Descent in Buildings (AM-190) 9781400874019

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Annals of Mathematics Studies Number 190

Descent in Buildings

Bernhard M¨uhlherr Holger P. Petersson Richard M. Weiss

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD 2015

Copyright © 2015 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire 0X20 1TW press.princeton.edu All Rights Reserved Library of Congress Cataloging-in-Publication Data M¨ uhlherr, Bernhard Matthias. Descent in buildings / Bernhard M¨ uhlherr, Holger P. Petersson, and Richard M. Weiss. pages cm. – (Annals of mathematics studies ; 190) Includes bibliographical references and index. ISBN 978-0-691-16690-2 (hardcover : alk. paper) – ISBN 978-0-691-16691-9 (pbk. : alk. paper) 1. Buildings (Group theory) 2. Combinatorial geometry. I. Petersson, Holger P., 1939– II. Weiss, Richard M. (Richard Mark), 1946– III. Title. QA174.2.M84 2015 516′ .13–dc23 2015008618 British Library Cataloging-in-Publication Data is available This book has been composed in LATEX Printed on acid-free paper ∞ The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed. Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

For Jacques

vi

“I refer to such cases as that of the art of enumeration, which, as Mr. Tylor clearly shews . . . , originated in counting the fingers, first of one hand and then the other, and lastly the toes.” Charles Darwin, The Descent of Man

Contents

Preface

xi

PART 1. MOUFANG QUADRANGLES

1

Chapter 1. Buildings

3

Chapter 2. Quadratic Forms

13

Chapter 3. Moufang Polygons

23

Chapter 4. Moufang Quadrangles

31

Chapter 5. Linked Tori, I

41

Chapter 6. Linked Tori, II

47

Chapter 7. Quadratic Forms over a Local Field

57

Chapter 8. Quadratic Forms of Type E6 , E7 and E8

69

Chapter 9. Quadratic Forms of Type F4

79

PART 2. RESIDUES IN BRUHAT-TITS BUILDINGS

83

Chapter 10. Residues

85

Chapter 11. Unramified Quadrangles of Type E6 , E7 and E8

91

Chapter 12. Semi-ramified Quadrangles of Type E6 , E7 and E8

93

Chapter 13. Ramified Quadrangles of Type E6 , E7 and E8

101

Chapter 14. Quadrangles of Type E6 , E7 and E8 : Summary

109

Chapter 15. Totally Wild Quadratic Forms of Type E7

115

viii

CONTENTS

Chapter 16. Existence

119

Chapter 17. Quadrangles of Type F4

129

Chapter 18. The Other Bruhat-Tits Buildings

137

PART 3.

141

DESCENT

Chapter 19. Coxeter Groups

143

Chapter 20. Tits Indices

153

Chapter 21. Parallel Residues

165

Chapter 22. Fixed Point Buildings

181

Chapter 23. Subbuildings

195

Chapter 24. Moufang Structures

205

Chapter 25. Fixed Apartments

217

Chapter 26. The Standard Metric

221

Chapter 27. Affine Fixed Point Buildings

233

PART 4.

241

GALOIS INVOLUTIONS

Chapter 28. Pseudo-Split Buildings

243

Chapter 29. Linear Automorphisms

251

Chapter 30. Strictly Semi-linear Automorphisms

259

Chapter 31. Galois Involutions

271

Chapter 32. Unramified Galois Involutions

275

PART 5.

285

EXCEPTIONAL TITS INDICES

Chapter 33. Residually Pseudo-Split Buildings

287

Chapter 34. Forms of Residually Pseudo-Split Buildings

297

CONTENTS

ix

Chapter 35. Orthogonal Buildings

303

Chapter 36. Indices for the Exceptional Bruhat-Tits Buildings

309

Bibliography

327

Index

333

Preface

Background: Groups of the form G(F ), where F is a field, G is an absolutely simple algebraic group isotropic over F and G(F ) denotes the group of F -rational points of G, include all the finite simple groups of Lie type (apart from the Ree and Suzuki groups) when the field F is finite as well as all simple non-compact Lie groups when F is R or C. For algebraically closed fields F , these groups are classified by Dynkin diagrams. For arbitrary fields F , they are classified starting over the separable closure of F by means of Galois descent. When G is exceptional, this classification reveals connections to octonion algebras, Jordan algebras and other, more exotic, algebraic structures. Writing about Armand Borel in [2], Jacques Tits said the following: The origin and circumstances of this collaboration are perhaps instructive: both Borel and I had already worked and published on the subject, which we approached with quite different backgrounds and aims; he was mainly influenced by Lie theory and algebraic geometry, and I by “synthetic” and projective geometry; he was primarily thinking in terms of tori and root systems, and I in terms of parabolic subgroups. The product of Tits’ combinatorial way of thinking was his theory of buildings. A building, roughly speaking, is a combinatorial/geometric structure modeled on the configuration of parabolic subgroups in an isotropic absolutely simple algebraic group. The theory of buildings provides a way of studying these algebraic groups in which the underpinnings from algebraic geometry are replaced by combinatorial notions. At the heart of Tits’ celebrated classification of spherical buildings of rank at least 3 in [55] is the result (Theorem 4.1.2) which says that a spherical building ∆ is uniquely determined by its local structure, by which is meant, roughly, the collection of rank 2 residues containing a given chamber. This result has as a consequence that the building ∆ satisfies a symmetry condition Tits called the Moufang property. This is equivalent to saying that ∆ possesses a root group datum as defined in [7, 6.1.1]. The conclusion of Tits’ classification is that apart from a few exceptions, the only thick irreducible spherical buildings of rank ℓ ≥ 3 are, in fact, the

xii

PREFACE

spherical buildings associated to the group of F -rational points of an absolutely simple group of F -rank ℓ. (The exact statement can be found in [65, 30.32].) Among the exceptions, the most interesting are the “mixed” buildings of type F4 . These buildings are defined over an inseparable extension K/F in characteristic 2 such that K 2 ⊂ F . Tits’ original classification was extended in [60] to include the rank 2 case under the assumption that the building ∆ is Moufang. This classification unearthed one previously unknown family of buildings, now usually called the quadrangles of type F4 . These generalized quadrangles are not associated with absolutely simple algebraic groups. It was shown in [39], however, that they can be found inside the “inseparable” buildings of type F4 mentioned in the previous paragraph by a process analogous to Galois descent. Tits’ classification of affine buildings Ξ of rank ℓ + 1 for ℓ ≥ 3 (in [57]) which relies heavily on his earlier work with Bruhat in [7], starts with the construction of the building at infinity ∆ = Ξ∞ of Ξ, a thick irreducible spherical building of rank ℓ. Being spherical and of rank at least 3, the building ∆ is Moufang and hence uniquely determined by a root group datum defined over a field F . The classification says then, roughly, (1) that the spherical buildings that arise in this way are those for which F is complete with respect to a discrete valuation and (2) that for each spherical building ∆ defined over such a field, there is a unique affine building Ξ whose building at infinity is ∆. This work was subsequently extended (in [65]) to the case that ℓ = 2 under the assumption that the building at infinity, whose rank is 2 in this case, satisfies the Moufang condition. In stark contrast to the spherical case, affine buildings are not uniquely determined by their local structure, just as a field complete with respect to a discrete valuation is not uniquely determined by its residue field. Nevertheless, the residues of affine buildings are, of course, a fundamental structural feature of these buildings. Goals: By a Bruhat-Tits building we mean a thick, irreducible affine building of rank ℓ + 1 whose building at infinity, a thick, irreducible spherical building of rank ℓ, satisfies the Moufang property as defined in 1.25. Thus a Bruhat-Tits building and a thick irreducible affine building are the same thing when ℓ ≥ 3. The original goal of this project was to determine the structure of the residues for all Bruhat-Tits buildings whose building at infinity is an exceptional Moufang quadrangle in order to fill in the last remaining gaps in Table 27.2 on page 291 of [65], where references to results describing the structure of the residues for all other Bruhat-Tits buildings can be found. In Parts 1 and 2 of this monograph we present our solution to this problem. The exceptional Moufang quadrangles are the generalized quadrangles (equivalently, the buildings of type B2 ) “of type E6 , E7 , E8 and F4 ” defined in [60, 16.6–16.7]. By [14, 5.3] and [60, 42.6], a generalized quadrangle of type Eℓ is precisely the spherical building associated to an absolutely

xiii

PREFACE

simple algebraic group whose index over a field F is • •

• •

•

•

for ℓ = 6, • •

•

•

•

•

•

for ℓ = 7 and • •

•

•

•

• ′

•

•

16 31 66 for ℓ = 8. These are the indices 2E6,2 , E7,2 and E8,2 in the notation of [53]. As mentioned above, the generalized quadrangles of type F4 , on the other hand, are not among the spherical buildings associated with absolutely simple algebraic groups. We make no restriction on the field of definition F in these investigations other than the necessary condition that it be complete with respect to a discrete valuation. In particular, we make no restriction on the characteristic of F . In fact, over half the cases in our classification of the possible configurations of residues occur only in the “wild” case, when the characteristic of the residue field F¯ is 2. In carrying out the classification of the possible configurations of residues, many of the ideas and results in [35] and [36] played an important role in guiding our intuition. Part 3 of this manuscript is the result of our efforts to explain and investigate this role. In it, we elaborate and, to some extent, extend ideas and results in [35] and [36] to develop a coherent “theory of descent” for buildings. Eventually, this theory, which applies to arbitrary buildings, became the centerpiece of this monograph. The split spherical buildings are those that correspond to an F -split simple algebraic group for some field F . In Parts 4 and 5, the notion of a pseudosplit building plays an important role. These are precisely the buildings that occur as subbuildings (in the sense of 23.5) of split spherical buildings. In the simply laced case, there are no pseudo-split buildings that are not split, but in the non-simply laced case, there are examples involving inseparable field extensions in characteristic 2 and 3. The buildings of type F4 defined over an inseparable extension K/F mentioned above are, for example, pseudo-split but not split. (These buildings are also the buildings attached to groups that are pseudo-split in the sense of [11, Definition 2.3.1], but only if at least one of the extensions K/F of F/K 2 is finite.) A building is called residually split or residually pseudo-split if all its proper residues are split or pseudo-split. Our goal in Parts 4 and 5 of these notes is to apply our results on descent to our classification of the possible configurations of residues in an exceptional Bruhat-Tits building. Our main ˜2 result is that every exceptional Bruhat-Tits building of type other than G is the fixed point building of a Galois action on a residually pseudo-split

xiv

PREFACE

Bruhat-Tits building. (This assertion is far from true if we replace “residually pseudo-split” by “residually split.”) Attached to each of these Galois actions is a Tits index. The Tits index consists of the Coxeter diagram of the pseudo-split building Ξ endowed with decorations which encode basic information about how the fixed point building is embedded in Ξ. In the final chapter of this monograph, we give tables of all the Tits indices which occur in these investigations. Our theory of descent gives a way to interpret the tables of indices in Tits’ Boulder and Corvallis notes ([53] and [56]) purely in the language of buildings. With this interpretation, our tables extend the tables in [56], which apply only to the case that the Bruhat-Tits building Ξ is locally finite. Context: This book can be seen as the third volume in a trilogy about Tits’ theory of buildings which begins with [62] and [65]. The first of these three monographs (The Structure of Spherical Buildings) is an introduction to the theory of buildings. It includes a proof that an irreducible spherical building of rank at least 3 is Moufang and is a kind of prequel to Part V of [60], where the classification of Moufang polygons (which had been carried out in Parts I–III of [60]) was used to give a new proof of Tits’ classification of irreducible Moufang spherical buildings. (The original proof of this result in [55] used a different strategy.) The second of these monographs (The Structure of Affine Buildings) contains a detailed proof of Tits’ classification of Bruhat-Tits buildings including those of rank 3. The basic strategy in this proof is to produce the building at infinity, a Moufang spherical building, and then to examine the extent to which the building at infinity determines the affine building uniquely via the notion of a valuation of a root datum. In both of these volumes, the primary focus is on root groups and commutator relations. The classification of semi-simple algebraic groups goes along different lines. Over an algebraically closed field, these groups are classified, up to isogeny, by Dynkin diagrams. The classification of semi-simple algebraic groups over arbitrary fields is then deduced from this result through the use of Galois descent; see, in particular, [53], [56] and [60, Chapter 42]. These results imply that Moufang and Bruhat-Tits buildings ought to have natural embeddings into residually pseudo-split buildings which can be described in purely combinatorial/geometric terms. The main goal of this third volume in our trilogy is to elaborate a theory of descent in buildings with the purpose of making this observation precise and to illustrate how this theory works when applied to the exceptional Bruhat-Tits buildings. Our theory of descent in buildings grew out of ideas which first appeared in [35] and [36] and were later applied in [39]. See also [9], where some of these ideas are developed and applied in another context. We mention also the monograph [63], a kind of companion to our trilogy, in which a class of algebraic structures related to the exceptional Moufang quadrangles is introduced and the classification of these algebraic structures is given.

PREFACE

xv

Organization: This monograph has two beginnings, one at the start of Part 1 and the other at the start of Part 3. Our goal in Parts 1 and 2 is to investigate the residues of Bruhat-Tits buildings. In Part 1 we give the necessary background for these investigations. We prove results about quadratic forms over fields complete with respect to a discrete valuation (in arbitrary characteristic) in Chapters 2 and 7, about Moufang quadrangles in Chapters 3 and 4, about invariants of Moufang sets (i.e. buildings of rank 1, roughly speaking) in Chapters 5 and 6 and about the quadratic forms which give rise to the exceptional Moufang quadrangles in Chapters 8 and 9. In Part 2 we carry out the classification of the possible configurations of residues in a Bruhat-Tits building whose building at infinity is an exceptional Moufang quadrangle. Our results in the case of quadrangles of type E6 , E7 and E8 are summarized in Chapter 14 and in this same chapter, the question of existence for all the different cases is reduced to a series of questions about quadratic forms. These questions are answered in Chapter 16, the case of quadrangles of type F4 is treated in Chapter 17 and the results in [65] about the residues in the remaining Bruhat-Tits buildings are reviewed in Chapter 18. In Part 3 we make a fresh start. Most readers, we expect, will want to begin at this point. Let ∆ be an arbitrary building and Γ an arbitrary subgroup of Aut(∆). The main result of Part 3 is 22.20 in which we prove that under minimal (and clearly necessary) conditions given in 22.19, the set ∆Γ of residues of ∆ stabilized by Γ form a thick building. We call a group Γ satisfying our conditions a descent group of ∆, and we call the building ∆Γ stabilized by a descent group Γ a form of ∆. The notion of a form of a building ∆ is thus the combinatorial analog of the notion of a form of an algebraic group. At the center of our theory of descent is the notion of a Tits index, a combinatorial version of the notion of an index in the relative theory of algebraic groups. In Part 4 we combine the results of Part 3 with the classification of Moufang spherical buildings. We introduce the notion of a pseudo-split spherical building in Chapter 28. In Chapter 29, we define the field of definition of an arbitrary Moufang spherical building ∆ and introduce the notion of a semi-linear automorphism of ∆. In Chapter 32, we show that a strictly semi-linear group of automorphisms Γ of order 2 of a pseudo-split spherical building ∆ is always a descent group of ∆, and we extend this result to a result about Bruhat-Tits buildings under the additional assumption that Γ is unramified in an appropriate sense. In Part 5 we combine the results of Part 2 and Chapter 32 to conclude that every exceptional Bruhat-Tits building is a form of a residually pseudo-split ˜ 2 we do this only modulo a conjecture.) Bruhat-Tits building. (In the case G In the final chapter, we display in a series of tables all the Tits indices which arise in this context.

xvi

PREFACE

Acknowledgments: We would like to express our gratitude to PierreEmmanuel Caprace for his advice and assistance with certain matters in this monograph. The first and third authors were partially supported in this project by DFG Grant 62201483. The third author was also partially supported by NSA Grant H98230-12-1-0230 and by the DFG through a Mercator Guest Professorship for the Winter Semester 2012-13 at the University of Giessen. The third author would also like to thank the Korea Institute of Advanced Study for its support and hospitality.

PART 1

Moufang Quadrangles

Chapter One Buildings We use this chapter to assemble a few standard definitions, fix some notation and review a few of the results about buildings and Moufang polygons which will be used most frequently in these notes. A summary of the basic facts about Coxeter groups and buildings with which we expect the reader to have some familiarity can be found, with references to proofs, in [65, 29.1-29.15]. These include the basic properties of roots, residues, apartments and projection maps. (We emphasize, however, that although we assume some familiarity with this background material, we have made every effort throughout these notes to include explicit references to the results in [60], [62], [65] and elsewhere each time they are applied.) When we refer to the type of a building ∆, we mean either the corresponding Coxeter diagram or, equivalently, the corresponding Coxeter system (W, S); see 19.2. The cardinality |S|, which we always assume to be finite, is called the rank of ∆. More generally, the rank of a J-residue of ∆ is |J| for each subset J of S. Root groups and the Moufang condition play a central role in this monograph. A root of a building is a root of one of its apartments. For a given root α of a building ∆, the corresponding root group Uα is the subgroup of Aut(∆) consisting of all elements that act trivially on each panel containing two chambers of α. Definition 1.1. As in [62, 11.7], we say that a building ∆ is Moufang (or satisfies the Moufang condition) if (i) it is thick, irreducible and spherical as defined in [62, 1.6 and 7.10]; (ii) its rank is at least 2; and (iii) for every root α, the root group Uα acts transitively on the set of all apartments containing α. We emphasize that if we say that a building is Moufang, we are implying that it is spherical, thick, irreducible and of rank at least 2. Nevertheless, when we say that a building is Moufang, we will sometimes also say explicitly that the building is spherical just to avoid any possible confusion. (In Chapter 24 we introduce the more general notion of a Moufang structure on a spherical building. See also [1, 8.3] and [44, Chapter 6, §4, and Chapter 11, §7] for other notions of a Moufang building. These other notions will not play any role in these notes.)

4

CHAPTER 1

Definition 1.2. Let ∆ be a building, let R be a residue of ∆ and let c be an arbitrary chamber of ∆. By [62, 8.21], there is a unique chamber z in R nearest c and (1.3)

dist(x, c) = dist(x, z) + dist(z, c)

for every chamber x ∈ R. This unique nearest chamber z is called the projection of c to R and is denoted by projR (c). The projection map projR : ∆ → R will play an important role in these notes starting in Chapter 21. Remark 1.4. A fundamental result of Tits says that an irreducible thick spherical building of rank at least 3 satisfies the Moufang condition as do all the irreducible residues of rank at least 2 of such a building. For a proof, see [62, 11.6 and 11.8]. Moufang sets. A building of type A1 —in other words, a building of rank 1—is only a set of cardinality at least 2 without any further structure, but the buildings of type A1 we will encounter come endowed with a group of permutations having special properties which led to the following definition introduced by Tits in [58]: Definition 1.5. A Moufang set is a pair (X, {Ux | x ∈ X}), where X is a set with |X| ≥ 3 and for each x ∈ X, Ux is a subgroup of Sym(X) (where we compose from right to left) such that the following hold: (i) For each x ∈ X, Ux fixes x and acts sharply transitively on X\{x}. (ii) For all x, y ∈ X and each g ∈ Ux , gUy g −1 = Ug(y) . The groups Ux for x ∈ X are called the root groups of the Moufang set. Definition 1.6. Let M = (X, {Ux | x ∈ X}) be a Moufang set and let G = hUx | x ∈ Xi.

By 1.5(i), the group G acts 2-transitively on X and by 1.5(ii), the root groups are all conjugate to each other in G. Let x, y be distinct elements of X. For each g ∈ Ux∗ := Ux \{1}, there exist a unique element µxy (g) in the double coset Uy gUy that interchanges x and y. Thus µ := µxy is a map from Ux∗ to G which depends on the choice of x and y. By [19, 3.1(ii)], the stabilizer Gxy is generated by the set {µ(g1 )µ(g2 ) | g1 , g2 ∈ Ux }.

Since Ux acts sharply transitively on X\{x}, the subgroup Gxy is isomorphic to the subgroup of Aut(Ux ) it induces. The tori of M are the conjugates in G of the subgroup Gxy . Since G acts 2-transitively on X, the tori are precisely the 2-point stabilizers in G.

5

BUILDINGS

Definition 1.7. Two Moufang sets (X, {Ux | x ∈ X}) and (X ′ , {Ux | x ∈ X ′ })

are isomorphic if there exists a bijection from X to X ′ that transports root groups to root groups (and such a bijection is called an isomorphism). Definition 1.8. Let M = (X, {Ux | x ∈ X}) and M′ = (X ′ , {Ux | x ∈ X ′ }) be two Moufang sets and let x, y be an ordered pair of distinct elements of X. An xy-isomorphism from M to M′ is a bijection ψ from X to X ′ inducing ′ an isomorphism ϕ from Ux to Uψ(x) such that (1.9)

′

′

ϕ(uµ(a)µ(b) ) = ϕ(u)µ (ϕ(a))µ (ϕ(b))

for all u ∈ Ux and all a, b ∈ Ux∗ , where µ = µxy and µ′ = µϕ(x)ϕ(y) are as in 1.6 with respect to M, respectively, M′ . If x1 , y1 is another ordered pair of distinct elements of X, then there is an element g in the group G defined in 1.6 mapping the ordered pair x, y to the ordered pair x1 , y1 and the composition of g with an xy-isomorphism from M to M′ is an x1 y1 -isomorphism from M to M′ . We will say that M and M′ are weakly isomorphic (and write M ≈ M′ ) if there is an xy-isomorphism from M to M′ for some choice of x, y in X (and hence for all choices of x, y in X), and we define a weak isomorphism from M to M′ to be an xy-isomorphism for some choice of x, y in X. The inverse of a weak isomorphism is a weak isomorphism as is the composition of two weak isomorphisms, and every isomorphism of Moufang sets is also a weak isomorphism. Remark 1.10. Let M, M′ , etc., be as in 1.8, let x, y be an ordered pair of distinct elements of X, let x′ , y ′ be an ordered pair of distinct elements of X ′ and suppose that ϕ is an isomorphism from Ux to Ux′ ′ such that (1.9) holds for all u ∈ Ux and all a, b ∈ Ux∗ with µ = µxy and µ′ = µx′ y′ . Then the map ψ from X to X ′ which sends x to x′ and y u to (y ′ )ϕ(u) for all u ∈ Ux is an xy-isomorphism from M to M′ . Notation 1.11. Let M = (X, {Ux | x ∈ X}) be a Moufang set, choose distinct points x, y in X, let µ = µxy be the map from Ux∗ to Aut(M) defined in 1.6, choose a ∈ Ux∗ and let m = µ(a). There exists a unique permutation ρ of Ux∗ such that ρ

y u = xm for all u ∈ Ux∗ . Therefore (1.12)

−1

um

ρ

y u = (y u )m

for all u ∈ Ux∗ since m interchanges x and y. We identify Ux with the set X\{x} via the map u 7→ y u , then we identify ρ with the permutation ρ y u 7→ y u of X\{x, y} and finally we extend ρ to a permutation of X by declaring that it interchanges x and y. Given these identifications, it follows

6

CHAPTER 1

from (1.12) that the permutations m and ρ of X are the same. In particular hUx , ρi = hUx , Uy i. Since this group acts transitively on X, it acts transitively on the set of root groups {Uz | z ∈ X}. It follows that M is uniquely determined by Ux and ρ (although ρ, of course, depends on the choice of a). We can thus set (1.13)

M = M(Ux , ρ).

This is the point of view taken in [17] and [19]. See 3.9 for examples of various families of Moufang sets described in terms of a single root group and a permutation of its non-trivial elements as in 1.11. Moufang polygons and root group sequences. A generalized n-gon (for n ≥ 2) is a building of type • n •

and a generalized polygon is a generalized n-gon for some n. See [62, 7.14 and 7.15] for an equivalent definition in terms of bipartite graphs. The classification of generalized n-gons satisfying the Moufang conditions (i.e. of Moufang polygons) was carried out in [60]. Moufang n-gons exist, in particular, only for n = 3, 4, 6 and 8. The classification says that each Moufang n-gon is uniquely determined by a root group sequence Ω as defined in [60, 8.7], and these root group sequences are, in turn, determined by certain algebraic data and isomorphisms x1 , . . . , xn from this algebra data to the root groups from which Ω is composed according to one of the nine recipes [60, 16.1–16.9]. Notation 1.14. In accordance with [65, 30.8], we will use the following names for the root group sequences obtained by applying the recipes [60, 16.1–16.9]: (i) T (K), where K is a field or a skew field or an octonion division algebra as defined in [60, 9.11]. (ii) QI (Λ), where Λ = (K, K0 , σ) is an involutory set as defined in [60, 11.1]. (iii) QQ (Λ), where Λ = (K, L, q) is a non-trivial anisotropic quadratic space as defined in 2.1 (see 2.14). (iv) QD (Λ), where Λ = (K, K0 , L0 ) is an indifferent set as defined in [60, 10.1]. (v) QP (Λ), where Λ = (K, K0 , σ, L, q) is an anisotropic pseudo-quadratic space as defined in [60, 11.17]. (vi) QE (Λ), where Λ = (K, L, q) is a quadratic space of type E6 , E7 or E8 as defined in 8.1.

BUILDINGS

7

(vii) QF (Λ), where Λ = (K, L, q) is a quadratic space of type F4 as defined in 9.1. (viii) H(Λ), where Λ = (J, F, #) is an hexagonal system as defined in [60, 15.15]. (ix) O(Λ), where Λ = (K, σ) is an octagonal system as defined in [60, 10.11]. Notation 1.15. We will say that a root group sequence is of of type T if it is isomorphic to a root group sequence in case (i) of 1.14, of type QI or of involutory type if it is isomorphic to a root group sequence in case (ii), of type QQ or of quadratic form type if it is isomorphic to a root group sequence in case (iii), etc. Among all the Moufang polygons, the exceptional Moufang quadrangles— those corresponding to a root group sequence of type QE or QF —are the most extraordinary. They will be the focus of our attention in Parts 2 and 5 of this monograph. Let c be a chamber of a Moufang spherical building ∆ and let E2 (c) denote the subgraph spanned by all the irreducible rank 2 residues of ∆. Another fundamental result of Tits ([62, 10.16]) says that ∆ is uniquely determined by E2 (c). The irreducible rank 2 residues containing c, which are in one-to-one correspondence with the edges of the Coxeter diagram of ∆, are Moufang polygons. Thus each of these residues is determined by a root group sequence. This leads to the notion of a root group labeling of the Coxeter diagram Π. In a root group labeling, the edges of Π are decorated with root group sequences and the vertices with isomorphisms identifying certain root groups of the root group sequences decorating the different adjacent edges. A description of the results of Tits’ classification of Moufang spherical buildings in terms of root group labelings is given in [65, 30.14]. In these notes we will apply the corresponding notation for these buildings as given in [65, 30.15]. Thus, in particular: Remark 1.16. In the notion in [65, 30.15], the Moufang quadrangles corresponding to the first eight cases of 1.14 are, in order, called: A2 (K), BI2 (Λ) Q D D P P E or CI2 (Λ), BQ 2 (Λ) or C2 (Λ), B2 (Λ) or C2 (Λ), B2 (Λ) or C2 (Λ), B2 (Λ) or E F F C2 (Λ), B2 (Λ) or C2 (Λ), and G2 (Λ). Remark 1.17. Let Ω′ be a subsequence of a root group sequence Ω as defined in [60, 8.17]. By [60, 7.4 and 8.1], the generalized polygon associated with Ω′ is a subbuilding of the generalized polygon associated with Ω. Suppose, for example, that Λ′ = (F, A, B) is an indifferent set. Then Λ := (F, F, F ) is an indifferent set containing Λ′ canonically as a “sub”indifferent set and by [60, 8.12] and the formulas in [60, 32.8], QD (Λ′ ) is a ′ subsequence of the root group sequence QD (Λ). Hence BD 2 (Λ ) is a subbuildD ing of B2 (Λ). As a second example, let Λ be the involutory set (E, F, σ), where E/F is a separable quadratic extension and σ is the non-trivial element of Gal(E/F ). Then Λ′ := (F, F, idF ) is canonically a “sub”-involutory

8

CHAPTER 1

set of Λ and by the formulas in [60, 32.6 and 32.9] QI (Λ′ ) is a subsequence of the root group sequence QI (Λ). Hence BI2 (Λ′ ) is a subbuilding of BI2 (Λ). It follows from [65, 30.16] that BIℓ (Λ′ ) is, in fact, a subbuilding of BIℓ (Λ) for all ℓ ≥ 3. Notation 1.18. Let ∆ be a Moufang spherical building, let Σ be an apartment of ∆, let Υ = ΥΣ be the set of all roots of Σ and let G = Aut(∆). We denote by G† the subgroup of G generated by all the root groups of ∆. By [62, 11.22], there exists a map [ µΣ : Uα∗ → G† α∈Υ

such that for each α ∈ Υ and for each non-trivial element g in the root group Uα , µΣ (g) is the unique element in the double coset U−α gU−α which maps Σ to itself. Here −α denotes the root of Σ opposite α (i.e. the complement of α in Σ regarded as a set of chambers). The wall of α is the set of all panels of ∆ containing one chamber in α and one in −α. If α ∈ Υ, then by [62, 3.13], there is a unique automorphism sα of Σ that stabilizes every panel in the wall of α and interchanges α with −α. We have sα = s−α and s2α = 1 for all α ∈ Υ. A reflection of Σ is an automorphism of the form sα for some α ∈ Υ. For each α ∈ Υ and each g ∈ Uα∗ , the element µΣ (g) induces sα on Σ (but is not necessarily of order 2). See 19.15 below. Notation 1.19. Let ∆ and G† be as in 1.18, let P be a panel of ∆ and let GP be the stabilizer of P in G† . We choose a chamber x in P and an apartment Σ containing x and let α denote the unique root of Σ containing x but not P ∩ Σ. By [62, 11.4], the root group Uα acts sharply transitively and, in particular, faithfully on P \{x}. Let Ux denote the image of Uα in Sym(P ) and let Ux+ denote the group generated by Uβ for all roots β of Σ containing x. If β is a root of Σ containing x other than α, then Uβ acts trivially on P . By [62, 11.11(ii)], the group Ux+ acts transitively on the set of apartments containing x. It follows that the permutation group Ux is independent of the choice of the apartment Σ. Thus gUx g −1 = Ug(x) for all x ∈ P and all g ∈ GP and hence the pair M∆,P := (P, {Ux | x ∈ P })

is a Moufang set as defined in 1.5 and

µΣ (g) = µxy (¯ g) for all g ∈ Uα , where µΣ is as in 1.18, y is the unique chamber of P ∩ α other than x, µxy is as in 1.6, g¯ denotes the image of g in Ux and µΣ (g) denotes the image of µΣ (g) in Sym(P ). Bruhat-Tits buildings. In these notes, we use the term “Bruhat-Tits building” in the sense introduced in [65]:

9

BUILDINGS

Definition 1.20. A Bruhat-Tits building is a thick irreducible affine building whose building at infinity is Moufang. The building at infinity of an affine building is constructed in [65, Chapter 8]. By 1.1, the building at infinity of a Bruhat-Tits building is spherical, irreducible and thick. Assumption 1.21. By [65, 27.4], there is no loss in generality if we assume that the building at infinity Ξ∞ of a Bruhat-Tits building Ξ is formed with respect to the complete system of apartments of Ξ (as defined in [65, 8.5]) and we will always do this in these notes. Let Ξ be a Bruhat-Tits building. The type of Ξ is an irreducible affine ˜ ℓ for some ℓ ≥ 2 and for X = A, B, . . . , F or G (see 20.41) Coxeter diagram X and the type of Ξ∞ is Xℓ . By [65, 27.5], the algebraic data corresponding to the Moufang building Ξ∞ is defined over a field or a skew-field or an octonion division algebra K which is complete with respect to a discrete valuation. Tits showed (see [65, 27.6]) that Ξ is uniquely determined by Ξ∞ and completed the classification of Bruhat-Tits buildings by determining exactly which Moufang buildings can appear as the building at infinity (see [65, 27.5]). Notation 1.22. We will apply the notation for Bruhat-Tits buildings given in the fourth column of Table 27.2 in [65] except that we suppress the reference to the valuation of K since we are assuming that the system of apartments A is complete, hence that the field or skew-field or octonion division algebra K is complete and hence that the discrete valuation of K is unique ˜ 2 (K) denotes the unique Bruhat-Tits (by [65, 23.15]). Thus, for example, A ˜ E (Λ) = C ˜ E (Λ) denotes the building whose building at infinity is A2 (K), B 2 2 unique Bruhat-Tits building whose building at infinity is BE2 (Λ) = CE2 (Λ), etc. ˜ ∗ (Λ) in the notation described in 1.22, then Ξ∞ is Remark 1.23. If Ξ = X ℓ obtained by simply removing the tilde. Note, however, that the spherical Coxeter diagrams Bℓ and Cℓ are the same for all ℓ ≥ 2 as are the affine ˜2 and C˜2 , but that the affine Coxeter diagrams B ˜ℓ and Coxeter diagrams B C˜ℓ are not the same when ℓ > 2. As a consequence, the inverse of the process of “deleting the tilde” is not so straightforward when X = B or C and ℓ > 2. Suppose, for example, that Q ∆ = BQ ℓ (Λ) = Cℓ (Λ)

for some ℓ ≥ 2 and some anisotropic quadratic space Λ = (K, L, q) such that K is complete with respect to a discrete valuation ν and 1 ∈ q(L). Then by [65, 19.23], the unique Bruhat-Tits building whose building at infinity is ∆ ˜ Q (Λ), where X = B if ν(q(L∗ )) = 2Z and X = C if ν(q(L∗ )) = Z. Similar is X ℓ results hold in the other cases; see [65, 27.2]. Definition 1.24. As observed in [65, 30.33], a Moufang building can be mixed as defined in [65, 30.24] (see also 28.3), algebraic or exceptional as defined in [65, 30.32] or classical as defined in [65, 30.30]. (If it is exceptional,

10

CHAPTER 1

it is automatically algebraic, if it is algebraic, then it is either exceptional or classical and if it is not algebraic, then it is either classical or mixed.) We will say that a Bruhat-Tits building is mixed, exceptional, classical, respectively, algebraic if its building at infinity is mixed, exceptional, classical, respectively, algebraic. Remark 1.25. Let F be a field complete with respect to a discrete valuation and let G be an absolutely simple algebraic group. If the F -rank of G is 1, then (∆, {Ux | x ∈ ∆}) is a Moufang set, where ∆ is the set of parabolic subgroups of G(F ) and Ux is the unipotent radical of x for each x ∈ ∆, and there exists a tree Ξ whose set Ξ∞ of ends is ∆ to which the action of the groups Ux can be extended. These trees together with Bruhat-Tits buildings in our sense are precisely the affine buildings that were investigated in [7] (together with certain non-discrete generalizations). The following result should have been formulated explicitly in [65]: Theorem 1.26. Let Ξ be a Bruhat-Tits building. Then every automorphism of ∆ := Ξ∞ is induced by a unique automorphism of Ξ. In other words, Aut(Ξ) and Aut(∆) are canonically isomorphic. Proof. By [65, 13.10 and 13.31], it suffices to show that any two valuations of the root datum of ∆ are equipollent. Let K (or {K, K op} or {K, E}) be the defining field of ∆ in the sense of [65, 30.29]. By [65, 27.5] K is complete with respect to a discrete valuation. As was observed in 1.22, the discrete valuation ν of K is unique. By [65, 19.4, 23.16, 24.9 and 25.5], the parameter system defining ∆ is ν-compatible as defined in the references in the second column of [65, Table 27.2]. By [65, 3.41(iii) and 16.4] combined with the results [65, 20.2(ii), 21.27(ii) and 22.16(ii)], it follows that any two valuations of the root datum of ∆ are equipollent as claimed. Remark 1.27. We allow ourselves, in light of 1.26, to identify the automorphism group of a Bruhat-Tits building with the automorphism group of its building at infinity. Note, however, that the Coxeter diagrams of Ξ and ∆ are, of course, different, and it can happen (see [65, 18.1]) that the isomorphism in 1.26 carries non-type-preserving automorphisms of Ξ to type-preserving elements of ∆. Simplicial complexes. In the original definition given in [55], a building is a simplicial complex, but in these notes (as in [62] and [65]), we view buildings as certain edge-colored graphs and the residues as certain subgraphs. See [62, 1.2 and 7.1] for the precise definitions. The vertices of these graphs are called chambers and when we write, for example, c ∈ ∆ or c ∈ R or c ∈ Σ or c ∈ α, we mean that

11

BUILDINGS

c is a chamber of the building ∆ or the residue R or the apartment Σ or the root α. In Chapters 26 and 27, however, where we work more closely with the notion of the building at infinity of an affine building, the notion of a building as a simplicial complex plays an important role. We use the rest of this chapter to fix some notation which we will need (only in those two chapters). Definition 1.28. A simplicial complex is a pair (V, S), where V is a set whose elements are called vertices and S is a subset of the power set of V whose elements are called simplices, such that (i) {v} ∈ S for all v ∈ V and (ii) all subsets of a simplex are also simplices. The dimension of a simplex is its cardinality minus one. The set V is generally identified with the set of simplices of dimension 0. Definition 1.29. Let B = (V, S) be a simplicial complex. A numbering of B is a surjective map from V to a set I (which we call the index set) such that the restriction of this map to each simplex is injective. A numbered simplicial complex is a simplicial complex endowed with a numbering. Definition 1.30. Let B = (V, S) and B ′ = (V ′ , S ′ ) be two simplicial complexes with numberings ν and ν ′ having index sets I and I ′ . A morphism from (B, ν) to (B ′ , ν ′ ) is a pair (ξ, σ), where ξ is a map from V to V ′ carrying simplices to simplices and σ is a map from I to I ′ such that ν ′ ◦ ξ = σ ◦ ν. An isomorphism from (B, ν) to (B ′ , ν ′ ) is a morphism (ξ, ν) such that ξ and ν are bijections and (ξ −1 , ν −1 ) is a morphism from (B ′ , ν ′ ) to (B, ν). We denote by Aut(B, ν) the group consisting of all isomorphisms from (B, ν) to itself. A subcomplex of (B, ν) is a numbered simplicial complex (B1 , ν1 ) whose vertex set is a subset of V and whose index set is a subset of I such that (incl, incl) is a morphism from (B1 , ν1 ) to (B, ν). Notation 1.31. Let Π be a Coxeter diagram with vertex set S, let n = |S|, let ∆ be a building of type Π, let V be the set of all maximal residues of ∆ and let ν be the map from V to S which sends a maximal residue whose type is J to the unique element s of S such that J = S\{s}. If R is a proper residue of ∆ and J ⊂ S is its type (so J = ∅ if R is a single chamber), then for each s ∈ S\J, there exists a unique (S\{s})-residue Rs such that R ⊂ Rs and by [62, 7.25], \ R= Rs . s∈S\J

For each residue R, we denote by AR the set of maximal residues containing R (so A∆ = ∅) and we set
∆# := (V, S), ν

12

CHAPTER 1

where S denotes the set of subsets AR of V for all residues R of ∆ (proper or not). Then ∆# is a numbered simplicial complex with index set S whose simplices of dimension k as defined in 1.28 correspond to residues of ∆ of rank n − k − 1 as defined in [65, 29.1]. In particular, every simplex of ∆# has dimension at most n − 1 and the chambers of ∆ (i.e. the minimal residues) correspond to the simplices of ∆# of dimension n − 1 (i.e. the maximal simplices). Remarks 1.32. Let ∆, ∆# , S and n be as in 1.31. Then the following hold: (a) The building ∆ can be reconstructed from ∆# : Two chambers are sadjacent in ∆ for some s ∈ S precisely when the intersection of the corresponding maximal simplices has dimension n − 2. (b) The correspondence residues of ∆

simplices of ∆#

is containment-reversing. (c) There is a canonical isomorphism from Aut(∆) to Aut(∆# ). (d) Apartments of ∆ correspond to certain subcomplexes of ∆# . More precisely, an apartment Σ of ∆ corresponds to the subcomplex (VΣ , SΣ ), where VΣ is the set of maximal residues of ∆ containing a chamber of Σ and SΣ is the set of simplices in S containing only elements of VΣ . In light of these observations, it is natural to think of ∆ and ∆# as the same object, simply seen from two points of view.

Chapter Two Quadratic Forms We use this chapter to assemble a few standard definitions and results about quadratic forms and polar spaces. Definition 2.1. A quadratic module is a triple Λ = (R, L, q) consisting of a commutative ring R with identity 1 = 1R , an R-module L and a quadratic form q on L, that is to say, a map q from L to R such that the map f from L × L to R given by (u, v) 7→ q(u + v) − q(u) − q(v) is bilinear and q(tu) = t2 q(u) for all u, v ∈ L and all t ∈ R. The symmetric bilinear form f will be denoted by ∂q. A quadratic module (R, L, q) (or a quadratic form q) is anisotropic if q(v) 6= 0 for all v ∈ L∗ := L\{0} and isotropic otherwise. In most cases, it causes no ambiguity to refer to q when Λ is meant and we will do this whenever it is convenient. A quadratic space is a quadratic module where the ring R is a field and thus L is a vector space over K. If Λ = (K, L, q) is a quadratic space, we set dimK (q) = dimK L. Remark 2.2. Let (R, L, q) be a quadratic module. If 2 is invertible in R, then q(u) = ∂q(u, u)/2 for all u ∈ L, so q is uniquely determined by ∂q. In particular, q is uniquely determined by ∂q for a quadratic space (K, L, q) when char(K) 6= 2. This is not true, in general, if char(K) = 2. The norm of an inseparable quadratic extension E/K, for example, is a non-zero quadratic form such that ∂q is identically zero. Definition 2.3. Let Λ = (R, L, q) and Λ′ = (R, L′ , q ′ ) be two quadratic modules over the same ring R. An isometry from Λ to Λ′ (or from q to q ′ ) is a bijective linear map η : L → L′ such that q = q ′ ◦ η. We say that Λ and Λ′ (or q and q ′ ) are isometric, and write Λ ∼ = Λ′ (or q ∼ = q ′ ), if there exists ′ ′ an isometry from Λ to Λ . A similarity from Λ to Λ (or from q to q ′ ) is an isometry from q to γq ′ for some invertible element γ of R; the factor γ is called the multiplier, or similarity factor, of the similarity. A similitude of Λ is a similarity from Λ to itself. Definition 2.4. Let Λ = (R, L, q) be a quadratic module. A subform of q is the restriction of q to a submodule of L. Every subform of q is, of course, itself a quadratic form over R. A quadratic submodule of Λ is a quadratic module (R, L′ , q ′ ) over R such that L′ is a submodule of L and q ′ is the restriction of q to L′ .

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CHAPTER 2

Definition 2.5. Let Λ = (R, L, q) and Λ′ = (R, L′ , q ′ ) be two quadratic modules over the same ring R. The orthogonal sum Λ ⊕ Λ′ of Λ and Λ′ is the quadratic module (R, M, q ⊕ q ′ ), where M = L ⊕ L′ and Q := q ⊕ q ′ is the quadratic form on M given by Q(v, v ′ ) = q(v) + q ′ (v ′ ) for all v ∈ L and all v ′ ∈ L′ . Note that


∂Q (v, 0), (0, v ′ ) = 0

for all v ∈ L and all v ′ ∈ L′ .

Notation 2.6. Let Λ = (R, L, q) be a quadratic module over R and suppose that S is a commutative ring with multiplicative identity 1S and π is a homomorphism from R to S mapping 1R to 1S . Using π to make S into an R-module, we form the tensor product LS = L ⊗R S. We then endow LS with the structure of an S-module in the usual way, so that (2.7)

s · (v ⊗ t) = (v ⊗ st)

for all v ∈ L and all s, t ∈ S. As is shown, for example, in [28, pp. 1.7-1.8], there exists a unique quadratic form on LS over S such that and

qS (v ⊗ s) = q(v) · s2 ∂qS (u ⊗ s, v ⊗ t) = ∂q(u, v) · st

for all u, v ∈ L and all s, t ∈ S. We will call the quadratic module ΛS := (S, LS , qS ) (or qS ) the scalar extension of Λ (or q) from R to S. In our applications of this construction, R will almost always be a subring of S with the same multiplicative identity and π will be the natural embedding; the one exception is in 2.9 below, where π is equally natural. For this reason, we do not mention π in our notation. Suppose that T is a third commutative ring with identity 1T and that π ′ is a homomorphism from S to T sending 1S to 1T . There is a canonical identification of (LS )T = (L ⊗R S) ⊗S T

with LT = L ⊗R T as T -modules (with tensor products formed using π, π ′ and their composition) where (v ⊗R s) ⊗S t = v ⊗R (st)

and v ⊗R t = (v ⊗R 1S ) ⊗S t

for all v ∈ L, s ∈ S and t ∈ T . Applying this identification, we have (2.8)

(qS )T = qT .

Example 2.9. Let Λ = (R, L, q) be a quadratic module and A an ideal in R. We denote by r 7→ r¯ (respectively, v 7→ v¯) the canonical map from R to ¯ := R/A (respectively, from L to L ¯ := L/AL). Then L ¯ is an R-module ¯ R under the well defined natural action (¯ r , v¯) 7→ rv

15

QUADRATIC FORMS

¯×L ¯ to L ¯ and the map q¯: L ¯→R ¯ given by from R q¯(¯ v ) := q(v)

¯ The quadratic module for all v ∈ L is a well defined quadratic form over R. ¯ := (R, ¯ L, ¯ q¯) is called the reduction of Λ modulo A. There is a natural Λ ¯ with L ¯ as R-modules ¯ identification of LR¯ = L ⊗R R such that v ⊗ r¯ = rv and v¯ = v ⊗ 1R¯

¯ is equal to the scalar for all v ∈ L and all r ∈ R. Given this identification, Λ ¯ extension ΛR¯ of Λ from R to R obtained by setting π equal to the map r 7→ r¯ in 2.6. Definition 2.10. A quadratic module Λ = (R, L, q) is strictly non-singular if the R-module L is free and of finite rank and the natural map
v 7→ x 7→ ∂q(x, v)

from L to its dual module is an isomorphism. This is equivalent to the condition that the matrix of ∂q relative to some (or any) basis of L is invertible. Definition 2.11. Let Λ = (K, L, q) be a quadratic space. The quadratic form q (or the quadratic space Λ) is non-singular if ∂q is non-degenerate, that is to say, if x ∈ L and ∂q(x, y) = 0 for all y ∈ L, then x = 0. Thus Λ is strictly non-singular if and only if it is non-singular and finite-dimensional. The quadratic form q is singular if it is not non-singular and totally singular if ∂q is identically zero. If q is anisotropic and char(K) 6= 2, then f (u, u) = 2q(u) 6= 0 for f = ∂q and for all u ∈ L∗ , so q is non-singular. It is not, however, true that anisotropic implies non-singular if char(K) = 2, as the example in 2.2 shows. See also 2.31 below. Example 2.12. Let q be the norm of a quadratic extension E/K, let f = ∂q and let a be an element of E not in K. We make E ⊗K E into a vector space over E by endowing it with the unique scalar multiplication such that (2.7) holds for all s, t, v ∈ E. Since the subset {1, a} of E is linearly independent over K, the subset {1 ⊗ 1, a ⊗ 1} of E ⊗K E is linearly independent over E. Therefore 1 ⊗ a − a ⊗ 1 6= 0 but qE (1 ⊗ a − a ⊗ 1) = a2 − f (1, a)a + q(a) = 0. Thus q is anisotropic but qE is not. Example 2.13. Let q be the norm of a quadratic extension E/K, let L/K be an extension such that L ∩ E = K and let F be the composite field LE. Then qL is the norm of the quadratic extension F/L. Remark 2.14. We will call a quadratic space non-trivial if it is of positive dimension. Note that by 2.11, the 0-dimensional quadratic form is both anisotropic and non-singular; this, it turns out, is a useful convention in these notes. In [60] and [65], however, we implicitly used the convention that “anisotropic” meant “non-trivial.” In particular, the building BQ ℓ (Λ) and the root group sequence QQ (Λ) exist only for non-trivial anisotropic quadratic spaces even though we did not say this explicitly in [60] and [65] (and probably forget to say it explicitly now and then in these notes).

