
Chapter9. Trees with Valuation 77
Chapter10. Wall Trees 89
Chapter11. Panel Trees 101
Chapter12. Tree-Preserving Isomorphisms 107
Chapter13. The Moufang Property at Infinity 119
Chapter14. Existence 131
Chapter15. Partial Valuations 147
Chapter16. Bruhat-Tits Theory 159
Chapter17. Completions 167
vi CONTENTS
Chapter18. Automorphisms and Residues 175
Chapter19. Quadranglesof QuadraticForm Type 189
Chapter20. Quadranglesof Indifferent Type 205
Chapter21. Quadranglesof Type E6,E7 and E8 209
Chapter22. Quadranglesof Type F4 221
Chapter23. Quadranglesof Involutory Type 229
Chapter24. Pseudo-QuadraticQuadrangles 239
Chapter25. Hexagons 261
Chapter26. Assorted Conclusions 275
Chapter27. Summaryof the Classification 289
Chapter28. Locally Finite Bruhat-TitsBuildings 297
Chapter29. AppendixA 321
Chapter30. AppendixB 343
Bibliography 361
Index 365
Preface
The main goal of this book is to present the complete proof of the clas-
sification of Bruhat-Tits buildings. By Bruhat-Tits building we mean an
affine building whose building at infinity satisfies the Moufang property.1
The proof we give is distilled from the contents of the first of the series of
fundamental articles [6]–[10] by Jacques Tits and Fran¸cois Bruhat and the
article [35] Tits wrote for the proceedings of a conference held in Como in
1984. A secondary goalof this book is to provide the reader with a detailed
approach to this remarkable literature.
Bruhat-Tits buildings arise in the study of algebraic groups over a field
with a discrete valuation.2 More precisely, there is a unique Bruhat-Tits
building ∆ in our sense of the term associatedto each pair (G,F), where F
isafieldcompletewithrespecttoadiscretevaluationandGisanabsolutely
simple algebraicgroupof F-rank (cid:1) at least 2.3 The classificationof Bruhat-
Tits buildings says,essentially,that all Bruhat-Tits buildings arise either in
this way or by some small variation on this theme.
The apartments of an affine building ∆ are Euclidean spaces of a fixed
dimension (cid:1) tiled by the fundamental domains of the action of an affine
Coxeter group. These fundamental domains are the chambers of ∆. The
building ∆ itself can be thought of as a set of apartments amalgamated
accordingto certainrulessothat, inparticular,everychamberis achamber
in many apartments.
Attached to each affine building is a “building at infinity” that is related
to the celestial sphere of a Euclidean space. The building at infinity is a
spherical building whose rank is the dimension (cid:1) of the apartments. As
indicated above, we say that an affine building is Bruhat-Tits if (cid:1) ≥ 2 and
itsbuildingatinfinitysatisfiestheMoufangproperty. TheMoufangproperty
means, roughly, that the automorphism group of the building at infinity is
rich enough to contain a special system of subgroups called a root datum.
The notion of a root datum is fundamental in the theory of buildings. In
particular, a spherical building satisfying the Moufang property is uniquely
1Thismeans,inparticular,thattherank(cid:1)ofthebuildingatinfinityisatleast2. All
buildingsareassumedtobethickandirreducibleinthefollowingdiscussion.
2The basic reference for the connection between algebraic groups defined over a field
withadiscretevaluationandaffinebuildingsisthearticle[33]Titswrotefortheproceed-
ingsofaconferenceheldinCorvallisin1977.
3In the literature, the term “Bruhat-Tits building” is often used to denote only the
affinebuildingsarisingfromsuchpairs(G,F)butincludingthosewhereGisofF-rank1.
viii PREFACE
determined by its root datum.
(For the affine building associated to a field F complete with respect to
a discrete valuation and an absolutely simple group G of positive F-rank (cid:1),
the building at infinity is just the spherical building associated with G and
F. This building always satisfies the Moufang property (assuming (cid:1) ≥ 2),
its apartments correspondto the maximal F-split tori of G and its root da-
tum consists,essentially,ofthe rootgroupsassociatedwith afixed maximal
F-split torus.)
Allsphericalbuildingsofrank(cid:1)≥3aswellasalltheirirreducibleresidues
ofrankatleast2satisfythe Moufangproperty.4 Thus ifthe apartmentsare
assumedtohavedimension(cid:1)atleast3,thenanaffinebuildingandaBruhat-
Tits building are the same thing.
The description of spherical buildings in terms of root data turns out
to be exactly the point of view needed to carry out the classification of
Bruhat-Tits buildings. In The Structure of Spherical Buildings [37] we gave
an introduction to the theory of spherical buildings including the proof of
Theorem 4.1.2 in [32] and its most important consequences. The present
monograph is intended as a sequel to [37].
