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Annals of Mathematics Studies Number 168 This page intentionally left blank 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.

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In The Structure of Affine Buildings , Richard Weiss gives a detailed presentation of the complete proof of the classification of Bruhat-Tits buildings first completed by Jacques Tits in 1986. The book includes numerous results about automorphisms, completions, and residues of these buildings. It al
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