16

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Notation 2.15. Let Λ = (R, L, q) be a quadratic module and let Ω = (R, M, Q) be a quadratic submodule of Λ. Then M ⊥ = {v ∈ L | ∂q(u, v) = 0 for all u ∈ M }

is a submodule of L; let Ω⊥ denote the corresponding quadratic submodule of Λ. If Ω is strictly non-singular, then L = M ⊕ M⊥ (by [30, 5.4.1], for example), and after the obvious identifications we have Λ = Ω ⊕ Ω⊥ in the sense of 2.5. In this case, we call M ⊥ (respectively, Ω⊥ ) the orthogonal complement of M (respectively, Ω). Notation 2.16. For each β ∈ K, let ξK,β denote the form x 7→ βx2

of dimension 1 on K. If q is an arbitrary 1-dimensional quadratic form, then q∼ = ξK,β for some β ∈ K and q is anisotropic if and only if β 6= 0. Remark 2.17. Every 2-dimensional anisotropic quadratic form over K is similar to the norm of a quadratic extension E/K (by [60, 34.2], for example). Proposition 2.18. Let Λ = (K, L, q) be a finite-dimensional, anisotropic quadratic space such that the dimension of the radical of ∂q is at most 1 and let n = dimK L and m = [n/2]. Then there exist elements α1 , . . . , αm of K ∗ and, if n is odd, an additional element β of K ∗ and quadratic forms q1 , . . . , qm over K such that qi is isometric to the norm of a separable quadratic extension Ei /K for each i ∈ [1, m], q∼ = α1 q1 ⊕ · · · ⊕ αm qm

if n is even and q∼ = α1 q1 ⊕ · · · ⊕ αm qm ⊕ ξK,β if n is odd, where ξK,β is as in 2.16. Proof. It follows from the hypotheses that there exists an orthogonal decomposition of q into the sum of 2-dimensional non-singular subspaces and at most one 1-dimensional subspace. The claim holds, therefore, by 2.16 and 2.17. Definition 2.19. Let Λ = (R, L, q) be a quadratic module. A hyperbolic pair of Λ (or of q) is a pair of elements u, v of L such that q(u) = q(v) = 0 and ∂q(u, v) = 1. A hyperbolic plane of Λ is a quadratic subspace (R, L′ , q ′ ) such that L′ is generated by a hyperbolic pair, which then forms a basis of L′ over R. For each hyperbolic pair u, v of q, we denote by H(u, v) the unique hyperbolic plane (R, L′ , q ′ ) of Λ such that L′ contains u and v. The hyperbolic planes of Λ are strictly non-singular. Thus Λ = H(u, v) ⊕ H(u, v)⊥

17

QUADRATIC FORMS

for every hyperbolic pair u, v by 2.15. A quadratic module is hyperbolic if it can be written as the orthogonal sum of finitely many hyperbolic planes. Hyperbolic quadratic modules are strictly non-singular and free of even rank and they remain hyperbolic under arbitrary scalar extensions. A hyperbolic quadratic space is a quadratic space that is hyperbolic as a quadratic module. Definition 2.20. We will say that a quadratic space (K, V, q) is unital if 1 ∈ q(V ). A pointed quadratic space is a quadratic space (K, V, q) together with a distinguished element 1 ∈ V called the base point such that 1 = q(1). Remark 2.21. If q is similar to the norm of a separable quadratic extension E/K, then (E, E ⊗K E, qE ) is isotropic (by 2.12) and hence hyperbolic (by [21, 7.13]). It follows that if q and all the notation are as in 2.18, then for every extension L/K such that L contains the fields E1 , . . . , Em and, if n is √ odd, also β, the quadratic form qL is either hyperbolic or the orthogonal sum a hyperbolic quadratic submodule and a 1-dimensional unital quadratic submodule. Definition 2.22. A quadratic space is split if it is either hyperbolic or the orthogonal sum of a hyperbolic quadratic submodule and a unital quadratic submodule of dimension 1. Let (K, V, q) be a quadratic space and let L be a field containing K. We will say that L/K is a splitting extension of q if the quadratic form qL is split. Typically, we will say that L is a splitting field of q when we really mean that L/K is a splitting extension of q; this should not cause any confusion. Note that q can have a splitting field only if it is finite-dimensional and either non-singular or char(K) = 2 and the dimension of the radical of ∂q is 1. If q is finite-dimensional, anisotropic and the radical of ∂q has dimension at most 1, then by 2.21, there always exist splitting fields. Definition 2.23. Let (K, V, q) be a finite-dimensional anisotropic quadratic space. We will say that E is a norm splitting field of q if E/K is a separable quadratic extension such that qE is hyperbolic. Definition 2.24. Let Λ = (K, V, q) be an anisotropic quadratic space. We say that Λ (or q) has a norm splitting if for some d ≥ 1 and some separable quadratic extension E/K, there exist non-zero α1 , . . . , αd in K ∗ such that (2.25)

q∼ = α1 N ⊕ · · · ⊕ αd N,

where N is the norm of the extension E/K. See [60, 12.18] for an equivalent definition. By [14, Lemma 4.2], a finite-dimensional non-singular anisotropic quadratic form q has a decomposition as in (2.25) for some separable quadratic extension E/K with norm N if and only if E is a norm splitting field of q as defined in 2.23. Remark 2.26. Let (K, V, q) be a quadratic space and suppose that (2.25) holds for some separable quadratic extension E/K with norm N and some α1 , . . . , αd ∈ K ∗ . Suppose, too, that L/K is an extension such that L ∩ E =

18

CHAPTER 2

K, let F be the composite field EL and let R be the norm of the extension F/L. Then F/L is a separable quadratic extension and by 2.13, qL ∼ = α1 R ⊕ · · · ⊕ αd R (but, of course, q might be anisotropic while qL is not). Definition 2.27. A quadratic space (K, V, q) is called round if the non-zero values of q are precisely the similarity factors of its similitudes. In other words, q is round if and only if it is unital and for each u ∈ V such that q(u) 6= 0, there exists a linear automorphism ψ of V such that q(ψ(v)) = q(u)q(v) for all v ∈ V . Notation 2.28. Let (K, V, q) be a quadratic space. As in [23, 7.4], we set q ⊗ hhαii = q ⊕ (−α)q for all α ∈ K ∗ and we define q ⊗ hhα1 , . . . , αn ii for α1 , . . . , αn ∈ K ∗ and n > 1 inductively to be
q ⊗ hhα1 , . . . , αn−1 ii ⊗ hhαn ii. If q is round (for example, if q is the norm of a quadratic extension), then by [21, 9.8], so is q ⊗ hhα1 , . . . , αn ii.

Proposition 2.29. Let (K, V, q) be a totally singular anisotropic quadratic space and let L/K be a finite separable extension. Then qL is anisotropic. Proof. Let n denote the degree of the extension L/K. We have L = K(a) for some a ∈ L, and by [29, Prop. 8.12] also L = K(a2 ). Hence {ai | i ∈ [0, n − 1]} and {a2i | i ∈ [0, n − 1]}

are both bases for L over K. Let u ∈ VL . Then u=

n−1 X i=0

vi ⊗ ai

for some v0 , . . . , vn−1 ∈ V . Since q is totally singular, so is qL . Hence qL (u) =

n−1 X

a2i q(vi ).

i=0

Thus if qL (u) = 0, then q(vi ) = 0 and therefore also vi = 0 for all i ∈ [0, n−1], so u = 0.

∗

∗

∗

We now assemble some results and definitions about the polar space associated to a regular quadratic form. We will need these results in Chapter 35.

19

QUADRATIC FORMS

Notation 2.30. Let V be a vector space over a commutative field K and let γ ∈ Aut(K). A γ-automorphism of V is an automorphism α of the additive group of V such that α(tv) = γ(t)α(v) for all t ∈ K and all v ∈ V . A semilinear automorphism of V is a γ-automorphism for some γ ∈ Aut(K); if V is non-trivial, then γ is uniquely determined by α and will be denoted by γα . We denote the group of all semi-linear automorphisms of V by ΓL(V ). The map α 7→ γα is a homomorphism from ΓL(V ) to Aut(K) with kernel GL(V ). A homothety of V is an automorphism of the form v 7→ tv for some t ∈ K ∗ . The group of all homotheties of V will be denoted by HT(V ). It is by definition the kernel of the natural maps from GL(V ) to PGL(V ) and from ΓL(V ) to PΓL(V ). We denote by β 7→ δβ the homomorphism from PΓL(V ) to Aut(K) induced by the homomorphism α 7→ γα . Notation 2.31. Let (K, V, q) be a quadratic space and let f = ∂q. As in [21], we set and

Def(q) = Rad(f ) = {v ∈ V | f (v, V ) = 0} Rad(q) = {v ∈ Rad(f ) | q(v) = 0}

and we say that q is regular if Rad(q) = 0. If q is anisotropic, then it is automatically regular. We call the subspace Def(q) the defect of q. It is trivial if and only if q is non-singular as defined in 2.11. Definition 2.32. Let (K, V, q) be a regular quadratic space. Suppose that there exists a subspace W of finite co-dimension in V such that the restriction of q to W is anisotropic. The Witt Decomposition Theorem ([21, 8.5]1 ) says that q can be decomposed into the orthogonal sum of a hyperbolic quadratic subform q0 and an anisotropic quadratic subform q an and that each of these subforms is unique up to isometry. The quadratic form q an is called the anisotropic part of q. Definition 2.33. Let (K, V, q) be a regular quadratic space such that the co-dimension of its defect is finite. We call a field L a pseudo-splitting field of q if L/K is a finite separable extension (so the restriction of qL to Def(q)L is anisotropic by 2.29) and the anisotropic part of qL is totally singular. Let W be a complement of Def(q) in V and let Q denote the restriction of q to W . The quadratic form Q is non-singular. By 2.21, there exists a field L such that L/K is a finite separable extension and QL is hyperbolic. Any such field (for any choice of W ) is a pseudo-splitting field of q. Note, too, that if char(K) = 2, then dimK (W ) = dimL (QL ) is even. Notation 2.34. Let (K, V, q) be a regular quadratic space. A subspace U of V is called isotropic if the restriction of q to U is isotropic. A subspace U 1 [21, 8.5] is an immediate consequence of [21, 8.3]. The proof of [21, 8.3] remains valid if it is assumed that W rather than V is finite dimensional. Thus the Witt Decomposition Theorem remains valid if the assumption that q is finite-dimensional is replaced by the assumption that there is an anisotropic subspace of finite co-dimension.

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of V is called totally isotropic if the restriction of q to U is trivial. The Witt index of (K, V, q) is the maximal dimension of a totally isotropic subspace. Notation 2.35. Let (K, V, q) be a quadratic space of positive dimension and let µ ∈ K ∗ . A µ-similitude of q is an element α of ΓL(V ) such that q(α(v)) = µγα (q(v)) for all v ∈ V (where γα is as in 2.30), in which case the scalar µα := µ is uniquely determined by α. A semi-linear similitude of q is a µ-similitude for some µ ∈ K ∗ . These include all similitudes of q as defined in 2.3 and, in particular, all the elements of HT(V ). We denote by ΓO(q) the group of all semi-linear similitudes of q. Thus GO(q) := GL(V ) ∩ ΓO(q) is the group of all similitudes of q. We also set PGO(q) = GO(q)/HT(V ) and PΓO(q) = ΓO(q)/HT(V ). Notation 2.36. Let V be a K-vector space of positive dimension. We denote by V(V ) the set of all subspaces of V . For each X ∈ V(V ), we denote by P(X) the set of 1-dimensional subspaces contained in X. Let S(V ) = {P(X) | X ∈ V(V )}.

The projective space P(V ) associated with V is the pair (P(V ), S(V )). The elements of P(V ) are called the points of P(V ), the elements of S(V ) are called the subspaces of P(V ) and an automorphism of P(V ) is a permutation of the set of points which maps S(V ) bijectively to itself. There is a natural homomorphism from ΓL(V ) to Aut(P(V )). The Fundamental Theorem of Projective Geometry says that if dimK V ≥ 3, then this homomorphism is surjective and its kernel is HT(V ). Notation 2.37. Let (K, V, q) be a regular quadratic space of Witt-index at least 1. We denote by V(q) the set of all totally isotropic subspaces of V of positive dimension. We put P(q) = P(V ) ∩ V(q) and S(q) = {P(X) | X ∈ V(q)}.

The polar space P(q) associated with q is the pair (P(q), S(q)). The elements of P(q) are called the points of P(q), the elements of S(q) are called the subspaces of P(q) and an automorphism of P(q) is a permutation of the set of points which maps S(q) bijectively to itself. There is a natural homomorphism from ΓO(q) to Aut(P(q)) which we denote by ϕq . Theorem 2.38. Let (K, V, q) be a regular quadratic space of dimension at least 5 and Witt index at least 2. Then the homomorphism ϕq is surjective and its kernel is HT(V ). Proof. We first observe that a quadratic form is the same thing as a (idK , 1)quadratic form as defined in [55, 8.2]. In particular, f := ∂q is a trace-valued (idK , 1)-hermitian form as defined in [55, 8.1]. Let κ denote the similarity class of q. Then the polar space Sκ obtained by applying [55, 8.4.2] to q is precisely P(q). Let π be the subset of P(V ) × P(V ) consisting of all pairs (hui, hvi) such that f (u, v) = 0. Then π is a polarity as defined in [55, 8.3.2] and the polar space Sπ defined in [55, 8.3.4] coincides with Sκ by the last remark in [55, 8.4.2].

QUADRATIC FORMS

21

We can thus apply [55, 8.6(II)] with π1 = π2 = π (so d1 > 3), S1 = S2 = Sκ and ϕ the identity map to conclude that ϕq is surjective and that its kernel is HT(V ). Notation 2.39. We call a subgroup H of ΓL(V ) strictly semi-linear if H ∩ GL(V ) = 1.

Proposition 2.40. Let V be a vector space over K of positive dimension, let G be a finite strictly semi-linear subgroup of ΓL(V ), let V G = FixV (G) and let F = FixK (γG ), where γG denotes the image of G under the map α 7→ γα defined in 2.30. Then the following hold: (i) V G is canonically a vector space over F and the canonical homomorphism ϕ from V G ⊗F K to V such that ϕ(v ⊗ t) = tv for all v ∈ V G and all t ∈ K is an isomorphism of K-vector spaces.

(ii) The map W 7→ W ∩ V G is an inclusion- and dimension-preserving bijection from the set of G-invariant non-zero subspaces of V to V(V G ) and its inverse is the map U 7→ hU iK .

Proof. This follows from the discussion in [49, §11.1, in particular, 11.1.6] combined with the observation that the Krull topology on G is the discrete topology since G is finite. Proposition 2.41. Let (K, V, Q) be a regular quadratic space of positive dimension and let G be a finite strictly semi-linear subgroup of ΓO(q) and let F and V G be as in 2.40. Then there exists a quadratic form q on V G such that the following hold: (i) qK is similar to Q. (ii) The map W 7→ W ∩ V G is an inclusion- and dimension-preserving bijection from the set of G-invariant subspaces of P(Q) to S(q) with inverse U 7→ hU iK .

Proof. By 2.40(ii), hV G iK = V . Thus Q(v) = 0 for all v ∈ V G would imply that Q(v) = 0 for all v ∈ V . Since Q is regular and of positive dimension, we can thus choose v0 ∈ V G such that Q(v0 ) 6= 0. Replacing Q by Q(v0 )−1 Q, we can assume that, in fact, Q(v0 ) = 1. For each α ∈ G, there exists µ ∈ K ∗ such that Q(α(w)) = µγα (Q(w)) for all w ∈ V , where γα is as in 2.35. Setting w = v0 , we have Q(v0 ) = µγα (1) = µ and hence µ = 1, and thus γα (Q(w)) = Q(w) for all w ∈ V G . We conclude that Q(V G ) ⊂ F . Thus the restriction of Q to V G yields a quadratic form q on V G over F . By 2.40(i), Q ∼ = qK . Thus (i) holds. If W is a totally isotropic subspace of V with respect to Q, then W ∩ V G is a totally isotropic subspace of V G with respect to q and if, conversely, U is a totally isotropic subspace of V G with respect to q, then hU iK is a totally isotropic subspace of V with respect to Q. By 2.40(ii), therefore, (ii) holds.

Chapter Three Moufang Polygons In this chapter, we assemble some basic facts about Moufang polygons and root group sequences. Notation 3.1. Suppose that ∆ is a Moufang n-gon for some n ≥ 3 (viewed as a chamber system as described in [62, 7.14–7.15]), let Σ be an apartment of ∆ (a circuit of length 2n) and let c be a chamber of Σ. Let α1 , α2 , . . . , α2n be the roots of Σ numbered either clockwise or counterclockwise so that c is the unique chamber in the intersection α1 ∩ · · · ∩ αn . Let Ui denote the root group Uαi for each i, let U+ = hU1 , U2 , . . . , Un i, let G = Aut(∆) and let G† be as in 1.18. The sequence Ω = (U+ , U1 , U2 , . . . , Un ) is a root group sequence as defined in [60, 8.7]. Definition 3.2. Let Ω = (U+ , U1 , U2 , . . . , Un ) be a root group sequence as defined in [60, 8.7]. The group U+ is called the trunk of Ω, the groups U1 , U2 , . . . , Un are called the terms of Ω and the number n of terms is called the length of Ω. The root group sequence (U+ , Un , Un−1 , . . . , U1 ), which we denote by Ωop , is called the opposite root group sequence of Ω. Thus U1 is both the first term of Ω and the last term of Ωop . See [60, 8.9] for the notion of an isomorphism of root group sequences and the notion of an anti-isomorphism of root group sequences. Remark 3.3. Let ∆, Σ, c, Ω, etc., be as in 3.1. The subgroup U+ is contained in the stabilizer G†c and the root group sequence Ω is uniquely determined, up to opposites, by the pair (c, Σ). We call Ω and Ωop the root group sequences of ∆ based at (c, Σ). By [60, 4.12], the group G† acts transitively on the set of ordered pairs (e, A), where A is an apartment of ∆ and e a chamber of A. It follows that the root group sequence Ω is— up to opposites and conjugation in G† —independent of the choice of the apartment Σ and the numbering of its roots. This fact justifies referring to Ω in 3.1 as the root group sequence of ∆. Remark 3.4. For each root group sequence Ω in 1.14, there is a unique Moufang polygon ∆ (obtained by applying the construction described in [60, 8.1]) such that Ω is isomorphic (as defined in [60, 8.9]) to a root group sequence of ∆. This is shown in [60, Chapter 32]. The classification of Moufang n-gons (as formulated in [60, Chapter 17]) says that, up to isomorphism, there are no other Moufang polygons.

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Hypothesis 3.5. Let ∆, Σ, c, Ω, n, etc., be as in 3.1. By 3.4, we can assume that the numbering of the roots of Σ has been chosen so that there is an isomorphism (as defined in [60, 8.9]) from Ω to one of the root group sequences on 1.14. We identify Ω with its image under this isomorphism. Thus Ω is the root group sequence defined by one of the recipes [60, 16.116.9] in terms of a suitable parameter system Λ and isomorphisms xi from some part of Λ to the root group Ui , one for each i ∈ [1, n]. Definition 3.6. Let ∆, Σ, and α1 , . . . , α2n be as in 3.5 and let µ1 and µn be as in [60, 6.1]. Thus for i = 1 or n, µi is the restriction to Ui∗ of the map µΣ defined in 1.18 and hence for each a ∈ Ui∗ , µi (a) is the unique element of the double coset ∗ ∗ U−α aU−α i i

that maps Σ to itself, where −αi = αi+n is the root in the apartment Σ opposite αi (and subscripts are to be read modulo 2n). The element µi (a) maps αj to α2i+n−j and hence µ (a)

Uj i

= U2i+n−j

for i = 1 or n, for all a ∈ Ui∗ and for all j ∈ [1, 2n]. By [60, 6.2], we have µi (a)−1 = µi (a−1 ) for all a ∈ Ui∗ . Notation 3.7. Let ∆, Σ, c, α1 and αn be as in 3.5. Let i = 1 or n, let P be the unique panel of ∆ containing c such that Ui acts non-trivially on P and let Hi be the subgroup of Aut(Ui ) induced by hµi (a)µi (b) | a, b ∈ Ui∗ i, where µi is as in 3.6. Thus Hi is, up to isomorphism, a torus (as defined in 1.6) of the Moufang set M∆,P defined in 1.19. We will call Hi the torus of ∆ at Ui or at P . Notation 3.8. Let i and the Moufang set M∆,P be as in 3.7. In keeping with 1.16, we denote this Moufang set by A1 (K) if i = 1 and Ω is as in case (i) of 1.14, we denote it by BI1 (Λ) if i = 1 and Ω is as in case (ii) of 1.14, we denote it by BQ 1 (Λ) if i = 4 and Ω is as in case (iii) of 1.14 and we denote it by BD 1 (Λ) if i = 1 and Ω is as in case (iv) of 1.14.

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Remark 3.9. Suppose that Ω = T (K) for some alternative division ring K, let µ = µ1 and let m = µ(x1 (1)). Let P and c be as in 3.7 with i = 1 and let d be the other chamber in P ∩ Σ. By [60, 32.5], we have (3.10)

µ(x1 (t)) = x4 (t−1 )x1 (t)x4 (t−1 )

and by the discussion at the beginning of [60, Chapter 32], x4 (t) = x1 (t)m for all t ∈ K. Letting ρ denote the unique permutation of U1∗ such that ρ

dx1 (t

)

= cx1 (t)

m

for all t ∈ K, we observe that the product x1 (ρ−1 (−t))m x1 (t)x1 (−ρ−1 (t))m ∈ U4 x1 (t)U4 (composed from left to right) interchanges the chambers c and d for all t ∈ K ∗ and hence must equal µ(x1 (t)). Comparing this product with (3.10), we conclude that ρ−1 (−t) = t−1 and thus ρ(t) = −1/t for all t ∈ K ∗ . Therefore A1 (K) ∼ = M(K, x 7→ −1/x) in the notation described in 1.11. Using the equations in [60, 32.6–32.9], we deduce by similar calculations that BI1 (K, K0 , σ) ∼ = M(K0 , x 7→ −1/x) for every involutory set (K, K0 , σ), ∼ BQ 1 (K, L, q) = M(L, x 7→ −x/q(x)) for every anisotropic quadratic space (K, L, q) and ∼ BD 1 (K, K0 , L0 ) = M(K0 , x 7→ 1/x) for every indifferent set (K, K0 , L0 ). Notation 3.11. Let i, j ∈ {1, n}, let Hi and µi be as in 3.6 and 3.7, let ∆′ be a second Moufang quadrangle, let ′ Ω′ = (U+ , U1′ , . . . , U4′ )

be the root group sequence obtained from a numbering of an apartment Σ′ , let µ′j be as in 3.6 applied to Σ′ , let P and P ′ be the panels obtained by applying 3.7 to the pair ∆, i, respectively, the pair ∆′ , j and let M := M∆,P and M := M∆′ ,P ′ . We will say that the torus Hi of ∆ at Ui is linked to the torus of ∆′ at Uj′ (as defined in 3.7) if there is an isomorphism ϕ from Ui to Uj′ such that ′

′

ϕ(wµi (a)µi (b) ) = ϕ(w)µj (ϕ(a))µj (ϕ(b)) for all w ∈ Ui and all a, b ∈ Ui∗ . By 1.10, such a map is the same thing as a weak isomorphism from M to M′ .

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Notation 3.12. Let Ω = (U+ , U1 , . . . , Un ) be an arbitrary root group sequence for some n ≥ 3. By [60, 8.11], there exists a Moufang n-gon ∆ and an apartment Σ of ∆ such that Ω is the root group sequence of ∆ determined by some numbering of the vertices of Σ. By [60, 7.5], the pair (∆, Σ) is uniquely determined by Ω. Let i = 1 or n and let µi be as in 3.6 applied to Σ. We will say that the root group sequence Ω is flexible at Ui if every automorphism ϕ of Ui such that (3.13) ϕ(wµi (a)µi (b) ) = ϕ(w)µi (ϕ(a))µi (ϕ(b)) for all w ∈ Ui and all a, b ∈ Ui∗ extends to an automorphism of the root group sequence Ω. The notion of flexibility is closely related to the property (Ind) introduced in [38]; see 3.17. ′ Remark 3.14. Let Ω = (U+ , U1 , . . . , Un ) and Ω′ = (U+ , U1′ , . . . , Un′ ) be isomorphic root group sequences for some n ≥ 3 and let ξ be an isomorphism from Ω to Ω′ . Let the pairs (∆, Σ) and (∆′ , Σ′ ) be as in [60, 8.11] applied to Ω and Ω′ . Thus Ω is the root group sequence of ∆ with respect to some numbering of the apartment Σ and Ω′ is the root group sequence of ∆′ with respect to some numbering of the apartment Σ′ . By [60, 7.5], there is an isomorphism from ∆ to ∆′ which induces the isomorphism ξ from Ω to Ω′ . Let i = 1 or n, let µ = µi be as in 3.6 applied to ∆, Σ and its numbering and let µ′ be defined analogously starting with Ω′ . Then ′ ′ ξ(wµ(a)µ(b) ) = ξ(w)µ (ξ(a))µ (ξ(b)) for all w ∈ Ui and all a, b ∈ Ui∗ by [60, 6.1].

Example 3.15. Let Ω = T (F ) for some commutative field F in the notation set out in 1.14 and let i = 1 or 3. By [60, 33.10], we have xi (u)µ(xi (s))µ(xi (t)) = xi (tsust) for all u ∈ F and all s, t ∈ F ∗ . Suppose that ϕ is an automorphism of Ui satisfying (3.13). We claim that this automorphism of Ui extends to an automorphism of Ω. Let ψ be the additive automorphism of F such that xi (u)ϕ = xi (ψ(u)) for all u ∈ F . There are automorphisms of Ω extending the maps xi (u) 7→ xi (tu) for all t ∈ F ∗ . Thus we can assume that ψ(1) = 1. Setting s = 1 in (3.13), we thus have ψ(tut) = ψ(t)ψ(u)ψ(t) for all t, u ∈ F . It follows then by Hua’s Theorem (see [27]) that ψ is an automorphism of F . Thus the maps xj (t) 7→ xj (ψ(t)) from Uj to Uj for each j ∈ [1, 3] extend to an automorphism of Ω. We conclude that Ω is flexible at both U1 and U3 . Proposition 3.16. Let Ω = T (F ) for some commutative field F , let i = 1 or 3 and let Mi denote the centralizer of Ui in the group of automorphisms ∗ of Ω. Then Mi acts sharply transitively on U4−i . Proof. It follows from [60, 3.7] that Mi acts faithfully on Uj for j = 4 − i. By [60, 37.10 and 37.12], the group Mi induces the permutation group {xj (t) 7→ xj (st) | s ∈ F ∗ } on Uj .

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27

The following is really just a reformulation of 3.15. Proposition 3.17. Let ∆ = A2 (F ) for some commutative field F and let P be a panel of ∆. Then every weak automorphism of the Moufang set M∆,P extends to an automorphism of ∆. Proof. Let c be a chamber of P , let Σ be an apartment containing c, let d be the other chamber of Σ in P and let Ω = (U+ , U1 , U2 , U3 ) be one of the two root group sequences of ∆ based at (Σ, c) chosen so that U1 acts non-trivially on P . Restriction to P is an isomorphism from U1 to the root group Uc of M := M∆,P . Let µ = µcd be as in 1.6 applied to M, let µ1 be as in 3.6 applied to ∆ and Σ and let U1 be identified with Uc via restriction to P . Then for all non-trivial a, b ∈ U1 , the action of µ(a)µ(b) on U1 is the same as the action of µ1 (a)µ1 (b), as was observed in 1.19. Now let ψ be a weak automorphism of M. We want to show that ψ extends to an isomorphism of ∆. Since the stabilizer of P in Aut(∆) acts 2-transitively on P , it suffices to assume that ψ fixes c and d. Thus ψ induces an automorphism of Uc . We denote this automorphism by ϕ. Then (3.13) holds for ϕ. By 3.15 and the conclusion of the previous paragraph, it follows that there is an automorphism ξ of Ω whose restriction to U1 is ϕ. By [60, 7.5], there exists a unique automorphism π of ∆ fixing c, P and Σ which induces the automorphism ξ on Ω and hence the automorphism ϕ on U1 . The permutation π −1 ψ of P centralizes Uc and fixes the chambers c and d. We conclude that ψ is the restriction of π to P . Proposition 3.18. Let F be a commutative field and let ∆ and ∆′ be two buildings both isomorphic to A2 (F ). Let P and P ′ be panels of ∆ and ∆′ and let ψ be a weak isomorphism from the Moufang set M∆,P to the Moufang set M∆′ ,P ′ . Then ψ extends to an isomorphism from ∆ to ∆′ . Proof. Since F is commutative, the root group sequence T (F ) is isomorphic to its opposite. It follows that we can choose an isomorphism π from ∆ to ∆′ that maps P to P ′ . Then β = π −1 ◦ ψ is a weak automorphism of the Moufang set M∆,P . By 3.17, we can choose an automorphism ξ of ∆ extending β. Let λ = π ◦ ξ. Then λ is an isomorphism from ∆ to ∆′ whose restriction to P is π ◦ β = ψ. Proposition 3.19. Let F be a commutative field, let ∆ be a spherical building satisfying the Moufang condition, let (W, S) be the corresponding Coxeter system, let s ∈ S, let J = S\{s} and let R be a J-residue of ∆. Suppose that there is a unique t in S that does not commute with s and that every {s, t}-residue of ∆ is isomorphic to A2 (F ). Then every automorphism of R inducing a permutation of J which fixes t extends to an automorphism of ∆. Proof. Let ξ be an automorphism of R, let θ be the unique permutation of S fixing s whose restriction to J is the permutation induced by ξ and suppose that θ fixes t. Let c be a chamber of R, let T be the unique {s, t}-residue containing c and let P = R ∩ T . Let ∆′ be a second copy of ∆ with the

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colors permuted using θ, so that we can then consider ξ as a color-preserving isomorphism from R to a residue R′ of ∆′ mapping c to a chamber c′ . Let T ′ be the unique {s, t}-residue of ∆′ containing c′ . The restriction of ξ to P induces an isomorphism from MR,P = M∆,P = MT,P to MR′ ,P ′ = M∆′ ,P ′ = MT ′ ,P ′ By 3.18, therefore, ξ can be extended to a map from T ∪ R to T ′ ∪ R′ whose restriction to T is an isomorphism from T to T ′ . By [62, 10.16], this extension of ξ has a unique extension to a special isomorphism from ∆ to ∆′ . We close this chapter with an example. Example 3.20. Let (K, L, q) be a non-trivial anisotropic quadratic space, let V = K 4 ⊕ L and let Q be the quadratic form on V over K such that Q(t1 , t′1 , t2 , t′2 , u) = t1 t′1 + t2 t′2 + q(u)

for all (t1 , t′1 , t2 , t′2 , u) ∈ V . Let Wi be the set of totally isotropic subspaces of V of dimension i for i = 1 and 2, let Q(V, Q) be the bipartite graph whose vertex set is the disjoint union W1 ∪W2 , where x ∈ W1 is adjacent to y ∈ W2 whenever x ⊂ y, and let ∆ be the corresponding chamber system of rank 2 as described in [63, 1.7-1.8]. By [61, 2.3.3], ∆ is a generalized quadrangle. We want to show that it has root group sequence isomorphic to the root group sequence QQ (K, L, q) in 1.14. For each isotropic vector u in V (i.e. for each vector u such that Q(u) = 0), we will denote simply by u the element of W1 it spans and for each pair of orthogonal isotropic vectors u and v, we will denote by uv the element of W2 they span. Next we set x1 = (1, 0, 0, 0, 0), x′1 = (0, 1, 0, 0, 0) as well as x2 = (0, 0, 1, 0, 0) and x′2 = (0, 0, 0, 1, 0) (in V ) and we identify L with its image in V under the map u 7→ (0, 0, 0, 0, u). Let Σ be the (unordered) circuit of Q(V, Q) containing the vertices x1 , x1 x′2 , x′2 , x′1 x′2 , x′1 , x′1 x2 , x2 , x1 x2

and let α = {x2 , x1 x2 , x1 , x1 x′2 , x′2 } and γ = {x1 x2 , x1 , x1 x′2 , x′2 , x′1 x′2 }

as well as

β = {x1 , x1 x′2 , x′2 , x′1 x′2 , x′1 } and δ = {x1 x′2 , x′2 , x′1 x′2 , x′1 , x′1 x2 }.

Considered as substructures of ∆, c := {x1 x′2 , x′2 } is a chamber, Σ is an apartment and α, . . . , δ are the four roots of Σ containing c.

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29

For all v, w ∈ L, let α˙ v be the linear automorphism of V that fixes x1 , x2 and x′2 , maps x′1 to x′1 − q(v)x1 + v and maps an arbitrary element u ∈ L to u − f (u, v)x1 , where f = ∂q, and let β˙w be the linear automorphism of V that fixes x1 , x′1 and x′2 , maps x2 to x2 − q(w)x′2 + w and maps an arbitrary element u ∈ L to u − f (u, w)x′2 . For all s, t ∈ K, let γ˙ t be the linear automorphism of V that fixes x1 and x′2 , acts trivially on L and maps x′1 to x′1 − tx′2 and x2 to x2 + tx1 and let δ˙s be the linear automorphism of V that fixes x′1 and x′2 , acts trivially on L and maps x1 to x1 + sx′2 and x2 to x2 − sx′1 . For all v, w ∈ L and all s, t ∈ K, the maps α˙ v , β˙ v , γ˙ t and δ˙s are all isometries of the quadratic space (K, V, Q) and hence induce automorphisms of Q(V, Q) which are simultaneously automorphisms of ∆. We denote these automorphisms by αv , βw , γt and δs . For each v ∈ L, the automorphism αv fixes x1 x2 , x1 and x1 x′2 as well as every vertex of Q(V, Q) adjacent to one of these three vertices and hence αv is contained in the root group Uα (by [60, 4.1]). Furthermore, the map v 7→ αv is an injective homomorphism from the additive group of L to Uα and the subgroup {αv | v ∈ L} acts transitively on the set of vertices adjacent to x′2 distinct from x1 x′2 . By [60, 3.7], Uα acts sharply transitively on this same set. It follows that, in fact, Uα = {αv | v ∈ L} ∼ = L. Similarly, we have Uβ = {βw | w ∈ L} ∼ = L, as well as Uγ = {γt | t ∈ K} ∼ =K and Uδ = {δs | s ∈ K} ∼ =K (where L and K here mean the additive group of L and K). Let ˆ+ = hUα , Uβ , Uδ , Uγ i. U Following the conventions in [60] that permutations are to be composed from left to right and [a, b] = a−1 b−1 ab, we then check that [δs , α−1 v ] = βsv γsq(v) for all s ∈ K and v ∈ L, [βw , α−1 v ] = γf (w,v) for all w, v ∈ L and [Uα , Uγ ] = [Uδ , Uγ ] = [Uβ , Uγ ] = [Uδ , Uβ ] = 1. Hence the maps x1 (s) 7→ δs , x2 (w) 7→ βw , x3 (t) 7→ γt and x4 (v) 7→ αv extend to an isomorphism from the group U+ generated by the root groups ˆ+ . We conclude that of the root group sequence QQ (K, L, q) to U ˆ + , Uδ , Uβ , Uγ , Uα ) (U is a root group sequence of ∆ isomorphic to QQ (K, L, q). Applying 1.16, we thus have ∆∼ = BQ 2 (K, L, q).

Chapter Four Moufang Quadrangles In this chapter we prove various results about Moufang quadrangles. These results, especially 4.10, will play a central role in Part 2 of this monograph. Hypothesis 4.1. We continue with the notation and identification in 3.5 and make the additional assumption that n = 4, so that Ω is one of the root group sequences in one of the six families (ii)–(vii) of 1.14 (but in places it will be advantageous to assume only that Ω is isomorphic to one of these root group sequences). By introducing the notions of a proper involutory set, a proper indifferent set and a proper anisotropic pseudo-quadratic space defined in [60, 35.3, 38.8 and 35.5], the six families of root group sequences in (ii)–(vii) of 1.7 can be made disjoint: Theorem 4.2. The root group sequence Ω is isomorphic to a root group sequence of exactly one of the following types: (i) QI (Λ) for some proper involutory set Λ = (D, D0 , τ ). (ii) QQ (Λ) for some non-trivial anisotropic quadratic space Λ = (K, L, q). (iii) QD (Λ) for some proper indifferent set Λ = (K, K0 , L0 ). (iv) QP (Λ) for some proper anisotropic pseudo-quadratic space Λ = (D, D0 , τ, L, q). (v) QE (Λ) for some quadratic space Λ of type E6 , E7 or E8 . (vi) QF (Λ) for some quadratic space Λ of type F4 . Proof. This holds by [60, 38.9]. Remark 4.3. In [60, 35.7–35.12] it is determined to what extent the algebraic structure Λ in 4.2 is an invariant of the corresponding Moufang quadrangle ∆. For example, if the bilinear form associated to an anisotropic quadratic space Λ = (K, L, q) is not identically zero, then QQ (Λ) ∼ = QQ (Λ′ ) ′ ′ ′ for some non-trivial anisotropic quadratic space Λ = (K , L , q ) if and only if Λ is similar to Λ′ . In particular, we can always assume in case (ii) of 4.2 that q is unital as defined in 2.20.

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Notation 4.4. Let ui ∈ Ui for i = 1 and 4. Then there exist unique ui ∈ Ui for i = 2 and 3 such that [u1 , u4 ] = u2 u3 . As in [60, 5.10], we set [u1 , u4 ]i = ui for i = 2 and 3. Remark 4.5. The root groups U1 and U3 are in each case conjugate to each other in G† as are the root groups U2 and U4 . In cases (i), (ii), (iii) and (vi) of 4.2, all the root groups are abelian. In case (iii) of 4.2, all the root groups are abelian even if the indifferent set Λ is not proper. In cases (iv) and (v) of 4.2, U2 and U4 are abelian but U1 and U3 are not, as is shown in [60, 38.10]. Remark 4.6. If Λ = (D, D0 , τ ) is an involutory set with τ 6= 1, then Λ is not proper if and only if either K is commutative or K is a quaternion division algebra, τ is its standard involution (as defined, for example, in [60, 9.6]) and K0 = Z(K). This is proved in [60, 11.3 and 23.23]. Remark 4.7. Let Λ = (D, D0 , σ, L, q) be an anisotropic pseudo-quadratic space with L 6= 0 and σ 6= 1, let Ω = QP (Λ) and let f be the skew-hermitian form associated with Λ. If D is commutative or char(D) 6= 2, then D0 is the set Dτ of fixed points of τ by [60, 11.2-11.3], hence f (v, v) 6= 0 for all v ∈ L∗ by [60, 11.19] and thus Λ is proper. Suppose that char(D) = 2. By [60, 21.16], there exists an additive subgroup B of D containing D0 such that (D, B, τ ) is an involutory set and one of the following holds: (i) Ω ∼ = QP (Λ′ ) for some proper anisotropic pseudo-quadratic space Λ′ = (D, B, τ, L′ , q ′ ) or (ii) B contains D0 properly and Ω ∼ = QI (D, B, τ ). In fact, (ii) holds if and only if f is identically zero and if (ii) holds, (D, B, τ ) is a proper involutory set by 4.6. Thus Ω is, up to isomorphism, either in case (i) of 4.2 or in case (iv), according to whether f is identically zero or not. Proposition 4.8. For i = 1 and 4, let Hi be the torus of ∆ at Ui as defined in 3.7. In other words, Hi is the subgroup of Aut(Ui ) induced by hµi (a)µi (b) | a, b ∈ Ui∗ i,

where µi is as in 3.6. Then the following hold:

(i) In case (iv) of 4.2 both H1 and H4 are non-abelian if the division ring D is non-commutative; if, however, D is commutative, then H4 is abelian, but H1 is abelian if and only if dimD L = 1. (ii) Both H1 and H4 are non-abelian in cases (i), (v) and (vi) of 4.2. (iii) In case (ii) of 4.2, H1 is always abelian, but H4 is abelian if and only if dimK L ≤ 2 or ∂q is identically zero. (iv) Both H1 and H4 are abelian in case (iii) of 4.2, even if the indifferent set Λ is not proper.

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Proof. These claims can be verified directly from the formulas in [60, 33.11– 33.15]. We give a few details, however, in those cases where it might not be so clear how to carry out this verification. Suppose first that we are in case (i) of 4.2. We want to show that H1 is nonabelian. Suppose, instead, that H1 is abelian. By [60, 33.13], it follows that [s, t]v[s, t]τ = v for all s, t, v ∈ D0∗ , where [s, t] = s−1 t−1 st. Since 1 ∈ D0 , we thus have [s, t][s, t]τ = 1 and hence [s, t]v[s, t]−1 = v for all s, t, v ∈ D0∗ . Let C = Z(D). Since Λ is proper, D is generated by D0 as a ring. It follows that [s, t] ∈ C for all s, t ∈ D0∗ , hence sC ∗ ∈ Z(D∗ /C ∗ ) for all s ∈ D0∗ and therefore Z(D∗ /C ∗ ) 6= 1. By [10, 3.9.3], however, Z(D∗ /C ∗ ) = 1. With this contradiction, we conclude that H1 is non-abelian. Suppose that we are in case (ii) of 4.2, so [60, 33.11] applies. Then by 6.8 below, H4 is abelian if and only if dimK L ≥ 2 or ∂q is identically zero. Suppose next that we are in case (iv) of 4.2, let f be as in [60, 11.19] and let T be as in [60, 11.24]. Note that L is a right vector space over D. By [60, 11.25], t 6= 0 if (a, t) ∈ T ∗ . Let ζ(a,t) (x) = x − at−1 f (a, x)

and ξ(a,t) (x) = ζ(a,t) (x)tτ for all (a, t) ∈ T ∗ and all x ∈ L. The map ζ(a,t) is D-linear and the form f is ζ(a,t) -invariant for all (a, t) ∈ T ∗ . By [60, 11.1(i) and 11.19], we have f (a, a) = t − tσ and thus (4.9)

ζ(a,t) (a) = at−1 tσ

for all (a, t) ∈ T . If (D, D0 , τ ) is proper, then H1 is non-abelian by case (i). Suppose that (D, D0 , τ ) is not proper. By 4.6, either D is commutative and τ 6= 1 or D is a quaternion division algebra, τ is its standard involution and D0 is its center. Suppose that D is commutative and dimD L > 1. By [60, 33.13], H1 is abelian if and only if the maps ξ(a,t) and ξ(b,s) commute for all (a, t), (b, s) ∈ T ∗ . Since D is commutative, it follows that H1 is abelian if and only if the maps ζ(a,t) and ζ(b,s) commute for all (a, t), (b, s) ∈ T ∗ . By [60, 11.3], D0 is the set Dτ := FixD (τ ). Thus if (a, t) ∈ T and a 6= 0, then by [60, 11.16(iii) and 11.19], also t − tσ = f (a, a) 6= 0 and thus by (4.9), the fixed point set of ζ(a,t) is hai⊥ (where ⊥ is defined with respect to f ). Since dimD L > 1, we can choose (a, t), (b, s) ∈ T such that a and b are non-zero and f (a, b) = 0. Thus (a + b, s + t) ∈ T ∗ . The map ζ(a,t) does not stabilize the subspace ha + bi and hence it does not stabilize the fixed point set ha + bi⊥ of the map ζ(a+b,s+t) . Therefore ζ(a,t) does not commute with ζ(a+b,s+t) . Hence H1 is non-abelian. By [60, 33.13], H1 is abelian if dimD L = 1. Suppose that D is quaternion, that τ is its standard involution and that F := D0 is its center. Let (a, t) ∈ T ∗ with a 6= 0, so t 6∈ F . Since t 6∈ F , there we can choose s ∈ D such that sτ ts does not commute with t. We

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have (as, sτ ts) ∈ T by [60, 11.6(ii)] and ζ(as,sτ ts) = ζ(a,t) . Thus the maps ξ(as,sτ ts) and ξ(a,t) do not commute by the choice of s. By (33.13), therefore, H1 is non-abelian. Now suppose that we are in case (v) of 4.2. Let E, v1 , . . . , vd for d = ˆ be the subspace of L 2 + 2ℓ−6 , ξ ∈ X and B1 be as in [60, 13.9], let L ˆ be the subspace of X spanned by ξ spanned by v1 = ǫ and v2 over E, let X ˆi = xi (X ˆ × K) for i = 1 and 3, let U ˆi = xi (L) ˆ for i = 2 and ξv2 over E, let U and 4 and let ˆ+ = U ˆ1 U ˆ2 U ˆ3 U ˆ4 ⊂ U+ . U ˆ X) ˆ ⊂L ˆ and h(a, X) ˆ 6= 0 for each non-zero a ∈ X ˆ By [60, 13.18–13.19], h(X, ˆ L) ˆ ⊂ L, ˆ aθ(a, ǫ) ⊂ X ˆ for all a ∈ X ˆ and and by [60, 13.9 and 13.40], θ(X, ˆ ·L ˆ ⊂ X, ˆ where h, θ, ǫ and the map (a, v) 7→ a · v are as in [60, 16.6]. By X [60, 8.7 and 32.10], therefore,
ˆ= U ˆ+ , U ˆ1 , U ˆ2 , U ˆ3 , U ˆ4 Ω

is a root group sequence. Furthermore, this root group sequence is an exˆ qˆ), where qˆ is the restriction tension (in the sense of [60, 21.5]) of QQ (K, L, ˆ By [60, 21.12], Ω ˆ ∼ of qˆ to L. = QP (Ξ) for some proper pseudo-quaternion algebra over a triple (D, K, σ), where D is a quaternion division algebra over K with standard involution σ. By the conclusion of the previous paragraph, it follows that H1 is non-abelian. Suppose, finally, that we are in case (vi) of 4.2. Then there is a subsequence of the root group sequence QF (Λ) isomorphic to QQ (Λ) that shares the same last term U4 . By case (ii), therefore, H4 is non-abelian. By [60, 28.45], it follows that H1 is also non-abelian. Proposition 4.10. Let ∆ and Ω = (U+ , U1 , U2 , U3 , U4 ) be as in 4.1, let Λ = (K, L, q) be a non-trivial anisotropic quadratic space, let 1 be a distinguished element of L such that q(1) = 1 and suppose that 1 is contained in the defect of q in those cases where the defect of q is non-trivial. Let K be identified with its image in L under the map t 7→ t · 1 and let Wi be a proper subgroup of Ui for i = 1 and 3 such that M := W1 U2 W3 U4 is a subgroup of U+ and ΩM := (M, W1 , U2 , W3 , U4 ) is a root group sequence isomorphic to QQ (Λ). Let κ be an isomorphism from ΩM to QQ (Λ) and let ΩM be identified with QQ (Λ) via κ. Then one of the following holds: (a) There is a multiplication on L extending the multiplication on K such that L/K is either a separable quadratic extension with norm q or L is a quaternion division algebra with center K and norm q, and there is an anisotropic pseudo-quadratic space Θ = (L, K, σ, X, Q) and an isomorphism from Ω to QP (Θ) mapping x4 (1) to x4 (1), where σ is either the unique non-trivial element of Gal(L/K) if dimK L = 2 or the standard involution of L if dimK L = 4.

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(b) Λ is of type Eℓ for ℓ = 6, 7 or 8, Ω ∼ = QE (Λ) and there is an isomorphism from Ω to QE (Λ) mapping x4 (1) to x4 (ǫ), where ǫ ∈ L is as in [60, 16.6]. (c) Λ is of type F4 , Ω ∼ = QF (Λ) and there is an isomorphism from Ω to QF (Λ) mapping x4 (1) to x4 (0, 1). (d) There is a multiplication on L extending the multiplication on K such that L/K is an inseparable field extension with u2 = q(u) ∈ K for all u ∈ L, and there is an anisotropic quadratic space Θ = (L, V, Q) whose associated bilinear form is not identically zero and an anti-isomorphism from Ω to QQ (Θ) mapping x4 (1) to x1 (1). (e) There is an isomorphism from Ω to QD (Θ) for some indifferent set Θ = (D, D0 , E0 ). In (a)–(b), ∂q is non-degenerate. In (c), ∂q is degenerate and in (d)–(e) it is, in fact, identically zero. Proof. Let f = ∂q and let µ4 be as in 3.6. Suppose first f is not zero. Let Yi for all i be as in [60, 21.1] and let N = Y1 U2 Y3 U4 . then Y3m = Y1 and W3m = W1 for all m ∈ µ4 (U4∗ ). Since f is not zero, we have [U2 , U4 ] 6= 1 by [60, 16.3] and hence Yi 6= Ui for i by [60, 21.20(ii)]. By [60, 21.4], therefore, N is a subgroup and

identically Note that identically = 2 and 4

ΩN := (N, Y1 , U2 , Y3 , U4 ) is a root group sequence; let Γ0 be the corresponding subquadrangle of ∆. Thus either Γ0 = ∆ (precisely when Y1 = U1 ), or ∆ is an extension of Γ0 as defined in [60, 21.6]. By [60, 7.4], the quadrangle ∆ can be identified with the quadrangle G(Σ, U+ , φ) defined in [60, 7.2]. Let Γ be the subgraph of ∆ that corresponds to the subgraph G(Σ, M, φ) of G(Σ, U+ , φ). Thus Γ contains the apartment Σ and by [60, 8.11], Γ is a Moufang quadrangle with root group sequence ΩM . In particular, Γ is isomorphic to BQ 2 (Λ). Since f is not identically zero, we have W3 = [U2 , U4 ] by [60, 16.3]. By [60, 21.3], either Y3 = U3 (i.e. [U1 , U3 ] = 1) or [U2 , U4 ] ⊂ Y3 . One way or the other, we thus have Wi ⊂ Yi for i = 1 and 3. Hence Γ ⊂ Γ0 . We have Γ 6= ∆ by hypothesis. Thus if Γ0 = Γ, then we can apply [60, 21.12] to conclude that either (a), (b) or (c) holds. The fact that the isomorphism can be chosen so that x4 (1) is mapped to x4 (1) in case (a) holds by [63, 8.1 and 11.13] as does the analogous assertion in case (b); in case (c) the fact that the isomorphism can be chosen so that x4 (1) is mapped to x4 (0, 1) follows from [60, 14.3 and 14.10] and the assumption that 1 lies in the defect of q. We suppose now that Γ is contained properly in Γ0 , equivalently, that Wi is contained properly in Yi for i = 1 and 3. In particular, [U2 , U4 ] = W3 6= Y3 . It follows from this that Γ0 is not of quadratic form type (i.e. as defined in [60, 16.3]). The quadrangle Γ0 is reduced as defined in [60, 21.2]. By

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[60, 21.8-21.9], therefore, we can identify ΩN with QI (F, B, σ) for some proper involutory set (F, B, σ). Thus the inclusion M ⊂ N gives rise to an injective homomorphism ϕ from the trunk of QQ (Λ) (as defined in 3.2) to the trunk of QI (F, B, σ) that sends the i-th term to the i-th term for all i ∈ [1, 4], bijectively for i = 2 or 4. By [60, 35.16], we can assume that ϕ(x1 (1)) = x1 (1). For each b ∈ F ∗ , there exists an automorphism of QI (F, B, σ) extending the automorphism x4 (a) 7→ x4 (ab) of U4 and acting trivially on U1 (by [60, 16.2] or [60, 37.33]). This means that we can also assume that ϕ(x4 (1)) = x4 (1). Next we let ξ be the map from K to B such that ϕ(x1 (t)) = x1 (ξ(t)) for all t ∈ K and we let ψ be the map from L to F such that ϕ(x4 (v)) = x4 (ψ(v)) for all v ∈ L. Since the restriction of ϕ to each root group is a homomorphism, the maps ξ and ψ are additive. Since ξ(1) = 1, we have (see 4.4) ϕ(x2 (v)) = ϕ([x1 (1), x4 (v)−1 ]2 ) = [x1 (1), x4 (ψ(v))−1 ]2 = x2 (ψ(v)) for all v ∈ L and since ψ(1) = 1, we have

ϕ(x3 (t)) = ϕ([x1 (t), x4 (1)−1 ]3 ) = [x1 (ξ(t)), x4 (1)−1 ]3 = x3 (ξ(t))

for all t ∈ K. We thus have x2 (ψ(tv)) = ϕ(x2 (tv)) = ϕ([x1 (t), x4 (v)−1 ]2 )


= [x1 (ξ(t)), x4 (ψ(v))−1 ]2 = x2 ξ(t)ψ(v)

and hence ψ(tv) = ξ(t)ψ(v) for all t ∈ K and all v ∈ L. In particular, ξ is multiplicative. Hence ξ(K) is a subfield of F contained in B. Next we observe that x3 (ξ(f (v, 1))) = ϕ(x3 (f (v, 1))) = ϕ([x2 (v), x4 (1)−1 ]) = [x2 (ψ(v)), x4 (1)−1 ] = x3 (ψ(v) + ψ(v)σ ) and hence ψ(v) + ψ(v)σ = ξ(f (v, 1)) for all v ∈ L. Thus if we identify this subfield with K via ξ, then Fσ = K, where Fσ is as defined in [60, 11.1]. By [60, 35.3], σ is not the identity and F = hBi. Since Y1 6= W1 (by assumption), we have K 6= B. By [60, 11.2], therefore, char(K) = 2, and by [60, 11.3], F is not commutative. By [60, 23.23] applied to the involutory set (F, Fσ , σ), we conclude that F is a quaternion division algebra, σ is its standard involution and K = Z(F ). Suppose now that Yi 6= Ui for i = 1 and 3. Thus ∆ is an extension of Γ0 . By [60, 21.11], there exist V and QV such that (F, B, σ, V, QV ) is an anisotropic pseudo-quadratic space and Ω ∼ = QP (F, B, σ, V, QV ). By [60, 16.17], there exist X and Q such that (F, K, σ, X, Q) is an anisotropic pseudo-quadratic space and ∼ QP (F, K, σ, X, Q). QP (F, B, σ, V, QV ) = Thus (a) holds.