Weassumethatthereaderhassomefamiliaritywiththebasicfactsabout
Coxetergroups,buildingsandtheMoufangpropertycontainedin[37].5 The
most important of these facts are summarized here in Appendix A (Chap-
ter 29). In Chapter 2 we assume some standard facts about root systems
which can be found in [3] or [17]. In Appendix B (Chapter 30), we sum-
marize the classification of Moufang spherical buildings of rank (cid:1) ≥ 2 (as
carried out in [32] and [36]) in terms of root data. This summary plays a
central role starting in Chapter 16.
Here is a brief overview of the organization of this monograph. In Chap-
ters 1–2 and 4–6, the basic properties and substructures of the apartments
ofanaffinebuildingarepresented. We introduceaffinebuildingsthemselves
and construct the building at infinity in Chapters 7–8. In Chapter 9 we
pause to examine the case of an affine building of rank 1.
An affine building of rank 1 is simply a tree without vertices of valency 1
or2togetherwithasetofsubgraphsofvalency2. Thesearetheapartments,
which are to be thought of as Euclidean spaces of dimension 1 tiled by the
intervals between successive integers. One purpose of Chapter 9 is to prove
an uncanny lemma (9.24) that draws a connection between trees and fields
4Thisisacorollaryofthefundamental resultTheorem 4.1.2of[32]. In[36],spherical
buildings of rank (cid:1) = 2, also known as generalized polygons, assumed to satisfy the
Moufang condition (hence: Moufang polygons) were classified. In Chapter 40 of [36]
the classification of Moufang polygons is used to give a revised proof of Tits’s famous
classification [32]of spherical buildings of rank (cid:1)≥3. Once itis established that such a
buildingaswellasallitsirreduciblerank2residuessatisfytheMoufangproperty(thisis
11.6and11.8in[37]),itremainsonlytoexaminehowtherootdataofMoufangpolygons
(i.e.rootdataofrank2)canbeassembledtoformtherootdatumofasphericalbuilding
ofhigherrank.
5TheessentialfactsneededareallcontainedinChapters1–3,Chapters7–9andChap-
ter11of[37].
PREFACE ix
with a discrete valuation. This is the first hint of the subtle connection
between Euclidean geometry and number theory that pervades the theory
of affine buildings.6
In Chapters 10 and 11 we return to the study of an affine building ∆ of
arbitrary rank and construct two whole forests of trees. These forests are
uniquelydeterminedbyafamilyoffunctionsdefinedoncertainsubstructures
of the building at infinity. In Chapter 12 we show that the affine building
∆ is uniquely determined by its building at infinity together with this “tree
structure.”
Atthispointweinvoke(forthefirsttime)theassumptionthatthebuilding
at infinity is Moufang. This means that the building at infinity is uniquely
determinedbyitsrootdatum. WeshowinChapter13howthetreestructure
of ∆ is uniquely determined by something called a valuation of this root
datum (a notionpreviouslyintroducedin Chapter 3), andinChapter 14 we
show that every root datum with valuation arises from an affine building
in this manner. This yields the conclusion that Bruhat-Tits buildings are
classified by root data with valuation.7
It is here that we invokethe classificationof sphericalbuildings satisfying
the Moufang condition. This means that we know all possible root data.
Each root datum, with some mild exceptions, is determined by algebraic
data defined over a field K.8 We show that a valuation of a given root
datum is uniquely determined by a discrete valuation of this field K (in the
number-theoretical sense of the term). We are thus left with the problem
of determining for eachrootdatum necessary and sufficient conditions for a
discrete valuationofthe field K to extend to a valuationof the rootdatum.
Thisproblemisreducedtoaprobleminrank2inChapters15–16andsolved
for each family of Moufang polygons in Chapters 19–25.
The building at infinity of a Bruhat-Tits building depends not only on
the building but also on the choice of a “system of apartments.” It is thus
necessarytodeterminewhentworootdatawithvaluationcorrespondtothe
same affine building but with perhaps two different systems of apartments.
We do this in Chapters 17–26 by studying “completions.” A good part of
Chapters 17–26 is also devoted to determining the structure of the residues
of a Bruhat-Tits building.
In Chapter 27 we summarize the conclusions of the classification in the
formof a listof the different families ofBruhat-Tits buildings together with
references to a few of their principal features. In Chapter 28 we describe
in more detail the classification of locally finite Bruhat-Tits buildings and
correlate the results with the tables in [33].
6Euclid himself would have taken great delight in this connection between his two
favoritetopics!
7See14.54fortheexact statementofthisresult.
8More accurately, K is, according to the case in question, a field, a skew field or an
octonion division algebra. For those spherical buildings arising from the F-points of an
absolutely simplealgebraic group (and are thus defined over F in the sense of algebraic
groups),F istheeitherthecenterofK ortheintersectionofthecenterofK withtheset
offixedpointsofaninvolutionofK.