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Suppose, finally, that Yi = Ui for i = 1 and 3. Thus Ω = ΩN . Let V = B/K. We give V the structure of a right vector space over F as described in [60, 11.2] and let Q be the map from V to F (defined only modulo K) given by Q(v + K) = v for all v ∈ B. Then Ξ = (F, K, σ, V, Q) is an anisotropic pseudo-quadratic space as defined in [60, 11.16] whose associated skew-hermitian form is identically zero. Furthermore, the maps xi (v) 7→ xi (v + K, v) for i = 1 and 3 and xi (u) 7→ xi (u) for i = 2 and 4 extend to an isomorphism from the root group sequence ΩN = QI (F, B, σ) to QP (Ξ). Thus (a) holds once again. (It is, however, more natural to think of Ω as QI (F, B, σ) in this case.) We turn now to the case that the bilinear form f is identically zero. Thus [U2 , U4 ] = 1 and char(K) = 2. If [U1 , U3 ] = 1, then (e) holds by 4.2. We can assume, therefore, that [U1 , U3 ] 6= 1. Suppose that ∼ QI (F, B, σ) (4.11) Ωop =

for some proper involutory set (F, B, σ). Let

H = hµ(x4 (u))µ(x4 (v)) | u, v ∈ L∗ i

and let D denote the subgroup of F ∗ generated by B ∗ . By [60, 11.3], F is not commutative. By [60, 35.3], therefore, D is non-abelian. By [60, 32.9], we find—taking note that it is Ωop and not Ω that appears in (4.11)—that the permutation group induced by H on U3 is isomorphic to the permutation group {u 7→ tu | t ∈ D} on K. Hence the commutator group [H, H] has no fixed points in U3∗ . Since ΩM ∼ = QQ (Λ), however, H induces an abelian group on W3 (by [60, 32.7]) and thus [H, H] centralizes W3 . With this contradiction, we rule out (4.11). By 4.2, we conclude that Ωop ∼ = QQ (F, V, Q)

for some anisotropic quadratic space (F, V, Q) whose associated bilinear form is not identically zero. Since char(K) = 2, the root group U4 is of exponent 2 and thus char(F ) = 2. By [60, 37.30], the automorphism group of QQ (F, V, Q) acts transitively on U1∗ . We can thus choose an antiisomorphism ϕ from Ω to QQ (F, V, Q) mapping x4 (1) to x1 (1). Let ξ be the additive bijection from L to F such that ϕ(x4 (v)) = x1 (ξ(v)) for all v ∈ L and let ψ be the additive injection from K to V such that ϕ(x1 (t)) = x4 (ψ(t)) for all t ∈ K. Since ϕ(x4 (1)) = x1 (1), we have ϕ(x3 (t)) = ϕ([x1 (t), x4 (1)]3 ) = [x1 (1), x4 (ψ(t))]2 = x2 (ψ(t))

for all t ∈ K. Therefore

x2 (ψ(tq(v))) = ϕ(x3 (tq(v))) = ϕ([x1 (t), x4 (v)]3 ) = [x1 (ξ(v)), x4 (ψ(t))]2 = x2 (ξ(v)ψ(t))

and hence (4.12)

ψ(tq(v)) = ξ(v)ψ(t)

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for all v ∈ L and all t ∈ K. Thus ξ(v)2 ψ(1) = ξ(v)ψ(q(v)) = ψ(q(v)2 ) and ξ(q(v))ψ(1) = ξ(q(v) · 1)ψ(1) = ψ(q(q(v) · 1)) = ψ(q(v)2 ) and hence ξ(v)2 = ξ(q(v)) for all v ∈ L. By (4.12) again, ξ(t)ψ(r) = ξ(t · 1)ψ(r) = ψ(q(t · 1)r) = ψ(t2 r)

for all r, t ∈ K. Hence ξ(s)ξ(t)ψ(1) = ψ(s2 t2 ) = ψ((st)2 ) = ξ(st)ψ(1) for all s, t ∈ K. It follows that the restriction of ξ to K = K ·1 ⊂ L is multiplicative. Thus (d) holds. Proposition 4.13. Let K, L, etc., be as in 4.10 and suppose that m := dimK L is finite. Then m = 2 or 4 in case (a), m = 4 + 2ℓ−5 in case (b), m = 4 + 2t for some t > 0 in case (c), m = 2t for some t ≥ 0 in case (d) and dimK L is even in case (e). Proof. In case (a) of 4.10, there is nothing to prove. We have m = 4 + 2ℓ−5 in case (b) by [60, 12.31]. The degree of a finite purely inseparable field extension is a power of the characteristic. It follows that m is a power of 2 in case (d) and by [60, 14.10], that m = 4 + 2t for some t > 0 in case (c). Suppose, finally, that we are in case (e), so Ω ∼ = QD (Θ) for some indifferent set Θ = (D, D0 , E0 ). The inclusion M ⊂ U+ gives rise to an injective homomorphism ϕ from the trunk of QQ (Λ) to the trunk of QD (Θ) mapping Ui to Ui for each i ∈ [1, 4], bijectively for i = 2 and 4. By [60, 16.3 and 35.18], we can assume that ϕ(xi (1)) = xi (1) for i = 1 and 4. Let ρi be the map from K to D0 for i = 1 and 3 such that ϕ(xi (t)) = xi (ρi (t)) for all t ∈ K and let ψi be the map from L to E0 for i = 2 and 4 such that ϕ(xi (v)) = xi (ψi (v)) for all v ∈ L. Applying ϕ to the identity [x1 (1), x4 (v)]2 = x2 (v),

we conclude that ψ2 = ψ4 and applying ϕ to the identity [x1 (t), x4 (1)]3 = x3 (t), we conclude that ρ1 = ρ3 . Let ρ = ρ1 and ψ = ψ4 . Applying ϕ to the identity [x1 (t), x4 (v)]2 = x2 (tv), we conclude, finally, that (4.14)

ψ(tv) = ρ(t)2 ψ(v)

for all t ∈ K and all v ∈ L. Now let ξ(t) = ρ(t)2 ⊂ D02 for all t ∈ K. It follows from (4.14) that F := ξ(K) is a subfield of D02 , that ξ is an isomorphism of fields from K to F and that the pair (ξ, ψ) is an isomorphism of vector spaces from (K, L) to (F, E0 ). Since W1 6= U1 , we have ρ(K) 6= D0 and hence F 6= D02 . Thus F is a subfield of D2 and [D2 : F ] > 1. By [60, 10.2], D2 ⊂ E0 . Now let w ∈ D0 . By [60, 10.1(i)], w2 ∈ D0 ⊂ E0 , so we can set v = ψ −1 (w2 ). Then x3 (w2 ) = [x1 (1), x4 (w2 )]3 = ϕ([x1 (1), x4 (v)]3 ) = x3 (ρ(q(v)))

MOUFANG QUADRANGLES

39

and hence w2 ∈ ρ(K). Therefore D04 ⊂ F . Since D0 generates D as a ring (by [60, 10.1(ii)]), it follows that D4 ⊂ F . Thus D2 /F is a purely inseparable extension. Hence [D2 : F ] = 2r for some positive integer r, so m = dimK L = dimF E0 = 2r · dimD2 E0 . Thus m is even. Proposition 4.15. Let ∆, Σ, W1 , U1 , 1 ∈ L, etc., be as in 4.10 and let m = µ(x4 (1)), where µ = µ4 is as in 3.6. Then in each case of 4.10, the group W1 is a normal subgroup of U1 , the quotient group X := U1 /W1 is abelian and X has the structure of a vector space over K such that [a, x4 (t · 1)−1 ]m 3 ≡ b (mod W1 ) for a, b ∈ U1 and t ∈ K if and only if ta ≡ b (mod W1 ). Proof. Suppose that we are in case (c) and let ϕ be the isomorphism from Ω to QF (Λ) in 4.10(c). By [60, 7.5], we can think of ϕ as an isomorphism from ∆ to the Moufang quadrangle corresponding to QF (Λ)—called BF 2 (Λ) in [65, 30.15]—which maps the apartment Σ to an apartment of BF (Λ), and by [60, 2 6.1] applied with respect to Σ and ϕ(Σ), we have ϕ(m) = µ(ϕ(x4 (1))) = µ(x4 (0, 1)). To prove the assertions, we can thus identify Ω with QF (Λ). Similar remarks hold in all the other cases. By [60, 32.6–32.11], x3 (v)m = x1 (v) for all v in the corresponding parameter group, which is T in case (a), S in case (b), X ⊕ K in case (c), V in case (d) and D0 in case (e). (Note that in case (c) we are writing X to denote the group called X0 in [60].) Therefore the claims follow from the commutator relations in [60, 16.3-16.7]. Proposition 4.16. Let K, L, etc., be as in 4.10, let m = dimK L, let X be as in 4.15 and let z = dimK X. If m and z are finite, then the following hold: (i) z = sm, where s is the dimension over L of the right vector space X in the anisotropic pseudo-quadratic space Θ in case (a) of 4.10. (ii) z = 2ℓ−3 in case (b). (iii) m = 4 + 2t for some t > 0 and z = 2t+2 in case (c). (iv) z = sm for some s ≥ 1 and W1 is a non-trivial subspace of the vector space V of co-dimension s over L in case (d). Proof. Suppose we are in case (a). We identify Ω with QP (Λ) via the isomorphism in 4.10(a). We can then identify the group X in 4.10(a) with the group X in 4.15. The group X in 4.10(a) is, in fact, a right vector space over L. Restricting scalars to K, we obtain the vector space structure on X in 4.15. Thus (i) holds with s = dimL X. In case (b), z = 2ℓ−3 by [60, 13.9]. Thus (ii) holds. Suppose we are in case (c), so W1 = x1 (0, K). By 4.13, m = 4 + 2t for some t > 0, where 2t = dimK F with respect to the

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scalar multiplication ∗ defined in [60, 14.10]. Identifying Ω with QF (Λ) via the isomorphism in 4.10(c), we have 1 = (0, 1) ∈ W ⊕ F , where W is the group called W0 in [60, 16.7], and hence t · 1 = t(0, 1) = (0, t ∗ 1) = (0, t2 ). We can identify the group X0 in 4.15 with the group called X0 in [60, 16.7]. The group X = X0 in [60, 16.7] is, in fact, a 4-dimensional vector space over F . If we denote by s × x the product of a vector x ∈ X by a scalar s ∈ F , then the map (t, x) 7→ t2 × x is the scalar multiplication from K × X to X defined in 4.15. Hence z = 4 · dimK F = 4 · 2t . Thus (iii) holds. Suppose, finally, that Ω is in case (d). We identify Ω with QQ (L, V, Q)op via the antiisomorphism in 4.10(d). Since [W1 , U4 ]3 = W3 , the group W1 is a subspace of the L-vector space V and X is thus a quotient of vector spaces over L. Restricting scalars to K, we obtain the vector space structure on X in 4.15. Hence z/m = dimL X. Thus (iv) holds. Remark 4.17. Suppose that we are in case (d) of 4.10 and that [W1 , U3 ] = 1. By [60, 16.3], this means that W1 is contained in the defect of Q. Hence the defect of Q is non-trivial and the co-dimension of the defect of Q is at most s = z/m. Since ∂Q is not identically zero, this co-dimension is also at least 2. Thus s ≥ 2 and dimL V ≥ 3. Proposition 4.18. Let K, L, etc., be as in 4.10. Then the following hold: (i) In case (a) of 4.10, the group U4 is abelian; the group U1 is abelian if and only if the skew-hermitian form associated with the anisotropic pseudo-quadratic space Θ in 4.10(i) is identically zero, in which case dimK L = 4, char(K) = 2 and there is an additive subgroup B of L such that (L, B, σ) is a proper involutory set and Ω ∼ = QI (L, B, σ). (ii) In case (b), the group U1 is non-abelian and the group U4 is abelian. (iii) In cases (c)–(e), the groups U1 and U4 are both abelian. Proof. This holds by 4.5 and 4.7. Proposition 4.19. Let H1 and H4 be as in 4.8, let K, L, m, z, etc., be as in 4.16 and suppose that m and z are finite. Then the following hold: (i) In case (a) of 4.10, H1 is abelian if and only if m = 2 and z = 1, and H4 is abelian if and only if m = 2. (ii) In cases (b) and (c), both H1 and H4 are non-abelian. (iii) In case (d), H4 is abelian, and if [W1 , U3 ] = 1, then H1 is non-abelian. (iv) In case (e), both H1 and H4 are abelian. Proof. This holds by 4.7, 4.8 and 4.17.

Chapter Five Linked Tori, I In Chapter 10 we will study pairs of residues R0 and R1 of a Bruhat-Tits building Ξ of type C˜2 . These residues are both Moufang quadrangles with a distinguished root group. Furthermore, there is a natural identification of the two root groups coming from the embedding in Ξ, and (as we will see in 10.19) the torus of R0 at its distinguished root group and the torus of R1 at its distinguished root group are linked as defined in 3.11. This observation will play an important role in our investigations. In preparation for applications of this observation, we investigate in this chapter and the next the consequences of the assumption that one Moufang set is weakly isomorphic to another. In this chapter, we consider the case that one of the two Moufang sets is BQ 1 (Λ) for some anisotropic quadratic space Λ (as defined in 3.8) and the other is BI1 (Ξ) for some involutory set Ξ. Our main results are 5.18–5.21. They will be applied in the proof of 13.15. Notation 5.1. Let Λ = (K, L, q) be a non-trivial anisotropic quadratic space and let f = ∂q. For each a ∈ L∗ , we let πa denote the reflection of L given by
πa (x) = x − f (x, a)/q(a) a for all x ∈ L. Note that for all a ∈ L∗ , πa (a) = −a, πa2 = 1, q(πa (b)) = q(b) for all b ∈ L and a⊥ := {x ∈ L | f (x, a) = 0} is the fixed point set of πa . We set

mΛ a,b (x) = q(a)/q(b) πa πb (x) for all a, b ∈ L∗ and for all x ∈ L.

Note that if a, b ∈ L∗ and q(b) = 1, then (5.2)

mΛ a,b (b) = f (a, b)a − q(a)b.

Notation 5.3. Let D be a division ring with center F such that its dimension over F is finite. By [31, 1.1], dimF D = n2 for some n (called the degree of D). We have n = 2 if and only if D is quaternion. For each involution τ of D, we set (5.4)

Dτ = {x ∈ D | xτ = x} and Dτ = {x + xτ | x ∈ D}.

The elements of Dτ are called traces with respect to τ . If τ is of the first kind (i.e. if F ⊂ Dτ ), then Dτ and Dτ are vector spaces over F . If τ is of

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the second kind, then k := F ∩ Dτ is a subfield of F , F/k is a separable quadratic extension and Dτ and Dτ are vector spaces over k. We will need some well known facts about involutions which we assemble in a few lemmas: Lemma 5.5. Let (D, D0 , τ ) be an involutory set. If char(D) 6= 2 or τ is of the second kind, then Dτ = D0 = Dτ . Proof. This holds by [60, 11.2 and 11.5]. Lemma 5.6. If τ is an involution of the first kind of D, then dimF Dτ = n(n − ǫ)/2 for ǫ = 1 or −1. If τ is an involution of the second kind and k = F ∩ Dτ , then dimk Dτ = dimk Dτ = n2 . Proof. This holds by [31, 2.6 and 2.17]. Lemma 5.7. Suppose that the degree n of D is 4 and that τ is an involution of the first kind. Then D is a biquaternion algebra. Proof. Since D has an involution of the first kind, it is, in particular, isomorphic over K to its opposite. This means that the image of D in the Brauer group is equal to its own inverse. The claim holds, therefore, by [31, 16.1]. Lemma 5.8. Suppose that D is quaternion and that char(D) = 2 and let σ be the standard involution of D. Then the following hold: (i) F = Dσ and dimF Dσ = 3. (ii) For each non-zero u ∈ Dσ , let xρu = uxσ u−1 for all x ∈ D. The map ρu is an involution of the first kind of D (for each non-zero u ∈ Dσ ), Dρu = uDσ = Dσ u−1 , Dρu = uDσ = F u and if v is another non-zero element of Dσ , then ρu = ρv if and only if F u = F v. (iii) Every involution of the first kind of D is of the form ρu for some nonzero u in Dσ . (iv) If τ is an involution of the first kind, then x2 ∈ F for all x ∈ Dτ ; furthermore, τ = σ if and only if Dτ ∩ F 6= 0. Proof. Assertions (i) and (ii) follow immediately from [60, 9.6] and assertion (iii) holds by [31, 2.7]. Let τ = ρu for some u ∈ Dσ . By (ii), we have Dτ = F u. Since u ∈ Dσ , we have (tu)2 = t2 · uuσ ∈ F for all t ∈ F . Furthermore, τ = σ if and only if Dτ = Dσ . Since Dτ = F u and Dσ = F , we conclude that τ = σ if and only if F ∩ Dτ 6= 0. Thus (iv) holds. Notation 5.9. Let Ξ = (D, D0 , τ ) be an involutory set (as defined in [60, 11.1]). We set −1 −1 mΞ xb a a,b (x) = ab

for all a, b ∈ D0∗ and for all x ∈ D0 .

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Hypothesis 5.10. For the remainder of this chapter, we make the following assumptions: (a) D, F and n are as in 5.3, n > 1, Ξ = (D, D0 , τ ) is an involutory set and if D is quaternion and τ is the standard involution of D, then D0 6= F . (b) K is a subfield of F such that m := [F : K] < ∞, K ⊂ D0 , D0 = KD0 and dimK D0 ≥ 3. In particular, dimK D = mn2 . (c) Λ = (K, D0 , q) is a pointed anisotropic quadratic space with base point 1 ∈ D0 (as defined in 2.20), f = ∂q, R = {v ∈ D0 | f (v, D0 ) = 0}, so R is the defect of q, and if R 6= 0, then 1 ∈ R. Ξ Λ Ξ (d) mΛ a,b = ma,b for all non-zero a, b ∈ D0 , where m and m are as in 5.1 and 5.9.

Lemma 5.11. The division ring D is generated as a ring by D0 . Proof. This holds by 5.10(a) and [60, 21.14 and 23.23]. Lemma 5.12. Let a, b ∈ D0 . Then the following hold: (i) a2 = f (a, 1)a − q(a); (ii) ab + ba = f (b, 1)a + f (a, 1)b − f (a, b). Proof. By (5.2), 5.9 and 5.10(d), we have Λ a 2 = a · 1 · a = mΞ a,1 (1) = ma,1 (1) = f (a, 1)a − q(a)

if a 6= 0. Thus (i) holds; (ii) follows by applying (i) to a + b, a and b. Proposition 5.13. Let R be as in 5.10(c). If R 6= 0, then F ∩ D0 = R, and if R = 0, then F ∩ D0 = K. Proof. Suppose that R 6= 0. Then char(D) = 2 (by 2.11). By 5.10(c), we have 1 ∈ R and hence f (a, 1) = 0 for all a ∈ D0 . By 5.12(ii), therefore, an element of D0 centralizes D0 if and only if it lies in R. By 5.11, the centralizer of D0 in D is F . Thus F ∩ D0 = R. We turn to the case that R = 0. We have K ⊂ F ∩ D0 by 5.10(b). Hence we only have to show F ∩ D0 ⊂ K. We treat the cases char(D) = 2 and char(D) 6= 2 separately. Suppose that char(D) = 2. Thus K ⊂ 1⊥ . Let c ∈ F ∗ ∩ D0 . By 5.12(ii), we have (5.14)

f (c, 1)x + f (x, 1)c + f (x, c) = xc + cx = 0

for all x ∈ D0 . If a ∈ 1⊥ \ K, then f (c, 1)a = f (a, c) ∈ K by (5.14) and thus f (a, c) = 0. Hence 1⊥ \ K ⊂ c⊥ . As dimK D0 ≥ 3 by 5.10(b), we have dimK 1⊥ ≥ 2 and therefore 1⊥ \ K spans 1⊥ . It follows that 1⊥ ⊂ c⊥ . Since

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R = 0, the subspaces 1⊥ and c⊥ are both of co-dimension 1 in D0 over K. Hence 1⊥ = c⊥ and thus 1 and c span the same 1-dimensional subspace of D0 . Thus c ∈ K. Therefore F ∩ D0 ⊂ K. Now assume that char(D) 6= 2 and choose c ∈ F ∩ D0 . Suppose that c ∈ 1⊥ . As dimK D0 ≥ 3 we can find a non-zero element a in c⊥ ∩ 1⊥ . By 5.12(ii), 2ac = ca + ac = 0. Hence c = 0. We conclude that F ∩ D0 ∩ 1⊥ = 0.

It follows that dimK (F ∩ D0 ) ≤ 1 because 1⊥ is a subspace of D0 of codimension 1 over K. Hence F ∩ D0 ⊂ K also in this case. Proposition 5.15. Suppose that D is quaternion, that char(D) = 2 and that τ is of the first kind but not the standard involution of D. Then X := K +Dτ is a K-subspace of dimension 1+m, where m is as in 5.10(b), and f (x, y) = 0 for all x, y ∈ X. Proof. Let σ be the standard involution of D. By 5.8(iii), there is a non-zero element u ∈ Dσ such that xτ = uxσ u−1 for all x ∈ D. By 5.8(ii), we have u 6∈ F (since τ 6= σ) and Dτ = F u. Hence K ∩ Dτ ⊂ F ∩ Dτ ⊂ F ∩ F u = 0, so dimK X = 1 + dimK (F u) = 1 + m. Let x ∈ Dτ . By 5.8(iv), we have x2 ∈ F and by 5.12(i), we have x2 = f (x, 1)x − q(x).

Since F ∩ Dτ = 0 and q(x) ∈ K ⊂ F , it follows that f (x, 1) = 0. Hence (5.16)

Dτ ⊂ 1 ⊥ .

Since Dτ = F u, we have xy = yx for all x, y ∈ Dτ . By 5.12(ii), therefore, f (x, 1)y + f (y, 1)x + f (x, y) = xy + yx = 0 for all x, y ∈ Dτ . By (5.16), we conclude that f (x, y) = 0 for all x, y ∈ X. Lemma 5.17. Suppose that D is quaternion and that τ is of the first kind and let σ be the standard involution of D. If R 6= 0, then char(D) = 2 and τ = σ. Proof. Suppose that R 6= 0. We have char(D) = 2 by 2.11, D0 ⊂ Dτ by [60, 11.1(i)] and dimF Dτ = 3 by 5.8(ii)-(iii). Let M denote the F -subspace of D spanned by D0 . If dimF M ≤ 2, it would follow that M is a subfield of D (since 1 ∈ D0 ). By 5.11, we conclude that dimF M = 3 and hence M = Dτ . By 5.10(c), 1 ∈ R. Hence x2 = q(x) ∈ K ⊂ F for all x ∈ D0 by 5.12(i). Since Dσ = {x ∈ D | x2 ∈ F }, it follows that D0 ⊂ Dσ . As Dσ is an F -subspace of D and Dτ is spanned by D0 as a vector space over F , we conclude that Dτ ⊂ Dσ . Since both Dτ and Dσ are of dimension 3 over F , they coincide. Hence the composition τ σ is an automorphism of D acting trivially on Dσ . As Dσ generates D as a ring (by 5.11), we conclude that τ = σ.

LINKED TORI, I

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We now come to the main results of this chapter. Proposition 5.18. Given 5.10, suppose that dimK D0 = 6, dimK D = 16 and R = 0. Then D is biquaternion, τ is of the first kind and D0 = Dτ . Proof. From mn2 = 16, we see that there are two possible cases, (n, m) = (4, 1) and (n, m) = (2, 4). Suppose (n, m) = (4, 1). Then F = K ⊂ D0 ⊂ D τ . Hence τ is of the first kind and as the degree of D is 4, it follows from 5.7 that D is biquaternion. By 5.6, we have dimK Dτ = dimF Dτ ≥ 6. Since Dτ ⊂ D0 , we obtain D0 = Dτ from the assumption dimK D0 = 6. It remains to show that the case (n, m) = (2, 4) is not possible. Assume that (n, m) = (2, 4). The intersection k := F ∩Dτ is a subfield of F such that dimk F ≤ 2. Since m = 4, we have k 6= K. As R = 0, we have F ∩ D0 = K by 5.13. Therefore F ∩D0 6= F ∩Dτ and thus D0 6= Dτ . By 5.5, we conclude that char(D) = 2 and τ is of the first kind. Since F ∩ D0 = K and K 6= F , we cannot have F ⊂ D0 . Thus τ is not the standard involution since otherwise F = Dτ ⊂ D0 . By 5.15, therefore, the K-subspace X := K + Dτ of D0 has dimension 5 and f (x, y) = 0 for all x, y ∈ X. As R = 0 and dimK D0 = 6, we obtain a contradiction. Proposition 5.19. Given 5.10, suppose that dimK D0 = 4, dimK D = 8 and R = 0. Then D is quaternion and τ is of the second kind. Proof. Since mn2 = 8 and n > 1, we have (n, m) = (2, 2) and D is a quaternion algebra. As R = 0, 5.13 yields K = F ∩ D0 . Since K 6= F , it follows that F is not contained in D0 . Suppose that τ is of the first kind. Then char(D) = 2, since otherwise F ⊂ Dτ = D0 by 5.5, and τ is not the standard involution, since otherwise F = Dτ ⊂ D0 . By 5.15, therefore, the K-subspace X := K + Dτ of D0 has dimension 3 and f (x, y) = 0 for all x, y ∈ X. As dimK D0 = 4 and R = 0, we obtain a contradiction. Hence τ is of the second kind. Proposition 5.20. Given 5.10, suppose that dimK D0 = 6, dimK D = 16 and R 6= 0. Then D is quaternion, τ is the standard involution and R = F . Proof. Since R 6= 0, we have char(D) = 2 and, by 5.10(c), K ⊂ R. By 5.13, we have R = F ∩ D0 . By 2.33, the co-dimension over K of R in D0 is even. Since dimK D0 is even, it follows that also dimK R is even. In particular, dimK R > 1. Therefore K is properly contained in R and hence also in F , so m > 1. Therefore (n, m) = (2, 4) since by 5.10(a), n > 1. Thus D is a quaternion algebra. Suppose that τ is of the second kind and put k = F ∩ Dτ . We have Dτ = D0 by 5.5 and hence dimk D0 = 4 by 5.6. Since K ⊂ k, it follows that dimK D0 is a multiple of 4. This contradicts the assumption that dimK D0 = 6. We conclude that τ is of the first kind. We can now apply 5.17 to conclude that τ is the standard involution. Hence F = Dτ ⊂ D0 , so R = F ∩ D0 = F .

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Proposition 5.21. Given 5.10, suppose that dimK D0 = 4, dimK D = 8 and R 6= 0. Then D is quaternion, τ is the standard involution and R = F . Proof. As in the proof of 5.20, char(D) = 2, F ∩ D0 = R, K is properly contained in R and m > 1. Hence (n, m) = (2, 2). In particular, D is a quaternion algebra. Suppose that τ is of the second kind. Then Dτ = D0 by 5.5 and dimK (F ∩ Dτ ) = m/2 = 1, but then K = F ∩ Dτ = F ∩ D0 = R.

With this contradiction, we conclude that τ is of the first kind. By 5.17, therefore, τ is the standard involution. Hence F = Dτ ⊂ D0 and therefore F = F ∩ D0 = R.

Chapter Six Linked Tori, II In this chapter we prove several more results about weak isomorphisms between Moufang sets arising from quadratic forms and involutory sets. Our main results are 6.13, 6.27 and 6.28. They will be needed in the proofs of 12.8, 12.9, 13.12 and 17.3. We would like to thank T. De Medts and R. Knop for some of the ideas used in the proofs we give. We fix a non-trivial anisotropic quadratic space Λ = (K, L, q) and let f ∗ as well as πa and a⊥ and for each a ∈ L∗ and mΛ a,b for all a, b ∈ L be as in 5.1. Let R denote the defect of q. Remark 6.1. Suppose that u and v are elements of L such that f (u, v) 6= 0. If char(K) = 2, then dimK hu, vi = 2 and hu, vi ∩ R = 0 and if char(K) 6= 2, then R = 0. Thus if dimK L ≥ 2 and R 6= L, then dimK L/R ≥ 2 (in every characteristic). Lemma 6.2. Let a, b ∈ L∗ . Then the following hold: (i) πa πb πa = ππa (b) . (ii) πa = πb if and only if ha, bi is 1-dimensional or a and b are both in R. Proof. By 5.1, we have  f (πa (x), b)  ·b πa πb πa (x) = πa πa (x) − q(b) f (x, πa (b)) =x− · πa (b) = ππa (b) (x) q(πa (b)) for all x ∈ L. Thus (i) holds. Suppose that πa = πb but that a 6∈ R. Choose c ∈ L such that f (a, c) = 1. From πa (c) = πb (c), it follows that −a/q(a) = −f (b, c)b/q(b). Hence ha, bi is 1-dimensional. Therefore (ii) holds. Lemma 6.3. Let a, b ∈ L be linearly independent. Then b ∈ a⊥ if and only if mΛ a,b (b) ∈ hbi. Proof. This holds by (5.2). Notation 6.4. We set ∗ T = TΛ = hmΛ a,b | a, b ∈ L i.

The elements in T are all similarities of Λ.

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Lemma 6.5. Suppose that dimK L ≥ 3 and let a, b ∈ L∗ . Then the following are equivalent: (i) ha, bi is 1-dimensional or a and b are both in R. (ii) mΛ a,b is contained in the center Z(T ) of T . (iii) πb (πa (c)) ∈ hci for all c ∈ L. ∗ Proof. Let µu,v = mΛ u,v for all u, v ∈ L and let ϕ = πb πa . Since dimK L ≥ 3, we can choose a non-zero element d ∈ a⊥ ∩ b⊥ . The element d is fixed by πa and πb and hence also by ϕ. Suppose that (i) holds. By 5.1 and 6.2(ii), µa,b (x) = (q(a)/q(b))x for all x ∈ L. Therefore (ii) holds. Suppose that (ii) holds. Then µa,b µc,b = µc,b µa,b for all c ∈ L∗ . It follows that πa πb πc = πc πb πc by 5.1 and thus

ϕ2 = πb πa πc πc πb πa = πb πa πc πa πb πc = ππb πa (c) πc = πϕ(c) πc for all c ∈ L∗ by 6.2(i). Setting c equal to the element d chosen above, we obtain ϕ2 = πϕ(d) πd = πd2 = 1. Therefore πc = πϕ(c) for all c ∈ L∗ . Since ϕ acts trivially on R, it follows from 6.2(ii) that hϕ(c), ci is 1-dimensional for all c ∈ L∗ . Thus (iii) holds. Assume, finally, that (iii) holds. It follows that for some t ∈ K ∗ , ϕ(c) = tc for all c ∈ L. Since ϕ(d) = d, we must have t = 1 and hence πa = πb . Thus (i) holds, again by 6.2(ii). Corollary 6.6. Suppose that dimK L ≥ 3 and let a ∈ L∗ . Then a ∈ R if and only if for all b ∈ L∗ .

2 (mΛ a,b ) ∈ Z(T )

∗ Proof. Again let µu,v = mΛ u,v for all u, v ∈ L . By 6.2(i), we have

µ2u,v = (q(u)/q(v))2 πu ππv (u) and hence (6.7)

µ2u,v ∈ Z(T ) if and only if µu,πv (u) = πu ππv (u) ∈ Z(T )

for all u, v ∈ L∗ . Suppose that a 6∈ R. It follows that we can choose b ∈ L such that f (a, b) = 1 and ha, bi is 2-dimensional. Then also ha, πb (a)i is 2-dimensional. By 6.5, therefore, µa,πb (a) 6∈ Z(T ). Hence µ2a,b 6∈ Z(T ) by (6.7). Now suppose that a ∈ R∗ . Thus a is fixed by πb for all b ∈ L∗ . Hence µa,πb (a) = µa,a = 1 and thus µ2a,b ∈ Z(T ) for all b ∈ L∗ , again by (6.7). Corollary 6.8. The group T = TΛ is abelian if and only if dimK L ≤ 2 or f is identically zero.

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Proof. If dimK L ≥ 3 and f is not identically zero, then by 6.5, T is nonabelian. Suppose, on the other hand, that dimK L ≤ 2. By [60, 34.2], L has the structure of a field containing K such that either L = K and q is similar to the quadratic form x 7→ x2 or L/K is quadratic and q is similar to the norm of this extension. If L/K is quadratic and separable, let σ be the non-trivial element in Gal(L/K), and let σ be the identity if L/K is trivial or inseparable. Then
q(a)πa (x) = q(a)x − f (x, a)a = aaσ x − xaσ + xσ a a = −a2 xσ

for all a, x ∈ L. Thus every element of T is of the form x 7→ tx for some t ∈ L∗ . In particular, T is abelian. Suppose, finally, that f is identically zero (but that dimK L is arbitrary). In this case, R = L, hence mΛ a,b ∈ Z(T ) for all a, b ∈ L∗ by 6.5 and so again, T is abelian.

Next we let FΛ be the subring of the endomorphism ring of the additive group of L generated by the set of all automorphisms ψ of this additive group such that mΛ ψ(a),a ∈ Z(T )

for all a ∈ L∗ . Let ξ : K → FΛ be the homomorphism of rings that sends an element t ∈ K to scalar multiplication by t. Lemma 6.9. Suppose that dimK L ≥ 3 and that R 6= L. Then the map ξ from K to FΛ is an isomorphism. ∗ Proof. Again let µu,v = mΛ u,v for all u, v ∈ L . Let ψ be an automorphism of the additive group of L such that µψ(a),a ∈ Z(T ) for all a ∈ L∗ . For each a ∈ L\R, there exists ta ∈ K ∗ such that ψ(a) = ta a by 6.5. We say that two elements a and b in L\R are linked if they are linearly independent and a + b 6∈ R. If a, b ∈ L\R are linked, then ta a + tb b = ta+b (a + b) and hence ta = ta+b = tb . Suppose that a and b are linearly independent elements of L\R that are not linked. Since dimK L ≥ 3 and q is anisotropic, we have |K| = ∞. Choose s ∈ K\{0, −1, −2} and let c = −a + sb. Then {c, c+a, c+b}∩R = ∅ and thus c is an element of L\R that is linked to both a and b. Therefore ta = tc = tb also in this case. By 6.1, it follows that t := ta is independent of the choice of a ∈ L\R. Hence ψ(a) = ψ(a + b) − ψ(b) = ta for all a ∈ R and all b ∈ L\R. Therefore ψ = ξ(t). We conclude that ξ is surjective.

Proposition 6.10. Let Λ = (K, L, q) and Λ′ = (K ′ , L′ , q ′ ) be anisotropic quadratic spaces. Suppose that the dimension of L at least 3 and that f := ∂q is not identically zero. Suppose, too, that ϕ is an additive bijection from L to L′ such that
Λ′ (6.11) ϕ mΛ a,b (w) = mϕ(a),ϕ(b) (ϕ(w)) ′

Λ for all w ∈ L and all a, b ∈ L∗ , where mΛ a,b and mϕ(a),ϕ(b) are as in 5.1. Then K ∼ = K ′ and ϕ is a ρ-linear similarity from Λ to Λ′ for some isomorphism ρ from K to K ′ .

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Proof. Let R be the defect of q and let R′ be the defect of q ′ . The map ϕ induces an isomorphism from TΛ′ to TΛ and hence from FΛ to FΛ′ . By 6.8 applied to Λ, the group T = TΛ is non-abelian. Thus by 6.8 applied to Λ′ , dimK ′ L′ ≥ 3 and R′ 6= L′ . It follows from 6.9, therefore, that ϕ induces an isomorphism ρ from K to K ′ such that (6.12)

ϕ(tv) = ρ(t)ϕ(v)

for all t ∈ K and all v ∈ L. Let f ′ = ∂q ′ and choose a ∈ L∗ . The subspace a⊥ is spanned by the elements it contains that are not in hai (since its dimension is at least 2). ′ Similarly, the subspace ϕ(a)⊥ (where ⊥′ is defined with respect to f ′ ) is spanned by the elements it contains which are not in hϕ(a)i. Let b ∈ L\hai. ⊥′ By 6.3, b ∈ a⊥ if and only if mΛ if and only a,b (b) ∈ hbi and ϕ(b) ∈ ϕ(a) ′ ⊥ ⊥′ if mΛ (ϕ(b)) ∈ hϕ(b)i. By (6.11), therefore, ϕ(a ) = ϕ(a) for all ϕ(a),ϕ(b) a ∈ L.

Now let u, v ∈ L∗ , let λ = q ′ ϕ(u) /ρ q(u) ∈ K ′ and let w be a non-zero ′ ′ element in u⊥ ∩ v ⊥ . Then ϕ(w) ∈ ϕ(u)⊥ ∩ ϕ(v)⊥ and hence mΛ u,v (w) = q(u)/q(v) · w

and ′

′ ′ mΛ ϕ(u),ϕ(v) (ϕ(w)) = q (ϕ(u))/q (ϕ(v)) · ϕ(w).

By (6.11) and (6.12), therefore,
ρ q(u)/q(v) · ϕ(w) = ϕ q(u)/q(v) · w) = q ′ (ϕ(u))/q ′ (ϕ(v)) · ϕ(w). Hence





q ′ ϕ(v) /ρ q(v) = q ′ ϕ(u) /ρ q(u) = λ.

Since u and v are arbitrary, we conclude that

q ′ ϕ(w) = λ · ρ q(w) for all w ∈ L.

Corollary 6.13. Let Λ = (K, L, q) and Λ′ = (K ′ , L′ , q ′ ) be two anisotropic quadratic spaces. Suppose that the dimension of L at least 3 and that f = ∂q is not identically zero. Suppose, too, that L = L′ as additive groups and that ′

Λ mΛ a,b = ma,b Λ′ ∼ ′ for all a, b ∈ L∗ , where mΛ a,b and ma,b are as in 5.1. Then K = K and Λ is ′ similar to Λ .

Proof. This holds by 6.10. Definition 6.14. Let Θ = (K, K0 , σ, L, q) be an anisotropic pseudo-quadratic space as defined in [60, 11.16], let f be the corresponding skew-hermitian form and let T be the group defined in

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LINKED TORI, II

[60, 11.24]. (Note that by [60, 11.16(iii)], the condition that K0 6= K if L0 6= 0 in [60, 11.24] is superfluous.) It follows from [60, 33.13] that for each non-trivial element (a, t) ∈ T , the map
(b, v) 7→ (b − at−1 f (a, b))tσ , tvtσ , which we denote by mΘ (a,t) , is an automorphism of T .

The following result and 6.17 below will be used only in 12.10, which we then apply in 33.4. Proposition 6.15. Let Θ = (K, K0 , σ, L, q) and Θ′ = (K ′ , K0′ , σ ′ , L′ , q ′ ) be two proper anisotropic pseudo-quadratic spaces, let T and T ′ be the groups obtained by applying [60, 11.24] to Θ and Θ′ and suppose that ϕ is an isomorphism from T to T ′ mapping the element (0, 1) ∈ T to the element (0, 1′ ) ∈ T ′ such that
Θ′ (6.16) ϕ mΘ (a,t) (w) = mϕ(a,t) (ϕ(w)) ′

Θ for all w ∈ T and all non-trivial (a, t) ∈ T , where mΘ (a,t) and mϕ(a,t) are as in 6.14. If dimK L > 2, then there exists an isomorphism ρ from K to K ′ mapping K0 to K0′ and a ρ-linear isomorphism ψ from L to L′ such that

ρ(v σ ) = ρ(v)σ for all v ∈ K, for all w ∈ L and

′



q ′ (ϕ(w)) ≡ ρ q(w) (mod K0 ) ϕ(b, v) = (ψ(b), ρ(v))

for all (b, v) ∈ T . Proof. By [67, 7.18], ϕ satisfies the conditions in [67, 7.16]. By [60, 21.14], every involutory set (D, D0 , τ ) with τ 6= 1 is either proper or D is commutative and D0 = {u ∈ K | uτ = u} or D is a quaternion division algebra, D0 is its center and τ is its standard involution. The claim holds, therefore, by [60, 35.5] and [67, 9.8 and 10.38]. Remark 6.17. Suppose in 6.15 that the involutory set (K, K0 , σ) is not proper. We claim that in this case, the hypothesis that Θ′ is proper follows from the other hypotheses of 6.15. Suppose, namely, that Θ′ is not proper but that the other hypotheses hold. Since Θ is proper, T is non-abelian. Hence also T ′ is non-abelian. By [60, 38.5], therefore, Θ′ is pre-proper as defined in [60, 35.5]. By [60, 11.22], we have char(K ′ ) = 2. By [60, 21.16], we can replace Θ′ by a proper anisotropic pseudo-quadratic space without changing K ′ and σ ′ but replacing K0′ by an additive subgroup K0′′ of K ′ that contains K0′ properly in such a way that the other hypotheses of 6.15 continue to hold. Now applying 6.15, we conclude that there exists an isomorphism ρ from K to K ′ that maps K0 to K0′′ such that ρ(v σ ) = ρ(v)σ

′

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for all v ∈ K. Since (K, K0 , σ) is not proper, we have K0 = Kσ , where Kσ is as in [60, 11.1]. It follows that K0′′ = Kσ′ ′ . Since Kσ′ ′ ⊂ K0′ by [60, 11.1], we obtain a contradiction to the assumption that K0′ is contained properly in K0′′ . Thus the claim holds. Our next goal is to prove 6.27 and 6.28. Notation 6.18. We fix an involutory set Ξ = (D, D0 , σ), where D is a quaternion division algebra and σ is its standard involution. ∗ Let Dσ be as in (5.4) and let mΞ a,b for all a, b ∈ D0 be as in 5.9. We denote the center of D by F and the trace and norm of D by T and N . If char(F ) = 2, then (6.19) Dσ = {a ∈ D | T (a) = 0} = {a ∈ D | a2 ∈ F } = {a ∈ D | a4 ∈ F }. Lemma 6.20. Let a, b ∈ Dσ and suppose that char(F ) = 2. Then the following hold: (i) ab = ba if and only if ab ∈ Dσ . (ii) If ab 6= ba, then {1, a, b, ab} is a basis for D over F . Proof. Since (ab)σ = ba, (i) holds. If ab 6= ba, then {1, a, b} is a linearly independent subset of Dσ and, by (i), ab 6∈ Dσ . Hence (ii) holds. Lemma 6.21. Let D′ be a second quaternion division algebra, let σ ′ be its standard involution, let T ′ be its trace and let F ′ be its center. Let a, b be non-commuting elements of Dσ and let a′ , b′ be non-commuting elements of ′ (D′ )σ . Suppose that char(F ) = 2 and that there is an isomorphism ϕ from F to F ′ mapping a2 to (a′ )2 , b2 to (b′ )2 and T (ab) to T ′ (a′ b′ ). Then there exists a unique isomorphism from D to D′ extending ϕ and mapping a to a′ and b to b′ . Proof. Let c = ab, c′ = a′ b′ , E = F (c) and E ′ = F ′ (c′ ). Then N (c) = ccσ = abba = a2 b2 and N ′ (c′ ) = c′ (c′ )σ = a′ b′ b′ a′ = (a′ )2 (b′ )2 . Thus ϕ maps the minimal polynomial of c to the minimal polynomial of c′ . Hence ϕ extends uniquely to an isomorphism ψ from E to E ′ mapping c to c′ . Since T (c) = T (ab) = ab + ba 6= 0, the extension E/F is separable. Hence also the extension E ′ /K ′ is separable. Since T (aσ c) = a2 b + ba2 = 0 and T (a) = 0, we have D ∼ = (E/F, a2 ) as defined in [60, 9.3]. Similarly, D′ ∼ = (E ′ /F ′ , (a′ )2 ). Therefore there is a unique extension of ψ to an isomorphism from D to D′ mapping a to a′ . Remark 6.22. By [60, 21.14 and 35.3], F 6= D0 if and only if D is generated by D0 as a ring. Thus if F 6= D0 , then the centralizer of D0 in D is F . By [60, 11.2], we have char(F ) = 2 if F 6= D0 . Since D0 ⊂ Dσ , we have (6.23)

for all a, b ∈ D0 .

T (ab) = (a + b)2 − a2 − b2

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LINKED TORI, II

For each u ∈ D∗ , let λu be the additive automorphism of D0 given by λu (x) = uxuσ for all x ∈ D0 . Lemma 6.24. If F 6= D0 , then the map u 7→ λu is an injective homomorphism of groups from D∗ to Aut(D0 ). Proof. Suppose that F 6= D0 . It suffices to show that the map u 7→ λu is injective. Suppose that uxuσ = x for some u ∈ D∗ and all x ∈ D0 . Thus uuσ = 1 and so ux = xu for all x ∈ D0 . By 6.22, D0 generates D as a ring and char(F ) = 2. Therefore, u ∈ Z(D) = F ⊂ Dσ , so u2 = uuσ = 1, and thus u = 1. ∗ Lemma 6.25. Suppose that D0 6= F and let κa,b = mΞ a,b for all a, b ∈ D0 and let

SΞ = hκa,b | a, b ∈ D0∗ i. Then F ∗ = {a ∈ D0∗ | κ2a,b ∈ Z(SΞ ) for all b ∈ D0∗ }. Proof. Let S = SΞ . If a ∈ F ∗ and b ∈ D0∗ , then b2 ∈ F and hence κ2a,b ∈ Z(S). Suppose, conversely, that a ∈ D0∗ and κ2a,b ∈ Z(S) for all b ∈ D0∗ . Choose b ∈ D0∗ . Then κ2a,b−1 commutes with κx,1 for all x ∈ D0∗ . Thus λ(ab)2 commutes with λx for all x ∈ D0∗ , so by 6.24, (ab)2 centralizes D0 . By 6.22, therefore, (ab)2 ∈ F . By (6.19), we have ab ∈ Dσ and hence ab = ba by 6.20(i). Thus a centralizes D0 and hence is contained in F . Lemma 6.26. Suppose that D0 6= F and let a, b ∈ D0∗ . Then ab = ba if and only if κa,1 κb,1 = κb,1 κa,1 , where κ is as in 6.25. Proof. We have κa,1 κb,1 = κb,1 κa,1 if and only if λab = λba and hence the claim holds by 6.24. Proposition 6.27. Let Λ = (K, L, q) be an anisotropic quadratic space, let R denote the defect of q and let Ξ = (D, D0 , σ) be an involutory set with D a quaternion division algebra and σ its standard involution such that D0 contains F := Z(D) properly. Suppose that L = D0 as additive groups and that Ξ mΛ a,b = ma,b Ξ for all a, b ∈ L∗ , where mΛ a,b and ma,b are as in 5.1 and 5.9. Then F = R and R has co-dimension 2 in L.

Proof. By hypothesis, the groups T = TΛ and S = SΞ defined in 6.4 and 6.25 are the same. By 6.25, S = T is non-abelian. By 6.8, therefore, dimK L ≥ 3 and R 6= L. By 6.6 and 6.25, F = R. By 6.22, we have char(K) = 2. Since 1 ∈ F = R, we have κa,1 = q(a)πa for all a ∈ L∗ = D0∗ , where κa,1 is as in 6.25. Choose a, b ∈ L∗ . By 6.2(i), the elements κa,1 = q(a)πa and κb,1 = q(b)πb of T commute if and only if πb = ππa (b) . By 6.2(ii), it follows

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that κa,1 and κb,1 commute if and only if f (a, b) = 0. By 6.26, therefore, ab = ba if and only if f (a, b) = 0. Let a ∈ L\R = D0 \F and choose b ∈ L such that f (a, b) 6= 0. Then ab 6= ba, so {1, a, b, ab} is a basis of D over F by 6.20(ii). It follows that the elements of a⊥ ∩ b⊥ commute with every element of D and hence ha, bi⊥ ⊂ F = R. By 6.1, dimK L/R ≥ 2. It follows that R has co-dimension 2 in L. Proposition 6.28. Let Ξ = (D, D0 , σ) and F be as in 6.18, let Ξ′ = (D′ , D0′ , σ ′ ) be a second involutory set with D′ a quaternion division algebra and σ ′ its standard involution and let F ′ = Z(D′ ). Suppose that D0 contains F properly and that D0′ contains F ′ properly. Suppose, too, that there is an additive bijection ρ from D0 to D0′ such that
Ξ′ ρ mΞ a,b (x) = mρ(a),ρ(b) (ρ(x)) ′

Ξ for all a, b ∈ D0∗ and all x ∈ D0 , where mΞ a,b and mρ(a),ρ(b) are as in 5.9. Then D ∼ = D′ .

Proof. The map ρ induces an isomorphism from S to S ′ , where S := SΞ and S ′ := SΞ′ are as in 6.25. By 6.25, therefore, ρ(F ) = F ′ . The triple (D′ , tD0′ , σ ′ ) is an involutory set for each t ∈ (F ′ )∗ . Let Ξ′1 := (D′ , tD0′ , σ ′ ) for t = ρ(1)−1 . Then Ξ′

′

1 mta,tb (tx) = tmΞ a,b (x) ′ ∗ ′ for all a, b ∈ (D0 ) and all x ∈ D0 . By replacing Ξ′ by Ξ′1 and ρ by the map x 7→ tρ(x), we can thus assume that ρ(1) = 1. For all a, b ∈ D0∗ , we have aba ∈ D0∗ and thus
′ ρ(aba)ρ(x)ρ(aba) = mΞ ρ(aba),ρ(1) ρ(x)
= ρ mΞ aba,1 (x)
Ξ Ξ = ρ mΞ a,1 mb,1 ma,1 (x)
′ Ξ′ Ξ′ = mΞ ρ(a),1 mρ(b),1 mρ(a),1 ρ(x)

= ρ(a)ρ(b)ρ(a)ρ(x)ρ(a)ρ(b)ρ(a)

for all x ∈ B. Hence

λρ(aba) = λρ(a)ρ(b)ρ(a) and therefore (6.29) ρ(aba) = ρ(a)ρ(b)ρ(a) ∗ for all a, b ∈ D0 by 6.24. By Hua’s Theorem (see [27]), it follows that the restriction of ρ to F is an isomorphism of fields from F to F ′ . Since D0 generates D as a ring, we can choose non-commuting elements a, b in D0 . By (6.23) and (6.29), ρ(a2 ) = ρ(a)2 , ρ(b2 ) = ρ(b)2 and ρ(T (ab)) = T ′ (ρ(a)ρ(b)), where T and T ′ are the trace maps of D and D′ . By 6.21, we conclude that ρ extends to an isomorphism from D to D′ .

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We conclude this chapter with one more small observation: Proposition 6.30. Let (K, L, q) be a pointed anisotropic quadratic space with base point 1 ∈ L as defined in 2.20 and let K be identified with its image in L under the map t 7→ t · 1. Suppose that ∂q is identically zero and that (6.31)

ma,1 mb,1 ∈ {mv,1 | v ∈ L∗ }

for all a, b ∈ L∗ , where mv,1 is as in 5.1. Then there is a unique multiplication on L making L into a field containing K as a subfield such that u2 = q(u) for all u ∈ L. Furthermore, the map a 7→ ma,1 is an injective homomorphism from the multiplicative group L∗ to the group of automorphisms of the additive group (L, +). Proof. By 5.1 and the assumption that ∂q = 0, we have mv,1 (x) = q(v)x for all v ∈ L∗ and all x ∈ L. By (6.31), therefore, (6.32)

q(a)q(b) ∈ q(L∗ )

for all a, b ∈ L∗ . It also follows from ∂q = 0 that the map q is additive and hence injective. By (6.32), therefore, there is a unique multiplication on L which we denote by juxtaposition such that q(a)q(b) = q(ab) for all a, b ∈ L∗ . This multiplication turns L into an integral domain whose multiplicative identity is the base point 1 ∈ L. If u ∈ L∗ , then q(q(u) · 1) = q(u)2 = q(u2 ),

and hence u2 = q(u). We also have

q(u)q(u/q(u)) = q(u)q(u)/q(u)2 = q(1) and thus u(u/q(u)) = 1 for all u ∈ L∗ and

q(t · 1)q(s · 1) = t2 s2 = q(ts · 1)

for all s, t ∈ K ∗ . Hence L is a field containing K as a subfield. Suppose that ∗ is a second multiplication on L that makes L into a field containing K as a subfield such that u ∗ u = q(u) for all u ∈ L. Then q(u ∗ v) = (u ∗ v) ∗ (u ∗ v) = (u ∗ u) ∗ (v ∗ v) = q(u) ∗ q(v) = q(u)q(v) = q(uv) and therefore u ∗ v = uv for all u, v ∈ L.

Chapter Seven Quadratic Forms over a Local Field Our goal in this chapter is to assemble various results about quadratic forms over a field complete with respect to a discrete valuation. Throughout this chapter we assume that K is a field of arbitrary characteristic which is complete with respect to a discrete valuation ν (but we ¯ is finite) and we adopt the usual do not assume that the residue field K convention that ν(0) = ∞. Notation 7.1. Let OK denote the ring of integers of K and let t 7→ t¯ be ¯ We will denote by O× the the natural map from OK to the residue field K. K group of units in OK . We fix a uniformizer p in K, that is to say, an element p of K such that ν(p) = 1. Proposition 7.2. Let (K, L, q) be an anisotropic quadratic space and let f = ∂q. Then the following inequalities hold: (i) ν(q(u + v)) ≥ min{ν(q(u)), ν(q(v))} and
(ii) ν(f (u, v)) ≥ ν(q(u)) + ν(q(v)) /2

for all u, v ∈ L.

Proof. Since K is complete, this holds by [65, 19.4] (which is a special case of [7, 10.1.15]). The following basic notion is due to Springer ([48]). Notation 7.3. Let Λ = (K, L, q) be an anisotropic quadratic space. We set Li = {u ∈ L | ν(q(u)) ≥ i}

for each i ∈ Z. Let i be an arbitrary integer. By 7.2(i), Li is an OK ¯ i = Li /Li+1 and let v 7→ v¯ denote the natural map submodule of L. Let L ¯ i . The map (t¯, v¯) 7→ tv from K ¯ ×L ¯ i to L ¯ i is well defined and from Li to L ¯ i into a vector space over K. ¯ Moreover, the map q¯i from L ¯ i to K ¯ makes L given by q¯i (¯ v ) = p−i q(v) ¯ i . If is well defined by 7.2(ii) and thus an anisotropic quadratic form on L i 6= 0, this quadratic form depends on the choice of the uniformizer p, but ¯ L ¯ i , q¯i ). Note that the automorphism only up to similarity. Let Λi = (K, v 7→ pv of L induces an isometry from the quadratic space Λi to the quadratic space Λi+2 . We call Λ0 and Λ1 the residual quadratic spaces of Λ.

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Proposition 7.4. If (K, L, q) is an anisotropic quadratic space of finite dimension, then ¯ 0 + dimK¯ L ¯ 1 = dimK L. dimK¯ L Proof. This holds by 7.3 and [65, 19.36]. Definition 7.5. Let (K, L, q) be an anisotropic quadratic space. We will say q is trim if 0 ∈ ν(q(L∗ )), that q is ramified if ν(q(L∗ )) = Z and that q is unramified if ν(q(L∗ )) = 2Z. It follows from the argument in [65, 19.17] that the following are equivalent: (i) q is trim. (ii) q is either ramified or unramified. ¯ 0 6= 0. (iii) L Thus, in particular, q is trim if and only if ν(q(L∗ )) = δZ for δ = 1 or 2; q ¯ 0 and L ¯ 1 are both non-trivial; is ramified if and only if the vector spaces L and q is unramified if and only if L1 = pL0 . Note, too, that if a subform of q is ramified, then so is q and, conversely, if q is unramified, then so is every positive-dimensional subform of q. In fact, it will be useful to make the convention that the trivial quadratic form is also unramified. Definition 7.6. Let (K, L, q) be a trim anisotropic quadratic space. We will say that q is tame if q¯0 and q¯1 are both non-singular. If q is tame, then q itself is also non-singular. Since q¯0 and q¯1 are anisotropic, q is automatically ¯ 6= 2. We will say that q is wild if the dimension of the tame if char(K) radical of ∂ q¯i is greater than 1 for i = 0 or 1 (or both) and that q is totally wild if ∂ q¯0 and ∂ q¯1 are both identically zero and at least one of the two has dimension greater than 1. Remark 7.7. Let E/K be a quadratic extension with norm N . Then N is an anisotropic quadratic form over K. The extension E/K is ramified ¯ = K) ¯ if and only if the quadratic form N is ramified. Note, however, (i.e. E ¯ K ¯ is a separable that the extension E/K is unramified if and only if E/ quadratic extension. This holds if and only if the quadratic form N is tamely unramified, i.e. both tame and unramified as defined in 7.5 and 7.6. We will say that the extension E/K is wild if the quadratic form N is wild (in which case N is automatically unramified). Remark 7.8. Let (K, L, q) be a ramified anisotropic quadratic space and let q ′ = q/q(v) for some v ∈ L∗ such that ν(q(v)) = 1. Then the quadratic space Λ′0 obtained by applying 7.3 to q ′ is similar to the quadratic space ¯ L ¯ 1 , q¯1 ). Λ1 = (K, Notation 7.9. Let (K, L, q) be an anisotropic quadratic space. We set |t| = exp(−ν(t))

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for all t ∈ K and kuk = exp(−ν(q(u))/2) for all u ∈ L. The map t 7→ |t| is an absolute value on K and by 7.2(i), u 7→ kuk is a norm on L with respect to this absolute value. By [32, XII, Prop. 2.2], for example, u 7→ kuk is, in fact, the unique norm on L up to equivalence. Proposition 7.10. Let (K, L, q) be an n-dimensional anisotropic quadratic space for some n ∈ N. Then L0 is free of rank n as an OK -module. Proof. There exist bases of L over K that are contained in L0 . Since OK is a PID (and, in particular, Noetherian), it thus suffices to show that there exists a basis e1 , . . . , en of L over K such that (7.11)

L0 ⊂ OK e1 + · · · + OK en .

Let u1 , . . . , un be an arbitrary basis of L and let λ1 , . . . , λn be the dual basis of HomK (L, K). By 7.9, the linear forms λ1 , . . . , λn are all continuous with respect to the norm u 7→ kuk. It follows that we can choose r ∈ Z such that λi (L2r ) ⊂ OK for all i ∈ [1, n]. Let ei = p−r ui for all i ∈ [1, n]. Now choose v ∈ L0 . Then v = α1 u1 + · · · + αn un = (pr α1 )e1 + · · · + (pr αn )en

for some α1 , . . . , αn ∈ K. We have pr v ∈ pr L0 = L2r and thus pr αi = pr λi (v) = λi (pr v) ∈ λi (L2r ) ⊂ OK for all i ∈ [1, n] by the choice of r. Remark 7.12. Suppose that (K, L, q) is a finite-dimensional anisotropic quadratic space and let ϕ denote the quadratic form on L0 over OK such that ϕ(u) = q(u) for all u ∈ L0 . By 7.10, L0 is a full rank lattice in L, that is to say, the rank of L0 over OK equals the dimension of L over K. There is thus a canonical isomorphism ψ from L0 ⊗OK K with L such that ψ(v ⊗ t) = tv for all v ∈ L0 and t ∈ K. Identifying L0 ⊗OK K with L via ψ (as vector spaces over K), we have q = ϕK . Proposition 7.13. Let (K, L, q) be an unramified anisotropic quadratic space of finite dimension and suppose that e1 , . . . , em is an OK -basis of L0 . Then
 ν q(t1 e1 + · · · + tm em ) = 2 min ν(ti ) | i ∈ [1, m] for all (t1 , . . . , tm ) ∈ K m .

¯0 Proof. Since e1 , . . . , em is an OK -basis of L0 , the set e¯1 , . . . , e¯m spans L ¯ over K and m = dimK L. Since q is unramified, it follows by 7.4 that the ¯ 0 . Thus if ti ∈ OK (respectively, ti ∈ OK × ) for set e¯1 , . . . , e¯m is a basis of L all (respectively, some) i ∈ [1, m], then
ν q(t1 v1 + · · · + tm vm ) = 0. The claim holds, therefore, by homogeneity.

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Proposition 7.14. Let q : L → K and q ′ : L′ → K be unramified finitedimensional anisotropic quadratic forms over K, let g : L × L′ → K be a bilinear form and let Q denote the quadratic form on M := L ⊕ L′ given by (7.15) Q(v, v ′ ) = q(v) + g(v, v ′ ) + pq ′ (v ′ ) ′ for all (v, v ) ∈ M , where p is as in 7.1. Then the following conditions are equivalent. (i) Q is anisotropic.
(ii) ν(g(v, v ′ )) > ν(q(v)) + ν(q ′ (v ′ )) /2 for all (v, v ′ ) ∈ M with v 6= 0 and v ′ 6= 0.

(iii) ν(g(e, e′ )) ≥ 1 for all (e, e′ ) ∈ B × B ′, where B is an arbitrary OK -basis of L0 and B ′ is an arbitrary OK -basis of L′0 . If these conditions hold, then  (7.16) ν(Q(v, v ′ )) = min ν(q(v)), ν(q ′ (v ′ )) + 1 , ¯0 ∼ ¯1 ∼ for all (v, v ′ ) ∈ M , Q = q¯0 and Q = (q ′ )0 .

Proof. Let f = ∂Q and choose non-zero v ∈ V and v ′ ∈ V ′ . Then
f (v, 0), (0, v ′ ) = g(v, v ′ ) for all (v, v ′ ) ∈ M . Hence (i) implies (ii) by 7.2(ii). Let e1 , . . . , em be an OK -basis of L0 , let e′1 , . . . , e′n be an OK -basis of L′0 and suppose that m n X X v= ti ei and v ′ = t′i e′i i=1

i=1

for t1 , . . . , t′n ∈ K. Assuming (iii), we have   m X n X 2ν(g(v, v ′ )) = 2ν  ti t′j g(ei , e′j ) i=1 j=1

 ≥ 2 min ν(ti ) + ν(t′j ) + 1 | i ∈ [1, m], j ∈ [1, n]   > 2 min ν(ti ) | i ∈ [1, m] + 2 min ν(t′j ) | j ∈ [1, n]

= ν(q(v)) + ν(q ′ (v ′ )) by 7.13. Hence (iii) implies (ii). Suppose now that (ii) holds. Then (iii) holds simply because it is a special case of (ii). Since q and q ′ are unramified, ν(q(v)) and ν(q ′ (v ′ )) are even. Hence


2ν g(v, v ′ ) ≥ ν q(v) + ν q ′ (v ′ ) + 2 by (ii)

′ ′ > ν q(v) + ν(pq (v ) 

≥ 2 min ν q(v) , ν pq ′ (v ′ ) . Therefore (7.16) holds. From (7.16) it follows that Q is anisotropic, M0 = L0 ⊕ L′0 , M1 = L2 ⊕ L′0 and M2 = L2 ⊕ L′2 , where for each i, Mi is defined (as in 7.3) with respect to Q, Li with respect to q and L′i with respect to q ′ . ¯0 ∼ ¯1 ∼ Thus (ii) implies (i) and (7.16) implies that Q = q¯0 and Q = (q ′ )0 .

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The next result is the converse of 7.14. Proposition 7.17. Let (K, M, Q) be an anisotropic quadratic space of finite dimension. Then there exist subspaces L and L′ of M , unramified anisotropic quadratic forms q and q ′ on L and L′ and a bilinear form g : L × L′ → K

¯0 ∼ ¯1 ∼ such that M = L ⊕ L′ and (7.15) holds. In particular, Q = q¯0 and Q = ′ (q )0 . ¯ 0 is a basis of Proof. Let {e1 , . . . , em } be a subset of M0 whose image in M ¯ ¯ ¯ 1 is M0 over K and let {em+1 , . . . , en } be a subset of M1 whose image in M ¯ ¯ ¯ a basis of M1 over K. (If M0 = 0, we set m = 0 and {e1 , . . . , em } = ∅; if ¯ 1 = 0, we set m = n and {em+1 , . . . , en } = ∅.) By [65, 18.34(ii)], M0 /M2 M ¯ This vector space has, canonically, the structure of a vector space over K. ¯ ¯ 0 . Thus the contains M1 as a subspace whose quotient is isomorphic to M natural image of the set B := {e1 , . . . , en } in M0 /M2 is a basis of M0 /M2 . By homogeneity, the set B is linearly independent over K. By 7.4, therefore, B is a basis of M . We let L be the subspace of M spanned by {ei | i ∈ [1, m]} and we let L′ be the subspace spanned by {ei | i ∈ [m + 1, n]}. Thus L = 0 if m = 0 and L′ = 0 if m = n and M = L ⊕ L′ in every case. Let q denote the restriction of Q to L, let q ′ denote the restriction of p−1 Q to L′ and let g denote the restriction of ∂Q to L × L′ . Then (7.15) holds. Let u be a nonzero element of L. Then there exist r ∈ Z and α1 , . . . , αm ∈ K such that pr u = α1 e1 + · · · + αm em and min{ν(αi ) | i ∈ [1, n]} = 0. Since e¯1 , . . . , e¯m is ¯ 0 over K, ¯ it follows that ν(q(pr u)) = 0 and hence that ν(q(u)) is a basis of M even. Similarly, ν(q ′ (v)) is even for every non-zero v ∈ L′ . By 7.5, therefore, q and q ′ are unramified. Proposition 7.18. Let (K, M, Q) be an anisotropic quadratic space of finite ¯ i is non-singular for i = 0 or 1. Then there dimension and suppose that Q exist unramified anisotropic quadratic forms q and q ′ over K such that Q ∼ = ¯0 ∼ ¯1 ∼ q ⊕ pq ′ . In particular, Q = q¯0 and Q = (q ′ )0 . ¯ 0 is nonProof. Replacing Q by pQ if necessary, we can assume that Q singular. By 7.17, there exists a subspace L of M such that the restriction ¯ 0 . Thus M ¯0 = L ¯ 0 . By 7.6, q is nonq of Q to L is unramified and q¯0 = Q singular. Hence M = L ⊕ L′ , where L′ is the orthogonal complement of L with respect to ∂Q. Let ϕ denote the restriction of Q to L′ and let q ′ = ϕ/p. ¯ 0 is orthogonal Thus Q = q ⊕ pq ′ . If u ∈ L′ ∩ M0 , then the image u ¯ of u in M ¯0 = M ¯ 0 with respect to ∂ Q ¯ 0 and thus u ¯ 0 is non-degenerate. to L ¯ = 0 since ∂ Q Hence L′ ∩ M0 ⊂ M1 . Hence if L′1 and L′2 are defined by applying 7.3 to q ′ , ¯0 ∼ ¯1 ∼ then L′1 ⊂ L′2 . Hence q ′ is unramified. By 7.14, Q = q¯0 and Q = (q ′ )0 . Proposition 7.19. Let q : L → K and q ′ : L′ → K be unramified anisotropic ¯0 ⊕ L ¯′ quadratic forms over K such that the quadratic form q¯0 ⊕ (q ′ )0 on L 0

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is anisotropic and let Q denote the quadratic form q ⊕ q ′ on M := L ⊕ L′ . Then Q is anisotropic and unramified,  (7.20) ν(Q(v, v ′ )) = min ν(q(v)), ν(q ′ (v ′ )) ¯ 0 = q¯0 ⊕ (q ′ )0 . for all v ∈ L and all v ′ ∈ L′ and Q

Proof. Choose non-zero v ∈ L and v ′ ∈ L′ . To prove that (7.20) holds for v and v ′ , it suffices to assume that ν(q(v)) = ν(q ′ (v ′ )). Let m = ν(q(v))/2. Since q is unramified, m is an integer. Replacing v and v ′ by v/pm and v ′ /pm , we can thus assume that ν(q(v)) = ν(q ′ (v ′ )) = 0. Since q¯0 ⊕ (q ′ )0 is anisotropic, we have q¯0 (¯ v ) + (q ′ )0 (¯ v ′ ) 6= 0 and hence ν(Q(v, v ′ )) = 0. Thus (7.20) holds. The remaining claims follow from this. Corollary 7.21. Let q : L → K, . . . , q ′ : L′ → K be a finite sequence of unramified anisotropic quadratic forms over K such that the quadratic form ¯0 ⊕ · · · ⊕ L ¯ ′ is anisotropic and let Q denote the quadratic q¯0 ⊕ · · · ⊕ (q ′ )0 on L 0 ′ form q ⊕ · · · ⊕ q on M := L ⊕ · · · ⊕ L′ . Then Q is anisotropic and unramified ¯ 0 = q¯0 ⊕ · · · ⊕ (q ′ )0 . and Q Proof. This holds by 7.19 and induction. Proposition 7.22. Let E/K be a ramified quadratic extension with norm N and suppose that α1 , . . . , αd are non-zero scalars such that the quadratic form q := α1 N ⊕ · · · ⊕ αd N is anisotropic. Then q is ramified and the quadratic forms q¯0 and q¯1 are similar. Proof. We can identify the vector space on which q lives with E d in such a way that for each u ∈ E ∗ , the map (v1 , . . . , vd ) 7→ (uv1 , . . . , uvd )

from E d to itself is a similitude of q with multiplier N (u). Since E/K is ramified, we can choose u ∈ E ∗ such that ν(N (u)) = 1. The corresponding ¯ 0 and similitude of q induces a similarity from q¯0 to q¯1 . In particular, both L ¯ L1 are non-trivial, which means that q is ramified. Proposition 7.23. Let (K, L, q) be an anisotropic quadratic space of finite dimension, let f = ∂q and let L′ and L′′ be subspaces of L that are orthogonal with respect to f , let q ′ be the restriction of q to L′ and let q ′′ be the restriction of q to L′′ . Suppose further that (q ′ )0 is non-singular. Then q¯0 ∼ = (q ′ )0 ⊕(q ′′ )0 if and only if (7.24)

dimK¯ (¯ q0 ) ≤ dimK¯ (q ′ )0 + dimK¯ (q ′′ )0 .

Moreover, (7.24) holds if L = L′ ⊕ L′′ and either q is unramified or there exists a ramified separable quadratic extension E/K such that E is a splitting field of both q ′ and q ′′ .

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Proof. We have L′0 = L′ ∩ L0 and L′′0 = L′′ ∩ L0 , so the images of L′0 and L′′0 ¯ 0 can be identified canonically with L ¯ ′0 under the natural map from L0 to L ′′ ′′ ′ ¯ . Under these identifications, L ¯ and L ¯ are orthogonal with respect and L 0 0 0 ¯ ′0 × L ¯ ′0 agrees with ∂(q ′ )0 and is thus to f¯0 := ∂ q¯0 . The restriction of f¯0 to L ¯′ ∩ L ¯ ′′ = 0. It follows that q¯0 ∼ non-degenerate. Hence L = (q ′ )0 ⊕ (q ′′ )0 if and 0 0 only if (7.24) holds. Now suppose that L = L′ ⊕ L′′ , so that

(7.25)

dimK (q) = dimK (q ′ ) + dimK (q ′′ ).

If q is unramified, then by 7.5, the subforms q ′ and q ′′ are also unramified and hence dimK (q) = dimK¯ (¯ q0 ) as well as dimK (q ′ ) = dimK¯ (q ′ 0 ) and ′′ ′′ dimK (q ) = dimK¯ (q 0 ) by 7.4. It follows from (7.25) that (7.24) holds if q is unramified. Suppose, instead, that there exists a ramified separable quadratic extension E/K such that E is a splitting field of both q ′ and q ′′ . By 2.24, the quadratic forms q, q ′ and q ′′ all satisfy the hypotheses of 7.22 with N the norm of the extension E/K. Thus by 7.22, dimK (q) = 2 dimK¯ (¯ q0 ), dimK (q ′ ) = 2 dimK¯ (q ′ )0 and dimK (q ′′ ) = 2 dimK¯ (q ′′ )0 . It again follows from (7.25) that (7.24) holds. ¯ be a separable quadratic extension. Then there Proposition 7.26. Let M/K ¯ = M (with exists a unique separable quadratic extension E/K such that E respect to the canonical extension of ν to E), and if N is the norm of the ¯0 is the norm of the extension M/K, ¯ where N ¯0 is extension E/K, then N ¯ ¯ the quadratic form on E over K obtained by applying 7.3 to N . Proof. This holds, for instance, by [22, §7, Thm. 1]. Our next goal is to prove 7.28. This important result was first proved by ¯ 6= 2 and by Tietze in [52] when Springer in [49] in the case that char(K) ¯ char(K) = 2 using a presentation of the Witt group. We first consider a special case: ¯ M, Q) is an anisotropic quadratic space Lemma 7.27. Suppose that (K, ¯ over K which is both non-singular and finite-dimensional. Then there exists an unramified anisotropic quadratic form q over K, unique up to isometry, such that q¯0 ∼ = Q. Moreover, q is non-singular. Proof. Let n = dimK¯ (Q) and m = [n/2]. By 2.18, there exist separable ¯ . . . , Fm /K ¯ with norms Q1 , . . . , Qm and elements quadratic extensions F1 /K, × × α1 , . . . , αm ∈ OK and, if n is odd, β ∈ OK such that ∼ Q=α ¯ 1 Q1 ⊕ · · · ⊕ α ¯ m Qm if n is even and

Q∼ ¯ 1 Q1 ⊕ · · · ⊕ α ¯ m Qm ⊕ ξK, =α ¯ β¯

if n is odd, where ξK, ¯ β¯ is as in 2.16. By 7.26, there exist unramified quadratic extensions E1 /K, . . . , Em /K with norms q1 , . . . , qm such that (qi )0 ∼ = Qi for all i ∈ [1, m]. Let q = α1 q1 ⊕ · · · ⊕ αm qm

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if n is even and q = α1 q1 ⊕ · · · ⊕ αm qn ⊕ ξK,β

if n is odd. By 7.21, q is anisotropic and unramified and q¯0 ∼ = Q. Now we suppose that (K, V, ψ) is an arbitrary unramified anisotropic quadratic space such that ψ¯0 ∼ = Q. By 7.6, ψ is tame and hence nonsingular. It remains only to show that ψ ∼ = q. We proceed by induction with respect to m. Suppose that m = 0. We can assume that n = 1 in this case. ¯ 6= 2 since Q is non-singular. By 2.16, ψ ∼ This forces char(K) = ξK,γ for some ∗ γ ∈ K . By Hensel’s Lemma ([41, Chapter 2, 4.6], for example), the natural × ¯ ∗ /(K ¯ ∗ )2 induces an isomorphism from O× /(O× )2 to map from OK to K K K ¯ ∗ /(K ¯ ∗ )2 . Since γ¯ = s¯2 β¯ for some s ∈ O× , it follows that γ = t2 β for some K K × t ∈ OK . Thus ψ ∼ = ξK,γ ∼ = ξK,β = q. Now suppose that m > 0 and choose v1 , v2 ∈ M0 such that the restriction ¯ 0 spanned by the images v¯1 and v¯2 of of Q to the subspace h¯ v1 , v¯2 i of M ¯ 0 is isometric to α v1 and v2 in M ¯ 1 Q1 . Let V1 denote the subspace hv1 , v2 i of V and let ρ denote the restriction of ψ to V1 . Since ψ is unramified, so is ρ. Since dimK¯ ρ¯0 ≥ 2, it follows from 7.4 that dimK V1 = 2 and ρ¯0 ∼ ¯1 Q1 . In particular, ρ is non-singular by 7.6. By 2.17, ρ is similar to =α the norm of a separable quadratic extension and by 7.26, therefore, ρ ∼ = δq1 × for some δ ∈ OK such that δ¯ ∈ α ¯1 Q1 (M1 ). By [43, Lemma 2], it follows that δ ∈ α1 q1 (E1∗ ). Thus ρ ∼ = α1 q1 . Let V2 = V1⊥ and let ϕ be the restriction of ψ to V2 . By 2.15, we have an ψ ∼ = ρ ⊕ ϕ. The quadratic form ϕ is, of course, anisotropic and unramified and since ψ is non-singular, so is ϕ. By 7.23, we have Q∼ ¯1 Q1 ⊕ ϕ¯0 . = ψ¯0 ∼ = ρ¯0 ⊕ ϕ¯0 ∼ =α Hence by Witt cancellation ([21, 8.4], for example), ϕ¯0 ∼ ¯ 2 Q2 ⊕ · · · ⊕ α ¯ m Qm =α

if n is even and

ϕ¯0 ∼ ¯ 2 Q2 ⊕ · · · ⊕ α ¯ m Qm ⊕ ξK, =α ¯ β¯ if n is odd. By induction, we conclude that ψ ∼ = q. Here, now, is the result of Springer and Tietze. Theorem 7.28. Suppose that Q0 and Q1 are anisotropic quadratic forms ¯ both non-singular and finite-dimensional. Then there exists an over K, anisotropic quadratic form q over K, unique up to isometry, such that q¯i ∼ = Qi for i = 0 and 1. Moreover, q is non-singular. Proof. By 7.27, there exist unramified anisotropic quadratic forms ϕ and ψ over K, unique up to isometry, such that ϕ¯0 ∼ = Q0 and ψ¯0 ∼ = Q1 , and, furthermore, ϕ and ψ are both non-singular. Let q = ϕ ⊕ pψ. Then q is also non-singular and by 7.14, q is anisotropic and q¯i ∼ = Qi for i = 0 and 1. If ρ is an arbitrary anisotropic quadratic form such that ρ¯i ∼ = Qi for i = 0 and 1, then ρ ∼ = q by 7.18 and the uniqueness of ϕ and ψ.

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Notation 7.29. Let X be a free OK -module of finite rank. Then L := X ⊗OK K is a finite-dimensional vector space over K and the inclusion OK ⊂ K gives rise to an embedding of X in L with respect to which X is a full rank OK lattice in L. Now suppose that ϕ : X → OK is a quadratic form over OK . ¯ to K ¯ and ϕK Applying 2.6, we obtain quadratic forms ϕK¯ from X ⊗OK K from L to K. Let ϕ¯ = ϕK¯ . By 2.9, there is a canonical identification of ¯ with X ¯ := X/pX as K-vector ¯ X ⊗OK K spaces with respect to which (7.30)

ϕ(¯ ¯ v ) = ϕ(v)

¯ for all v ∈ X, where v¯ denotes the image of v in X. Remark 7.31. Let (K, L, q) be a finite-dimensional anisotropic quadratic space and suppose that q is unramified. Let X = L0 and let ϕ be as in 7.6. ¯ 0 as defined in 7.3 coincides with X ¯ as defined in Then L1 = L2 = pL0 , so L 7.29. By 7.12, we can identify q with ϕK and by (7.30), we can identify ϕ¯ with q¯0 . Proposition 7.32. Let X, L, ϕ, ϕ¯ = ϕK¯ and q := ϕK be as in 7.29 and suppose that ϕ¯ is anisotropic. Then q is anisotropic and unramified, X = L0 and q¯0 = ϕ. ¯ Proof. Since ϕ¯ is anisotropic, we have ϕ(v) 6= 0 for each v ∈ X\pX. Since every element u of L∗ satisfies tu ∈ X\pX for some t ∈ K, it follows that ϕ is anisotropic. Therefore also q is anisotropic. Since q is an extension of ϕ, we have X ⊂ L0 , where L0 is as defined in 7.3. We claim that, in fact, X = L0 . Suppose that v is a non-zero element of L0 . Then q(v) ∈ OK and u := pn v ∈ X\pX for some integer n. If n < 0, then v = p−n u ∈ X, so we may assume that n ≥ 0. Thus q(u) = ϕ(u) = ϕ(¯ ¯ u) 6= 0

by (7.30). Hence q(u) = p2n q(v) ∈ p2n OK is a unit in OK , so n = 0 and v = u ∈ X. Thus X = L0 as claimed. Hence q¯0 = ϕ. ¯ Thus dimK¯ (¯ q0 ) = dimK¯ (ϕ) ¯ = dimK L. By 7.4, therefore, q is unramified. Proposition 7.33. Let ϕ, ϕ¯ and q := ϕK be as in 7.29 and suppose that ϕ¯ is hyperbolic. Then ϕ and q are also hyperbolic. Proof. Suppose that ϕ is anisotropic. For every v ∈ L there exists t ∈ K ∗ such that tv ∈ X (where L and X are as in 7.29). It follows that q is anisotropic. Let X # = {x ∈ X | ∂q(x, X) ⊂ OK }.

Then X ⊂ L0 ⊂ X # . We want to show that X # ⊂ X. Let x ∈ X # . Then y := pn x ∈ X\pX for some n ∈ Z. If n < 0, then x = p−n y ∈ X, so we can assume that n ≥ 0. If n > 0, then ∂q(y, X) ∈ pX and therefore ¯ = 0 Since ϕ¯ is hyperbolic and hyperbolic forms are non-singular, ∂ ϕ(¯ ¯ y , X)

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it follows that y¯ = 0. With this contradiction, we conclude that n = 0 and hence x ∈ X. Therefore X # ⊂ X and thus X = L0 . It follows that q¯0 = ϕ. ¯ Since q¯0 is anisotropic (by 7.3) and ϕ¯ is hyperbolic, this is impossible. We conclude that ϕ is isotropic. We can thus choose u ∈ X\pX such that ϕ(u) = 0. Let x 7→ x ¯ be the ¯ := X/pX. Then ϕ(¯ natural map from X to X ¯ u) = 0. Thus there exists v ∈ X such that u ¯, v¯ is a hyperbolic pair for ϕ. ¯ Therefore γ := ∂ϕ(u, v) ∈ × OK . Hence u, γ −1 v − γ −2 ϕ(v)u is a hyperbolic pair in X for ϕ. Let H be the corresponding hyperbolic plane in X. Then X = H ⊕ H⊥ (by 2.19) and ¯ is orthogonal to h¯ the image of H⊥ in X u, v¯i with respect to ∂ ϕ. ¯ It follows now by induction with respect to the rank of X over OK that the restriction of ϕ to X ′ is hyperbolic. Hence ϕ is hyperbolic. Since scalar extensions of hyperbolic quadratic forms are hyperbolic, q is also hyperbolic. Proposition 7.34. Let (K, L, q) be a round anisotropic quadratic space over × ¯ ∗ ) and let Q K, suppose that α is an element of OK satisfying α ¯ 6∈ q¯0 (L 0 denote the quadratic form q ⊗ hhαii on the vector space M := L ⊕ L as ¯0 ∼ defined in 2.28. Then Q is anisotropic and Q αii. = q¯0 ⊗ hh¯ Proof. It will suffice to show that (7.35)

 ν(Q(u, v)) = min ν(q(u)), ν(q(v))

for all u, v ∈ L. To prove that (7.35) holds for all u, v ∈ L, it will suffice to consider u, v ∈ L∗ such that ν(q(u)) = ν(q(v)). Since q is round, there exists a similitude of q with multiplier q(u)−1 . Extending diagonally, we obtain a similitude ψ of Q with the same multiplier mapping the subspace (L, 0) to itself. Thus Q(ψ(x)) = q(u)−1 Q(x) for all x ∈ M and q(ψ(x)) = q(u)−1 q(x) for all x ∈ L. By replacing u by ψ(u) and v by ψ(v), we can thus assume that ν(q(u)) = ν(q(v)) = 0. Since q is round, so is q¯0 (by [23, 7.10]). Therefore × ¯ ∗ ), there exists w ∈ OK such that q¯0 (w)¯ ¯ q0 (¯ v ) = q¯0 (¯ u). Since α ¯ 6∈ q¯0 (L 0 it follows that q¯0 (¯ u) 6= α ¯ q¯0 (¯ v ). Thus Q(u, v) = q(u) − αq(v) implies that ν(Q(u, v)) = 0 = ν(q(u)). We conclude that (7.35) holds for all u, v ∈ L. Note that 7.34 holds by 7.19 in the special case that q is unramified. ¯ be a separable quadratic extension with norm Proposition 7.36. Let M/K ¯ let Q denote the orthogonal sum R, let β1 , . . . , βd be elements of K, β1 R ⊕ · · · ⊕ βd R and suppose that Q is anisotropic. Let N be as in 7.26, let α1 , . . . , αd be × arbitrary elements of OK such that α ¯ i = βi for all i ∈ [1, d] and let q denote the orthogonal sum α1 N ⊕ · · · ⊕ αd N.

Then q is the unique unramified anisotropic quadratic form such that q¯0 ∼ = Q. Proof. By 7.21, q is anisotropic and unramified and q¯0 = Q. Uniqueness holds by 7.28.

QUADRATIC FORMS OVER A LOCAL FIELD

67

In the remaining results of this chapter, we focus on the behavior of quadratic forms under field extensions. Proposition 7.37. Let (K, M, Q) be an anisotropic quadratic space of finite dimension and let E/K be a field extension. Suppose that E is complete with respect to a discrete valuation νE which extends the discrete valuation ν of K. ¯ 0 )E¯ and (Q ¯ 1 )E¯ are both anisotropic. Suppose, too, that the quadratic forms (Q ¯ i )E¯ for i = 0 Then the quadratic form q := QE is also anisotropic and q¯i ∼ = (Q and 1. Proof. Suppose first that Q is unramified and let ϕ : M0 → OK be the quadratic form over OK obtained from the restriction of Q to M0 as in 7.6. Then ¯0 (7.38) ϕ¯ = Q and Q = ϕK by 7.31. Let ψ = ϕOE be the scalar extension of ϕ from OK to OE (as defined in 2.6). By several applications of (2.8), we have QE = (ϕK )E = ϕE = (ϕOE )E = ψE

and ψ¯ = ψE¯ = (ϕOE )E¯ = ϕE¯ = (ϕK¯ )E¯ = ϕ¯E¯ . ¯ By 7.32 ¯ 0 )E¯ by (7.38). Since (Q ¯ 0 )E¯ is anisotropic, so is ψ. Hence ψ¯ = (Q ¯ 0) ¯ . applied to ψ, QE = ψE is anisotropic and unramified and (QE )0 ∼ = (Q E Thus the claim holds if Q is unramified. We turn now to the general case. Let L, L′ , q, q ′ and g be as in 7.17. Identifying M with L ⊕ L′ , we have ′ QE (v, v ′ ) = qE (v) + gE (v, v ′ ) + pqE (v ′ )

for all (v, v ′ ) ∈ (L ⊕ L′ )E = ME (where gE is, of course, the extension by scalars of the bilinear form g from K to E) as well as ¯0 ∼ ¯1 ∼ (7.39) Q = q¯0 and Q = (q ′ )0 . By the conclusion of the previous paragraph, the quadratic forms qE and (q ′ )E are both anisotropic and unramified and
′ ) ∼ (q ′ ) (7.40) (qE )0 ∼ q0 )E¯ and (qE = (¯ 0 = 0 E ¯.

Let e1 , . . . , em be an OK -basis of L and let e′1 , . . . , e′n be an OK -basis of L′ . By 7.14 applied to Q, we have ν(g(ei , e′j )) ≥ 1 for all i ∈ [1, m] and all j ∈ [1, n]. Since the valuation νE on E agrees with the valuation ν on K, we have νE (gE (ei , e′j )) ≥ 1 for all i ∈ [1, m] and all j ∈ [1, n]. By 7.14 applied to ′ ) . QE , we conclude that QE is anisotropic, (QE )0 = (qE )0 and (QE )1 = (qE 0 ∼ ∼ ¯ ¯ By (7.39) and (7.40), it follows that (QE )0 = (Q0 )E¯ and (QE )1 = (Q1 )E¯ . Remark 7.41. If E/K is a finite extension, then E is complete with respect to a discrete valuation extending ν if and only if E/K is unramified. If F = K(β) is a simple transcendental extension of K, then by [4, Chapter VI, Prop. 10.2], there exists a unique discrete valuation νF on F extending ν such that ν(β) = 0, the canonical image β¯ of β in E¯ is transcendental over ¯ ¯ and E ¯ = K( ¯ β). K

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Corollary 7.42. Let Λ = (K, M, Q) be a finite-dimensional anisotropic quadratic space, let β be a transcendental over K, let νF be the unique extension of ν to a valuation of F := K(β) such that νF (β) = 0 and let E be the completion of F with respect to νF . Then QE is also anisotropic. Proof. The valuation νF has a unique extension to a discrete valuation νE ¯ (formed with respect to νE ) is a simple transcendental of E. By 7.41, E ¯ By [21, Lemma 7.15], therefore, the quadratic forms (Q ¯ 0 )E¯ extension of K. ¯ 1 )E¯ are both anisotropic. Hence by 7.37, QE is also anisotropic. and (Q Corollary 7.43. Suppose that (K, M, Q) and E/K satisfy the hypotheses of 7.37 and that Q is trim. Then Q is ramified over K if and only if QE is ramified over E. ¯ 1 )E . In particular, Proof. Since Q is trim, so is QE . By 7.37, (QE )1 ∼ = (Q ¯ (QE )1 has positive dimension if and only if (Q1 )E . Thus QE is ramified if and only if Q is. Proposition 7.44. Let (K, M, q) be an anisotropic quadratic space of finite dimension and suppose that L/K is a finite extension such that (¯ qi )L¯ is hyperbolic for i = 0 and 1. Then qL is hyperbolic. Proof. By 7.18, it suffices to consider the case that q is unramified. Let ϕ : M0 7→ OK be the quadratic form over OK obtained from the restriction of q to M0 as in 7.6 and let ψ = ϕOL . Then q = ϕK and q¯0 = ϕ¯ by 7.31. By several applications of (2.8), therefore, we have qL = (ϕK )L = ϕL = (ϕOL )L = ψL and (¯ q0 )L¯ = ϕ¯L¯ = (ϕK¯ )L¯ = ϕL¯ = (ϕOL )L¯ = ψL¯ . Since (¯ q0 )L¯ is hyperbolic, so is ψL¯ . By 7.33, therefore, qL = ψL is also hyperbolic. Proposition 7.45. Let D be a composition algebra over K with norm N ¯ be the residue field of D as defined and trace T as defined in [65, 30.17], let D ¯ in [65, 9.22] and suppose that dimK¯ D ≥ 2 and T¯ 6= 0. Then there exists an unramified quadratic extension L/K such that the quadratic form NL is hyperbolic. ¯ has dimension at least 2 and its trace is non-zero, it contains Proof. Since D ¯ By [43, Lemma 3], there exists an a separable quadratic extension M/K. ¯ = M and L is a subalgebra unramified quadratic extension L/K such that L of D. Then L is a splitting field of D and hence NL is hyperbolic.

Chapter Eight Quadratic Forms of Type E6, E7 and E8 We begin this chapter by recalling the definition of a quadratic form of type Eℓ as given in [60, 12.31]: Definition 8.1. A quadratic space Λ = (K, L, q) is of type Eℓ for ℓ = 6, 7 or 8 if the following hold: (i) dimK L = 2d, where d = 2 + 2ℓ−6 . (ii) q is anisotropic. (iii) q has a norm splitting as defined in 2.24. In other words, there exist a separable quadratic extension E/K with norm N and elements α1 , . . . , αd of K ∗ such that q∼ = α1 N ⊕ · · · ⊕ αd N. (iv) α1 · · · α4 6∈ N (E) if ℓ = 7 and −α1 · · · α6 ∈ N (E) if ℓ = 8. Note that if ℓ = 7 in 8.1, then the quaternion algebra (8.2)

D = (E/K, α1 · · · α4 )

is division by [60, 9.4] and by [60, 12.28], the image [D] of D in the Brauer group Br(K) is the Hasse invariant (or, in [21, §14], the Clifford invariant) of q. In particular, D is independent of the decomposition of L in 8.1(iii). Remark 8.3. By [14, Thm. 5.3], condition (iv) in 8.1 can be replaced by (iv′ ) The Hasse invariant of q is non-trivial if ℓ = 7 and trivial if ℓ = 8. Remark 8.4. It follows from [21, 13.1 and 13.4] that the discriminant extension of any quadratic form having a norm splitting as displayed in 8.1(iii) is E/K if d is odd and trivial if d is even. Thus if q, ℓ and E/K are as in 8.1, then the discriminant extension of q is E/K if ℓ = 6 and trivial if ℓ = 7 or 8. If ℓ = 7 or 8, the extension E/K is not, in general, an invariant of q. Remark 8.5. Suppose that q is a quadratic form of type Eℓ for ℓ = 7 or 8 over K, let d be as in 8.1(i) and suppose that q∼ = β1 N ′ ⊕ · · · ⊕ βd N ′

is an arbitrary norm splitting of q as defined in 2.24. Thus β1 , . . . , βd are non-zero elements of K and N ′ is the norm of a separable quadratic extension E ′ /K. By [14, Thm. 5.1(i)] and [60, 12.28], the Hasse invariant of q is [D′ ] for D′ = (E ′ /K, (−1)d/2 β1 · · · βd ). It follows by 8.3(iv′ ) that 8.1(iv) holds with β1 , . . . , βd in place of α1 , . . . , αd .

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Remark 8.6. Let q, α1 , . . . , α6 , E/K and N be as in 8.1 with ℓ = 8. Let D1 , D2 and D3 be the quaternion algebras (E/K, −α2 /α1 ), (E/K, −α4 /α3 ) and (E/K, −α6 /α5 ) as defined in [60, 9.3]. Let Ni be the norm of Di for all i ∈ [1, 3]. Then ∼ α1 N1 ⊕ α3 N2 ⊕ α5 N3 . (8.7) q= Since q is anisotropic, D1 , D2 and D3 are division algebras by [60, 9.4]. By 8.1(iv), (−α2 /α1 ) · (−α4 /α3 ) · (−α6 /α5 ) ∈ N (E)

and thus [D1 ] + [D2 ] + [D3 ] = 0 in the Brauer group Br(K). Conversely, if D1 , D2 and D3 are three quaternion division algebras over K such that [D1 ] + [D2 ] + [D3 ] = 0 in the Brauer group Br(K) and α1 , α3 and α5 are elements of K ∗ such that the quadratic form q defined as in (8.7) is anisotropic, then q is of type E8 . Similarly, a quadratic form of type E7 over K can be characterized as the anisotropic orthogonal sum of two quadratic forms similar to the norms of quaternion division algebras D1 and D2 over K such that [D1 ] + [D2 ] + [D3 ] = 0 in Br(K) for a third quaternion division algebra D3 . In this case, [D3 ] is the Hasse invariant of the quadratic form. Finally, a quadratic form of type E6 can be characterized as the anisotropic sum of two quadratic forms, one similar to the norm of a quaternion division algebra D over K and the other similar to the norm of a separable quadratic extension E/K such that E is a subalgebra of D over K. Remark 8.8. By [60, 12.32, 12.34 and 12.35], there exist fields of arbitrary characteristic over which there exist quadratic forms of type E6 , E7 and E8 . By [60, 12.37], if there exists a quadratic form of type Eℓ over a field K for ℓ = 7 or 8, then there also exists quadratic forms of type Eℓ−1 over K. By [60, 12.38], on the other hand, there exist fields over which there exist quadratic forms of type E6 but none of type E7 and others over which there exist quadratic forms of type E7 but none of type E8 . Proposition 8.9. Let (K, L, q) be a quadratic space of type Eℓ for ℓ = 6, 7 or 8, let E be a norm splitting field of q and let u, v ∈ L. Then there exist subspaces W1 , . . . , Wd of L for d = 2 + 2ℓ−6 which are pairwise orthogonal with respect to ∂q such that the restriction of q to Wi is similar to the norm of the extension E/K and u, v ∈ W1 + W2 . Proof. By [60, 12.15], we can choose a norm splitting map T of (K, L, q) as defined in [60, 12.14] such that for all non-zero w ∈ L, the restriction of q to hw, T (w)i is similar to the norm of E/K. Let W be a 3-dimensional subspace of L containing u, v and T (u) and let v ′ be a non-zero vector in W ∩ hu, T (u)i⊥ . Since E/K is separable, we have v ∈ W = hu, T (u), v ′ i. By [60, 12.18(iv)], there exist v1 , . . . , vd ∈ L such that d = 2 + 2ℓ−6 , v1 = u, v2 = v ′ and the subspaces hvi , T (vi )i are pairwise orthogonal. Proposition 8.10. Let (K, L, q) be a quadratic space of type Eℓ for ℓ = 6, 7 or 8 and let W be a subspace of V such that the restriction of q to W is

QUADRATIC FORMS OF TYPE E6 , E7 AND E8

71

similar to the norm of a quaternion division algebra D1 over K and let Q denote the restriction of q to W ⊥ . Then the following hold: (i) If ℓ = 6, then Q is similar to the norm of E/K, where E/K is the discriminant extension of q. (ii) If ℓ = 7, then Q is similar to the norm of the unique quaternion division algebra D2 such that [D1 ] + [D2 ] is the Hasse invariant of q. (iii) If ℓ = 8, then Q is a quadratic form of type E7 whose Hasse invariant is [D1 ]. Proof. Let Q1 denote the restriction of q to W . Since the discriminant of Q1 is trivial, it follows by 8.4 and [21, 13.4] that Q is the norm of the discriminant extension of q if ℓ = 6, so (i) holds, and that Q has trivial discriminant if ℓ ≥ 7. Let ℓ = 7 and let D3 be the unique quaternion division algebra over K such that [D3 ] is the Hasse invariant of q. By [14, 3.8], Q is similar to the norm of the unique quaternion division algebra D2 such that [D2 ] = [D1 ] + [D3 ]. Thus (ii) holds. If ℓ = 8, then Q has Hasse invariant equal to [D1 ], again by [14, 3.8]. It follows by [14, 4.12(ii)] that Q is a quadratic form of type E7 . Thus (iii) holds. Hypothesis 8.11. We assume for the rest of this chapter that K is complete ¯ and t 7→ t¯ be as in with respect to a discrete valuation ν and let p, OK , K 7.1. (For the existence of quadratic forms of type E6 , E7 and E8 under this assumption, see 8.35 and 16.7.) Hypothesis 8.12. Let Λ = (K, L, q) be a quadratic space of type Eℓ for ℓ = 6, 7 or 8, let S be the group obtained by applying [60, 16.6] to Λ and let Z denote the subgroup {(0, t) | t ∈ K} of S. (Note that we use additive notation for S even though it is not abelian.) By [60, 38.10], Z is both the center and the derived group of S. Notation 8.13. Let X := X0 be the vector space obtained by applying [60, 16.6] to Λ. Thus, in particular, (8.14)

dimK X = 2ℓ−3 .

Let 1 be the distinguished element of L∗ called ǫ in [60, 16.6], so q(1) = 1, let (a, x) 7→ ax, θ and h be the maps from X × L to X, from X × L to L and from X × X to L in [60, 16.6], let π be the map from X to L given by π(a) = θ(a, 1) and let
ω(a, t) = ν q(π(a) + t) for all (a, t) ∈ S. The map (a, x) 7→ ax is bilinear.

Proposition 8.15. The quadratic space Λ = (K, L, q) is exceptionally νcompatible as defined in [65, 21.22].

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Proof. By [51, §2, Cor. 2], there exists a Bruhat-Tits building Ξ whose building at infinity ∆ = Ξ∞ is the exceptional quadrangle BQ 2 (Λ). Let Σ be an apartment of ∆. By [65, 13.30 and 16.4(i)] applied to Ξ, there is a valuation φ of the root datum of ∆ with respect to Σ and a positive integer δ such that φ(x1 (0, t)) = δν(t) for all t ∈ K. The claim holds, therefore, by [65, 21.27(i)]. See [66] for an alternate proof in the cases ℓ = 6 and ℓ = 7.
Proposition 8.16. ω − (a, t) = ω(a, t) for all (a, t) ∈ S.

Proof. Let (a, t) ∈ S. By [63, 11.10], −(a, t) = (−a, −t + g(a, a)) for all (a, t) ∈ S, where g is as in [63, 1.17(C3)] (or [60, 13.26]). By [60, 13.35], π(−a) = π(a) for all a ∈ X, where π is as in 8.13. By [60, 13.47(i)] if char(K) 6= 2 and by [60, 13.42 and 13.45(i)] if char(K) = 2, therefore, q(π(a) + t) = q(π(−a) − t + g(a, a)).

Proposition 8.17. Let S and Z be as in 8.12 and for each i, let Si = {(a, t) ∈ S | ω(a, t) ≥ i}

and Zi = Z ∩ Si , where ω is as in 8.13. Then Si is a subgroup of S for each i and S4 + Z0 is a normal subgroup of S0 . Proof. By 8.15, 8.16 and [60, 21.18], Si is a subgroup of S for each i. It follows that the commutator group [S0 , S4 ] is contained in Z0 . Therefore S4 + Z0 is a normal subgroup of S0 . Let s(a, t) = (sa, s2 t) for all s ∈ K and all (a, t) ∈ S. By [60, 13.35], we have (8.18)

and hence (8.19)

q(π(sa) + s2 t) = s4 q(π(a) + t)
ω s(a, t) = 4ν(s) + ω(a, t)

for all s ∈ K and all (a, t) ∈ S. ¯ be as in 8.11. If (a, t) ∈ S0 , s ∈ OK and s1 ∈ pOK , then Let p, OK and K
(s + s1 )a, (s + s1 )2 t − (sa, s2 t) ∈ (s1 a, s21 t) + Z

and by (8.19), ((s + s1 )a, (s + s1 )2 t) ∈ S0 , (sa, s2 t) ∈ S0 and (s1 a, s21 t) ∈ S4 . Hence
(s + s1 )a, (s + s1 )2 t − (sa, s2 t) ∈ (s1 a, s21 t) + Z0 since S0 is a subgroup. It follows that the map
(8.20) s, (a, t) 7→ (sa, s2 t)
¯ × S0 /(S4 + Z0 ) to S0 /(S4 + Z0 ) that gives induces a map from K

S0 /(S4 + Z0 ) ¯ the structure of a vector space over K. The same map (s, (a, t)) 7→ s(a, t) gives S/Z the structure of a vector space over K, and the map (a, t) 7→ a induces an isomorphism from this vector space to X.

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QUADRATIC FORMS OF TYPE E6 , E7 AND E8

Proposition 8.21. Let ∞ \

Y =

(Sm + Z).

m=1

Then Y = Z.

Proof. Let (a, s) ∈ Si and (b, t) ∈ Sj for some i, j and let u = f (h(a, b), 1), where f = ∂q and h is as in 8.13. Then [(a, s), (b, t)] = (0, u).
By 7.2(ii), ν(u) ≥ ν q(h(a, b)) /2. By 8.15 and [65, 21.22(ii)], therefore, ν(u) ≥ (i + j)/4.

It follows that Y is contained in the center of S. Since Z is the center of S, we conclude that Y = Z. Our proof of the next result is similar to the proof of [65, 18.34(iii)]. Theorem 8.22. dimK¯ S0 /(S4 + Z0 ) = dimK S/Z = dimK X, where S, Z, S0 , S4 and Z0 are as in 8.12 and 8.17. Proof. Let V = S0 /(S4 + Z0 ) and let B = {ui | i ∈ M } for some index set M be a set of elements of S0 whose image in V is a ¯ The set B is linearly independent modulo Z (over K) since basis over K. otherwise we could choose scalars si ∈ K almost all zero such that X si u i ∈ Z 0 i∈M

and

 min ν(si ) | i ∈ M = 0.

Since S/Z is finite-dimensional, it follows that |B| is finite. We can thus assume that M = [1, n] for some n, i.e. that B = {u1 , u2 , . . . , un }. It will suffice now to show that B spans S modulo
Z. Choose w = (a, t) ∈ S ∗ . By (8.18), ω sw = 4ν(s) + ω(w) for all s ∈ K. Let k1 be the largest integer such that p−k1 w ∈ S0 . Thus ω(w) − 4k1 ∈ [0, 3] (1) (1) and ω(w) < 4(k1 + 1). There exist scalars s1 , . . . , sn ∈ OK and v ∈ S4 such that n X (1) p−k1 w − si u i ∈ v + Z 0 . i=1

k1

Let w1 = p v. Then ω(w1 ) ≥ 4k1 + 4 > ω(w) and w − w1 ∈

n X i=1

(1)

pk1 si ui + Z.

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If w1 = 0, then w is in the span of B modulo Z. We can thus assume that w1 6= 0. By induction and the conclusion of the previous paragraph, we can assume that there exist elements k1 , k2 , k3 , . . . of Z, elements w1 , w2 , w3 , . . . of S ∗ and elements (1)

(2)

(3)

si , si , si , . . . of OK for all i ∈ [1, n] such that

w − wr ∈ xr + Z

for (8.23)

xr =

n X

(1)

pk1 si

(2)

+ pk2 si

i=1

as well as (8.24) and (8.25)

(r)


+ · · · + pkr si

ui

ω(w) < ω(w1 ) < ω(w2 ) < · · · < ω(wr ) ω(wr−1 ) − 4kr ∈ [0, 3]

for all r ≥ 1 (where w0 = w). By (8.24) and (8.25), (8.26)

k1 ≤ k2 ≤ k3 ≤ · · ·

and no value is repeated more than four times in the sequence (8.26). Hence (8.27)

lim kr = ∞.

r→∞

(r)

For each i ∈ [1, n] and each r ≥ 1, let zi denote the coefficient of ui in (r) (8.23). By (8.27), (zi )r≥1 is a Cauchy sequence (for each i ∈ [1, n]) and hence has a limit fi ∈ K. Let n X x= fi ui . i=1

Then

w − x ∈ wr + (xr − x) + Z

for all r ≥ 1. For each m ≥ 1, there exists N ≥ 1 such that for all r ≥ N , (r) wr ∈ Sm and (zi − fi )ui ∈ Sm for all i ∈ [1, n], so also xr − x ∈ Sm . Thus ∞ \ w−x∈ (Sm + Z). m=1

Hence w − x ∈ Z by 8.21. We conclude that w is contained in the subspace spanned by B modulo Z.

¯ i = Si /(Si+1 + Zi ) for each i. Then X ¯ i is isomorProposition 8.28. Let X phic to a quotient of subspaces of V for each i ∈ [0, 3] and dimK X =

3 X i=0

¯i. dimK¯ X

QUADRATIC FORMS OF TYPE E6 , E7 AND E8

75

Proof. Let Wi = (Si + Z0 )/(S4 + Z0 ) for i ∈ [0, 4]. Then W0 = S0 /(S4 + Z0 ), W4 = 0, Wi is a subspace of W0 and Wi /Wi+1 ∼ = (Si + Z0 )/(Si+1 + Z0 )
∼ = Si / Si ∩ (Si+1 + Z0 ) ¯i =X

for all i ∈ [0, 3].

¯ i for each i, where X ¯ i is as in 8.28. Notation 8.29. Let mi = dimK¯ X Choose s ∈ K ∗ , let the maps π, h, g, φ and θ be as in 8.13 and [60, 16.6] ˆ = sh, gˆ = sg, φˆ = hφ and θˆ = sθ. Next let Sˆ be the group and let π ˆ = sπ, h obtained by replacing g with gˆ in the definition of S in [60, 16.6], let
ω ˆ (a, t) = ν q(ˆ π (a) + t) ˆ+ be the group obtained by replacing h, φ and θ for all (a, t) ∈ Sˆ and let U ˆ φˆ and θ. ˆ Then in the commutator relations defining U+ in [60, 16.6] by h, ˆ the maps xi (a, t) 7→ the map (a, t) 7→ (a, st) is an isomorphism from S to S, xi (a, st) for i = 1 and 3, x2 (u) 7→ x2 (su) and x4 (u) 7→ x4 (u) extend to an ˆ+ and isomorphism from U+ to U ω ˆ (a, st) = ω(a, st) + 2ν(s). Applying this transformation for a suitable s, we can thus assume from now on that (8.30) mi > 0 for i = 0 or 1. Proposition 8.31. Let φ be as in [60, 16.6]. Then for each v ∈ L∗ , the map ξv given by
ξv (a, t) = av, tq(v) + φ(a, v) for all (a, t) ∈ S is an automorphism of S.

Proof. Let 1 ∈ L, U1 and x1 be as in [60, 16.6] and let mv = µ(x4 (v))µ(x4 (1)) for some v ∈ L∗ . We have ¯ 1 = 1 by [60, 12.45] and a · 1 = a and φ(a, 1) = 0 for all a ∈ X by [60, 12.52 and 13.32]. By [60, 32.10], therefore,
x1 (a, t)mv = x1 − av, tq(v) + φ(a, v) for all (a, t) ∈ S. Since the map (a, t) 7→ (−a, t) is an automorphism of S, we conclude that ξv is one too. Proposition 8.32. Suppose that 1 ∈ ν(q(L∗ )) and for each i, let mi be as in 8.29. Then mi = mi+2 for all i. Proof. Let v be an element of L such that ν(q(v)) = 1 and let ξv be as in 8.31. By [65, 21.10(ii)], ξv maps Si to Si+2 and Zi to Zi+2 (where Si and ¯ ¯ i to X ¯ i+2 for Zi are as in 8.17) and hence induces a K-linear map from X each i. Let v¯ be as in [60, 12.45] (where ǫ ∈ L is as in [60, 16.6]). By [60, 12.52–12.53] (or [63, 1.3 and 1.17(A3)], where ǫ is called 1), (av)¯ v = q(v)a ¯ i+2 to X ¯ i for for all a ∈ X. Thus if u = v¯/q(v), then ξu induces a map from X ∼ ¯ ¯ i+2 each i which is the inverse of the map induced by ξv . Therefore Xi = X for all i.

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∗

∗

∗

We use the rest of this chapter to prove some more special properties of quadratic forms of type E5 (defined in 8.33), E6 , E7 and E8 which we will need later on. Definition 8.33. We will say that a quadratic space (K, V, q) (or a quadratic form q) is of type E5 if q is anisotropic, of dimension 4 and has a norm splitting as defined in 2.24. By [60, 9.3], for example, this is the same thing as saying that q is similar to the norm of a quaternion division algebra and by [60, 20.28], this quaternion division algebra is unique. ¯ for ℓ = 5, Proposition 8.34. Let Q be a quadratic form of type Eℓ over K 6, 7 or 8. Then there exists a unique unramified quadratic form q over K such that q¯0 = Q. Proof. This holds by 7.27. ¯ for ℓ = 5, Proposition 8.35. Let Q be a quadratic form of type Eℓ over K 6, 7 or 8 and let q be as in 8.34. Then q is also of type Eℓ . ¯ R is the norm Proof. We have Q = β1 R ⊕ · · · ⊕ βd R, where β1 , . . . , βd ∈ K, ¯ of a separable quadratic extension M/K and d = 2 if ℓ = 5, d = 1 + 2ℓ−6 if ℓ > 5, β1 · · · β4 6∈ R(M ) if ℓ = 7 and −β1 · · · β6 ∈ R(M ) if ℓ = 6. By 7.36, there exist E/K, N and α1 , . . . , αd such that q∼ = α1 N ⊕ · · · ⊕ αd N.

Thus q is of type Eℓ if ℓ = 5 or 6. If ℓ = 7, then α1 · · · α4 6∈ N (E) because β1 · · · β4 6∈ R(M ), so q is of type Eℓ also in this case. Suppose, finally, that ℓ = 8. By a change of basis, we can assume that β6 = −1/β1 · · · β5 . By 7.36, therefore, we can assume that α6 = −1/α1 · · · α5 . Hence q is of type Eℓ also in this last case.

Proposition 8.36. Suppose that Λ = (K, L, q) is a ramified anisotropic quadratic space such that q¯0 is similar to q¯1 . Then the following hold: (i) If q is of type E7 , then q¯0 is not of type E5 . (ii) If q is of type E8 , then q¯0 is not of type E6 . Proof. Suppose that q¯0 is of type Eℓ for ℓ = 5 or 6. By 8.34, there exists a unique unramified quadratic form ρ over K such that ρ¯0 = q¯0 . Applying 7.14 and 7.28 with ρ in place of q and q ′ and g = 0, we conclude that q is isometric to ρ ⊕ βρ for some β ∈ K such that ν(β) = 1. By 8.35, ρ is of type Eℓ . Thus ρ = α1 N ⊕ · · · ⊕ αd N , where N is the norm of an unramified × separable quadratic extension E/K, α1 , . . . , αd ∈ OK , d = 2 if ℓ = 5 and d = 3 if ℓ = 6. Thus q∼ = α1 N ⊕ α2 ⊕ βα1 N ⊕ βα2 N

QUADRATIC FORMS OF TYPE E6 , E7 AND E8

77

if ℓ = 5 and q∼ = α1 N ⊕ α2 ⊕ α3 N ⊕ βα1 N ⊕ βα2 N ⊕ βα3 N if ℓ = 6. We have (α21 α22 β)2 ∈ K 2 ⊂ N (E) and hence by 8.5, q is not of type E7 if ℓ = 5. Let ℓ = 6. Then −(α1 α2 α3 )2 β 3 6∈ N (E) since ν(−(α1 α2 α3 )2 β 3 ) is odd but ν(N (E ∗ )) = 2Z. Hence q is not of type E8 , again by 8.5. Proposition 8.37. Let q be a ramified quadratic form of type E6 and sup¯ pose that q¯0 is similar to the norm of a quaternion division algebra C over K ¯ and that q¯1 is similar to the norm of a separable quadratic extension B/K. ¯ Then B is isomorphic to a subalgebra of C over K. Proof. We can assume that q¯0 equals the norm of C. By [43, Thm. 1], there ¯ ∼ exists a unique unramified quaternion algebra D over K such that D = C. Let ρ denote the norm of D. Then ρ¯0 = q¯0 . By 7.26, there exists a unique ¯ = B and q¯1 = αN ¯0 for unramified quadratic extension E/K such that E × some α ∈ OK , where N is the norm of E/K. Let Q = ρ ⊕ pαN , where p is ¯i ∼ as in 8.11. By 7.14, Q is anisotropic and Q = q¯i for i = 0 and 1. By 7.28, ∼ therefore, q = Q. By 8.4, the discriminant extension of ρ is trivial. Thus the discriminant extension of q is E/K. By 8.4, it follows that E is a norm splitting field of q. Thus qE is hyperbolic, so by Witt cancellation, E is a norm splitting field of D. Thus E is isomorphic to a subfield of D and hence ¯ = C. E¯ = B is isomorphic to a subfield of D Proposition 8.38. Let Λ = (K, L, q) be a ramified quadratic space of type E7 and suppose that for i = 0 and 1, q¯i is similar to the norm of a quaternion ¯ Then C0 and C1 are not isomorphic but contain division algebra Ci over K. ¯ a common separable quadratic extension of their center K. Proof. By [43, Thm. 1], there exist unique quaternion division algebras D ¯ ∼ ¯ ∼ ¯ and H ¯ are as and H over K such that D = C0 and H = C1 , where D defined in [65, 9.22]. Let ρ and ξ be the norms of D and H. Applying 7.14 and 7.28 with ρ in place of q, ξ in place of q ′ and g = 0, we conclude that q is similar to ρ ⊕ βξ for some β ∈ K such that ν(β) = 1. By 8.3, there exists a quaternion division algebra J over K such that [J] is the Hasse invariant of q. Applying [14, 3.8 and 3.9] to ρ ⊕ βξ, we obtain [J] = [D] + [H] in Br(K). By [47, Lemma 5.33], J¯ is also a quaternion division algebra over ¯ By 7.45, D, H and J have unramified splitting extensions. Thus by [47, K. Thm. 5.7], ¯ + [H] ¯ = [J] ¯ [C0 ] + [C1 ] = [D] ¯ in Br(K). Hence C0 is not isomorphic to C1 , but by [14, 3.5], there is a ¯ such that M is a subalgebra of both C0 separable quadratic extension M/K and C1 . Proposition 8.39. Let Λ = (K, L, q) be a quadratic space of type E7 , let D be as in 8.2, let ρ be a quadratic form that is similar to the norm of D, let

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β, F , νF and E be as in 7.42, let νE denote the unique extension of νF to E and let Q = qE ⊕ βρE . Then qE is a quadratic form of type E7 , Q is a quadratic form of type E8 and  (8.40) νE (Q(u, w)) = min νE (qE (u)), νE (ρE (w)) for all u ∈ LE and all w ∈ DE .

Proof. By 7.42, both qE and ρE are anisotropic. Let u ∈ LE and w ∈ DE . To show that (8.40) holds for u and w, it suffices to assume that νE (qE (u)) = νE (ρE (w)). Replacing q and ρ by q/p and ρ/p if necessary, we can assume that, in fact, νE (qE (u)) = νE (ρE (w)) = 0. By [21, 19.6], the quadratic form ¯ E )0 (qE )0 ⊥ β(ρ is anisotropic. Hence νE (Q(u, w)) = 0. Thus (8.40) holds. It follows from (8.40) that Q is anisotropic. By 8.6, therefore, Q is of type E8 and its subform qE is of type E7 .

Chapter Nine Quadratic Forms of Type F4 The Moufang quadrangles of type F4 were discovered in the course of carrying out the classification of Moufang polygons in [60]. These quadrangles gave rise to the notion of a quadratic form of type F4 . In this chapter we prove some results about these quadratic forms which we will need in Chapter 17. We begin by recalling the definition ([60, 14.1]): Notation 9.1. A quadratic space Λ = (K, L, q) is of type F4 if char(K) = 2, q is anisotropic and • for some separable quadratic extension E/K with norm N , • for some subfield F of K containing K 2 viewed as a vector space over K with respect to the scalar multiplication (t, s) 7→ t2 s for all (t, s) ∈ K × F and • for some α ∈ F ∗ and some β ∈ K ∗ , q is similar to the quadratic form qK on E ⊕ E ⊕ F given by (9.2)

qK (u, v, s) = β −1 (N (u) + αN (v)) + s

for all (u, v, s) ∈ E ⊕ E ⊕ F . We do not assume that dimK F is finite. Note that the defect of qK is (0, 0, F ) and the co-dimension of its defect is 4. We set f = ∂q and fK = ∂qK . Notation 9.3. Let E/K, K/F , α and β be as in (9.1). Let W = E ⊕ E (which is called W0 in [60, 14.6]), let D be the composite field E 2 F , let X = D ⊕ D (which is called X0 in [60, 14.5]) and let qˆ be the quadratic form ˆ := X ⊕ K given by on the F -vector space L (9.4)

qˆ(x, y, t) = α(N (x) + β 2 N (y)) + t2

ˆ where N here denotes the restriction of N to D. We will for all (x, y, t) ∈ L, sometimes write qF in place of qˆ and we will denote its bilinear form by fF or fˆ. ˆ := (F, L, ˆ qF ) Notation 9.5. Let Λ be as in (9.1). The quadratic space Λ ˆ is a quadratic space of defined in 9.3 is called the dual of Λ. By [60, 14.13], Λ ˆ and the decomposition type F4 . Applying the recipe for the dual in (9.1) to Λ ˆ is (K 2 , E 2 ⊕E 2 ⊕F 2 , q 2 ) and hence isometric (9.4), we find that the dual of Λ to the original quadratic space Λ.

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Remark 9.6. Let q be as in (9.2). Neither the quadratic extension E/K nor the constants α and β nor the decomposition (9.2) are invariants of the similarity class of q. By [60, 28.44], however, the similarity class of the ˆ = (F, L, ˆ qF ) defined in (9.4) and, in particular, the field F quadratic space Λ ˆ is, up to similarity, are invariants of the similarity class of q. In particular, Λ independent of the choice of E/K, α and β and the decomposition (9.2). Proposition 9.7. Let Λ = (K, L, q), F and f be as in 9.1 and suppose that f (w, w′ ) = 1 for some w, w′ ∈ L. Then there exist w1 , w1′ ∈ L and ξ ∈ F ∗ such that f (w1 , w1′ ) = ξ, q(w1 ) = ξq(w), q(w1′ ) = ξq(w′ ) and hw, w′ i is orthogonal to hw1 , w1′ i with respect to f .

ˆ be as in Proof. We identify q with the quadratic form qK in (9.2) and let L 9.3. Let W denote the subspace E ⊕ E of L and let X denote the subspace ˆ (These are the subspaces called W0 and X0 in [60, 14.5– D ⊕ D of L. 14.6].) Let q1 be the restriction of q to the subspace W of L and let f1 be the restriction of f to W . We have w = (b, s) and w′ = (b′ , s′ ) for some b, b′ ∈ W and some s, s′ ∈ F . Thus (9.8)

f1 (b, b′ ) = f (w, w′ ) = 1.

Let Θ and ψ be the maps defined as in [60, 14.15-14.16], let a be a non-zero element of X, let (9.9)

t = f1 (Θ(a, b), b′ ),

let ξ = qˆ(a, t), let and let


w1 = Θ(a, b) + tb, ξs + ψ(a, b)


w1′ = Θ(a, b′ ) + tb′ , ξs′ + ψ(a, b′ ) .

ˆ ⊂ F and, since qˆ is anisotropic, ξ is Thus w1 , w1′ ∈ W ⊕ F = L, ξ ∈ qˆ(L) non-zero. By [60, 14.18(i)], we have (9.10)

f1 (Θ(a, c), c′ ) = f1 (c, Θ(a, c′ ))

and (9.11)

f1 (Θ(a, c), c) = 0

for all c, c′ ∈ W . By (9.8), (9.9), (9.10) and (9.11), hw, w′ i is orthogonal to hw1 , w1′ i with respect to f . Next we observe that f (w1 , w1′ ) = f1 (Θ(a, b) + tb, Θ(a, b′ ) + tb′ ) = f1 (Θ(a, b), Θ(a, b′ )) + t2 f1 (b, b′ ) = f1 (Θ(a, Θ(a, b′ )), b) + t2

QUADRATIC FORMS OF TYPE F4

81

by (9.8) and three applications of (9.10), and f1 (Θ(a, Θ(a, b′ )), b) + t2 = qˆ(a, t) by (9.8) and [60, 14.18(iv)]. Therefore f (w1 , w1′ ) = ξ. Finally, we have
q(w1 ) = q Θ(a, b) + tb, ξs + ψ(a, b)
= q1 Θ(a, b) + tb + ξs + ψ(a, b)

= q1 (Θ(a, b)) + t2 q1 (b) + ξs + ψ(a, b) by (9.11) = qˆ(a, t) · q(b, s) by [60, 14.18(xi)] = ξq(w)

and

q(w1′ )

= ξq(w′ ) by a similar argument.

Corollary 9.12. Suppose that f (w, w′ ) 6= 0 for some w, w′ ∈ L. Then E/K, α, β and the similarity from q to qK in 9.1 can be chosen so that the images of w and w′ are both contained in the subspace (E, 0, 0) of E ⊕ E ⊕ F .

Proof. Replacing w′ by w′ /f (w, w′ ), we can assume that f (w, w′ ) = 1. Let q˜ denote the restriction of q to the subspace hw, w′ i. There exists a quadratic extension E/K and an element β ∈ K such that the quadratic space (K, hw, w′ i, q˜) is isometric to (K, E, β −1 N ), where N is the norm of the extension E/K. Since f (w, w′ ) 6= 0, the extension E/K is separable. The claim holds now (with α = ξ) by 9.7.

Proposition 9.13. Suppose that K is complete with respect to a discrete valuation ν and that F is closed with respect to ν, and let E/K and D/F be as in 9.1 and 9.3. Then the extension E/K is unramified if and only if the extension D/F is unramified. Proof. Let δ˜ be as in [65, 22.8] and let νF denote the restriction of ν/δ˜ to F . Then νF is a valuation of F and F is complete with respect to νF , so νF is the unique valuation on F . In particular, whether or not D/F is unramified does not depend on the choice of a valuation. We denote by N and T the norm and trace of the extension E/K. The restrictions of N and T to D are the norm and trace of the extension D/F . Suppose that the extension E/K is unramified. Then there exists δ ∈ E such that E = K(δ) and ν(N (δ)) = ν(T (δ)) = 0. We have D = F (δ 2 ), N (δ 2 ) = N (δ)2 and T (δ 2 ) = T (δ)2 , so ν(N (δ 2 )) = ν(T (δ 2 )) = 0. It follows that ¯ F¯ is non-zero and hence that D/F is unramified. Suppose, the trace of D/ conversely, that D/F is unramified. Then there exists γ ∈ D such that D = F (γ) and ν(N (γ)) = ν(T (γ)) = 0. Since E = K(γ), it follows that E/K is also unramified. Proposition 9.14. Let K, q, F and f be as in 9.7 and for each i, let q¯i be as in 7.3 and f¯i = ∂ q¯i . Suppose that K is complete with respect to a discrete valuation ν, that F is closed with respect to ν and that f¯i 6= 0 for i = 0 or 1. Then E/K, α, β and the similarity from q to qK in 9.1 can be chosen so that the extension E/K is unramified.

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Proof. Since f¯i 6= 0, we can choose w, w′ ∈ L such that

ν(q(w)) = ν(q(w′ )) = ν(f (w, w′ )) = i.

By 9.12, we can assume in (9.2) that w, w′ are elements of E such that ν(N (w)) = ν(N (w′ )) = ν(T (w, w′ )) = j for j = 0 or 1, where N and T are the norm and trace of the extension ¯j ) ≥ 2 and ∂ f¯j 6= 0. By 7.4, it follows that E/K is E/K. Therefore dimK¯ (N unramified. ˆ = (F, L, ˆ qˆ) be as in 9.1 and 9.3 Proposition 9.15. Let Λ = (K, L, q) and Λ and suppose that K is complete with respect to a discrete valuation ν and that F is closed with respect to ν. Then q is totally wild if and only if qˆ is totally wild. Proof. As observed in 9.13, the dual of qˆ is isometric to q. It thus suffices to show that if q is not totally wild, then neither is qˆ. Suppose q is not totally wild. By 9.14, we can assume that the extension E/K in (9.2) is unramified. By 9.13, therefore, qˆ is, in fact, not totally wild.

PART 2

Residues in Bruhat-Tits Buildings

Chapter Ten Residues We now turn our attention to the residues of a Bruhat-Tits building whose building at infinity is an exceptional quadrangle. Let ∆, Σ, c, α1 , . . . , α2n , Ui and Ω = (U+ , U1 , . . . , Un ) be as in 3.1. We suppose, in addition, that n = 4 and that we are in case (v) or case (vi) of 4.2. Thus ∆ is an exceptional Moufang quadrangle and either Ω = QE (Λ)

for some quadratic space Λ = (K, L, q) of type Eℓ for ℓ = 6, 7 or 8 (“case Eℓ ”) or Ω = QF (Λ)

for some quadratic space Λ = (K, L, q) of type F4 (“case F4 ”). Hypothesis 10.1. From now on, we assume as well that there is an affine building Ξ of type ˜2 = C

•..............................................•..........................................•....

whose building at infinity is ∆. Thus Ξ is a Bruhat-Tits building in the sense of 1.20 (and [65, 13.1]) and we apply the identification of Aut(Ξ) with Aut(∆) described in 1.26. Remark 10.2. Recall that we are assuming in these notes that Ξ∞ is formed with respect to the complete system of apartments of Ξ; see 1.21. (Note, in particular, that the structure of the residues does not depend on the choice of a system of apartments.) By [65, 27.5], therefore, the field K is complete with respect to a discrete valuation ν and in the F4 -case that the subfield F of K in defined in [60, 14.3] is closed with respect to this valuation. By [65, 23.15], the valuation ν of K is unique. Remark 10.3. The converse is also valid: If Λ is an arbitrary quadratic space of type Eℓ for ℓ = 6, 7 or 8 or of type F4 over a field K that is complete with respect to a discrete valuation, and if in the F4 -case the subfield F defined in [60, 14.3] is closed with respect to this valuation and if ∆ is the corresponding Moufang quadrangle of type Eℓ or F4 (which exists by [60, 17.8]), then there always exists a unique affine building Ξ such that ∆ is the building at infinity of Ξ with respect to its complete system of apartments. Existence holds by [65, 27.2] (and 8.15 if Λ is of type Eℓ for ℓ = 6, 7 or 8; see also [51, §2, Cor. 2]) and uniqueness holds by [65, 27.6]. See 34.16.

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Applying the notation in 1.22, we have ˜ E (Λ) or C ˜ E (Λ) Ξ=B 2 2 in the case Eℓ and ˜ F (Λ) or C ˜ F (Λ) Ξ=B 2 2 in the case F4 . Notation 10.4. In the case Eℓ , let δΨ be the number called δΛ in [65, 21.25], let δΛ be the number called δ in [65, 19.17] (see 7.5) and let δ ∗ = δΨ /δΛ . In the case F4 , let qK and qF be as in [65, 22.3 and 22.5], let δ˜ be as in [65, ˜ F /δK . By [65, 21.26 22.9], let δK and δF be as in [65, 22.10] and let δ ∗ = δδ ∗ ∗ and 22.11], δ = 1 or 2 in both cases. If δ = 1, we will say that we are in the long case and if δ ∗ = 2, we will say that we are in the short case. Notation 10.5. Suppose we are in the case Eℓ . In this case, we can assume that Ω and x1
, . . . , x4 are as in [60, 16.6]. For i = 1 and 3, let ϕi (xi (a, t)) = ν q(π(a) + t) /δΨ for all (a, t) ∈ S and for i = 2 and 4, let ϕi (xi (u)) = ν(q(u))/δΛ for all u ∈ L, where the group S is as in 8.15. Notation 10.6. Suppose we are in the case F4 . In this case, we can assume that Ω and x1 , . . . , x4 are as in [60, 16.7]. Let ϕi (xi (b, s)) = qK (b, s)/δK for all (b, s) ∈ W ⊕ F = L for i = 2 and 4 and let ϕi (xi (a, t)) = qF (a, t)/δF for ˆ for i = 1 and 3, where (K, L, qK ) and (F, L, ˆ qF ) are all (a, t) ∈ X ⊕ K = L as in 9.1 and 9.3 and δK and δF are as in 10.4. Remark 10.7. In both cases Eℓ and F4 and for all i ∈ [1, 4], the map ϕi from Ui∗ to Z defined in 10.5 and 10.6 is surjective. By [65, 16.4, 21.27 and 22.16], there is a root map ι as defined in [65, 3.12] and a unique valuation ϕ of the root datum of ∆ based at Σ with respect to ι as defined in [65, 3.21] which agrees with ϕ1 on U1∗ and ϕ4 on U4∗ . By [65, 13.29], this valuation agrees also with ϕ2 on U2∗ and ϕ3 on U3∗ . By [65, 8.27], there is a unique apartment A of Ξ whose apartment at infinity is Σ. By [65, 13.30], we can assume that ϕ = ϕR , where R is an irreducible residue of ∆ of rank 2 (and thus a gem of Ξ as defined in [65, 7.3]) containing chambers of the apartment A (so R ∩ A is both a residue of A and, by [62, 8.13(i)], an apartment of R) and ϕR is as in [65, 13.8]. Notation 10.8. Let Ui,k = {u ∈ Ui | ϕi (u) ≥ k} for each i ∈ [1, 4] and each k ∈ Z. By [65, 18.20], Ui,k+1 is a normal subgroup of Ui,k for each i and each k. Let ¯i,k = Ui,k /Ui,k+1 U for all i and all k.

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We now consider the standard embedding of the apartment A in the Euclidean plane which takes the intersection A∩R, where R is as in 10.7, to the set of eight triangles containing the origin. Here is a piece of this embedding, over which we have superimposed a basis of the root system Φ = B2 = C2 consisting of a short vector u and a long vector v: ....... . ............... .......... ... ... ............ ... .... ..................................... ...... . ................................ .................................. .. ... ... ... ..... ................................ .. . .. . ... .............. . . . ........ ....... .............................. ..... ............................. .......... ....... ... ... ............... .. .. . . . . . . . . .. .................. ..... .. ...... . .... . .. ............................. ....... .... .. . ................... ..... ... ... ...... ... ... . ... ... . . . . . . . . .............................. ..... .... .. ... ...... .... ... ... . . . .... ... .... .... .. .... ..... .... . . . . . . . ... ... .. ..... ... .. ... .... . .. ....... .

•

(See Figure 2.2 in [65] for the “bigger picture.”) The bullet is at the origin in our picture, the vector u points at (1, 0) and the vector v points at (−1, 1). The chambers in this picture are the closed triangles. A solid, dashed or dotted panel of A is a pair of triangles sharing a segment which is solid, dashed or dotted. In particular, we are using “solid,” “dotted” and “dashed” in our picture in place of “1,” “2” and “3” as names for the three vertices of the Coxeter diagram C˜2 . (Note that the vector u is covering up a dotted segment and the vector v is covering up two solid segments.) Notation 10.9. Let R0 = R and let R1 denote the residue of Ξ containing the eight chambers in our picture that contain the point (0, −1) and let C be the lightly shaded triangle in R0 ∩R1 . Thus R0 and R1 are the two irreducible residues of rank 2 of Ξ containing the chamber C. Let P = R0 ∩ R1 . Thus P is the unique panel of Ξ containing both C and the chamber (i.e. triangle) to its left in our picture. The group G† defined in 1.18 acts transitively on the set of chambers of Ξ. Thus every irreducible rank 2 residue of Ξ is in the same G† -orbit at R0 or R1 . Main Goal 10.10. The main goal of Chapters 11–17 of these notes is— given 10.1—to determine the structure of the residues R0 and R1 . Notation 10.11. For i = 0 and 1, let βi be the unique root of the apartment Ri ∩ A of Ri containing the four chambers to the right of the vertical line through the origin in our picture. Let Φ+ be the set of positive roots of the root system Φ with respect to the basis {u, v}. For all w ∈ Φ+ , we let aw denote the half-plane containing w whose wall is perpendicular to w and passes through the origin. Thus the more darkly shaded chamber in our picture, which we denote by C0 , is the unique chamber in R0 ∩ au ∩ av . We can assume that the chamber c of Σ in 3.1 is the parallel class of sectors of Ξ (see [65, 8.9]) containing the sector σ(R0 , C0 ) of the apartment A as defined in [65, 4.1]. It follows that either
ι(α1 ), . . . , ι(α4 ) = (u, 2u + v, u + v, v)

88 or

CHAPTER 10


ι(α1 ), . . . , ι(α4 ) = (v, u + v, 2u + v, u),

where the roots α1 , . . . , α4 are the roots of the apartment Σ of ∆ such that Ui = Uαi and the root map ι is as in 10.7 (see [65, 3.12]). For each w ∈ Φ+ , let a′w be the half-plane containing aw whose wall passes through the point (0, −1). Thus a′u = au and the more lightly shaded chamber (which we named C in 10.9) in our picture is the unique chamber in R1 ∩ a′u ∩ a′v . Remark 10.12. By [65, 16.3], ι is long (respectively, short) as defined in [65, 16.12] if ι(α1 ) = v (respectively, ι(α1 ) = u). Thus by [65, 21.27(ii) and 22.16(ii)], ι(α1 ) = v in the long case (as defined in 10.4) and ι(α1 ) = u in the short case. By [65, 13.18], Ui,0 is the pointwise stabilizer in Ui of the half-plane aι(ai ) for each i ∈ [1, 4] and Ui,1 is the pointwise stabilizer in Ui of the half-plane a′ι(ai ) for each i ∈ [1, 4] such that ι(ai ) 6= u. Let (0)

U+ = U1,0 U2,0 U3,0 U4,0 and K = U1,1 U2,1 U3,1 U4,1 . It follows from 10.8 and [65, 3.21(V2)] that (0) (0) U+ is a subgroup of U+ and that K is a normal subgroup of U+ . Thus Ui,0 ∩ K = Ui,1 for each i ∈ [1, 4], so the natural image of Ui,0 in the quotient (0) (0) ¯+ U := U+ /K

¯i,0 . Let can be identified with U (10.13)

(0) ¯ ¯ 0 = (U ¯+ ¯2,0 , U ¯3,0 , U ¯4,0 ). Ω , U1,0 , U

In the long case, where ι(α4 ) = u by 10.12, we set (1)

U+ = U1,1 U2,1 U3,1 U4,0 and M = U1,2 U2,2 U3,2 U4,1 . This time it follows from 10.8 and [65, 3.21(V2)] (1) (1) that U+ is a subgroup of U+ and that M is a normal subgroup in U+ . Thus Ui,1 ∩ M = Ui,2 for each i ∈ [1, 3], so the natural image of Ui,1 in the quotient ¯ (1) := U (1) /M U + + ¯i,1 , and U4,0 ∩ M = U4,1 , so the natural image of can be identified with U (1) ¯ ¯4,0 . Let U4,0 in U+ can be identified with U (10.14)

¯ 1 = (U ¯ (1) , U ¯1,1 , U ¯2,1 , U ¯3,1 , U ¯4,0 ). Ω +

In the short case, where ι(α1 ) = u by 10.12, we set (1)

U+ = U1,0 U2,1 U3,1 U4,1 and M = U1,1 U2,2 U3,2 U4,2 . Again it follows from 10.8 and [65, 3.21(V2)] (1) (1) that U+ is a subgroup of U+ and that M is a normal subgroup in U+ .

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Thus Ui,1 ∩ M = Ui,2 for each i ∈ [2, 4], so the natural image of Ui,1 in the quotient ¯ (1) := U (1) /M U + + ¯i,1 , and U1,0 ∩ M = U1,1 , so the natural image of can be identified with U (1) ¯ ¯1,0 . Let U1,0 in U+ can be identified with U (10.15)

¯ 1 = (U ¯ (1) , U ¯1,0 , U ¯2,1 , U ¯3,1 , U ¯4,1 ). Ω +

By [65, 18.18], the residues R0 and R1 are both Moufang. The following result describes their root group sequences in terms of the root group sequence Ω of ∆. ¯ 0 be as in (10.13) in both the long and short Proposition 10.16. Let Ω ¯ ¯ 1 be as in (10.15) in cases, let Ω1 be as in (10.14) in the long case and let Ω ¯ the short case. Then for i = 0 and 1, Ωi is a root group sequence isomorphic to a root group sequence of Ri with respect to the apartment A ∩ Ri of Ri all of whose terms fix the chamber C. Proof. This is proved in [65, 18.25 and 18.28]. Main Goal 10.17. By [60, 7.5], a Moufang polygon is uniquely determined by any one of its root group sequences. We can thus reformulate 10.10 by saying that our main goal in Chapters 11–17 is—given 10.1—to determine ¯ 0 and Ω ¯ 1 defined in all the possibilities for the two root group sequences Ω 10.16. Notation 10.18. Let I = 1 in the short case and let I = 4 in the long case ¯I,0 is the unique term (as defined in 3.2) (as defined in 10.4). By 10.11, U ¯ 0 and Ω ¯ 1 have in common and if J = 5 − I, that the root group sequences Ω ¯J,0 is the term at the other end of the root group sequence Ω ¯ 0 from then U ¯I,0 (i.e. one is the first term and the other is the last term) and U ¯J,1 is the U ¯ 1 . Let µ = µI be as defined in 3.6 with respect term at the other end of Ω to the apartment Σ of ∆, let i = 0 or 1 and let µ ¯ be as defined in 3.6 with respect to Ri , the apartment Ri ∩ A of Ri and the root βi of Ri ∩ A defined in 10.11. Let UI◦ = {a ∈ UI∗ | ϕI (a) = 0},

where ϕI is as defined in 10.5, and let a ∈ UI◦ . By [65, 18.17 and 18.20], a induces on Ri a non-trivial element ai of the root group of Ri corresponding to the root βi and by [65, 3.25], µ(a) induces the element µ ¯(ai ) on Ri . The following remark will play a fundamental role in our investigations. Remark 10.19. Let UI◦ and µ = µI be as in 10.18. For all a, b ∈ UI◦ , let ¯I,0 induced by the product µ(a)µ(b) and ma,b denote the automorphism of U let HI = hma,b | a, b ∈ UI◦ i.

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¯I,0 as defined in 3.7. For this Thus HI is the torus of both R0 and R1 at U reason, we call HI the joint torus of Ξ. For given a, b ∈ UI◦ , the product µ(a)µ(b) ¯I,0 ) which can be calculated in both Ω ¯ 0 and Ω ¯1 induces an element of Aut(U ¯I,0 ) we using the formulas in Chapter 32 of [60]. The element ma,b ∈ Aut(U obtain by such calculations must be independent, however, of whether we ¯ 0 or in Ω ¯ 1. choose to calculate in Ω We conclude this chapter with one last comment: ¯ j,i Remark 10.20. Let Wj = xj (0, K) and Wj,i = Wj ∩ Uj,i and let W ¯ denote the image of Wj,i in Uj,i for j = 1 and 3 and for all i, where Uj,i and ¯j,i are as in 10.8. By [65, 19.35(i)], U ¯ (0) := W ¯ 1,0 U ¯2,0 W ¯ 3,0 U ¯4,0 W + (0)

is a subgroup of U+ and ¯ ∗ := (W ¯ (0) , W ¯ 1,0 , U ¯2,0 , W ¯ 3,0 , U ¯4,0 ) Ω 0 + ¯ 0 ), where Λ ¯ 0 is as in 7.3. If is a root group sequence isomorphic to QQ (Λ ∗ ¯ ¯ ¯ ¯ W1,0 6= U1,0 , we can apply 4.10 to the pair (Ω0 , Ω0 ). Note, too, that the ¯1,0 /W ¯ 1,0 is endowed with the structure of a vector space over K ¯ quotient U as described in 4.15 and that if we are in case Eℓ for ℓ = 6, 7 or 8, then ¯1,0 /W ¯ 1,0 , where m0 is as in 8.29. m0 = dimK¯ U

Chapter Eleven Unramified Quadrangles of Type E6, E7 and E8 In this chapter, we begin to consider the case that the building at infinity of the Bruhat-Tits building Ξ is a Moufang quadrangle of type E6 , E7 and E8 . We continue with all the assumptions and notation in Chapters 8 and 10. In particular: Hypothesis 11.1. The triple Λ = (K, L, q) is a quadratic space of type Eℓ ¯ 0, for ℓ = 6, 7 or 8, K is complete with respect to a discrete valuation ν, Λ ¯ ¯ ¯ Λ1 , f0 := ∂ q¯0 and f1 := ∂ q¯1 are as in 7.3, δΛ and δΨ are as in 10.4, R0 and ¯ 0 and Ω ¯ 1 are the two root R1 are the two residues of Ξ defined in 10.9 and Ω group sequences defined in 10.16. Proposition 11.2. One of the following three cases holds: (i) δΛ = δΨ = 2. (ii) δΛ = 1 and δΨ = 2. (iii) δΛ = δΨ = 1. Proof. This holds by [65, 21.26]. Definition 11.3. We will say that Ξ is an unramified quadrangle if 11.2(i) holds, a semi-ramified quadrangle if 11.2(ii) holds and a ramified quadrangle if 11.2(iii) holds (even though it is really the building at infinity of Ξ rather than Ξ itself which is a quadrangle). Our goal in this chapter is to treat the first of these three cases; we turn to the other two in Chapters 12 and 13. Theorem 11.4. Suppose that δΛ = δΨ = 2 and that R0 and R1 are not both ¯ 0 ) and one of the following holds: ¯1 ∼ indifferent. Then Ω = QQ (Λ ¯ 0 is of type Eℓ and Ω ¯0 ∼ ¯ 0 ). (i) Λ = QE (Λ

¯ 0 is of type F4 and Ω ¯0 ∼ ¯ 0 ). (ii) Λ = QF (Λ

(iii) ℓ = 7, f¯0 = 0 and there is an anisotropic quadratic space Θ0 = ¯ 0, U ¯1,0 , Q) such that (L ¯0 ∼ Ω = QQ (Θ0 )op , ¯ 0 /K ¯ is a purely inseparable extension of degree 8 such that where L 2 ¯ 0 , Def(Q) 6= 0 and the co-dimension in U ¯1,0 of q¯0 (u) = u for all u ∈ L the defect of Q is 2.

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Proof. By [60, 16.6], we can assume that q is unital. By 10.4, we are in the long case and thus I = 4 by 10.18. By 7.4 and (8.1)(i), we have (11.5)

¯ 0 = dimK L = 4 + 2ℓ−5 . dimK¯ L

Since δΨ = 2 we have ω(S ∗ ) = 2Z, where S and ω are as in 8.12 and 8.13. Therefore Si ⊂ Si+1 and hence mi = 0 for all odd i, where Si and mi are as in 8.17 and 8.29. Therefore (11.6)

m0 + m2 = dimK X = 2ℓ−3 and m0 > 0

by (8.14), 8.28 and (8.30). ¯ ∗ and W ¯ j,i be as in 10.20 and let Let Ω 0 ¯ ∗1 := (W ¯ +(1) , W ¯ 1,1 , U ¯2,1 , W ¯ 3,1 , U ¯4,0 ), Ω ¯ (1) := W ¯ 1,1 U ¯2,1 W ¯ 3,1 U ¯4,0 . The maps xi (t) 7→ xi (t/p) for i = 1 and where W + ¯ ∗1 to 3, x2 (v) 7→ x2 (v/p) and x4 (v) 7→ x4 (v) induce an isomorphism from Ω ∗ ∗ ¯ . Hence Ω ¯ is a root group sequence isomorphic to QQ (Λ ¯ 0 ). If m2 = 0, Ω 0 1 ¯1,1 = W ¯ 1,1 and hence Ω ¯1 = Ω ¯ ∗ = QQ (Λ ¯ 0 ). If m2 > 0, we it follows that U 1 ¯ 1,1 6= U ¯1,1 and can apply 4.10 also to the pair (Ω ¯ 1, Ω ¯ ∗1 ). Furthermore, have W ¯1,1 /W ¯ 1,1 , where U ¯1,1 /W ¯ 1,1 is endowed with the structure of a m2 = dimK¯ U ¯ as in 4.15. vector space over K ¯ 0, Ω ¯ ∗0 ). Since dimK¯ L ¯ 0 is greater than 4, We now apply 4.10 to the pair (Ω ¯ 0 is not in case (a) of 4.10. Suppose that Ω ¯ 0 is in case (b) we know that Ω ¯ 0 is of type Eℓ or F4 . It follows that m0 = 2ℓ−3 , or case (c) of 4.10. Thus Λ in case (b) by 4.16(ii) and in case (c) by 4.16(iii) and (11.5). By (11.6), ¯1 ∼ ¯ 0 ). therefore, m2 = 0. As we observed above, m2 = 0 implies that Ω = QQ (Λ We conclude that (i) or (ii) holds. ¯ 0 is in case (d) of 4.10. In particular, [U ¯2,0 , U ¯4,0 ] = 1, Now suppose that Ω ¯ 0 is a power of 2. Thus ℓ = 7 and dimK¯ L ¯0 = 8 so f¯0 = 0, and by 4.13, dimK¯ L ¯ 1,0 , U ¯3,0 ] = 1. By 4.17, it follows that by (11.5). Since [W1 , U3 ] = 1, also [W ¯1 ∼ ¯ 0 ), m0 ≥ 16. By (11.6), therefore, m0 = 16 and m2 = 0, so again Ω = QQ (Λ and s = 2 in 4.17, so the defect of Q is of co-dimension 2. Thus (iii) holds. ¯ 0 is in case (e) of 4.10, so R0 is indifferent. In parSuppose, finally, that Ω ¯ ¯1 ∼ ¯ 0 ) and hence also R1 is indifferent ticular, f0 = 0. If m2 = 0, then Ω = Q(Λ by [60, 38.4]. Since R0 and R1 are not both indifferent by assumption, we ¯ 1 . By 4.8(iv) aphave m2 > 0. This means that we can apply 4.10 to Ω plied to R0 , the joint torus HI defined in 10.19 is abelian, and by (11.5), ¯ 0 > 2. By 4.19, it follows that Ω ¯ 1 cannot be in cases (a), (b) or (c) of dimK¯ L ¯ ¯ 0 is a power of 2 4.10. Suppose that Ω1 is in case (d) of 4.10. Then dimK¯ L ¯ (by 4.13), so ℓ = 7 and dimK¯ L0 = 8. Since [W1 , U3 ] = 1, it follows by 4.17, ¯1 that m2 ≥ 16. This is impossible, however, by (11.6). We conclude that Ω is in case (e) of 4.10, but this implies that R1 is indifferent.

Chapter Twelve Semi-ramified Quadrangles of Type E6 , E7 and E8 We continue with all the assumptions and notation in 11.1. In this chapter, we analyze the case (ii) of (11.2). Thus δΛ = 1 and δΨ = 2. By 10.4, we are in the short case and hence I = 1 by 10.18. Our goal in this chapter is to prove 12.7–12.9. Let mi for each i be as in 8.29. Since δΨ = 2, we have mi = 0 for all odd i. Hence m0 + m2 = dimK X = 2ℓ−3 by (8.14) and 8.28. Since δΛ = 1, we have m0 = m2 by 8.32. Therefore (12.1)

m0 = 2ℓ−4 .

¯ i for i = 0 and 1. Thus Let ni = dimK¯ L (12.2)

n0 + n1 = dimK L = 4 + 2ℓ−5

by 7.4 and (8.1)(i). Since δΛ = 1, (12.3)

both n0 and n1 are positive. ∗ ¯ Let Ω0 and Wj,i be as in 10.20. Then

¯ (1) , W ¯ 1,0 , U ¯2,1 , W ¯ 3,1 , U ¯4,1 ) ¯ ∗ := (W Ω 1 + ¯ 1 ), where is a root group sequence isomorphic to QQ (Λ ¯ +(1) := W ¯ 1,0 U ¯2,1 W ¯ 3,1 U ¯4,1 . W

¯ 1,0 6= U ¯1,0 , we can apply 4.10 to both pairs, (Ω ¯ 0, Ω ¯ ∗ ) and (Ω ¯ 1, Ω ¯ ∗ ). Since W 0 1 It follows from 4.13 that (12.4)

n0 and n1 are both even

(since n0 and n1 cannot both be equal to 1) and from 4.15 that ¯1,0 /W ¯ 1,0 . (12.5) m0 = dimK¯ U Remark 12.6. As described in [60, 16.6], the commutator relations defining the root group sequence QE (Λ)—but not the quadrangle ∆ itself (see [60, 27.20])—depend on the choice of the distinguished element 1 of L (which is called ǫ in [60, 16.6]). Replacing 1 by an element v ∈ L such that ν(q(v)) = 1 before we apply [60, 16.6] to write down the commutator relations to which we apply 10.5 to define the maps ϕi has the effect of interchanging R0 with R1 up to isomorphism. We are now ready to consider the three cases ℓ = 6, ℓ = 7 and ℓ = 8. We begin with the case ℓ = 6.

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Theorem 12.7. Suppose that ℓ = 6, that δΛ = 1 and δΨ = 2 and that R0 and R1 are not both indifferent. Then after interchanging R0 and R1 —as explained in 12.6—if necessary, n0 = 4, n1 = 2 and one of the following holds: (i) For i = 0 and 1, f¯i := ∂ q¯i (where q¯i is as in 7.3) is non-degenerate and there is an anisotropic pseudo-quadratic space ¯ i , K, ¯ σi , Vi , Qi ) Θi = (L such that ¯i ∼ Ω = QP (Θi ),

¯ 0 has the structure of a quaternion division algebra over K ¯ with where L ¯ 0 , dimL¯ V0 = 1, L ¯ 1 /K ¯ is norm q¯0 , σ0 is the standard involution of L 0 a separable quadratic extension with norm q¯1 , σ1 is the non-trivial ele¯ 1 /K) ¯ and dimL¯ V¯1 = 2. Furthermore, L ¯ 1 is isomorphic ment of Gal(L 1 ¯ 2. to a subfield of L (ii) f¯0 is non-degenerate but f¯1 = 0, there is a proper involutory set ¯ 0 , B, σ) Θ0 = (L such that ¯0 ∼ Ω = QI (Θ0 ),

¯ 0 is a quaternion division algebra over K ¯ with norm q¯0 , σ is where L its standard involution and there is an anisotropic quadratic space ¯ 1, U ¯1,0 , Q) Θ1 = (L such that ¯1 ∼ Ω = QQ (Θ1 )op ,

¯ 1 /K ¯ is an inseparable quadratic extension, Def(Q) 6= 0 and the where L ¯ 1 of the defect of Q is 2. co-dimension over L Proof. By (12.2), n0 + n1 = 6. Hence we can assume that n0 = 4 and n1 = 2 ¯ 0, Ω ¯ ∗0 ) and by (12.3), (12.4) and 12.6. We now apply 4.10 to the pairs (Ω ¯ 1, Ω ¯ ∗ ). Suppose first that Ω ¯ 1 is in case (a) of 4.10. Since n1 = 2, it follows (Ω 1 that there is an anisotropic pseudo-quadratic space ¯ 1 , K, ¯ σ1 , V1 , Q1 ) Θ1 = (L such that ¯1 ∼ Ω = QP (Θ1 ),

¯ 1 /K ¯ is a separable quadratic extension with norm q¯1 and σ1 is the where L ¯ 1 /K). ¯ By 4.16(i), dimL¯ V1 = m0 /n1 and hence non-trivial element of Gal(L 1 ¯ 0 cannot be in cases (b) or (c) of by (12.1), dimL¯ 1 V1 = 2. Since n0 = 4, Ω 4.10. By [60, 11.3 and 11.19], the skew-hermitian form associated with Q1 is non-degenerate. By 4.8 applied to R1 , therefore, the joint torus HI defined

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¯ 0 cannot be in 10.19 is non-abelian. By 4.19 applied to R0 , it follows that Ω ¯ 0 is also in case (a). Thus by 4.16(i) and in cases (d) or (e) of 4.10. Hence Ω and (12.1), as well as 8.37, we conclude that (i) holds. ¯ 0 is in case (d) of 4.10. Thus U ¯1,0 is abelian and there Suppose next that Ω ¯ 0, U ¯1,0 , ρ) such that is an anisotropic quadratic space Θ0 = (L ¯0 ∼ Ω = QQ (Θ1 )op .

¯ 1 of W ¯ 1,0 in U ¯1,0 is m0 /n0 = By 4.16(iv) and (12.1), the co-dimension over L ¯ 1,0 , U ¯3,1 ] = 1, and thus 4.17 yields a contradic1. Since [W1 , U3 ] = 1, also [W ¯ 0 is not in case (d). tion. We conclude that Ω ¯ 1 cannot be in cases (b) or (c) of 4.10. Suppose next that Since n1 = 2, Ω ¯1,0 is abelian and there is an anisotropic it is in case (d). Thus f¯1 = 0, U ¯ 1, U ¯1,0 , Q) such that quadratic space Θ1 = (L ¯1 ∼ Ω = QQ (Θ1 )op

¯ 1 /K ¯ is an inseparable quadratic extension. By 4.16(iv) and (12.1), the and L ¯ 1 of W ¯ 1,0 in U ¯1,0 is m0 /n1 = 2. By 4.17 (which we can co-dimension over L ¯1,0 is apply because [W1 , U3 ] = 1), the co-dimension of the defect of Q in U of dimension 2. By 4.8(iii) applied to R1 , therefore, the joint torus HI is non-abelian. It follows now by 4.18, 4.19 and the conclusion of the previous ¯ 0 is in case (a) of 4.10 and (ii) holds. paragraph that Ω ¯ 1 is in case (e) of 4.10. Thus R1 is indifferent. It remains to assume that Ω ¯0 By 4.8(iv), the joint torus HI is abelian. It follows by 4.19(i)-(ii) that Ω ¯0 cannot be in cases (a), (b) or (c) of 4.10. We have already ruled out that Ω ¯ is in case (d). Hence Ω0 is also in case (e) and R0 is also indifferent. Theorem 12.8. Suppose that ℓ = 7, that δΛ = 1 and δΨ = 2 and that R0 and R1 are not both indifferent. Then after interchanging R0 and R1 if necessary, one of the following holds: ¯i,0 is non(i) n0 = n1 = 4 and for i = 0 and 1, f¯i is non-degenerate, U abelian and there is anisotropic pseudo-quadratic space ¯ i , K, ¯ σi , Vi , Qi ) Θi = (L such that ¯i ∼ Ω = QP (Θi ),

¯ i has the structure of a quaternion division algebra over K ¯ where L ¯ with norm q¯i , σi is the standard involution of L0 and dimL¯ i Vi = 2. ¯ 0 and L ¯ 1 are distinct Furthermore, the quaternion division algebras L ¯ but contain a common separable quadratic extension of their center K. (ii) n0 = n1 = 4, f¯0 = 0 but f¯1 is non-degenerate, there is an anisotropic quadratic space ¯ 0, U ¯1,0 , Q) Θ0 = (L such that ¯0 ∼ Ω = QQ (Θ0 )op ,

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¯ 0 /K ¯ is an inseparable quadratic extension of degree 4 such that where L ¯ 20 ⊂ K, ¯ Def(Q) 6= 0 and the co-dimension of the defect of Q in U ¯1,0 L is 2 and there is a proper involutory set ¯ 1 , B, σ) Θ1 = (L such that ¯1 ∼ Ω = QI (Θ1 ),

¯ 1 is a quaternion division algebra over K ¯ with norm q¯1 and σ where L is its standard involution. (iii) n0 = n1 = 4, f¯i = 0 for i = 0 and 1 and there is an anisotropic quadratic space ¯ 0, U ¯1,0 , Q) Θ = (L such that ¯0 ∼ ¯1 ∼ Ω =Ω = QQ (Θ)op ,

¯ 0 /K ¯ is a purely inseparable extension of degree 4 such that where L ¯ 2 ⊂ K, ¯ Def(Q) 6= 0 and the co-dimension of the defect of Q is 2. L 0 ¯ 0 is of type E6 , (iv) n0 = 6, n2 = 2, f¯0 and f¯1 are both non-degenerate, Λ ¯0 ∼ ¯ 0) Ω = QE (Λ and there is an anisotropic pseudo-quadratic space ¯ 1 , K, ¯ σ, V, Q) Θ1 = (L such that ¯1 ∼ Ω = QP (Θ1 ),

¯ 1 /K ¯ is a separable quadratic extension with norm q¯1 , σ1 is the where L ¯ 1 /K) ¯ and dimL¯ V = 4. non-trivial element of Gal(L 1 ¯ 0 is of type F4 , (v) n0 = 6 and n1 = 2, Λ ¯0 ∼ ¯ 0 ), Ω = QF (Λ f¯1 = 0 and there is a quadratic space

¯ 1, U ¯1,0 , Q) Θ1 = (L ¯ 0 in the sense of 9.5 such that of type F4 similar to the dual of Λ ¯1 ∼ ¯ 1 )op Ω = QQ (Θ ¯ 1 /K ¯ is an inseparable quadratic extension with norm q¯1 . and L

Proof. By (12.1), m0 = 8. By (12.2)–(12.4) and 12.6, we can assume that either n0 = 6 and n1 = 2 or n0 = n1 = 4. Suppose first that n0 = n1 = 4. ¯ 0 nor Ω ¯ 1 can be in cases (b) or (c) of 4.10. Suppose that By 4.13, neither Ω ¯ ¯ 0 and Ω ¯ 1 are in case (a), in which case U1,0 is non-abelian. By 4.18, both Ω

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f¯0 and f¯1 are both non-degenerate. By 4.16(i) and 8.38, we conclude that (i) holds. ¯1,0 is abelian. Suppose, too, that Ω ¯ 0 is in case We can thus suppose that U ¯ 0, U ¯1,0 , Q) (d). Thus f¯0 = 0, there exists an isotropic quadratic space Θ = (L such that ∼ QQ (Θ)op ¯0 = Ω

¯ 0 /K ¯ 0 is a purely inseparable field extension of degree 4 such that and L 2 ¯ ¯ ¯ 0 of W ¯ 1,0 in U ¯1,0 is m0 /n0 = 2. L0 ⊂ K0 . By 4.16(iv), the co-dimension over L ¯ By 4.17 and (12.1), therefore, the defect of Q in U1,0 is of co-dimension 2. ¯1 Hence by 4.19(iii), HI is non-abelian. By 4.18 and 4.19, it follows that Ω ¯ 1 is in case (a), then (ii) holds by 4.18(i) and if Ω ¯1 is in case (a) or (d). If Ω is in case (d), then by 6.13 and [60, 16.11], (iii) holds. By 12.6, we can thus ¯ 0 nor Ω ¯ 1 is in case (d). assume from now on that neither Ω ¯ 0 and Ω ¯ 1 are in case (e), then R0 and R1 are both indifferent. If If both Ω ¯ 1 is in case (a), then by 4.18(i) and 4.19(i), H1 is non-abelian and hence Ω ¯ 1 is not in case (e). By 12.6, it remains only to consider that by 4.19(iv), Ω ¯ 0 and Ω ¯ 1 are both in case (a). Then for i = 0 and 1, there exists case that Ω ¯ i , Bi , σi ) such that a proper involutory set Θi := (L ¯i ∼ Ω = QQ (Θi ),

¯ i is a quaternion division algebra and σi is its standard involution. where L ¯ 0 and L ¯ 1 are isomorphic quaternion division algebras. By 6.28 and 10.19, L ¯ 0 and L ¯ 1 . Hence By 4.10(a), q¯0 and q¯1 are, up to similarity, the norms of L they are similar to each other. By 8.36, however, this is impossible. ¯ 0 is not in cases (a) or (d) Suppose now that n0 = 6 and n1 = 2. By 4.13, Ω ¯ ¯ 0 is in case (b). By of 4.10 and Ω1 is not in cases (b) or (c). Suppose that Ω ¯ ¯ 4.18, it follows that U1,0 is non-abelian and Ω1 is in case (a). In particular, f¯0 is non-degenerate for both i = 0 and i = 1. By 4.16(i), therefore, (iv) ¯ 0 is in case (c). By 4.18(iii) applied to R0 , U ¯1,0 is holds. Suppose that Ω ¯ abelian. By 4.18(i), therefore, Ω1 cannot be in case (a). By 4.19(ii) applied ¯ 1 cannot be in case (e). to R0 , HI is non-abelian. By 4.19(iv), therefore, Ω ¯ ¯ We conclude that Ω1 is in case (d). In particular, f1 = 0. By 12.11 below, we ¯ 0 is in case (e). By 4.18(iii) conclude that (v) holds. Suppose, finally, that Ω ¯ ¯ 1 cannot be in case and 4.19(iv), U1,0 and HI are both abelian. Therefore Ω (a) by 4.18(i) and since [W1 , U3 ] = 1, it cannot be in case (d) by 4.19(iii). ¯ 1 is also in case (e) and thus both R0 and R1 are indifferent. Hence Ω Theorem 12.9. Suppose that ℓ = 8, that δΛ = 1 and δΨ = 2 and that R0 and R1 are not both indifferent. Then after interchanging R0 and R1 if necessary, n0 = 8, n1 = 4 and one of the following holds: ¯ 0 is of type E7 , (i) f¯0 and f¯1 are non-degenerate, Λ ¯0 ∼ ¯ 0) Ω = QE (Λ

and there is an anisotropic pseudo-quadratic space ¯ 1 , K, ¯ σ, V, Q) Θ1 = (L

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such that ¯1 ∼ Ω = QP (Θ1 ),

¯ 1 is a quaternion division algebra over K ¯ with norm q¯1 , σ is where L ¯ 1 and dimL¯ V = 4. the standard involution of L 1 ¯ 0 is of type F4 , (ii) Λ ¯0 ∼ ¯ 0 ), Ω = QF (Λ

f¯1 = 0 and there is a quadratic space

¯ 1, U ¯1,0 , Q) Θ1 = (L ¯ 0 in the sense of 9.5 such that of type F4 similar to the dual of Λ ¯1 ∼ Ω = QQ (Θ1 )op ,

¯ 1 /K ¯ is a purely inseparable extension of degree 4 and L ¯ 2 ⊂ K. ¯ where L 1 (iii) f¯0 = 0, there is an anisotropic quadratic space ¯ 0, U ¯1,0 , Q) Θ0 = (L such that ¯0 ∼ Ω = QQ (Θ0 )op ,

¯ 0 /K ¯ is a purely inseparable extension degree 8 with L ¯ 20 ⊂ K, ¯ where L ¯ ¯ Def(Q) 6= 0 and the co-dimension of the defect of Q in U1,0 is 2, f1 is non-degenerate and there is a proper involutory set ¯ 1 , B, σ) Θ1 = (L such that ¯1 ∼ Ω = QI (Θ1 ),

¯ 1 is a quaternion division algebra over K ¯ with norm q¯1 and σ where L is its standard involution. Proof. By 12.6, we can assume that n0 ≥ n1 . Thus by (12.3)–(12.4), we have ¯ 0 is in case (e) of 4.10. Then U ¯1,0 is n0 = 6, 8 or 10. Suppose first that Ω abelian by 4.18(iii) and, by 4.18(iv), the joint torus HI is abelian. By 4.18(i) ¯ 1 is not in case (a) and by 4.19(ii)–(iii), if n1 6= 4 and 4.19(i) if n1 = 4, Ω ¯ ¯ 1 is also in Ω1 is not in case (b), (c) or (d) (since [W1 , U3 ] = 1). Therefore Ω case (e), so R0 and R1 are both indifferent. We can thus suppose from now ¯ 0 is not in case (e). on that Ω ¯ 0 is in case (b) with Now suppose that n0 6= 8. By 4.13, n0 = 6 and Ω ℓ = 6 or in case (c). By 4.16(ii)–(iii) and 12.5, it follows in both cases that m0 = 8, whereas m0 = 16 by (12.1). With this contradiction, we conclude that n0 = 8 and n1 = 4. ¯ 0 cannot be in case (a) of 4.10 and Ω ¯ 1 cannot be in case (b) By 4.13, Ω ¯ or in case (c). Suppose that Ω0 is in case (b). By 4.18(ii), it follows that

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99

¯1,0 is non-abelian and hence that Ω ¯ 1 is in case (a). In particular, f¯i is U non-degenerate for both i = 0 and 1. By 4.16, we conclude that (i) holds. ¯ 0 is in case (c). Then U ¯1,0 is abelian, the coNext we suppose that Ω dimension of the defect of q¯0 is 4 and the joint torus HI is non-abelian by ¯ 1 cannot be in case (a). 4.19(ii). By 4.18(i), 6.27 and 10.19, it follows that Ω ¯ 1 cannot be in case (e). Thus Ω ¯ 1 is in case (d). In particular, By 4.19(iv), Ω f¯1 = 0. By 12.11 below, we conclude that (ii) holds. ¯ 0 is in case (d). Thus f¯0 = 0 and there exists an Suppose, finally, that Ω ¯ 0, U ¯1,0 , Q) such that anisotropic quadratic space Θ0 = (L ¯0 ∼ Ω = QQ (Θ0 )op . By 4.17, the co-dimension of the defect of the quadratic form Q is m0 /n0 = 2. ¯1,0 is abelian and by 4.19(iii), H1 is non-abelian. By 4.13, Ω ¯1 By 4.18(iii), U ¯ cannot be in cases (b) or (c) and by 4.19(iv), Ω1 cannot be in case (e). ¯ 1 is in case (d). There thus exists an anisotropic quadratic Suppose that Ω ¯ ¯1,0 , Q1 ) such that space Θ1 = (L1 , U ¯1 ∼ Ω = QQ (Θ1 )op . By 4.17, the co-dimension of the defect of the quadratic form Q1 is 4 and hence Θ1 cannot be similar to Θ0 . By 6.13 and 10.19, however, Θ1 must be ¯ 1 is not in case similar to Θ0 . With this contradiction, we conclude that Ω ¯ 1 is in the one remaining case, (a), and so (iii) holds by (d). Therefore Ω 4.18(i). ¯ 1 (with its structure as a field or a quaternion Remark 12.10. Let Θ1 and L division algebra) be as in 12.8(iv) or 12.9(i). The quadratic form q¯0 is of type E6 in the first of these two cases and of type E7 in the second. Let ρ be the ˜ in the first case, where γ is as in [64, anisotropic pseudo-quadratic form γ Q ˜ 3.17] and Q is as in [64, eq. (34)] applied to q¯0 , and let ρ be the anisotropic ˆ obtained by applying [64, eq. (42)] to q¯0 in the pseudo-quadratic form Q second case (see also [64, 5.4]). Let Θρ be the corresponding anisotropic pseudo-quadratic space and let Ωρ be the root group sequence obtained by applying [60, 16.5] to Θρ . The anisotropic pseudo-quadratic space Θρ is ¯ σ), where either q¯0 is of proper and defined over an involutory set (D, K, ¯ type E6 , D/K is the discriminant extension of q¯0 and σ 6= 1 or q¯0 is of type ¯ whose image in Br(K) ¯ is the E7 , D is a quaternion division algebra over K Hasse invariant of q¯0 and σ is the standard involution of D. As is shown explicitly in [15], the torus at the first term of Ωρ is linked (as defined in ¯ 0 . The torus at the first term of Ω ¯ 1 is 3.11) to the torus at the first term of Ω ¯ also linked to the torus at the first term of Ω0 (by 10.19). It follows from [60, 35.19], 6.15 and 6.17 that Θ1 and Θρ are similar pseudo-quadratic spaces as ¯ 1 is isomorphic to D and hence L ¯ 1 /K ¯ defined in [60, 11.27]. In particular, L ¯ ¯ is the discriminant extension of q¯0 in 12.8(iv) and the image of L1 in Br(K) is the Hasse invariant of q¯0 in 12.9(i). Remark 12.11. In 12.8(v) and 12.9(ii), the elements of the torus HI can ¯ 0 using [60, 32.11] or in Ω ¯ 1 using [60, 32.7]. By 6.13 and be calculated in Ω

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¯ 1, U ¯1,0 , Q) is similar to 10.19, we conclude that the quadratic space Θ1 = (L ¯ 0 as defined in 9.5 in these two cases. In the dual of the quadratic space Λ particular, Θ1 is a quadratic space of type F4 and the co-dimension of the defect of Q is 4.

Chapter Thirteen Ramified Quadrangles of Type E6, E7 and E8 We continue with all the assumptions and notation in 11.1. In this chapter, we consider case (iii) of 11.2. Thus δΛ = δΨ = 1. By 10.4, we are in the long case and hence I = 4 by 10.18 (as in 11.4). Our goal in this chapter is to prove 13.15. Let mi for all i be as in 8.29. Since δΛ = 1, we have m0 = m2 and m1 = m3 by 8.32. By (8.14) and 8.28, therefore, m0 + m1 = 2ℓ−4 .

(13.1)

Let π be as in 8.13. Since π(0) = 0, we have q(π(0) + t) = t2 for all t ∈ K. Therefore Z1 ⊂ S2 and hence by 8.28, ¯ 1 = S1 /(S2 + Z1 ) = S1 /S2 . (13.2) X Since S1 /S2 is non-trivial (because δΨ = 1), we conclude that (13.3)

m1 > 0. ∗

For each (a, t) ∈ S , let ζ(a,t) be the map from L to itself given by ζ(a,t) (v) = θ(a, v) + tv

for all v ∈ L, where θ is as in 8.13. By [63, 1.17(C)], the map ζ(a,t) is ¯ ¯ linear and by [65, 21.10(i)], ζ(a,t) induces a similarity from
Λi to Λi+j for ∗ each (a, t) ∈ S and for each i, where j = ν q(π(a) + t) . Since δΨ = 1, it ¯ 0 to Λ ¯ 1 and from Λ ¯ 1 to follows that there exist injective similarities from Λ ¯ ¯ ¯ Λ0 . Hence Λ0 and Λ1 have the same dimension and thus ¯ 0 is similar to Λ ¯ 1. (13.4) Λ By 7.4 and (8.1)(i), therefore, (13.5)

¯ 0 = 2 + 2ℓ−6 . n0 = dimK¯ L

Remark 13.6. Let u and v be elements of L∗ such that ν(q(u)) = 1 and ν(q(v)) = 0. By 7.2(ii), ν(f (u, v)) ≥ 1. We have [x2 (u), x4 (v)−1 ] = x3 (0, f (u, v))

and


 ν q π(0) + f (u, v) = ν(f (u, v)2 ) ≥ 2.

¯2,1 and U ¯4,0 in the root group sequence Ω ¯ 1 commute Hence the root groups U elementwise. It follows by 4.2 that either ¯1 ∼ (13.7) Ω = QI (Θ1 )op

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for some proper involutory set Θ1 = (D, D0 , τ ), or (13.8)

¯1 ∼ Ω = QQ (Θ1 )op

¯ 0, U ¯1,1 , Q) or R1 is indifferent. for some anisotropic quadratic space Θ1 = (L By 4.8(ii)-(iii), we conclude that (13.9)

if HI is non-abelian, then (13.7) holds

and (13.10)

if HI is abelian, then either (13.8) holds or R1 is indifferent.

Lemma 13.11. When ℓ = 7, q¯0 is not similar to the norm of a quaternion ¯ and when ℓ = 8, q¯0 is not a quadratic form of type division algebra over K E6 . Proof. This holds by 8.36 and (13.4). Proposition 13.12. Suppose that δΛ = δΨ = 1 and that m0 > 0. Then ℓ = 7 or 8 and R0 and R1 are both indifferent. ¯ 0, Ω ¯ ∗ ). By 4.13 and (13.5), Proof. By 10.20, we can apply 4.10 to the pair (Ω 0 2 + 2ℓ−6 is even. Therefore ℓ = 7 or 8. ¯ 0 is in case (e) of 4.10. Suppose first that ℓ = 7, so We claim that Ω ¯ 0 cannot be in case (a) of 4.10. By 4.13, Ω ¯ 0 is not in n0 = 4. By 13.11, Ω ¯ 0 is in case (d), then since [W1 , U3 ] = 1, 4.17 implies cases (b) or (c). If Ω ¯ 0 = 8. By (13.1) and (13.3), this is impossible. Hence that m0 ≥ 2 · dimK¯ L ¯ 0 is in case (e). Ω ¯ 0 is in case (b), (c) or (e) Now suppose that ℓ = 8, so n0 = 6. By 4.13, Ω ¯ 0 is not in case (b). Suppose that Ω ¯ 0 is in case (c). Thus q¯0 is and by 13.11, Ω a quadratic form of type F4 . Hence the defect of q¯0 is of co-dimension 4. By 6.27 and 10.19, it follows that (13.7) does not hold. By 4.19(ii), however, HI ¯0 is not abelian. Thus 13.9 yields a contradiction. We conclude again that Ω must be in case (e). This proves our claim. Thus R0 is indifferent (whether ¯ = 2. ℓ = 7 or ℓ = 8). In particular, char(K) It remains only to show that R1 is also indifferent. Suppose that R1 is not indifferent. By 4.8(iv) applied to R0 , the joint torus HI is abelian. Therefore ¯ = 2, the (13.8) holds by (13.10). Since R1 is not indifferent and char(K) ¯1,1 as a co-dimension of the quadratic form Q and hence the dimension of U ¯ 0 must be at least 2. It follows from [60, 16.3] that the vector space over L sets ¯4,0 ]3 [x1 (a, t), U ¯ ∗ into disjoint for all (a, t) ∈ S such that ω(a, t) = 1 form a partition of U 3,1 ∗ ¯ subgroups, at least 2 of these sets are needed to cover U3,1 and for each (a, t) ∈ S such that ω(a, t) = 1, the map (13.13)

u 7→ [x1 (a, t), u]3

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103

¯4,0 to [x1 (a, t), U ¯4,0 ]3 ∪ {0}, where x1 (a, t) is an additive surjection from U ¯1,1 , ω is as in 8.13 and 0 is the identity denotes the image of x1 (a, t) in U ¯3,1 . By [60, 16.6], it follows that the sets element of U Ξ(a,t) := {(av, q(v)t + φ(a, v)) | v ∈ L∗ and ν(q(v)) = 0} for (a, t) ∈ S such that ω(a, t) = 1 are subgroups forming a partition of the set of non-zero elements of X1 = S1 /S2 and at least two of these sets are distinct, where (av, q(v)t + φ(a, v)) denotes the image
of (av, q(v)t + φ(a, v)) in S1 /S2 . Choose (a, t) ∈ S ∗ such that ν q(π(a) + t) = 1 and let W = Ξ(a,t) ∪ {0}. Then W is a subspace of ¯ 1 over K ¯ with respect to the scalar multiplication given in (8.20). Since X the map in (13.13) is an additive surjection, so is the map v¯ 7→ (av, q(v)t + φ(a, v))

¯ 0 to W . By [63, 1.17(A1) and 4.14], this map is K-linear ¯ from L and by [65, 21.10(ii)], it is injective. By (13.5), therefore, dimK¯ W = 2 + 2ℓ−6 . There exist (a′ , t′ ) ∈ S such that ω(a′ , t′ ) = 1 and Ξ(a′ ,t′ ) is disjoint from Ξ(a,t) . Hence m1 ≥ 2 · dimK¯ W = 4 + 2ℓ−5 .

By (13.1) and our assumption that m0 > 0, we have m1 < 2ℓ−4 . It follows that ℓ = 8, m1 ≥ 12 and hence (13.14)

m0 ≤ 4.

Now choose (a, t) ∈ S such that ω(a, t) = 0. By [63, 1.17(A1) and 4.14] and [65, 21.10(ii)] again, the map v¯ 7→ (av, q(v)t + φ(a, v))

¯ ¯ 0 to X ¯ 0 , where this time (av, q(v)t + φ(a, v)) is a K-linear injection from L ¯ 0 . From this it follows, however, denotes the image of (av, q(v)t+φ(a, v)) in X ¯ 0 = 6, in contradiction to (13.14). Thus also R1 is that m0 ≥ dimK¯ L indifferent. Here, now, is the main result of this chapter. Theorem 13.15. Suppose that δΨ = 1 and that R0 and R1 are not both indifferent. Then ¯0 ∼ ¯ 0 ), Ω = QQ (Λ there exists an involutory set Θ1 := (D, D0 , τ ) such that ¯1 ∼ Ω = QI (Θ1 )op ,

¯ ⊂ Z(D) ∩ D0 and dimK¯ D0 = 2 + 2ℓ−6 , and exactly one of the where K following holds:

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¯ τ is of the first (i) ℓ = 6, D is a quaternion division algebra, Z(D) = K, kind and f¯0 = 6 0.

¯ is a separable (ii) ℓ = 7, D is a quaternion division algebra, Z(D)/K ¯ quadratic extension, τ is of the second kind and f0 is non-degenerate.

¯ is an (iii) ℓ = 7, char(K) = 2, D is a quaternion division algebra, Z(D)/K inseparable quadratic extension, τ is of the first kind and the defect of q¯0 has dimension 2. (iv) ℓ = 7, char(K) = 2, D is commutative, τ is of the second kind, f¯0 is ¯ is a purely inseparable extension of degree 4 identically zero and D0 /K 2 ¯ such that D0 ⊂ K.

¯ τ is of the (v) ℓ = 8, D is a biquaternion division algebra, Z(D) = K, first kind, f¯0 is non-degenerate and D0 = Dτ , where Dτ is as defined in (5.4).

¯ is a (vi) ℓ = 8, char(K) = 2, D is a quaternion division algebra, Z(D)/K ¯ τ is of purely inseparable extension of degree 4 such that Z(D)2 ⊂ K, the first kind and the defect of q¯0 has dimension 4. Proof. By (13.1) and 13.12, we have m0 = 0 and m1 = 2ℓ−4 . ¯ 0 ). Suppose first that f¯0 = 0. This ¯0 = Ω ¯ ∗0 ∼ By 10.20, therefore, Ω = QQ (Λ ¯ means that char(K) = 2 and R0 is indifferent. Thus R1 is not indifferent. By 4.8(iv), the joint torus HI is abelian, so by (13.10), there exists an anisotropic ¯ 0, U ¯1,1 , Q) such that (13.8) holds. By [60, 37.30], quadratic space Θ1 = (L ¯ we can, in fact, identify Ω1 with QQ (Θ1 )op so that ¯1 is the identity of the ¯ 0 . Let ∗ denote the scalar multiplication from L ¯0 × U ¯1,1 to U ¯1,1 . By field L [60, 16.6], (13.16)

u ¯ ∗ (a, t) = (au, tq(u) + φ(a, u))

for all (a, t) ∈ S1 and u ∈ L0 , where S1 and L0 are as in 7.3 and 8.17, (a, t) ¯ 0 . By denotes the image of (a, t) in S1 /S2 and u ¯ denotes the image of u in L [60, 13.32], (13.17) and by [60, 13.32 and 13.35], (13.18)

φ(a, r · 1) = 0

φ(a, ru) = r2 φ(a, u) = φ(ra, u)

× for all r ∈ K and all u ∈ L. Thus if r, s ∈ OK , then by (13.17),

rs ∗ (a, t) = (ars, t(rs)2 )

= r¯ ∗ (as, ts2 ) = r¯ ∗ s¯ ∗ (a, t) ¯ is a subfield of L ¯ 0 . Let u be for all (a, t) ∈ S1 , from which it follows that K an element of L such that ν(q(u)) = 0. Let ρ denote the map (13.19)

v 7→ f (1, v) · 1 + v

RAMIFIED QUADRANGLES OF TYPE E6 , E7 AND E8

105

from L to itself and let z = uρ . Then q(z) = q(u). By [60, 12.45 and 12.52], we have (au)z = q(u)a and by [60, 13.32 and 13.66], we have (13.20) φ(a, u)q(u) = φ(au, uρ ). Since f¯0 is identically zero, we have z¯ = u ¯. Thus

u ¯∗ u ¯ ∗ (a, t) = z¯ ∗ u¯ ∗ (a, t) = z¯ ∗ (au, tq(u) + φ(a, u))

= ((au)z, tq(u)2 + φ(a, u)q(u) + φ(au, uρ )) = (q(u)a, tq(u)2 ) = q(u) ∗ (a, t)

¯ We conclude that L ¯ 0 /K ¯ is a for all (a, t) ∈ S1 and hence u ¯2 = q(u) ∈ K. ¯ 20 ⊂ K. ¯ It follows that purely inseparable extension of dimension n0 with L n0 is power of 2. By (13.5), therefore, n0 = 4 and ℓ = 7. Hence m1 = 8 by ¯1,1 = X ¯ 1 . Therefore (13.16). By (13.2), U ¯1,1 = m1 /n0 = 2. dimL¯ U 0

¯1 = U ¯1,1 making it into a field By [60, 34.2(ii)], there is a multiplication on X ¯ containing a subfield we can identify with L0 such that the quadratic form Q ¯1,1 is similar to the norm of the quadratic extension X ¯ 1 /L ¯ 0 . Since R1 is on U not indifferent, this extension is separable. Let τ be the non-trivial element ¯1 , L ¯ 0 , τ ) is an involutory set and by [60, 38.2], in its Galois group. Then (X ¯1, L ¯ 0 , τ ). QQ (Θ1 ) ∼ = QI (X

Thus (iv) holds. ¯ 0 ≥ 3, We assume from now on that f¯0 is not identically zero. Since dimK¯ L it follows from 4.8(iii) that the joint torus HI is non-abelian, so by 13.9, there exists a proper involutory set Θ1 = (D, D0 , τ ) such that ¯1 ∼ Ω = QI (Θ1 )op . Let F = Z(D) and let n be the degree of D. Since Θ1 is proper, we have n > 1 and D0 generates D as a ring. Thus 5.10(a) holds. Let 1 be the distinguished element of L introduced in [60, 16.6] and let ¯1 ¯ 0 . By [60, 35.16], we can identify Ω ¯ 1 with QI (Θ1 )op and be its image in L ¯ 0 , so that ¯ thus, in particular, D0 with L 1 is the multiplicative identity of D0 (after replacing Θ1 by an isotope if necessary). Let ∗ denote multiplication in ¯ 1 = S1 /S2 for all (a, t) ∈ S1 D. Let (a, t) denote the image of (a, t) in D = X ¯ 0 for all u ∈ L0 . Then and let u ¯ denote the image of u in D0 = L u ¯ ∗ (a, t) = (au, tq(u) + φ(a, u))

× for all u ∈ L0 and all (a, t) ∈ S1 . If r, s ∈ OK , then by (13.17),

rs ∗ (a, t) = (ars, t(rs)2 )

= r¯ ∗ (as, ts2 ) = r¯ ∗ s¯ ∗ (a, t)

for all (a, t) ∈ S1 , from which it follows that ¯ ⊂ D0 K

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is a subfield of D, and by (13.18), u¯ ∗ r¯ ∗ (a, t) = u ¯ ∗ (ar, tr2 )

= (aur, tr2 q(u) + r2 φ(a, u))

= r¯ ∗ (au, tq(u) + φ(a, u)) = r¯ ∗ u¯ ∗ (a, t) for all (a, t) ∈ S1 , all r ∈ OK and all u ∈ L0 , from which it follows that ¯ ⊂ F . By (13.18), we also have K (¯ r u¯) ∗ (a, t) = r¯ ∗ u ¯ ∗ (a, t) for all (a, t) ∈ S1 , all r ∈ OK and all u ∈ L0 , where r¯u ¯ denotes the scalar product of r¯ with u ¯ defined in 7.3. Hence r¯u ¯ = r¯ ∗ u ¯ for all r ∈ OK and all u ∈ L0 . Therefore ¯D ¯ 0 ⊂ D0 K

and ¯0 ≥ 3 dimK¯ (D0 ) = dimK¯ L

¯ 0 < ∞ and hence as well as dimK¯ D0 = dimK¯ L

m := dimK¯ F < ∞.

¯ in place of K. In particular, we have Thus 5.10(b) holds with K (13.21)

dimK¯ D = mn2 .

¯ 0 = D0 denote the defect of q¯0 . The distinguished element 1 of Let R ⊂ L L in [60, 16.6] used to construct the root group sequence Ω is arbitrary. We can thus assume that if R 6= 0, then ¯ 1 ∈ R. Since q(1) = 1, also q¯0 (¯1) = ¯1. ¯ 0 = (K, ¯ L ¯ 0 , q¯0 ) in place of Λ = (K, D0 , q), ¯1 in Thus 5.10(c) holds with Λ ¯ = 2 if R 6= 0. Thus place of 1 and f¯0 in place of f . Note, too, that char(K) 5.10(d) holds by 10.19 and [60, 33.11] as well as [60, 33.13], where we set (D, D0 , τ ) in place of (K, K0 , σ) and 0 in place of L0 , so that the group T in [60, 33.13] is {(0, t) | t ∈ D0 }. (Note that tτ = t for all t ∈ D0 by [60, 11.1(i)].) We have thus checked that all the conditions in 5.10 hold. Now suppose that ℓ = 6, so dimK¯ D0 = n0 = 3 by (13.5) and ¯1,1 ) = dimK¯ (X ¯ 1 ) = m1 = 4 dimK¯ D = dimK¯ (U by (13.2) and (13.16). Hence mn2 = 4 by (13.21), so m = 1 and n = 2. This ¯ ⊂ D0 ⊂ D τ means that D is a quaternion division algebra over F and F = K τ (where D is as defined in (5.4)), so τ is of the first kind. Thus (i) holds. We assume now that ℓ ≥ 7. Suppose that R 6= 0. Then by 6.27, R = F and (13.22)

the co-dimension of Def(q0 ) is 2.

Choose u ∈ L such that ν(q(u)) = 0 and assume that u ¯ ∈ F . Let ρ be the map defined in (13.19) and let z = uρ . Then q(z) = q(u) and by [60, 12.45

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107

and 12.52], we have (au)z = q(u)a. Since u ¯ ∈ F = R, we have z¯ = u¯. Thus by (13.20),

u ¯∗ u ¯ ∗ (a, t) = z ∗ u¯ ∗ (a, t) = z¯ ∗ (au, tq(u) + φ(a, u))

= ((au)z, tq(u)2 + φ(a, u)q(u) + φ(au, uρ )) = (q(u)a, tq(u)2 ) = q(u) ∗ (a, t) ¯ We conclude that if R 6= 0, then and hence u¯2 = q(u) ∈ K. (13.23)

¯ F¯ 2 ⊂ K.

We can now finish the proof. If ℓ = 7, then by 5.19 and 5.21 as well as (13.22) and (13.23), either (ii) or (iii) holds. If ℓ = 8, then by 5.18 and 5.20 as well as (13.22) and (13.23), either (v) or (vi) holds.

Chapter Fourteen Quadrangles of Type E6 , E7 and E8: Summary Our goal in this chapter is to give a summary of the different cases that occur in 11.4, 12.7, 12.8, 12.9 and 13.15. The list of cases given in 14.3 below is to be interpreted using the following ad hoc conventions (1)–(6): (1) In each case we give a triple of numbers. The number in the middle is ¯I,0 over K, ¯ where I is as in 10.18. The first the dimension of the root group U ¯J,0 and and the third number number is the dimension of the root group U ¯ is the dimension of the root group UJ,1 , where J = 5 − I. If one of these root groups is an image of the non-abelian root group U1 of Ω and Z(U1 ) has a non-trivial image in this root group, we give the dimension in the form 1 + k, where k is the dimension of the image of the corresponding quotient ¯ (See, however, 14.4 below.) group over K. ¯ i ) for i = 0 and i = 1. Thus d = d0 + d1 (2) d = dimK (L) and di = dimK¯ (L and d0 > 0. If I = 1, then d0 and d1 are the first and third entries in the indicated triple of dimensions. If I = 4, then d0 is the second entry in the indicated triple (and either d1 = 0 or d1 = d0 ). (3) f¯0 = ∂ q¯0 and f¯1 = ∂ q¯1 , where q¯0 and q¯1 are as in 7.3. An expression of the form f¯i 6= 0 indicates that f¯i is neither identically zero nor non-degenerate. The case (1.iii.a) in 14.3 (see 14.1) is, however, an exception: In this case f¯0 ¯ = 2. is non-degenerate unless char(K) (4) In each case, there is an expression of the form X0 + X1 . If I = 1, then ¯ op and the symbol X1 the symbol X0 describes the root group sequence Ω 0 ¯ 1 . If I = 4, the symbol X0 describes the describes the root group sequence Ω ¯ 0 and the symbol X1 describes the root group sequence root group sequence Ω op ¯ Ω1 . In this fashion the last term of the root group sequence described by X0 and the first term of the root group sequence described by X1 both correspond to the root group UI,0 in the “middle.” (5) The meaning of the various symbols is as follows: Qk means “of type QQ (as defined in 1.15) with respect to an anisotropic quadratic form of dimension k with defect of dimension at most 1.” Qm k means “of type QQ with respect to an anisotropic quadratic form of dimension k with defect of co-dimension m.”

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Qm ∗ means “of type QQ with respect to an anisotropic quadratic form with non-trivial defect of co-dimension m but of undetermined dimension as explained in 14.4.” Ik means “of type QI with respect to an involutory set (D, B, τ ), where D is a quaternion division algebra whose center Z(D) is of dimension k ¯ τ is the standard involution of D and B contains Z(D) properly.” over K, Iks (for s = 1 or 2) means “of type QI with respect to an involutory set (D, B, τ ), where D is a division algebra whose center is of dimension k ¯ τ is an involution of the second kind (so B is automatically the over K, τ set D of fixed points), D is commutative if s = 1 and D is quaternion of s = 2.” I b means “of type QI with respect to an involutory set (D, B, τ ), where ¯ and τ is an involution of the D is a biquaternion division algebra over K first kind.” Pk means “of type QP with respect to an anisotropic pseudo-quadratic space of dimension k defined over an involutory set (D, F, σ), where D is a quaternion division algebra, F is its center and σ is its standard involution.” Pk◦ means “of type QP with respect to an anisotropic pseudo-quadratic space of dimension k defined over an involutory set (E, F, σ), where E/F is a separable quadratic extension and σ is the non-trivial element in its Galois group.” Eℓ (for ℓ = 6, 7 or 8) means “of type QE with respect to a quadratic space of type Eℓ .” F4 means “of type QF .” (6) For each symbol X in (5), a root group sequence is X op if its opposite is X. Definition 14.1. We refer to each of the twenty-three cases of 14.3 by a triple of “coordinates.” Thus, for example, “(1.iii.a)” refers to the case where ℓ = 6 and d0 = 3. A case is termed generic if it can occur in any characteristic. The generic cases are precisely the cases in 14.3 whose triple has last coordinate a: case (1.i.a), case (1.ii.a), etc. In all the non-generic ¯ must be 2 (but the characteristic of K can be cases, the characteristic of K either 2 or 0). Remark 14.2. The unramified cases in 14.3 (as defined in 11.3) are those whose label has second coordinate i and the ramified cases are those whose label is of the form (1.iii.∗), (2.iv.∗) or (3.iii.∗). The remaining cases are

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111

semi-ramified. The unramified cases are those described in 11.4. The semiramified are those described in 12.7 for ℓ = 6, 12.8 for ℓ = 7 and 12.9 for ℓ = 8. The unramified cases are those described in 13.15. Here now is our summary: Theorem 14.3. Under Hypothesis 11.1 and the conventions (1)–(6) given ¯ 0 and Ω ¯ 1 are both indifferent or above, either the root group sequences Ω the various dimensions, types, etc., are as indicated in exactly one of the following twenty-three cases: (1) ℓ = 6, so d = 6, and: (i) (1 + 8, 6, 1) with I = 4, so d0 = 6 and d1 = 0, and: (a) E6 + (Q6 )op with f¯0 non-degenerate (generic). (b) F4 + (Q4 )op with f¯0 6= 0. 6

(ii) (4, 1 + 4, 2) with I = 1, so d0 = 4 and d1 = 2, and: (a) (P1 )op + P2◦ with f¯0 and f¯1 non-degenerate (generic). (b) (I1 )op + (Q2 )op with f¯0 non-degenerate and f¯1 = 0. ∗

(iii) (1, 3, 4) with I = 4, so d0 = 3, q¯1 is similar to q¯0 and: (a) Q3 + I1 with f¯0 = 6 0 (generic). (2) ℓ = 7, so d = 8, and: (i) (1 + 16, 8, 1) with I = 4, so d0 = 8 and d1 = 0, and: (a) E7 + (Q8 )op with f¯0 non-degenerate (generic). (b) F4 + (Q4 )op with f¯0 6= 0. 8

(c) (Q2∗ )op + (Q08 )op with f¯0 = 0.

(ii) (4, 1 + 8, 4) with I = 1, so d0 = d1 = 4, and: (a) (P2 )op + P2 with f¯0 and f¯1 non-degenerate (generic). (b) Q2 + I1 with f¯0 = 0 and f¯1 non-degenerate. (c)

∗ 2 Q∗

+ (Q2∗ )op with f¯0 = 0 and f¯1 = 0.

(iii) (6, 1 + 8, 2) with I = 1, so d0 = 6 and d1 = 2, and: (a) (E6 )op + P4◦ with f¯0 and f¯1 non-degenerate (generic). (b) F4 + (Q4 )op with f¯0 6= 0 and f¯1 = 0. ∗

(iv) (1, 4, 8) with I = 4, so d0 = 4, q¯1 is similar to q¯0 and:

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(a) Q4 + (I22 )op with f¯0 non-degenerate (generic). (b) Q2 + (I2 )op with f¯0 6= 0. 4

(c) Q04 + (I81 )op with f¯0 = 0.

(3) ℓ = 8, so d = 12, and: (i) (1 + 32, 12, 1) with I = 4, so d0 = 12 and d1 = 0, and: (a) E8 + (Q12 )op with f¯0 non-degenerate (generic). (b) F4 + (Q4 )op with f¯0 6= 0. 12

(ii) (8, 1 + 16, 4) with I = 1, so d0 = 8 and d1 = 4, and: (a) (E7 )op + P4 with f¯0 and f¯1 non-degenerate (generic). (b) F4 + (Q4 )op with f¯0 6= 0 and f¯1 = 0. ∗

(c)

Q2∗

+ I1 with f¯0 = 0 and f¯1 non-degenerate.

(iii) (1, 6, 16) with I = 4, so d0 = 6, q¯1 is similar to q¯0 and: (a) Q6 + I b with f¯0 non-degenerate (generic). (b) Q2 + I4 with f¯0 6= 0. 6

¯ i is of type Qk∗ for i = 0 or 1 Remark 14.4. In each case of 14.3 where Ω ¯ i has the structure of and for some k, we have δΛ = 1 and δΨ = 2, f¯i = 0, L ¯2 ⊂ K ¯ ⊂L ¯ i, a field such that L i ¯i ∼ Ω = QQ (Υ)

¯ i and the for some anisotropic quadratic space Υ defined over this field L dimension of the defect of Υ equals ¯ dimL¯ 2i K. This dimension is undetermined in the sense that, within a given case of 14.3, it depends on the choice of q. In particular, the “1” in a symbol like “1+4” should not be interpreted as the dimension of anything. Proposition 14.5. Suppose under Hypothesis 11.1 that R0 and R1 are both indifferent. Then f¯0 and f¯1 are both identically zero. ¯i,α for i ∈ [1, 4] and α ∈ [0, 1] be the terms in the root group Proof. Let U ¯ α defined in (10.13) and either (10.14) or (10.15). The groups U ¯2,0 sequence Ω ¯ 0 . Suppose that f¯0 6= 0 and and U4,0 are the second and fourth terms of Ω choose u, v ∈ L such that ν(q(u)) = ν(q(v)) = ν(f (u, v)) = 0. Then x2 (u) ¯2,0 and U ¯4,0 whose commutator x3 (0, f (u, v)) is a and x4 (v) are elements of U ¯ non-trivial element of U3,0 . This implies, however, that R0 is not indifferent. Therefore f¯0 = 0.

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113

We want to show now that f¯1 = 0. It suffices to assume that δΛ = 1 (where δΛ is as in 10.4), since otherwise q¯1 is 0-dimensional. Suppose that I = 4 (where I is as in 10.18). By 10.4, it follows that δΨ = δΛ = 1. Thus Ξ is a ramified quadrangle as defined in 11.3. By (13.4), therefore, f¯1 = 0. ¯2,1 and U ¯4,1 are the second Suppose, instead, that I = 0, so δΨ = 2. Then U ¯ 1 . If f¯1 6= 0, then we can choose u, v ∈ L such that and fourth terms of Ω ν(q(u)) = ν(q(v)) = ν(f (u, v)) = 1, but then x2 (u) and x4 (v) are elements ¯2,1 and U ¯4,1 whose commutator x3 (0, f (u, v)) is a non-trivial element of of U ¯3,1 (by [65, 21.29]). This is impossible since R1 is indifferent. Therefore U f¯1 = 0 also when I = 0. Notation 14.6. If D is a quaternion division algebra over K with trace T , ¯ the division algebra and by T¯ the trace map defined in [65, we denote by D ¯ is a quaternion division algebra over K ¯ and T¯ is its 26.15]. Thus either D ¯ ¯ ¯ ¯ K ¯ is trace or D/K is a separable quadratic extension and T is its trace or D/ 2 ¯ ¯ ¯ an inseparable extension of degree 2 or 4 with D ⊂ K and T is identically zero. Proposition 14.7. Let Λ be of type E7 , let D be the quaternion division algebra over K whose image in Br(K) is the Clifford invariant of q, let T and T¯ be as in 14.6 and suppose that f¯0 and f¯1 are both identically zero. Then Ξ is in case (2.i.c), (2.ii.c) or (2.iv.c) of 14.3 if T¯ 6= 0 and R0 and R1 are both indifferent if T¯ = 0. Proof. We give the proof of this result in Chapter 15; see 15.4 and 15.6. Proposition 14.8. Let ∆ = (K, L, q) be a quadratic space of type Eℓ for ℓ = 6, 7 or 8. Suppose that 1 ∈ q(L) and that K is complete with respect ¯ L ¯ i , q¯i ) and f¯i := ∂ q¯i for i = 0 and 1 be to a discrete valuation, let Λi = (K, ¯ as in 7.3 and let T be as in 14.7 if ℓ = 7. Then exactly one of the following holds: (i) dimK¯ Li and f¯i for i = 0 and 1 are as in one of the twenty-three entries in 14.3 (i.e. zero, non-zero or non-degenerate) and T¯ 6= 0 if this entry is (2.i.c), (2.ii.c) or (2.iv.c). (ii) f¯0 and f¯1 are both identically zero and, if ℓ = 7, T¯ is also identically zero. Proof. This holds by 14.3, 14.5 and 14.7 applied to the Bruhat-Tits building Ξ obtained by applying 10.3 to ∆. Existence 14.9. To show that all twenty-three cases in 14.3 really occur, it suffices by 10.3 to construct quadratic spaces of type E6 , E7 and E8 over a field that is complete with respect to a discrete valuation as in each of the twenty-three cases of 14.8(i). We do this in Chapter 16. Conjecture 14.10. Let Λ be of type Eℓ for ℓ = 6, 7 or 8 and suppose that R0 and R1 are both indifferent. We conjecture that the dimensions of the

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root groups of R0 and R1 are always as one of the cases in the list above (where R0 and R1 are not both indifferent). For example, if ℓ = 8, then the dimension triples (2, 1 + 16, 10) and (6, 1 + 16, 6) should both be impossible. Remark 14.11. We observe that 14.3 together with 17.3 below and the results 19.35, 20.10, 23.27 and 24.58 in [65] reveal a hierarchy among the different types of Moufang quadrangles listed in 4.2 with respect to which for ∆ a Moufang quadrangle of a given type, the irreducible rank 2 residues of the corresponding Bruhat-Tits building Ξ can only be Moufang quadrangles of the same or lower type in this hierarchy. At the top of this hierarchy are, in order, the types E8 , E7 and E6 . These are followed by the types F4 and pseudo-quadratic at the same level, then the types involutory followed by quadratic form and finally indifferent at the bottom. Remark 14.12. The quadrangles of type F4 , which are not among the spherical buildings associated with absolutely simple algebraic groups, were discovered in the course of carrying out the proof of [60, 21.12]. In fact, they were overlooked in early versions of this proof and could well have been overlooked altogether in the classification of Moufang polygons. It is thus particularly satisfying to see these quadrangles (or at least some of them) arise in 14.3.

Chapter Fifteen Totally Wild Quadratic Forms of Type E7 In this chapter, we assume that (K, L, q) is a totally wild quadratic form of type E7 as defined in 7.6. Let D be the quaternion division algebra over K whose image in Br(K) is the Clifford invariant of q and let T and T¯ be as in 14.6. Our goal is to prove 14.7. We assume as usual that 1 ∈ q(L), choose a base point 1 ∈ L such that q(1) = 1 and identify K with its image in L under the map t 7→ t · 1. Proposition 15.1. q(π(u ∗ a) + N (u)t) = N (u)2 q(π(a) + t) for all (a, t) ∈ S and all u ∈ D, where ∗ is the map from D × X to X given in [64, 3.6] and N is the norm of D.

ˆ be the pseudo-quadratic form on X to D defined in [64, eq. Proof. Let Q ˆ 1 be the pseudo-quadratic form on X to D defined at the top (42)] and let Q of page 212 of [64]. Suppose first that char(K) 6= 2. Then f (π(a), t) = 0 by [60, 13.41] and thus q(π(a) + t) = q(π(a)) + t2 for all (a, t) ∈ S. We also ˆ have T (Q(a)) = 0 for all a ∈ X. By [64, 5.3(i)], therefore, ˆ q(π(a) + t) = N (Q(a) + t) ˆ is equal to Q ˆ 1 modulo K. Since also for all (a, t) ∈ S. By [64, 5.4], Q ˆ 1 (a)) = 0 for all a ∈ X, it follows that, in fact, Q ˆ is simply equal to Q ˆ 1. T (Q ˆ ∗ a) = uQ(a)uσ for all a ∈ X and all u ∈ D. Therefore Hence Q(u ˆ ∗ a) + N (u)t) = N (u)2 · N (Q(a) ˆ q(π(u ∗ a) + N (u)t) = N (Q(u + t)

and thus q(π(u ∗ a) + N (u)t) = N (u)2 q(π(a) + t) for all (a, t) ∈ S and all u ∈ D. ˜ 1 and Q ˜ 2 be as in [64, (37)-(38)], Now suppose that char(K) = 2 and let Q ˜ ˜ so f (π(a), 1) = Q1 (a) + Q2 (a) for all (a, t) ∈ S by [64, (17), (41) and 5.2]. By [64, 5.3(ii)], therefore, q(π(a) + t) = G(a, t), where ˆ ˜ 1 (a) + t)(Q ˜ 2 (a) + t) (15.2) G(a, t) := N (Q(a)) + (Q for all (a, t) ∈ S. By [64, (28) and (37)–(42)] and a long calculation with coordinates1 (similar to the calculation needed to prove [64, 5.4]), we have (15.3)

G(u ∗ a, N (u)t) = N (u)2 G(a, t)

1 Here are some details. Let E/K and α , . . . , α be as in 8.1. By [60, 16.6], we can 1 4 assume that α1 = 1. We let si = αi for all i ∈ [2, 4] and set s234 = s2 s3 s4 . Thus D = (E/K, s234 ). Choose a ∈ X and u ∈ D. There exist unique v, w ∈ E such that u = v + ew, where e = e2 is as in [60, 9.2]. By the formulas [64, (28) and (37)–(42)], one calculates that σ ˜ P (u ∗ a) = v2 P (a) + s234 (w σ )2 P (a)σ + Q(a)vw

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for all (a, t) ∈ S and all u ∈ D. Hence q(π(u ∗ a) + N (u)t) = N (u)2 q(π(a) + t) for all (a, t) ∈ S and all u ∈ D also when char(K) = 2.

Proposition 15.4. If T¯ 6= 0, then the residues R0 and R1 are not both indifferent. ¯ i = Si /(Si+1 + Zi ) for all i be as in 8.28. We set α = 0 if Proof. Let X ¯ 0 6= 0. If X ¯ 0 = 0, then X ¯ 1 6= 0 by (8.30) and we set α = 1. If α = 1, then X ¯ α is as in (10.14). Let (a, t) be Ξ is ramified as defined in 11.2 and hence Ω ¯ α is non-trivial. an element of Sα whose image in X Suppose that T¯ 6= 0. It follows that we can choose a unit u in the ring of integers OD of D such that T (u) = 1. Hence N (u + 1) = N (u). By 15.1, therefore, we have

(15.5) q π(u ∗ a) + N (u)t = q π((u + 1) ∗ a) + N (u + 1)t and the elements π(v ∗ a) + N (v)t for v = 1, u and u + 1 all lie in Lα (as defined in 7.3). By [60, 13.37], we have     π((u + 1) ∗ a) + N (u + 1)t = π(u ∗ a) + N (u)t   + π(a) + t + h(a, u ∗ a) + s

for some s ∈ K. Thus also h(a, u ∗ a) + s ∈ Lα . Since q is exceptionally ν-compatible as defined in [65, 21.22], we have


ν q(π(a) + t) + ν q(π(u ∗ a) + N (u)t) ν q(h(a, u ∗ a)) ≥ =α 2 and hence h(a, u ∗ a) ∈ Lα . Therefore also s ∈ Lα . Now let v 7→ v¯ be the ¯ α . Since the map q¯α from L ¯ α to K ¯ is additive, we natural map from Lα to L have

q¯α π((u + 1) ∗ a) + N (u + 1)t = q¯α π(u ∗ a) + N (u)t

+ q¯α π(a) + t + s + q¯α h(a, u ∗ a) . and

˜ i (u ∗ a) = N (v)Q ˜ i (a) + s234 N (w)Q ˜ j (a) + s234 T (vwP (a)) Q for (i, j) = (1, 2) and (i, j) = (2, 1), where N and T are the norm and trace of D and the ˜ and P are as in [64, (39) and (41)]. From these identities, it then follows that functions Q ˜ ∗ a) = N (u)Q(a) ˜ Q(u and    ˜ 2 + s234 T (vwP (a))2 s234 N (P (u ∗ a)) + N (u)2 N (P (a)) = s234 N (v)N (w)Q(a)  ˜ + N (u)Q(a)T (vwP (a))

˜ 1 (u ∗ a) + N (u)t Q ˜ 2 (u ∗ a) + N (u)t = Q

˜ 1 (a) + t)(Q ˜ 2 (a) + t) + N (u)2 (Q

for all t ∈ K. Thus (15.3) holds.

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By (15.5), therefore, q¯α (π(a) + t + s) = q¯α (h(a, u ∗ a)). Thus if h(a, u ∗ a) ∈ Lα+1 , then π(a) + t + s ∈ Lα+1 and hence (a, t) ∈ Sα+1 + Zα .

¯ α is non-trivial, we conclude that, in fact, Since the image of (a, t) in X
ν q(h(a, u ∗ a)) = α.

Thus the three elements π(a) + t, π(u ∗ a) + N (u)t and h(a, u ∗ a) all have ¯ α . Since non-trivial images in L [x1 (a, t), x3 (u ∗ a, N (u)t)−1 ] = x2 (h(a, u ∗ a)),

¯1,α , U ¯3,α ] 6= 0, where U ¯1,α and U ¯3,α are terms in the it follows by 8.16 that [U root group sequence (α) ¯ ¯ α = (U ¯+ ¯2,α , U ¯3,α , U ¯4,0 ) Ω , U1,α , U

defined in (10.13) if α = 0 or (10.14) if α = 1. By 10.16, therefore, the residue Rα is not indifferent. It follows from 15.4 that if T¯ 6= 0, then in 14.3 either (2.i.c), (2.ii.c) or (2.iv.c) holds. To conclude the proof of 14.7, it remains only to show that also the converse of 15.4 holds. To do this, we take a completely different tack. Proposition 15.6. If T¯ = 0, then the residues R0 and R1 are both indifferent. Proof. Let Q, E and νE be as in 8.39. Thus E is complete with respect to νE , Q is a quadratic form of type E8 and qE is a subform of Q of type E7 . By 10.3, we can apply the results of Chapter 10 to Q. Let Ω◦ , S ◦ , h◦ and π ◦ be the root group sequence, the group and the maps obtained by applying the recipe [60, 16.6] to Q and let Ξ◦ be the corresponding Bruhat¯ ◦ and Ω ¯ ◦ be the root group sequences obtained by Tits building. Next let Ω 0 1 ◦ applying 10.16 to Ω . We are assuming that q is totally wild and that T¯ = 0. By (8.40), therefore, the bilinear forms ∂Q0 and ∂Q1 are both identically ¯ ◦ and Ω ¯ ◦ are zero. By 14.3(3), it follows that the root group sequences Ω 0 1 both indifferent. By [42, 4.12], we can think of the trunk of Ω as a subgroup of the trunk of Ω◦ in such a way that the i-th term of Ω is a subgroup of the i-th term of Ω◦ and we can think of the algebraic data defining Ω as contained in the algebraic data defining Ω◦ so that the maps h, π, etc., appearing in the commutator relations in [60, 16.6] are simply restrictions of the functions h◦ and π ◦ to X × X and X × L. ¯ ◦ is indifferent, there are no pairs (a, t), (b, s) ∈ S ◦ such that Since Ω 0


νE Q(π ◦ (a) + t) = νE Q(π ◦ (b) + s) = νE Q(h◦ (a, b)) = 0.

118 Hence there are no pairs (a, t), (b, s) ∈ S such that


ν q(π(a) + t) = ν q(π(b) + s) = ν q(h(a, b)) = 0.

CHAPTER 15

Hence R0 is indifferent. It follows that Ξ is neither in case (2.i.c) nor in case (2.ii.c) of 14.3. Thus we only have to exclude that Ξ is in case (2.iv.c). We can thus assume that Ξ is ramified. It follows that Ξ◦ is also ramified. ¯ ◦1 is indifferent, there are no pairs (a, t), (b, s) ∈ S ◦ such that Since Ω


νE Q(π ◦ (a) + t) = νE Q(π ◦ (b) + s) = νE Q(h◦ (a, b)) = 1. Hence there are no pairs (a, t), (b, s) ∈ S such that


ν q(π(a) + t) = ν q(π(b) + s) = ν q(h(a, b)) = 1. Hence also R1 is indifferent.

This completes the proof of 14.7.

Chapter Sixteen Existence Our goal in this chapter is to prove the following: Theorem 16.1. There exist Bruhat-Tits buildings as described in each of the twenty-three cases of 14.3. As was observed in 14.9, it will suffice to show that quadratic forms as in each of the twenty-three cases of 14.8(i) exist. We will treat the generic unramified cases in 16.11, the generic semiramified cases in 16.12 and the generic ramified cases in 16.15—see 7.6, 11.3 and convention (5) at the beginning of Chapter 14—and then the wild unramified cases in 16.16, the wild semi-ramified cases in 16.18 and, finally, the wild ramified cases in 16.22. In preparation for the proof of 16.1, we assemble in 16.2–16.10 a few definitions and simple observations about quadratic forms. Definition 16.2. A quadratic form over a field K is called a 1-fold Pfister form if it is isometric to either the hyperbolic plane or to the norm of a separable quadratic extension E/K. A quadratic form q over a field K is called an n-fold Pfister form for some n > 1 if there exist α1 , . . . , αn−1 ∈ K ∗ and a 1-fold Pfister form N over K such that q is isometric to the 2n dimensional quadratic form N ⊗ hhα1 , . . . , αn−1 ii

as defined in 2.28. By Pfister form, we will always mean a quadratic form that is an n-fold Pfister form for some n ≥ 1. Remark 16.3. Pfister forms are round (by [21, Cor. 9.9]). By [21, Cor. 9.10], a Pfister form is isotropic if and only if it is hyperbolic. Anisotropic Pfister forms have, in particular, a norm splitting as defined in 2.24. A 2-fold Pfister form is the same thing as the norm of a quaternion algebra (cf. 8.33) and a 3-fold Pfister form is the same thing as the norm of an octonion algebra. Remark 16.4. For every n ≥ 1 there exist fields over which there exist anisotropic n-fold Pfister forms. Here is an example: Let n ≥ 2, let D/F be an arbitrary separable quadratic extension, let N be the norm of D/F , let K = F (α1 , . . . , αn−1 ) with α1 , . . . , αn−1 algebraically independent over F and let E = DK = DK . Then E/K is a separable quadratic extension whose norm is NK . Now let m ∈ [2, n] and let q denote the m-fold Pfister form NK ⊗K hhα1 , . . . , αm−1 ii.

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Thus if m > 2, then q = q ′ ⊗K hhαm−1 ii = q ′ ⊕ (−αm−1 )q ′ , where q ′ = NK ⊗K hhα1 , . . . , αm−2 ii. It follows by [21, 19.6] and induction on m that q is anisotropic. The following is the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt: Proposition 16.5. Let F be an arbitrary field. Then there exists a field K of characteristic 0 complete with respect to a discrete valuation such that ¯ = F , and there exists a field L of characteristic equal to the characteristic K ¯ = F. of F complete with respect to a discrete valuation such that L Proof. For the existence of K when char(F ) > 0, see [45, 4.3.3]. For L it suffices to take the field F ((x)) of Laurent series over F . Remark 16.6. Let n ≥ 1. By 16.4 and 16.5, there exist fields K complete with respect to a discrete valuation with the property that there ¯ both with exist anisotropic n-fold Pfister forms over the residue field K, ¯ char(K) = 0 and with char(K) = char(K). Remark 16.7. By 8.8 and 16.5, there exist fields K complete with respect to a discrete valuation with the property that there exist quadratic forms of ¯ both with char(K) = 0 and type E6 , E7 and E8 over the residue field K, ¯ with char(K) = char(K). Definition 16.8. Let K be a field complete with respect to a discrete val¯ = 2. Then ν(2), which equals ∞ unless the uation ν such that char(K) characteristic of K is 0, is called the absolute ramification index of K. Remark 16.9. Let K be a field complete with respect to a discrete valu¯ = 2 but char(K) = 0. Let p ∈ K be a ation ν and suppose that char(K) uniformizer, i.e. an element such that ν(p) = 1. We have 2 = pr α for some × r > 0 and some α ∈ OK . Let m > 1, let E = K(α), where α is a root of m the polynomial g = x − p, let N be the norm of the extension E/K and let ω(u) = ν(N (u)). Then g is irreducible over K by Eisenstein’s Criterion, ¯=K ¯ (by [22, §6, ω is a valuation of E, E is complete with respect to ω, E Thm. 1(i)]) and ω(2) = rω(p) = rm. This means that for given n ≥ 1, the field K in 16.6 can be chosen to have arbitrarily large absolute ramification index. Remark 16.10. Let F be an arbitrary field and let K = F (α, β, κ, λ) with α, β, κ and λ algebraically independent over F . Let g1 = x2 − α, g2 = x2 − β and g3 = x2 − αβ if char(K) 6= 2 and let g1 = x2 + x + α, g2 = x2 + x + β and g3 = x2 + x + α + β if char(F ) = 2. Next let Li be the splitting field of gi over K and let Ni be the norm of the extension Li /K for all i ∈ [1, 3] (in

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EXISTENCE

all characteristics). Then the polynomials g1 , g2 and g3 are irreducible over K and the quadratic form κN1 ⊕ λN2 ⊕ N3 over K is anisotropic (by [21, 19.6]). We now begin with the actual proof of 16.1. In each case we will produce a field K complete with respect to a discrete valuation ν and quadratic forms over K having the desired properties. For given K and ν, we let OK , t 7→ t¯ and p be as in 7.1. We begin with the generic unramified cases of 16.1: Proposition 16.11. There exist quadratic forms as in the cases (1.i.a), (2.i.a) and (3.i.a) of 14.3 in arbitrary characteristic. Proof. By 16.7, we can choose K such that there exist quadratic forms of ¯ The claim holds, therefore, by 8.35. type E6 , E7 and E8 over K. Next come the generic semi-ramified cases: Proposition 16.12. There exist quadratic forms as in the cases (1.ii.a), (2.ii.a), (2.iii.a) and (3.ii.a) of 14.3 in arbitrary characteristic. Proof. By 16.7, we can choose K such that there exists a quadratic form Q ¯ By 8.35, there is a unique unramified quadratic form q of type E8 over K. of type E8 over K such that q¯0 ∼ = Q. We have q∼ = α1 N ⊕ · · · ⊕ α6 N, where N is the norm of an unramified separable quadratic extension E/K × and α1 , . . . , α6 are elements of OK such that −α1 · · · α6 ∈ N (E). As observed in [60, 12.37], we have α1 α2 α3 α4 6∈ N (E). Note, too, that pα1 α2 α3 α4 6∈ N (E) since ν(pα1 · · · α4 ) = 1 6∈ 2Z = ν(N (E ∗ )). Let q1 = α1 N ⊕ α2 N ⊕ (pα3 )N, let q2 = α1 N ⊕ α2 N ⊕ (pα3 )N ⊕ (pα4 )N, let q3 = α1 N ⊕ α2 N ⊕ α3 N ⊕ (pα4 )N and let q4 = α1 N ⊕ α2 N ⊕ α3 N ⊕ α4 N ⊕ (pα5 )N ⊕ (pα6 )N. By 7.14, qi is ramified and anisotropic for each i ∈ [1, 4]. Thus q1 is of type E6 , q2 and q3 are of type E7 and q4 is of type E8 . By 7.14 and 7.23 again, the residue forms of these quadratic forms are non-singular and have dimensions as in cases (1.ii.a), (2.ii.a), (2.iii.a) and (3.ii.a) of 14.3.

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For the ramified case, we will need some additional preparation. We first observe that the polynomial g = x2 − px + p is irreducible over K by Eisenstein’s Criterion and if E is its splitting field, then E/K is ramified (because ν(N (γ)) = 1, where γ is a root of g and N is the norm of E/K). ¯ = 2. Let E be the splitting field of Proposition 16.13. Suppose char(K) × the polynomial x2 − px + p ∈ K[x], let µ = 1 − pβ for some β ∈ OK such 2 ¯ and let D be the that the polynomial h := x − x + β¯ is irreducible over K quaternion algebra (E/K, µ). Then D is a tamely ramified division algebra, ¯ is the splitting field of h over K. ¯ and D Proof. Let N be the norm, T the trace and let σ be the standard involution of D as defined in [60, 9.6]. We have D = {u + ev | u, v ∈ E},

where e is an element of E ⊥ := {v ∈ E | T (v σ E) = 0} such that N (e) = −µ, and multiplication in D is given by formulas in [60, 9.1], where e is called e2 . Let ξ be a root of x2 − px + p in E ⊂ D. We have N (ξ) = T (ξ) = p and hence T (ξ −1 ) = T (ξ)/N (ξ) = 1. Let θ = (1 + e)ξ −1 ∈ D and let L denote the subring K[θ] of D. Then T (θ) = T ((1 + e)ξ −1 ) = T (ξ −1 ) = 1 and N (θ) = N (1 + e)N (ξ −1 ) = (1 − µ)p−1 = β.

Thus H := x2 − x + β is the minimal polynomial of θ over K. Since OK ¯ the polynomial H is is a principal ideal ring and h is irreducible over K, ¯ is the splitting field of irreducible over K. Thus L is a subfield of D and L ¯ Since h is separable, the extension L/K is unramified. h over K. Let α = 2 − p and z = e(α + ξ). Thus T (z) = 0 and ν(α) ≥ 1. Calculating modulo K ∗ , we have T (θσ z) ≡ T ((ξ −1 )σ (α + ξ)) ≡ T (ξ(α + ξ)).

Since ξ 2 = pξ − p, we have

T (ξ(α + ξ)) = T (ξ(α + p) − p) = T (2ξ − p) = 2T (ξ) − 2p = 0.

Hence z ∈ L⊥ = {w ∈ D | T (wσ L) = 0}. By [60, 20.17], it follows that D∼ = (L/K, −N (z)). Finally, we observe that −N (z) = µN (α + ξ) = µ(α2 + αT (ξ) + N (ξ)) = µ(α2 + αp + p) ∈ p + p2 OK

and hence ν(−N (z)) = 1. By [43, Prop. 3], therefore, D ∼ = (L/K, −N (z)) is ¯ ∼ ¯ a tamely ramified division algebra with D = L.

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Proposition 16.14. Let D1 , . . . , Dt be quaternion division algebras over K with norms N1 , . . . , Nt and suppose that E ⊂ Di for all i ∈ [1, t], where E/K is a ramified separable quadratic extension. Suppose, furthermore, that × α1 , . . . , αt are elements of OK such that the quadratic form Q=α ¯ 1 R1 ⊕ · · · ⊕ α ¯ t Rt ¯ ¯ i /K ¯ over K is anisotropic, where Ri is the norm of the quadratic extension D for all i ∈ [1, t]. Then the quadratic form q = α1 N1 ⊕ · · · ⊕ αt Nt over K is also anisotropic and q¯0 = Q. Proof. Let N be the norm of the extension E/K. There exists an element u ∈ E such that ν(N (u)) = 1. Since N is the restriction of Ni to E, we also have ν(Ni (u)) = 1 for all i ∈ [1, t]. Let (v1 , . . . , vt ) be a non-zero vector in D1 × · · · × Dt and let m = min{ν(Ni (vi )) | i ∈ [1, t]}. Then min{ν(Ni (u−m vi )) | i ∈ [1, t]} = 0.

Since Q is anisotropic, we have and since


ν q(u−m v1 , . . . , u−m vt ) = 0

q(u−m v1 , . . . , u−m vt ) = N (u)−m q(v1 , . . . , vt ), we conclude that ν(q(v1 , . . . , vt )) = m. It follows that q is anisotropic and that q¯0 = Q. We can now treat the generic ramified cases: Proposition 16.15. There exist quadratic forms as in the cases (1.iii.a), (2.iv.a) and (3.iii.a) of 14.3 in arbitrary characteristic. Proof. By 16.5 and 16.10, we can assume that there exist α, β, κ and λ in × OK with the properties (a) and (b):

(a) The polynomials g1 = x2 − α ¯ , g2 = x2 − β¯ and g3 = x2 − α ¯ β¯ are 2 ¯ all irreducible if char(K) 6= 2 and the polynomials g1 = x − x + α ¯, ¯ if g2 = x2 − x + β¯ and g3 = x2 − x + α ¯ + β¯ are all irreducible over K ¯ = 2. Let Mi be the splitting field of gi over K ¯ for all i ∈ [1, 3]. char(K)

¯ 2 ⊕ R3 is anisotropic over K, ¯ where (b) The quadratic form Q := κ ¯ R1 ⊕ λR ¯ Ri denotes the norm of the extension Mi /K for all i ∈ [1, 3].

¯ 6= 2. Let E be a field containing K such that Suppose now that char(K) √ E/K is a ramified quadratic extension, for example, E = K( p), and let ¯ ∗ )2 = N ¯0 (E) ¯ and hence α 6∈ N be the norm of this extension. Then α ¯ 6∈ (K N (E), so the quaternion algebra D1 := (E/K, α) is a division algebra. Since ¯ 1 = M1 . Similarly, D2 := (E/K, β) E/K is ramified, so is D1 and thus D and D3 := (E/K, αβ) are ramified quaternion division algebras such that

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¯ 2 = M2 and D ¯ 3 = M3 . Now suppose that char(K) ¯ = 2. In this case we D let E be the splitting field of the polynomial x2 − px + p over K. Then E/K is a ramified separable extension. Let N be its norm. Let µ1 = 1 − pα, µ2 = 1−pβ, µ3 = µ1 µ2 and ξ = α+β −pαβ. Then ξ¯ = α ¯ + β¯ and µ3 = 1−pξ. Let Di denote the quaternion algebra (E, µi ) for all i ∈ [1, 3]. By 16.13, Di ¯ i = Mi for all i ∈ [1, 3]. is a ramified division algebra such that D Thus in all characteristics, we have found ramified quaternion division algebras D1 , D2 and D3 over K containing a common subfield E such that ¯ i = Mi for all i ∈ [1, 3] and E/K is a ramified separable extension, D [D1 ] + [D2 ] + [D3 ] = 0 in Br(K). Let N be the norm of the extension E/K, let Ni be the norm of Di for all i ∈ [1, 3] and let q = κN1 ⊕ λN2 ⊕ N3 . By (b) and 16.14, q is anisotropic and q¯0 = Q. By 8.6, therefore, q is of type E8 . Thus q is as in case (3.iii.a) of 14.3. By [60, 12.37], it follows that κN1 ⊕ λN2 is a quadratic form of type E7 as in case (2.iv.a) and κN1 ⊕ λN is a quadratic form of type E6 as in case (1.iii.a). Next we treat the wild unramified cases. Proposition 16.16. There exist quadratic forms as in the cases (1.i.b), (2.i.b) and (3.i.b) of 14.3 as well as quadratic forms in the case (2.i.c) with the extra condition in 14.8(i) in both characteristic 0 and in characteristic 2. ¯ = 2 (but without a restriction Proof. By 16.6, we can choose K with char(K) on char(K) itself) such that there exists an anisotropic 5-fold Pfister form (16.17)

Q = S ⊗ hhµ2 , µ3 , µ4 ii,

¯ and µ2 , µ3 , µ4 where S is the norm of a quaternion division algebra B over K ¯ Since char(K) ¯ = 2, we can choose a subfield L are non-zero elements of K. ¯ is an inseparable quadratic extension. of B such that L/K By [43, Thm. 1], there exists a unique unramified quaternion division ¯ = B and by [23, 11.2], there exists a subfield algebra D over K such that D ¯ = L. In E of D such that E/K is a separable quadratic extension and E particular, the extension E/K is wild but D is not wild. By [60, 20.17], we have D = (E/K, α1 ) for some α1 ∈ K ∗ . Since D is unramified, we can × ¯ 2 = L2 , then [23, 9.7] with d = 0 would assume that α1 ∈ OK . If α ¯1 ∈ E imply that D is wild. By replacing α1 by an element of α1 N (E), we may thus assume that α ¯ 1 = 1. Let Sˆ denote the restriction of S to L ⊂ B and let ˆ := S ⊕ µ2 Sˆ ⊕ µ3 Sˆ ⊕ µ4 Sˆ ⊕ (µ1 µ2 µ3 µ4 )S. ˆ Q

ˆ is a subform of Q. Hence Q ˆ is also anisotropic. Since L/K ¯ is By (16.17), Q ˆ inseparable and ∂S is non-degenerate, the defect of Q has dimension 8.

125

EXISTENCE × Choose α2 , α3 , α4 ∈ OK such that α ¯ i = µi for all i ∈ [2, 4] and let

q = N ⊕ (−α1 )N ⊕ (−α2 )N ⊕ (−α3 )N ⊕ (−α4 )N ⊕ (−α1 · · · α4 )N = N ′ ⊕ (−α2 )N ⊕ (−α3 )N ⊕ (−α4 )N ⊕ (−α1 · · · α4 )N,

where N is the norm of the extension E/K and N ′ is the norm of D. By ˆ 7.21, therefore, q is an unramified anisotropic quadratic form with q¯0 = Q. Let q1 be the subform N ⊕ (−α1 )N ⊕ (−α2 )N ⊕ (−α3 )N

of q, let q2 be the subform

N ⊕ (−α1 )N ⊕ (−α2 )N

of q1 and let q3 be the subform

N ⊕ (−α2 )N ⊕ (−α3 )N ⊕ (−α4 )N ⊕ (−α1 · · · α4 )N

of q. By 8.6, q is of type E8 . By [60, 12.37], it follows that q1 and q3 are of type E7 and q2 is of type E6 . Thus q is as in case (3.i.b) of 14.3, q1 is as in case (2.i.b), q2 is as in case (1.i.b) and q3 is as in case (2.i.c) with the extra condition in 14.8(i). We turn now to the wild semi-ramified cases. Proposition 16.18. There exist quadratic forms as in the cases (1.ii.b), (2.ii.b), (2.iii.b), (3.ii.b) and (3.ii.c) of 14.3 as well as in the case (2.ii.c) with the extra condition in 14.8(i) in both characteristic 0 and in characteristic 2. Proof. We continue where we left off in the proof of 16.16. Let let

q ′ = N ⊕ (−α1 )N ⊕ (−α2 )N ⊕ (−α3 )N ⊕ (−pα4 )N ⊕ (−pα1 · · · α4 )N,

q ′′ = pN ⊕ (−pα1 )N ⊕ (−α2 )N ⊕ (−α3 )N ⊕ (−α4 )N ⊕ (−α1 · · · α4 )N,

let

let let and let

q1′ = pN ⊕ (−pα1 )N ⊕ (−α2 )N ⊕ (−α3 )N, q1′′ = N ⊕ (−α1 )N ⊕ (−α2 )N ⊕ (−pα3 )N, q2′ = N ⊕ (−α1 )N ⊕ (−pα2 )N

q3′ = (−α2 )N ⊕ (−α3 )N ⊕ (−pα4 )N ⊕ (−pα1 · · · α4 )N. ˆ it follows from 7.14 that q ′ , q ′′ , q ′ , q ′′ , q ′ and q ′ are all Since q¯0 = Q, 1 1 2 3 anisotropic. Hence by 8.6, q ′ and q ′′ are of type E8 , q1′ , q1′′ and q3′ are of type E7 and q2′ is of type E6 . Thus by 7.14 again, q ′ is as in case (3.ii.b) of 14.3, q ′′ is as in case (3.ii.c), q1′ is as in case (2.ii.b), q1′′ is as in case (2.iii.b), q2′ is as in case (1.ii.b) and q3′ is as in case (2.ii.c) with the extra condition in 14.8(i).

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For the wild ramified cases we will need some further preparation. Definition 16.19. Let Λ = (K, L, q) be an anisotropic quadratic space. Suppose that Λ is both round and pointed with base point 1 ∈ L (as defined in 2.20 and 2.27), let f = ∂q and let T (u) = f (u, 1) for all u ∈ L. Following [23, 8.1, 8.4 and 8.8], we set  texp(q) = min ν(T (u)) | u ∈ L0 , where L0 is as in 7.3, we set

 ω(q) = min ν(T (u)) − ν(q(u))/2 | u ∈ L

× and for each α ∈ OK , we set  nexpq (α) = sup d ≥ 0 | α ∈ (1 − pd β)q(L0 \L1 ) for some β ∈ OK .

(Note that α = (1 − p0 β)q(1) with β = 1 − α ∈ OK for all α ∈ OK .) The number texp(q) is called the trace exponent of the pointed quadratic space (Λ, 1) and the number nexpq (α) is called the norm exponent of the element α; either of these numbers could, of course, equal ∞. Proposition 16.20. Let E/K be a ramified separable quadratic extension with norm N and canonical base point 1 ∈ E, let r be the trace exponent of N as defined in 16.19 with respect to the base point 1, let e be the absolute ramification index of K as defined in 16.8 and suppose that 2r < e. Then −1 ∈ N (E). Proof. Let ω and nexpN be as in 16.19. It follows from [23, 8.5(a)] that × ω(N ) ≤ r. Hence 2ω(N ) ≤ 2r < e. Since −1 = 1 − pe β for some β ∈ OK , we have nexpN (−1) ≥ e. By [23, 8.10], therefore, −1 ∈ N (E).

¯ = 2, let E be the splitting field of the polyProposition 16.21. Let char(K) 2 ¯ be a separable quadratic extension. nomial x − px + p over K and let M/K × Then there exists an element α ∈ OK such that α ¯ = 1 and D := (E/K, α) ¯ = M. is a quaternion division algebra over K containing E such that D × Proof. Choose β ∈ OK such that the field M is the splitting field of the polynomial

¯ x2 − x + β, let α = 1 − pβ and let D denote the quaternion algebra (E/K, α). By 16.13, ¯ = M. D is a division algebra and D Here, finally, come the wild ramified cases: Proposition 16.22. There exist quadratic forms as in the cases (2.iv.b) and (3.iii.b) of 14.3 as well as in the case (2.iv.c) with the extra condition in 14.8(i) in both characteristic 0 and in characteristic 2.

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Proof. By 16.9, we can choose K such that its absolute ramification index (as defined in 16.8) is at least 3 and there exists an anisotropic 4-fold Pfister form (16.23) Q = W ⊗ hhµ1 , µ2 , µ3 ii ¯ where W is the norm of a separable quadratic extension M/K ¯ and over K, ¯ Let B be the quaternion division µ1 , µ2 , µ3 are non-zero elements of K. ¯ µ1 ) and let R be the norm of B. Thus R = W ⊗ hhµ1 ii and algebra (M/K, hence (16.24)

Q = R ⊗ hhµ2 , µ3 ii.

Let E be as in 16.21 and let N and T be the norm and trace of the extension E/K. If γ ∈ E is a root of the polynomial x2 − px + p, then ν(N (γ)) = ν(T (γ)) = 1. Thus texp(N ) ≤ 1 by 16.19. By 16.20, therefore, (16.25)

−1 ∈ N (E).

× Choose α and define D as in 16.21, let S be the norm of D, choose κ ∈ OK such that µ1 = κ ¯ , let C denote the octonion algebra (D, κ) as defined in [60, 9.11] and let Sˆ be the norm of C as defined in [60, 9.10]. Thus (16.26) Sˆ = S ⊗ hhκii.

¯ and B is a division algebra, it Since S¯0 is the norm of the extension M/K ¯ ¯ follows that κ ¯ = µ1 6∈ S0 (D). By 7.34, the norm Sˆ of C is anisotropic (so C ¯ K, ¯ κ ¯ µ1 ) = B. is a division algebra by [60, 9.9(v)]) and C¯ = (D/ ¯ ) = (M/K, Let D1 = D = (E/K, α), D2 = (E/K, κ) and D3 = (E/K, λ) for λ = ακ ¯=κ (so λ ¯ since α ¯ = 1). By (16.25) and (16.26), we have Sˆ = N ⊕ (−α)N ⊕ (−κ)N ⊕ (λ)N = N ⊕ (−α)N ⊕ (−κ)N ⊕ (−λ)N.

Thus the norms N ⊗ hhκii and N ⊗ hhλii of D2 and D3 are subforms of Sˆ and hence anisotropic. We conclude that D1 , D2 and D3 are division algebras such that [D1 ] + [D2 ] + [D3 ] = 0 in Br(K). ¯ = κ ¯ so both D ¯ 2 /K ¯ and D ¯ 3 /K ¯ are inseparable We have λ ¯ 6∈ S¯0 (D), ¯2 = D ¯ 3 = K( ¯ √µ1 ) by 7.34. Let Wi quadratic extensions and, in fact, D ¯ i /K ¯ for all i ∈ [1, 3]. Since D1 = D and denote the norm of the extension D ¯ = M , we have W1 = W , where W is the non-singular quadratic form D ¯ √µ1 ) into B = (M, µ1 ). In in (16.23). There is a natural embedding of K( particular, W1 , W2 and W3 are all subforms of R = W ⊗ hhµ1 ii. By (16.24), therefore, ˆ := W1 ⊕ µ2 W2 ⊕ µ3 W3 Q is a subform of Q and hence anisotropic (with defect of dimension 4). Now let q = S 1 ⊕ µ2 S 2 ⊕ µ3 S 3 ,

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where S1 = S is the norm of D = D1 and Si is the norm of Di for i = 2 and 3. Thus (Si )0 = Wi for each i ∈ [1, 3]. By 16.14, q is anisotropic and ˆ By 8.6, q is of type E8 and the subforms S1 ⊕ µ2 S2 and µ2 S2 ⊕ µ3 S3 q¯0 = Q. are of type E7 . Thus q is as in case (3.iii.b) of 14.3, the subform S 1 ⊕ µ2 S 2 is as in case (2.iv.b) and the subform µ2 S2 ⊕ µ3 S3 = µ2 N ⊕ (−µ2 κ)N ⊕ µ3 N ⊕ (−µ3 ακ)N is as in case (2.iv.c) with the extra condition in 14.8(i). This concludes the proof of 16.1.

Chapter Seventeen Quadrangles of Type F4 In this chapter, we turn to the case that the building at infinity ∆ of the Bruhat-Tits building Ξ is a Moufang quadrangle of type F4 . The main result of this chapter is 17.3. See also 17.12. Hypothesis 17.1. Let Λ = (K, L, q) be a quadratic space of type F4 , let ˆ = (F, L, ˆ qˆ) be as in 9.5 qK be as in (9.2), let q be identified with qK , let Λ and suppose that K is complete with respect to a discrete valuation ν and F is closed with respect to ν. Let ∆ be the Moufang quadrangle corresponding ˜ F (Λ), let Λ ¯ 0, Λ ¯ 1, to the root group sequence Ω = QF (∆) (see 4.2), let Ξ = B 2 f¯0 := ∂ q¯0 and f¯1 := ∂ q¯1 be as in 7.3, let δK , δF , δ˜ and δ ∗ be as in 10.4, let ϕi for i ∈ [1, 4] be as in 10.5, let R0 and R1 be the two residues of Ξ defined ¯ 0 and Ω ¯ 1 be the two root group sequences defined in 10.16. in 10.9 and let Ω A consequence of our first result is that we can assume that we are in the long case (as defined in 10.4): ˆ Proposition 17.2. Let I be as in 10.18. Replacing Ω = QF (Λ) by QF (Λ) has the effect of replacing I by 5 − I without altering ∆ or Ξ.

ˆ op . This means (by [60, Proof. By [60, 28.45], QF (Λ) is isomorphic to QF (Λ) ˆ does not alter ∆ or Ξ 7.5] and [60, 27.5–27.6]) that replacing Ω by QF (Λ) (up to isomorphism). ˆ has the effect of replacing K by K ˆ := F and F by Replacing Λ by Λ 2 ˆ F := K . If M is one of these fields, we denote by νM the restriction of ν to M divided by the index of ν(M ) in Z. Thus νK = ν and ˜ ν ˆ (Fˆ ∗ ) = νF ((K 2 )∗ ) = νK ((K 2 )∗ )/δ˜ = (2/δ)Z K

by [65, 22.9] and hence b˜ ˜ δ := |Z/νKˆ (Fˆ ∗ )| = 2/δ.

ˆ is isometric to Λ. Thus if we let δ ˆ and δ ˆ be defined by The dual of Λ K F ˆ then δ ˆ = δF and δ ˆ = δK . Therefore applying [65, 22.10] to Λ, K F b˜ ∗ ˜ δb∗ := δδ ˆ = 2δK /(δδF ) = 2/δ . Fˆ /δK Thus δ ∗ = 1 if and only if δb∗ = 2. The claim holds, therefore, by 10.18.

Theorem 17.3. Suppose that Λ is of type F4 and that R0 and R1 are not ˆ and I be as defined in 9.5 both indifferent, let δK be as in 10.4 and let Λ ˆ and 10.18. Then after replacing Ω by QQ (Λ) if necessary so that I = 4, as is allowed by 17.2, one of the following holds:

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¯ 0 is of type F4 , δK = 2, (i) Λ ¯0 ∼ ¯ 0) Ω = QF (Λ and ¯1 ∼ ¯ 0 ). Ω = QQ (Λ ¯ 0 is non-trivial and of co-dimension 2, δK = 1, (ii) The defect of Λ ¯0 ∼ ¯ 0) Ω = QQ (Λ and there exists a proper involutory set Θ1 = (C, C0 , σ) such that ¯1 ∼ Ω = QI (Θ1 )op , where C is a quaternion division algebra over F and σ is its standard involution. Proof. By [65, 22.11], we have ˜ F ≤ 2. δK ≤ δ˜ ≤ δδ Since I = 4, we are in the long case by 10.18. Let p ∈ K be a uniformizer ¯ ∗0 and W ¯ j,i be as in 10.20 and let for ν = νK , let Ω ¯ ∗ := (W ¯ (1) , W ¯ 1,1 , U ¯2,1 , W ¯ 3,1 , U ¯4,0 ), Ω 1 + ˜ F = δK . It follows that ¯ (1) := W ¯ 1,1 U ¯2,1 W ¯ 3,1 U ¯4,0 . By 10.4, δδ where W + (17.4)

δF = 1 and either δ˜ = δK = 1 or δ˜ = δK = 2.

Let xi and ϕi for i ∈ [1, 4] be as in 10.6. Suppose first that δ˜ = δK = 2. Thus, in particular, νF is the restriction of νK /2 to F . By [60, 16.7], the maps xi (0, t) 7→ xi (0, t/p) for i = 1 and 3, x2 (b, s) 7→ x2 (b/p, s/p2 ) and x4 (b, s) 7→ x4 (b, s) extend to an automorphism ψ of the group W1 U2 W3 U4 . We have ϕi (xi (0, t/p)) = ϕi (xi (0, t)) − 1

for i = 1 and 3 and all t ∈ K because δ˜ = 2 and

ϕ2 (x2 (b/p, s/p2 )) = ϕ2 (x2 (b, s)) − 1

¯ ∗1 is a root group sequence and the autobecause δK = 2. It follows that Ω ¯ ∗ to Ω ¯ ∗. morphism ψ induces an isomorphism from Ω 1 0 ′ ′ ˆ such that Suppose that (a, t) and (a , t ) are two elements of X ⊕ K = L ′ ′ ′ ′ ϕ1 (x1 (a, t)) = ϕ3 (x3 (a , t )) = 1. Then [x1 (a, t), x3 (a , t )] = x2 (0, fˆ1 (a, a′ )), ˆ and where fˆ1 denotes the restriction of fˆ to the subspace X of L,
ϕ2 x2 (0, fˆ1 (a, a′ )) = νK (fˆ1 (a, a′ )) = 2 · νF (fˆ1 (a, a′ )). By 7.2(i) applied to qˆ, we have νF (fˆ1 (a, a′ )) ≥ 1 and hence ϕ2 (x2 (0, fˆ1 (a, a′ ))) ≥ 2.

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Therefore (17.5)

¯1,1 , U ¯3,1 ] = 1. [U

Suppose now that f¯0 is not identically zero. By 9.14, we can assume that the extension E/K in 9.1 is unramified. Since δK = 2, both ν(α) and ν(β) must be even. By 9.13, also D/F is unramified, where D is as in 9.3. By ¯ ⊂ (9.4), we conclude that fˆ0 is not identically zero. Since (0, 0, K) = (0, 0, K)

ˆ 0 6= (0, 0, K). ¯ Thus U ¯1,0 6= W1,0 , ¯ 0 is in the radical of fˆ0 , it follows that L L ∗ ¯ ¯ ¯ so we can apply 4.10 to the pair (Ω0 , Ω0 ). Since f0 is not identically zero ¯ 0 can only be in case (c) of 4.10. Thus Λ ¯ 0 is but also not non-degenerate, Ω of type F4 and ∼ QF (Λ ¯0 = ¯ 0 ). Ω ¯1,1 6= W ¯ 1,1 . Since Ω ¯ ∗1 ∼ Suppose that U = Ω∗0 , we can apply 4.10 to the pair ∗ ¯ 1, Ω ¯ ). Since f¯0 is neither identically zero nor non-degenerate, we deduce (Ω 1 ¯ 1 is also in case (c) of 4.10. This is impossible, however, by (17.5). that Ω ¯1,1 = W ¯ 1,1 and therefore Hence U ¯1 = Ω ¯ ∗1 ∼ ¯ ∗0 ∼ ¯ 0 ). Ω =Ω = QQ (Λ Thus (i) holds.

Now suppose that f¯0 = 0. If fˆ0 6= 0, then by 9.14 applied to qˆ, we can assume that the extension D/F in 9.1 is unramified, but then 9.13 implies that E/F is unramified and hence f¯0 6= 0. Thus fˆ0 = 0. By [60, 16.7], ¯1,0 , U ¯3,0 ] and [U ¯2,0 , U ¯4,0 ] are both trivial. In other words, R0 is therefore, [U indifferent. Suppose that R1 is not indifferent. By (17.5), it follows that ¯2,1 , U ¯4,0 ] 6= 1. Since δK = δ˜ = 2 and δF = 1, this means that there exist [U (b, s), (b′ , s′ ) ∈ W ⊕ F = L such that νK (q(b, s)) = 2, νK (q(b′ , s′ )) = 0 and
νF qF (0, f1 (b, b′ )) = νF (f1 (b, b′ )2 ) = νK (f1 (b, b′ )) = 1, where f1 denotes the restriction of f to W = E ⊕ E ⊂ L. Thus

νK (q(b/p, s/p2 )) = νK (f1 (b/p, b′ )) = νK (q(b′ , s′ )) = 0. This contradicts the fact that f¯0 = 0. Hence also R1 is indifferent. This contradicts our hypothesis that R0 and R1 are not both indifferent. Next we suppose that δ˜ = δK = 1. Thus, in particular, νF equals the restriction of νK = ν to F . Suppose that (b, s) and (b′ , s′ ) are elements of L∗ such that ν(q(b, s)) = 1 and ν(q(b′ , s′ )) = 0. By 7.2(ii), ν(f1 (b, b′ )) ≥ 1

and hence ϕ3 (x3 (0, f1 (b, b′ ))) = νF (ˆ q (0, f1 (b, b′ ))) = νK (f1 (b, b′ )2 ) ≥ 2. Therefore ¯2,1 , U ¯4,0 ] = 1. (17.6) [U It follows from 4.2 and 4.8 (just as in 13.6) that if the joint torus HI is non-abelian, then ¯1 ∼ (17.7) Ω = QI (Θ1 )op

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for some proper involutory set Θ1 = (C, C0 , σ). Suppose that f¯0 is not identically zero. By 9.13 and 9.14, we can assume that both extensions E/K and D/K are unramified. By 4.8(iii) applied to R0 , the joint torus HI is non-abelian. Therefore (17.7) holds. Let 1 = ¯ 0 . By [60, 35.16], (0, 1) ∈ L and let ¯ 1 be the image of this element 1 in L ¯ 1 mapping x4 (1) to x4 (¯1) we can find an isomorphism η from QI (Θ1 )op to Ω (after replacing Θ1 by a similar involutory set in the sense of [60, 11.9]). Let ˆ 1 such that η(x1 (a)) = x1 (ρ(a)) for all a ∈ C ρ be the map from C to L ¯ 0 such that η(x4 (b)) = x4 (ψ(b)) for all and let ψ be the map from C0 to L ¯ b ∈ C. Then ψ(1) = 1. Applying η to the identity [x1 (a), x4 (1)]3 = x3 (a), we find that η(x3 (a)) = x3 (ρ(a)) for all a ∈ C. Applying η to the identity [x1 (a), x4 (b)]3 = x3 (b ∗ a), we thus have [x1 (ρ(a)), x4 (ψ(b))]3 = x3 (ρ(b ∗ a)) for all a ∈ C and all b ∈ C0 , where ∗ denotes multiplication in C. For each ¯ 0 and for each (a, t) ∈ L ˆ 1 , let (a, t) (b, s) ∈ L0 , let (b, s) denote its image in L ˆ 1 . If we now identify C with L ˆ 1 via ρ and C0 with L ¯0 denote its image in L via ψ, it follows by [60, 16.7] that (b, s) ∗ (a, t) = (Υ(a, b) + sa, q(b, s)t + ω(a, b))

ˆ 1 , where Υ is as in [60, 14.15] and ω is for all (b, s) ∈ L0 and all (a, t) ∈ L the map called ν in [60, 14.16]. In particular, (17.8)

(0, s) ∗ (a, t) = (sa, st)

ˆ 1 . It follows that {(0, s) | s ∈ OF } ⊂ C0 for all s ∈ OF and all (a, t) ∈ L is subring of C isomorphic to F¯ . We denote this subfield by Z. Since the maps Υ and ω are F -linear in the first variable, we have (0, r) ∗ (b, s) ∗ (a, t) = (b, s) ∗ (0, r) ∗ (a, t) ˆ 1 . Since Θ1 is proper, the for all (b, s) ∈ L0 and all r ∈ OF and all (a, t) ∈ L subgroup C0 generates C as a ring. Therefore Z is contained in the center of C. ˆ denote the orthogonal sum ND + β 2 ND , where ND is the norm of Let Q ˆ is a subform of qˆ. Since ν(β 2 ) is even the extension D/F . Thus (9.4), αQ ˆ is unramified, so and D/F is unramified, it follows by [43, Prop. 4] that Q ˆ 0 is non-degenerate. ν(ˆ q (X ∗ )) = 2Z + ν(α), and Q Suppose that ν(α) is even, in which case ν(ˆ q (X ∗ )) = 2Z. We also have ν(ˆ q (0, t)) = ν(t2 ) ∈ 2Z

for all t ∈ K ∗ . Since δF = 1, it follows that there exist a ∈ X and t ∈ K such that ν(ˆ q (a, 0)) = ν(ˆ q (0, t)) = 0 and ν(ˆ q (a, t)) > 0. Hence (a, 0) is in the ˆ 0 is non-degenerate, this is impossible. We conclude radical of fˆ0 . Since Q that ν(α) is odd and hence (17.9)

ν(ˆ q (X ∗ )) = 2Z + 1.

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ˆ such that ν(ˆ Choose (a, t) ∈ L q (a, t)) = 1. Since ν(ˆ q (0, t)) is even, it ˆ 1 . By (17.8), it follows by (17.9) that ν(ˆ q (a, 0)) = 1 and (a, t) = (a, 0) in L ˆ 1 = C is of dimension 4 over the subfield Z. Since Z lies in follows that L the center of C and the involutory set Θ1 is proper, we conclude that C is a quaternion division algebra and Z is the center of C. Since Z ⊂ C0 , the involution σ is of the first kind. By [60, 35.8], we can thus assume that σ is, in fact, the standard involution of C. ¯ 1,0 6= U ¯1,0 , so we can apply 4.10 to the pair Suppose, in addition, that W ∗ −1 −1 ¯ ¯ (Ω0 , Ω0 ). Since ν(β ) or ν(αβ ) is even and E/K is unramified, it follows from (9.2) that f¯0 is neither non-degenerate nor identically zero. We must be, therefore, in case (c) of 4.10 again. In particular, the co-dimension of the defect of q¯0 is 4. Since α is odd, however the quadratic form N ⊕ αN is ¯ 1,0 = U ¯1,0 . Hence ramified. With this contradiction, we conclude that W ¯0 = Ω ¯ ∗0 ∼ ¯ 0) Ω = QQ (Λ and the co-dimension of the defect of q¯0 is 2 (as it must be by 6.27). Thus (ii) holds. Suppose now that f¯0 is identically zero but that fˆ0 is not. By 9.12 applied ˆ we can assume that the extension D/F in 9.1 is unramified. By 9.13, to Λ, therefore, the extension E/K is also unramified. It follows from (9.2) and the assumption that f¯0 is identically zero that ν(α) is even. Furthermore, ¯ 0 is non-degenerate, νF (β 2 ) = ν(β 2 ) = 2 · ν(β) is even. By [43, Prop. 4], ∂ Q where Q denotes the restriction of qˆ to X. Therefore ν(ˆ q (X ∗ )) = 2Z. We 2 ∗ also have ν(ˆ q (0, t)) = ν(t ) ∈ 2Z for all t ∈ K . Since δF = 1, it follows that there exist a ∈ X and t ∈ K such that ν(ˆ q (a, 0)) = ν(ˆ q (0, t)) = 0 ˆ ¯ 0 is nonand ν(ˆ q (a, t)) > 0. Hence (a, 0) is in the radical of f0 . Since ∂ Q degenerate, this is impossible. It thus remains only to consider the case that f¯0 and fˆ0 are both identically ¯1,0 , U ¯3,0 ] = 1 and [U ¯2,0 , U ¯4,0 ] = 1 (by [60, 16.7]). This means zero. Thus [U ˆ that R0 is indifferent. Suppose there exist elements (a, t) and (a′ , t′ ) of L ′ ′ ′ ′ ˆ ˆ such that ν(ˆ q (a, t)) = ν(ˆ q (a , t )) = 1 and ν(q(0, f (a, a ))) = ν(f (a, a )) = 1. By 9.12, it follows that we can assume that the extension D/F in 9.1 is unramified. Hence by 9.13, also the extension E/K is unramified. It follows from (9.2) and the assumption that f¯0 is identically zero that ν(α) is even. This implies by (9.4), however, that fˆ0 is not identically zero. With this ¯1,1 , U ¯3,1 ] = 1. By (17.6), it follows that contradiction, we conclude that [U R1 is also indifferent. Proposition 17.10. The residues R0 and R1 in 17.1 are both indifferent if and only if q is totally wild. Proof. By 9.15 and 17.2, we can assume that I = 4. If q is totally wild, it therefore follows from 17.3 that R0 and R1 are both indifferent. Suppose that q is not totally wild. By 9.14, there exist E/K, α and β as in (9.2) such that E/K is unramified. Let D/F and qˆ be as in 9.3. By 9.13, D/F

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¯2,0 , U ¯4,0 ] 6= 1 and is also unramified. If ν(α) is odd, then f¯0 6= 0, hence [U ˆ therefore R0 is not indifferent. If ν(α) is even, then f 0 6= 0 by (9.4), hence ¯1,0 , U ¯3,0 ] 6= 1 and this also implies that R0 is not indifferent. [U Proposition 17.11. Let Λ = (K, L, q) be a quadratic space of type F4 over a field K that is complete with respect to a discrete valuation ν such that 1 ∈ q(L), let F ⊂ K be as in 9.6, let δ ∗ and δK be as in 10.4 and let ¯ L ¯ i , q¯i ) and f¯i := ∂ q¯i for i = 0 and 1 be as in 7.3. Suppose that Λi = (K, ∗ δ = 1 and that the subfield F of K is closed with respect to ν. Then exactly one of the following holds: (i) The defect of q¯0 is of co-dimension 4 and δK = 2. (ii) The defect of q¯i is of co-dimension 2 for both i = 0 and 1 and δK = 1. (iii) f¯0 and f¯1 are both identically zero. Proof. This holds by applying 17.3 (where I = 4 because δ ∗ = 1) to the Bruhat-Tits building Ξ obtained by applying 10.3 to Λ. 17.12. Existence. We claim that both cases in 17.3 really occur. To show this, it suffices by 10.3 to construct two quadratic spaces Λ = (K, L, q) satisfying the following: Λ is of type F4 satisfying the hypotheses of 17.11 (so, in particular, δ ∗ = 1), the bilinear form f¯0 := ∂ q¯0 , where q¯0 is as in 7.3, is not identically zero and δK = 1 in the one case and δK = 2 in the other. We begin with the case δK = 2. Let α, β and γ be algebraically independent over the field F4 with four elements, let K = F2 (α, β)((γ)), let E = F4 (α, β)((γ)), let N be the norm of the quadratic extension E/K and let ν be the unique valuation of E such that ν(F4 (α, β)) = 0 and ν(γ) = 1. Thus K is complete with respect to ν. Next we set F = F2 (α, β 2 )((γ 2 )). Since K 2 ⊂ F , we can regard F as a vector space over K with scalar multiplication ∗, where t ∗ s = t2 s for all t ∈ K and all s ∈ F . This makes the identity map on F into a quadratic form on F over K. Let L = E ⊕ E ⊕ F and let q be the quadratic form on L over K given by q(u, v, s) = β(N (u) + αN (v)) + s for all (u, v, s) ∈ L. By [60, 14.22], the restriction Q of q to F4 (α, β) ⊕ F4 (α, β) ⊕ F2 (α, β 2 )

is an anisotropic quadratic space over F2 (α, β). It follows from this that q itself is anisotropic and ν(q(L∗ )) = 2Z. Therefore Λ = (K, L, q) is a quadratic space of type F4 such that q¯0 is isometric to Q and δK = 2. In particular, we have f¯0 6= 0. Since F is finite-dimensional over K with respect to ∗, it is closed with respect to ν. Furthermore δ˜ = 2 and hence δF = δ ∗ = 1 by [65, 22.11], where δ˜ and δF are as in 10.4. We turn now to the case that δK = 1. Let K, E, N and ν be as in the previous example, but this time we set F = F2 (α, β 2 )((γ)). Again we have K 2 ⊂ F , so we can again consider F as a vector space over K with respect

QUADRANGLES OF TYPE F4

135

to the scalar multiplication ∗, where t ∗ s = t2 s for all t ∈ K and all s ∈ F . We let q be the quadratic form on L = E ⊕ E ⊕ F given by q(u, v, s) = β(N (u) + γN (v)) + s for all (u, v, s) ∈ L. As before, F is closed with respect to ν and Λ = (K, L, q) is a quadratic space of type F4 with f¯0 6= 0, but this time δK = δ˜ = 1. Let ˆ qˆ) be the dual quadratic space as defined in 9.4. Then L ˆ = D ⊕ D ⊕ K, (F, L, where D = F4 (α2 , β 2 )((γ 2 )), and qˆ(x, y, t) = γ(N (x) + β −2 N (y)) + t2 ˆ Thus qF = 1, so again δ ∗ = 1. Thus both cases in 17.3 for all (x, y, t) ∈ L. really occur, as claimed above.

Chapter Eighteen The Other Bruhat-Tits Buildings In this chapter, we summarize the results in [65] about the residues of Bruhat-Tits buildings other than those associated with the exceptional Moufang quadrangles considered in Chapters 11–17. Let Ξ be a Bruhat-Tits building of rank ℓ + 1 endowed with its complete system of apartments and let ∆ = Ξ∞ . Then ℓ ≥ 2 (by 1.20) and by 1.22, we can assume that for some suitable parameter system Λ, either ˜ ℓ (Λ) and ∆ ∼ Ξ∼ =X = Xℓ (Λ) for X = A, D with ℓ ≥ 4, E with ℓ = 6, 7 or 8, F with ℓ = 4 or G with ℓ = 2 or ˜ X (Λ) and ∆ ∼ ˜ X (Λ) or C Ξ∼ = BX (Λ) = CX (Λ) =B ℓ

ℓ

ℓ

ℓ

for X = I, Q, P, E, F or D with ℓ = 2 in the last three cases. In every case, we can assume that Λ is complete with respect to a discrete valuation in an appropriate sense. Let Π denote the Coxeter diagram of Ξ and let S denote the vertex set of ˜ ℓ for X = A, B, C, D, E, F or G. A vertex of the Coxeter Π. Thus Π = X ˜ diagram Xℓ is called special if its deletion yields the spherical diagram Xℓ . We call a J-residue of the building Ξ a gem if J is the complement in S of a special vertex; see 10.7. By [65, 18.18], the gems of Ξ are all Moufang. An irreducible residue of Ξ is always a residue of a gem except when Xℓ = G2 ˜ 2 joined by a simple bond. By [65, and J consists of the two vertices of G 25.33], these J-residues are also Moufang. By [65, 26.39], there is only one Aut(Ξ)-orbit of gems and hence all the ˜ ℓ 6= C˜ℓ (and thus, in gems of Ξ are isomorphic to each other as long as X ˜ ˜ ˜ ˜ ˜ particular, Xℓ 6= C2 = B2 ). If Xℓ = Cℓ , there are at most two Aut(Ξ)-orbits of gems by [65, 18.5]. Most often, there are, in fact, exactly two orbits in this case and usually gems in different orbits are not isomorphic to each other. When X2 = B2 = C2 and ∆ is an exceptional Moufang quadrangle, then the two residues R0 and R1 which were the subject of our investigations in Chapters 11–17 are gems. In particular, there are always two non-isomorphic types of gems in this case. We now describe the structure of the gems of Ξ in each case. Suppose ˜ ℓ (K) for X = D or E and for first that Xℓ is simply laced. In this case, Ξ = X some field or skew-field or octonion division algebra K. (In fact, K can be an octonion division algebra only if Xℓ = A2 and K can be non-commutative only if X = A.) By [65, 18.30], the gems of Ξ are all isomorphic to the ¯ spherical building Xℓ (K).

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Now suppose that ∆ = CQ ℓ (Λ) for some anisotropic quadratic space Λ = (F, V, q). In this case, we can replace Λ by a similar quadratic space so that 1 ∈ ν(q(L)) (by [60, 35.17]). Since F is complete with respect to a discrete ˜ Q (Λ) and the gems of valuation, we can apply 7.3–7.5. By [65, 19.23], Ξ = B 2 Q ¯ Ξ are all isomorphic to Bℓ (Λ0 ) and if Λ is unramified and if Λ is ramified, ˜ Q (Λ) and the gems associated with one special vertex of Π ˜ are then Ξ = C 2 ¯ 0 ) and those associated with the other special vertex are isomorphic to BQ ( Λ ℓ ¯ isomorphic to BQ ℓ (Λ1 ). It can happen in the ramified case that the gems form a single Aut(Ξ)-orbit, but, in general, there are two. Suppose next that ∆ = CIℓ (Λ) for some involutory set Λ = (K, K0 , σ), genuine or honorary (as defined in [60, 38.11] and [65, p. 229]). The involutory set Λ is defined to be ramified if ν(K0∗ ) 6= Z and unramified otherwise. ¯ = (K, ¯ K ¯ 0, σ ¯1 Let Λ ¯ ) be as defined in [65, 23.22] and, if Λ is unramified, let Λ ¯ and Λ ¯ 1 are involutory sets and by [65, 23.27], be as in [65, 23.23]. Then Λ ˜ I (Λ) and the gems of Ξ are all isomorphic to CI (Λ) ¯ if Λ is ramified Ξ=B ℓ ℓ I ˜ and if Λ is unramified, then Ξ = Cℓ (Λ), the gems associated with one special ˜ are isomorphic to CI (Λ) ¯ and the gems associated with the other vertex of Π ℓ ¯ 1 ). It can happen in the unramified special vertex are isomorphic to CIℓ (Λ case that the gems form a single Aut(Ξ)-orbit, but, in general, there are two. ¯ is commutative if Λ is ramified and genuine. If Λ is honBy [65, 23.24], K ¯ is either commutative or a quaternion orary and ramified, however, then K division algebra (by [65, 26.20]). Now suppose that ∆ = CP ℓ (Λ) for some anisotropic pseudo-quadratic space Λ = (K, K0 , σ, L, q). The pseudo-quadratic space Λ is defined to be ramified if ν(q(L) + K0∗ ) 6= Z ¯ be the anisotropic pseudo-quadratic space and unramified otherwise. Let Λ ¯ 0 over K ¯ described in [65, 24.50]. If with underlying right vector space L ¯ 1 be the anisotropic pseudo-quadratic space with underlying ν(K0∗ ) = Z, let Λ ¯ 1 over K ¯ described in [65, 24.54]. If ν(K0∗ ) 6= Z (which right vector space L ¯ to be commutative) but Λ is nevertheless unramified, let Λ ¯ ′1 = forces K ¯ L ¯ 1 , q1 ) be the anisotropic quadratic space described in [65, 24.55]. By (K, ˜ P (Λ) and all the gems of Ξ are isomorphic to CP (Λ) ¯ if Λ is [65, 24.58], Ξ = B ℓ ℓ P ˜ ramified; if Λ is unramified, then Ξ = Cℓ (Λ); if Λ is unramified and ν(K0∗ ) = ˜ are isomorphic to CP (Λ) ¯ Z, the gems associated with one special vertex of Π ℓ P ¯ and the gems associated to the other special vertex are isomorphic to Cℓ (Λ1 ); and if Λ is unramified and ν(K0∗ ) 6= Z, then the gems associated with one ˜ are isomorphic to CP (Λ) ¯ and the gems associated to the special vertex of Π ℓ Q ¯′ other special vertex are isomorphic to Cℓ (Λ1 ). By [65, 24.59], furthermore, ¯ 0 if Λ is ramified; dimK L = dimK¯ L ¯ 0 + dimK¯ L ¯ 1 if Λ dimK L = dimK¯ L ∗ ′ ¯ ¯ is unramified and ν(K0 ) = Z; and dimK L = dimK¯ L0 + dimK¯ L1 if Λ is unramified and ν(K0∗ ) 6= Z. If ∆ = BD 2 (Λ) for some indifferent set Λ, then by [65, 20.10], all the gems of Ξ are also indifferent quadrangles. Suppose next that Xℓ = F4 . By [65, 26.12–26.13], the parameter system Λ that determines ∆ is a composition algebra as defined in [65, 30.17], by

THE OTHER BRUHAT-TITS BUILDINGS

139

which is meant a pair (K, F ) such that one of the following holds: (i) K is a field of characteristic 2 and F is a proper subfield containing K 2. (ii) K = F is a field. (iii) K is a field and F is a subfield such that K/F is a separable quadratic extension. (iv) K is a quaternion division algebra and F is its center. (v) K is an octonion division algebra and F is its center. ¯ := (K, ¯ F¯ ) is also a composition algebra, dimF¯ K ¯ By [65, 26.15], the pair Λ ¯ equals dimF K or (dimF K)/2, the gems of Ξ are all isomorphic to F4 (Λ) ¯ is in case (i), (x − 1) or (x). and if Λ is in case (x) for x = i, ii, . . . , v, then Λ ¯ = 4 or 8 and either K ¯ Thus, for example, if Λ is in case (v), then dimF¯ K 2 ¯ ¯ ¯ ¯ ¯ ¯ is commutative, char(K) = 2 and K ⊂ F or F is the center of K and K is either a quaternion division algebra or an octonion division algebra. Suppose, finally, that Xℓ = G2 . By [65, 25.25 and 25.27], the parameter system Λ that determines ∆ is an hexagonal system as defined in [60, 15.15] (better known in the literature as a quadratic Jordan division algebra of degree 3). By [60, 17.6], this means Λ is a triple (K, J, #), where # is a map from J ∗ to itself and one of the following holds: (i) J is a field of characteristic 3, K is a proper subfield containing J 3 and x# = x2 for all x ∈ J ∗ . (ii) J = K is a field and x# = x2 for all x ∈ J ∗ . (iii) J is a field, K is a subfield such that J/K is a separable cubic extension with norm N and x# = N (x)/x for x ∈ J ∗ . (iv) J is either an associative division algebra of degree 3 with center K or J is the set of fixed points of an involution τ of the second kind of an associative division algebra of degree 3 with center E containing K such that K = E τ . In both cases dimK J = 9 and x# = N (x)/x for all x ∈ J ∗ , where N is the norm of the division algebra. (v) J is a vector space of dimension 27 over K and # is as described in [60, 15.29] or [60, 15.34]. In this case, Λ is called an Albert algebra. ¯ be the Jordan division algebra over K ¯ = (K, ¯ J, ¯ #) ¯ obtained by apLet Λ plying [65, 25.28] to Λ. By [65, 25.29–25.30], the gems of Ξ are all isomor¯ dimK¯ J¯ equals dimK J or (dimK J)/3, if Λ is in case (x) for phic to G2 (Λ), ¯ is in case (i), (x − 1) or (x) and if Λ is in case (v) x = i, ii, . . . , v, then Λ

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¯ is in case (iv), then J is a skew-field (rather than the set of fixed and Λ points of an involution of a skew-field). Thus, for example, if Λ is an Al¯ bert algebra, then dimK¯ J¯ = 9 or 27 and either J¯ is a field containing K, ¯ or J¯ is a skew-field of dimension 9 over K ¯ with char(K) = 3 and J¯3 ⊂ K ¯ ¯ and x# ¯ (x)/x for all x ∈ J¯∗ or Λ ¯ is itself an Albert division norm N =N ˜ 2 (Λ) also has irreducible residues of rank 2 that are algebra. The building G not contained in a gem. By [65, 25.33], these residues are all isomorphic to ¯ if dimK¯ J¯ = dimK J and if dimK¯ J¯ = (dimK J)/3, then J¯ is a field A2 (K) ¯ or a skew-field and these residues are all isomorphic to A2 (J).

PART 3

Descent

Chapter Nineteen Coxeter Groups Our goal in Part 3 of this monograph is to develop a theory of descent for buildings. The central result of this theory is 22.20. In this chapter, we assemble various results about Coxeter groups which we will need. Notation 19.1. Let W be an arbitrary group with a distinguished set of generators S containing only elements of order 2, let MS denote the free monoid on the set S and let ℓ : MS → N be the length function. For each word f = s · · · t in MS we denote by rf the product s · · · t in W . (We will, however, almost always write s in place of rs for a word s ∈ MS of length 1 when it is clear from the context that we are in W rather than MS .) For w ∈ W , we put ℓS (w) = min{ℓ(f ) | rf = w}.

Thus, in particular, ℓS (1) = 0. A word f of MS is called a reduced representation of an element w of W if rf = w and ℓS (rf ) = ℓ(f ). Let MW,S denote the set of unordered pairs {s, t} of distinct elements in S such that st has finite order. For all {s, t} ∈ MW,S , we denote this finite order by mst , we let p(s, t) ∈ MS denote the word · · · stst of length mst ending in t comprised entirely of s’s and t’s but having neither ss nor tt as subword and we set pst = rp(s,t) ∈ W . Thus, for example, p(s, t) = st if mst = 2 mst and p(s, t) = tst if mst = 3. The product pst p−1 ts equals the image of (st) mst in W and hence pst = pts in W for all {s, t} ∈ MW,S . We call or (ts) two words f, g ∈ MS adjacent if there exist {s, t} ∈ MW,S and words h1 , h2 (possibly empty) such that f = h1 p(s, t)h2 and g = h1 p(t, s)h2 . Two words f, g ∈ MS are called homotopic if there is a sequence of words f0 , f1 , . . . , fs in MS such that f = f0 , fi is adjacent to fi−1 for all i ∈ [1, s] and fs = g. A word in MS is called p-reduced if it is not homotopic to a word of the form f1 ssf2 . (Note that in [62, 4.1], a word is defined to be reduced if it is p-reduced in the sense we give here. See 19.3(B).) Definition 19.2. Let (W, S) be a Coxeter system. Thus W is a group, S ⊂ W is a distinguished set of generators of W containing only elements of order 2 and W admits the presentation hS | s2 = 1 for all s ∈ S and (st)mst = 1 for all {s, t} ∈ MW,S i,

where MW,S is as in 19.1. See [1, §§2.1–2.4] for several equivalent definitions of a Coxeter system. The cardinality |S| of S is called the rank of (W, S).

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In these notes we only consider Coxeter systems of finite rank. An automorphism of (W, S) is an automorphism of the group W which stabilizes the set S. There is thus a canonical isomorphism from Aut(W, S) to Aut(Π), where Π is the associated Coxeter diagram with vertex set S. A Coxeter system (W, S) is spherical if |W | < ∞ and irreducible if Π is connected. A Coxeter group is a group W such that (W, S) is a Coxeter system for some subset S ⊂ W. We now give a characterization of Coxeter groups which can be found (somewhat hidden) in the proof of the main result of [54]. It first appeared in explicit form in [34]. We will apply this characterization in the proof of 20.32. Proposition 19.3. Let W , S, ℓS , MW,S as well as mst and pst for s, t ∈ S such that {s, t} ∈ MW,S be as in 19.1. Then the following are equivalent: (i) (W, S) is a Coxeter system. (ii) The following two conditions hold: (a) ℓ(ws) 6= ℓ(w) for each s ∈ S and each w ∈ W .

(b) If s, t are distinct elements of S such that ℓ(ws) = ℓ(w)−1 = ℓ(wt), then mst 6= ∞ and ℓ(wpst ) = ℓ(w) − mst . (iii) The following two conditions hold: (A) Any two reduced representations of w ∈ W are homotopic.

(B) A word f ∈ MS is a reduced representation of rf if and only if it is p-reduced.

Proof. If (W, S) is a Coxeter system, then (a) holds by [62, 3.6] and (b) holds by [44, Thm. 2.16]. Thus (i) implies (ii). We show next that (ii) implies (iii). Suppose that (a) and (b) hold. Let w ∈ W . For the proof of (A), we proceed by induction on ℓ(w). If ℓ(w) = 0, then w = 1 and w has only one reduced representation (namely, the empty one). Suppose now that ℓ(w) > 0 and let f, g ∈ MS be reduced representations of w. Let s be the last letter of f and let t be the last letter of g, so that f = f1 s and g = g1 t for some f1 , g1 ∈ MS . Then ℓ(ws) = ℓ(w) − 1 = ℓ(wt), f1 is a reduced representation of ws and g1 is a reduced representation of wt. We distinguish two cases. Suppose first that s = t. Then f1 and g1 are both reduced representations of ws. By induction, therefore, f1 and g1 are homotopic. It follows that f and g are also homotopic. Now suppose that s 6= t. By (b), we have mst 6= ∞ and ℓ(wpst ) = ℓ(w) − mst . Let v = wpst and choose a reduced representation h of v. Then hp(t, s) and hp(s, t) are reduced representations of w which are adjacent in MS as defined in 19.1. By the conclusion of the previous paragraph, f and hp(t, s) are homotopic as are g and hp(s, t). We conclude that f and g are homotopic. Thus (A) holds.

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A reduced representation of some element of W is, of course, p-reduced (as defined in 19.1). Suppose, conversely, that f is a p-reduced word which is not a reduced representation of w := rf . Thus ℓ(w) < ℓ(f ). We can assume that f is chosen among all such words to have minimal length and let s be the last letter of f . Then f = gs, where g is a p-reduced word of length ℓ(f ) − 1. By the choice of f , the word g is a reduced representation of v := rg . Since f is not a reduced representation of w = vs, we have ℓ(vs) 6= ℓ(v) + 1. By (a), therefore, ℓ(w) = ℓ(v) − 1. Thus there exists a word h of length ℓ(v) − 1 such that hs is a reduced representation of v. By (i), hs is homotopic to g and hence hss is homotopic to gs = f . This contradicts the assumption that f is p-reduced. Thus (B) holds. We conclude that (ii) implies (iii). We now show that (iii) implies (i). Suppose that (A) and (B) hold. Let ˆ let Π be graph with vertex set s 7→ sˆ be a bijection from S to a set S, ˆ S where vertices sˆ and tˆ are joined by an edge whenever s and t do not commute in W . By labeling each edge {ˆ s, tˆ} of Π with mst , we turn Π into ˆ , S) ˆ be the corresponding Coxeter system and a Coxeter diagram. Let (W ˆ → W be the canonical homomorphism from W ˆ to W mapping sˆ let ϕ : W ˆ having the same to s for each s ∈ S. Let w ˆ and vˆ be two elements of W ˆ image w under ϕ, let f and gˆ be representations in MSˆ of w ˆ and vˆ which ˆ , S) ˆ and let f and are reduced representations with respect to the pair (W g be the words fˆ and gˆ with the “hats” removed from each of the factors. Since the words fˆ and gˆ are reduced representations of w ˆ and vˆ, the words f and g are both p-reduced representations of w. By (B), f and g are reduced representations of w with respect to (W, S). By (A), therefore, f and g are homotopic. Therefore w ˆ = vˆ. Hence ϕ is injective. Thus (W, S) is a Coxeter system. Notation 19.4. Let (W, S) be a Coxeter system with length function ℓ = ℓS . For each w ∈ W , we set J − (w) = {s ∈ S | ℓ(ws) < ℓ(w)}

and J + (w) = {s ∈ S | ℓ(ws) > ℓ(w)}.

By 19.3(a), S = J − (w) ∪ J + (w) for all w ∈ W .

Notation 19.5. Let (W, S) be a Coxeter system. If J ⊂ S and WJ = hJi, then (WJ , J) is also a Coxeter system (by [62, 4.6]) and its length function is the restriction of the length function ℓS to WJ (by [62, 4.8]). A subset J of S is called spherical if the Coxeter system (WJ , J) is spherical. If (WJ , J) is spherical for some J ⊂ S, then WJ has a unique longest element with respect to J (by [62, 5.5]). We denote this longest element by wJ . Since wJ−1 has the same length as wJ , the element wJ must have order 2. Remark 19.6. Suppose that (W, S) itself is a spherical Coxeter system, so that it has a longest element wS . By [62, 5.11], the map s 7→ wS swS defines an automorphism of the corresponding Coxeter diagram Π which stabilizes

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each connected component and acts non-trivially on a given connected component if and only if that component is of type Am for some m ≥ 2, Dm or I2 (m) (including I2 (3) = A2 ) for some odd m ≥ 3 or E6 . Notation 19.7. Let (W, S) be a Coxeter system, let Π be the corresponding Coxeter diagram and let Σ = ΣΠ (or Σ(W,S) ) be the Coxeter chamber system of Π as defined in [62, 2.4]. Thus Σ is the Cayley graph on the group W with respect to the generating set S with the edge {w, ws} endowed with the “color” (or “type”) s for each w ∈ W and each s ∈ S. Let δ : W × W → W be the Weyl-distance function on Σ given by δ(u, v) = u−1 v for all u, v ∈ W . We call the vertices of Σ chambers. A gallery is a sequence (x0 , x1 , . . . , xm ) of chambers such that {xi−1 , xi } is an edge for each i ∈ [1, m] and the type of a gallery (x0 , x1 , . . . , xm ) is the word s1 · · · sm ∈ MS such that si = x−1 i−1 xi for each i ∈ [1, m]. By [62, 2.5], there is a gallery of type f from a chamber u to a a chamber v if and only of rf = u−1 v. Thus the ordinary distance dist(u, v) between two chambers u and v of Σ is ℓ(δ(u, v)). By [62, 8.11], (Σ, δ) is, up to isomorphism, the unique thin building of type (W, S). Let J ⊂ S. A J-residue of Π is a subgraph of Σ whose vertex set is a left coset of the subgroup WJ defined in 19.5 together with all the edges joining two vertices of this coset (whose color is then automatically in J). If x ∈ W , then the vertices of the J-residue xWJ are precisely the vertices that can be reached from x by a gallery whose type is in the free monoid MJ . Every J-residue is isomorphic to Π(WJ ,J) , the Coxeter chamber system of (WJ , J). A residue of Π is a J-residue for some J ⊂ S. If R is a J-residue, then J is the type of R and |J| is the rank of R. We will make frequent use of the following observations. Proposition 19.8. Let (W, S) and Σ = ΣΠ be as in 19.7. Then the following hold: (i) If J ⊂ S and f is a reduced representation of an element of WJ , then f lies in the free monoid MJ . (ii) If w ∈ W and J ⊂ S, then there is a unique element wJ of minimal length in the J-residue of Σ containing w and J ⊂ J + (wJ ). (iii) If w ∈ W and J ⊂ J + (w), then w = wJ and ℓ(wv) = ℓ(w) + ℓ(v) for all v ∈ WJ . (iv) If ℓ(wv) = ℓ(w) + ℓ(v) for some w, v ∈ W , then J − (v) ⊂ J − (wv). (v) If w ∈ W and J ⊂ J − (w), then WJ is finite, w = wJ wJ and ℓ(w) = ℓ(wJ ) + ℓ(wJ ), where wJ is as in 19.5. (vi) If w ∈ W and J ⊂ J − (w), then ℓ(wu) + ℓ(u) = ℓ(w) for all u ∈ WJ . Proof. Assertion (i) holds by [62, 3.24]. Let w ∈ W , let R be the J-residue of Σ containing w for some J ⊂ S (as defined in [62, 1.2]) and let wJ = projR (1),

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where projR is as in 1.2. Thus dist(1, z) = dist(1, wJ ) + dist(wJ , z)

(19.9) and hence

ℓ(z) = ℓ(wJ ) + ℓ((wJ )−1 z)

(19.10)

for all z ∈ R by 1.3. If s ∈ J, then wJ s ∈ R\{wJ }, hence ℓ(wJ s) > ℓ(wJ ) by (19.9) and therefore s ∈ J + (wJ ). Thus (ii) holds. Now suppose that J ⊂ J + (w). By (19.9), every chamber in R other than wJ is adjacent to chambers of R that are nearer to 1. It follows that w = wJ . Thus by (19.10), ℓ(wv) = ℓ(w) + ℓ(v) for all v ∈ WJ . Thus (iii) holds. If ℓ(wv) = ℓ(w) + ℓ(v) for some v ∈ W and s ∈ J − (v), then ℓ(wvs) ≤ ℓ(w) + ℓ(vs) = ℓ(w) + ℓ(v) − 1 = ℓ(wv) − 1.

Thus (iv) holds. Assertion (v) is proved in [44, Thm. 2.16]. Suppose now that J ⊂ J − (w) and let u ∈ WJ . By (v), WJ is finite, (wJ )−1 w = wJ

(19.11) and

(19.12) ℓ(wJ ) = ℓ(w) − ℓ(wJ ), and by [62, 5.7], we have −ℓ(wJ ) + ℓ(wJ u) = −ℓ(u).

(19.13) Hence

ℓ(wu) = ℓ(wJ ) + ℓ((wJ )−1 · wu) by (19.10) = ℓ(wJ ) + ℓ(wJ u) = ℓ(w) − ℓ(wJ ) + ℓ(wJ u)

Thus (vi) holds.

= ℓ(w) − ℓ(u)

by (19.11) by (19.12)

by (19.13).

Remark 19.14. If θ ∈ Aut(W, S) and w ∈ W , then ℓ(θ(w)) = ℓ(w) and θ(J ♯ (w)) = J ♯ (θ(w)) for ♯ ∈ {+, −}. Notation 19.15. Let (W, S) and Σ = ΣΠ be as in 19.7. The group W acts on Σ by left-multiplication and, in fact, we can identify W with the group of all type-preserving automorphisms of Σ (by [62, 2.8]). A panel of Σ is simply an edge (or, more properly, a residue of rank 1). A root of Σ is a set of chambers of the form {u ∈ Σ | dist(u, x) < dist(u, y)}

for some panel {x, y}. By 19.3(a), the complement −α of a root α is also a root. The wall of a root α is the set of panels containing exactly one chamber in α. A reflection of (W, S) (or of Σ) is an element of W conjugate to an element of S. For each root α, there is (by [62, 3.11]) a unique reflection sα in W interchanging α with its complement −α (in its action by leftmultiplication). By [62, 3.14], the reflection sα stabilizes every panel in the wall of α (which is also the wall of −α). See [40, 2.3] for the definition of the arctic and antarctic regions of a root.

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Proposition 19.16. Let (W, S) and Σ := ΣΠ be as in 19.7, let α be a root of Σ and let P be a panel in the wall of α. Suppose that (W, S) is spherical, let R be the arctic region of α, let X be the set of roots of Σ whose wall contains panels contained in R and let M be the subgroup of W generated by the reflections sβ for all β ∈ X. Then M stabilizes R and acts transitively on the set of panels in the wall of α of the same type as P . Proof. Let s ∈ S be the type of the panel P . Let A be the set of chambers of α contained in an s-panel in the wall of α and let B be the stabilizer of A in W . By [62, 3.11], a type-preserving automorphism of Σ that maps one chamber in A to another must stabilize α. It follows that B stabilizes α and acts transitively on A. The reflections sβ for β ∈ X are type-preserving and interchange two chambers of R. Therefore M stabilizes R. Let T be the unique residue opposite R. If u ∈ R and γ is a minimal gallery from u to projT (u), then by [40, 2.6(iv)], γ is of odd length and the unique panel in the middle is contained in the wall of α. It follows (again by [62, 3.11]) that an automorphism of Σ stabilizing R must also stabilize α. Therefore M ⊂ B. It will suffice, therefore, to show that B ⊂ M . Let g ∈ B. Since B stabilizes α, it also stabilizes R. Each panel in R lies in the wall of a root in the set X and hence M contains a reflection interchanging the two chambers in any given panel in R. By [40, 2.6(i)], R is a residue. In particular, R is connected. Hence the group M acts transitively on R. Therefore there exists h ∈ M such that gh−1 fixes a chamber of R. Since W acts sharply transitively on Σ, we conclude that g = h ∈ M . The following will be used in the proof of 21.8. Proposition 19.17. Let Σ be as in 19.15, let {d, e} and {u, v} be two panels of Σ whose types are s and t and suppose that k := dist(d, u) = dist(e, v) and dist(d, v) = dist(e, u) = k + 1. Let γ be a minimal gallery from d to u, let f be its type and let w = rf . Then t = w−1 sw and there is a gallery of type f from e to v. Proof. Let α denote the root {x ∈ Σ | dist(x, d) < dist(x, e)}. Then d and u are both contained in α, and e and v are both contained in its complement −α. The panels {d, e} and {u, v} are thus both contained in the wall of α. Hence the reflection sα interchanges u and v as well as d and e. Since sα is type-preserving, γ ′ := sα (γ) is a gallery of type f from e to v and the concatenations (d, γ ′ ) and (γ, v) are both minimal galleries from d to v, the first of type sf and the second of type f t. By 19.7, therefore, sw = rsf = d−1 v = rf t = wt.

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Notation 19.18. Let (W, S) be a Coxeter system and let w ∈ W . By [62, 4.1–4.4], the set of letters appearing in a reduced representation of w does not depend on the choice of the reduced representation. We call this set the support of w and denote it by supp(w). By [62, 4.11], J − (w) ⊂ supp(w). Thus if J ⊂ S is spherical and wJ is as in 19.5, then J + (wJ ) ∩ J = ∅, hence J = J − (wJ ) and therefore (19.19)

supp(wJ ) = J.

Proposition 19.20. Let (W, S) be a Coxeter system and let J = supp(w) for some w ∈ W . Suppose that tw = ws for some s, t ∈ S not both in J. Then s = t and s commutes with every element of J. Proof. Replacing w by w−1 if necessary, we can assume that t 6∈ J. Let f be the type (as defined in [62, 1.1]) of a minimal gallery from 1 to w in Σ = ΣΠ . By [62, 2.5], w = rf and there are galleries of type tf and type f s from 1 to tw = ws in Σ. By 19.3(B), the word tf is a reduced representation of w (since t 6∈ J). Therefore supp(tw) = {t} ∪ J by (19.19) and f s, which has the same length as tf , is also a reduced representation of w, so supp(ws) ⊂ supp(w) ∪ {s}. Since t ∈ supp(tw), it follows that s = t. By 19.3(iii), therefore, sf and f s are homotopic. Since s 6∈ supp(w), this is only possible if s commutes with every letter in f . Definition 19.21. We will call a subset J of S a component of S if the subdiagram ΠJ of Π spanned by J is a connected component of Π. We will call a subset J of S a direct summand of (W, S) if it is a union of components of S. If J ⊂ S is arbitrary, we denote by D(J) the smallest direct summand of (W, S) containing J. Proposition 19.22. Let (W, S) be spherical and suppose that J is a subset of S such that wJ wS = wS wJ . Then supp(wS wJ ) = D(S\J). Proof. Let w1 = wS wJ and J1 = wS JwS . Then J1 ⊂ S by 19.6 and hence wJ = wS wJ wS = wJ1 . Thus supp(wJ ) = supp(wJ1 ). Hence J = J1 by (19.19). Since wJ JwJ = J, we conclude that w1 Jw1 = J. Next we let K = supp(w1 ) and J ′ = S\J. Since wJ wS = wS wJ , we have 2 w1 = 1. Since J ⊂ J − (wS ), we have w1 = wJ and ℓ(wS ) = ℓ(w1 ) + ℓ(wJ ) by 19.8(v). Thus (19.23)

J ⊂ J + (w1 )

by 19.8(ii) and S = supp(wS ) = supp(w1 ) ∪ supp(wJ ) = K ∪ J, so (19.24)

J ′ ⊂ K.

Now let s ∈ S\K. By (19.24), s ∈ J. Thus w1 sw1 ∈ w1 Jw1 = J ⊂ S. By 19.20, therefore, [s, K] = 1. It follows that K is a direct summand of (W, S). Hence D(J ′ ) ⊂ K.

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Let L = K\D(J ′ ). Since D(J ′ ) is a direct summand of (WK , K), so is L. Hence w1 = uv for unique elements u ∈ WD(J ′ ) and v ∈ WL , and ℓ(w1 ) = ℓ(u) + ℓ(v). By 19.8(iv), we have J − (v) ⊂ J − (w1 ). Since J ′ ⊂ D(J ′ ), however, we have L ⊂ J and hence J − (v) ⊂ L ⊂ J + (w1 ) by (19.23). Therefore J − (v) = ∅. Thus v = 1 and hence w1 = u ∈ WD(J ′ ) . By 19.8(i), we conclude that K ⊂ D(J ′ ). Notation 19.25. Let (W, S), Π and Σ = ΣΠ be as in 19.7. Suppose that (W, S) is spherical, let opΣ denote the map sending each element of Σ to its unique opposite and let σ denote the permutation s 7→ wS swS of S described in 19.6. By [62, 5.11], two chambers of Σ are s-adjacent for some s ∈ S if and only if their images under opΣ are σ(s)-adjacent. In particular, the image of a residue of Σ under opΣ is again a residue. Two residues of Σ are called opposite if they are interchanged by opΣ . By [62, 5.2], no root of Σ contains a pair of opposite chambers. Proposition 19.26. Let (W, S) and Σ = ΣΠ be as in 19.7 and suppose that (W, S) is irreducible and spherical. Let R, T and T ′ be proper residues such that T and T ′ are opposite in Σ. Then R ∩ T and R ∩ T ′ cannot both be non-empty. Proof. Let J be the type of R and let K be the type of T . Suppose there exists a chamber u in T ∩ T ′ . Then opΣ (u) is also in T ∩ T ′ . Since residues are convex, it follows from [62, 5.4] that T = T ′ = Σ. Since T is proper, we conclude that T ∩ T ′ = ∅. Since R is proper, we have J 6= S. Suppose now that R ∩ T and R ∩ T ′ are both non-empty. We choose a chamber c ∈ R ∩ T ′ and let e = projT (c) (as defined in 1.2). By [62, 3.25], e ∈ R. Let γ be a minimal gallery from c to e, let f be the type of γ, and let w = rf . Replacing R by a smaller residue containing c and e if necessary, we can assume that J = supp(w) (without sacrificing the observation that J 6= S). Since T ∩ T ′ = ∅, the element w is non-trivial and hence J 6= ∅. Let d = opΣ (c). By [62, 5.14(ii)], d is opposite e in T and by [62, 3.22], γ can be extended to a minimal gallery from c to d whose type g is in MJ∪K . Since c and d are opposite in T , we have rg = wwK and since c and d are opposite in Σ, we also have rg = wS . Thus wS = wwK . By (19.19), we conclude that J ∪ K = S, so s ∈ S\J implies that s ∈ K. By 19.6, we also have wKw−1 = wS (wK KwK )wS ⊂ S. Thus if s ∈ S\J, then t := wsw−1 ∈ S, and hence s commutes with every element of J by 19.20. This is impossible, however, since (W, S) is irreducible and neither J nor S\J is empty. Proposition 19.27. Let α be a root of Σ and let T be the arctic region of α. Then α contains every proper residue of Σ which contains a chamber of T. Proof. Suppose that R is a residue of Σ containing a chamber of T and a chamber of −α. Then R contains an adjacent pair of chambers, one in α and one in −α. The unique reflection sα of Σ interchanging α and −α

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151

interchanges these two chambers and is type-preserving. Hence sα maps R to itself. Since sα interchanges α and −α, it also interchanges T and the antarctic region of α. It follows that R contains a chamber of the antarctic region of α. By [40, 2.6(vi)], the arctic and antarctic regions of α are opposite residues. By 19.26, we conclude that R = Σ.

Chapter Twenty Tits Indices In this chapter we introduce the notion of a Tits index and the notion of the relative Coxeter diagram of a Tits index. Our main goal is to give a proof of 20.32. Definition 20.1. A Tits index is a triple T = (Π, Θ, A) consisting of (i) a Coxeter diagram Π with vertex set S, (ii) a subgroup Θ of Aut(W, S), where (W, S) is the Coxeter system corresponding to Π, and (iii) a subset A of S stabilized by Θ such that for each s ∈ S\A, the subset Js := Θ(s) ∪ A is spherical and ws Aws = A, where ws := wJs is the longest element in WJs and Θ(s) denotes the Θ-orbit containing s. We call (W, S) (or, equivalently, Π) the absolute type of T; see 20.34. Remark 20.2. Note that in 20.1(iii), Js and ws depend only on the orbit Θ(s) and not on the element s itself. Convention 20.3. A Tits index T = (Π, Θ, A) is called anisotropic if A = S, (where S is as in 20.1(i)) and isotropic otherwise. Since we are only interested in isotropic Tits indices in these notes, we make A 6= S part of the definition of a Tits index in order not to have to repeat the adjective “isotropic” over and over. Note that it follows from this convention that A is spherical. Later we will display Tits indices following the conventions in [53]. See 34.2 for a description of these conventions. Note that when we say that the subset A is stabilized by Θ in 20.1(iii), we mean that A is mapped to itself by Θ, but that it is allowed that Θ acts non-trivially on A. (We always use the word “stabilize” in this sense in these notes.) Hypothesis 20.4. For the rest of this chapter, we assume that T = (Π, Θ, A) is a Tits index as defined in 20.1 and 20.3, that (W, S) is the Coxeter system corresponding to Π and that ℓ is the length function of (W, S). In addition, we fix the following notation:

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(i) For each s ∈ S\A, s˜ := ws wA , where wA is the longest element of WA and ws is as in 20.1(iii), and ˜ := hSi ˜ ⊂ W. (ii) S˜ := {˜ s | s ∈ S\A} and W Remark 20.5. By 20.2, the element s˜ in 20.4(i) depends only on the orbit Θ(s) and not on the element s itself. Remark 20.6. For each Θ-invariant subset J of S, let ΠJ denote the subdiagram of Π spanned by J, let ΘJ denote the subgroup of Aut(ΠJ ) induced by Θ and let TJ = (ΠJ , ΘJ , A ∩ J). If A is a proper subset of J, then TJ is also a Tits index as defined in 20.1 and 20.3. The triple TJ is not, in general, a Tits index if A 6⊂ J; see, however, 20.7 and 22.20(iv). Remark 20.7. Let J, ΘJ and TJ be as in 20.6, suppose that K := A\J is a direct summand of S and let L = A ∩ J. Then wA = wL wK and wΘ(s)∪A = wΘ(s)∪L wK for all s ∈ S\A. Therefore TJ is a Tits index, the set S˜ is contained in WJ and if we replace T by TJ in 20.4 and (W, S) by ˜ , S) ˜ remains unchanged. (WJ , J), the pair (W Remark 20.8. By 19.6, 20.1(iii) and 20.4(i), s˜A˜ s = A for all s˜ ∈ S˜ and −1 ˜ hence wAw = A for all w ∈ W . Remark 20.9. Let s, s˜, etc., be as in 20.4. Since s 6∈ A, we have ws 6∈ WA by 19.8(i). Therefore ws 6= wA and hence s˜ 6= 1. Definition 20.10. A Tits index (Π, Θ, A) is quasi-split if A = ∅ and split if, in addition, Θ = 1. Note that if A = ∅, condition 20.1(iii) holds if and only if all Θ-orbits are spherical. Lemma 20.11. Let s ∈ S\A. Then the following hold: (i) s˜, ws and wA are pairwise commuting elements of order 2 and ℓ(ws ) = ℓ(˜ s) + ℓ(wA ). ˜ ⊂ CW (wA ) = {w ∈ W | wwA = wA w}. In particular, W ˜ ⊂ Fix(Θ) = {w ∈ W | θ(w) = w for all θ ∈ Θ}. (ii) W (iii) If w ∈ Fix(Θ), then s ∈ J ♯ (w) if and only if Θ(s) ⊂ J ♯ (w) for ♯ ∈ {+, −}. (iv) s ∈ J − (˜ s).

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Proof. By 20.1(iii), ws Aws = A. Therefore ws wA ws = wA . Since ws is an involution, it follows that the elements s˜, wA and ws commute pairwise. Since wA and ws are involutions and s˜ 6= 1 (by 20.9), the order of s˜ is 2. Furthermore, A ⊂ Js ⊂ J − (ws ). By 19.8(v), therefore, s˜ = (ws )A

(20.12)

and ℓ(ws ) = ℓ(˜ s) + ℓ(wA ). Thus (i) holds. Since A and Js are Θ-invariant, both wA and ws lie in Fix(Θ). Hence also s˜ lies in Fix(Θ). Thus (ii) holds. By 19.14, also (iii) holds. By (i), we have s˜ 6= 1. Thus s˜ ∈ WJs implies that Js 6⊂ J + (˜ s). By 19.8(ii) and (20.12), on the other hand, we have A ⊂ J + (˜ s). Hence Θ(s) 6⊂ J + (˜ s). By (ii) and (iii), therefore, s ∈ J − (˜ s). Thus (iv) holds. ˜ and s ∈ S\A. Then the following hold: Lemma 20.13. Let w ∈ W (i) If ℓ(ws) = ℓ(w) + 1, then ℓ(w˜ s) = ℓ(w) + ℓ(˜ s) and s ∈ J − (w˜ s). (ii) If ℓ(ws) = ℓ(w) − 1, then ℓ(w˜ s) = ℓ(w) − ℓ(˜ s) and s ∈ J + (w˜ s). (iii) A ⊂ J + (w). Proof. We have A ⊂ S = J + (1). By induction, it will thus suffice to show that (i), (ii) and A ⊂ J + (w˜ s)

(20.14)

hold for given w and s under the assumption that A ⊂ J + (w). Suppose first that ℓ(ws) = ℓ(w) + 1. By 20.11(ii)–(iii), we have Θ(s) ⊂ J + (w). Since A ⊂ J + (w) by assumption, we have Js ⊂ J + (w). By 19.8(iii), it follows that ℓ(wv) = ℓ(w) + ℓ(v) for all v ∈ WJs . In particular, we have ℓ(w˜ s) = ℓ(w) + ℓ(˜ s) and ℓ(wws ) = ℓ(w) + ℓ(ws ). −

By 20.11(iv), s ∈ J (˜ s) and thus s ∈ J − (w˜ s) by 19.8(iv). We also have ℓ(w˜ s · wA ) = ℓ(wws ) = ℓ(w) + ℓ(ws ) = ℓ(w) + ℓ(˜ s) + ℓ(wA ) by 20.11(i) and hence ℓ(w˜ s · wA ) = ℓ(w˜ s) + ℓ(wA ). Therefore, ℓ(w˜ s · t) = ℓ(w˜ swA · wA t)

≥ ℓ(w˜ s) + ℓ(wA ) − ℓ(wA t) > ℓ(w˜ s)

for t ∈ A. Hence A ⊂ J + (w˜ s). Thus (i) and (20.14) hold under the assumption that ℓ(ws) = ℓ(w) + 1.

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Suppose now that ℓ(ws) = ℓ(w) − 1. By 20.11(ii)–(iii) again, we have (20.15)

Θ(s) ⊂ J − (w).

By 19.8(iii) and the assumption that A ⊂ J + (w), we have ℓ(wwA ) = ℓ(w) + ℓ(wA ). We also have (20.16)

A ⊂ J − (wA ).

˜ , wA ] = 1. In particular, wwA = wA w, so By 20.11(i), [W ℓ(wA w) = ℓ(wwA ) = ℓ(w) + ℓ(wA ). By 19.8(iv), (20.15) and (20.16), therefore, Js = A ∪ Θ(s) ⊂ J − (wwA ). By 19.8(v), it follows that w˜ s = wwA ws = (wwA )Js . Hence Js ⊂ J + (w˜ s) by 19.8(ii). By 19.8(iii), finally, we obtain ℓ(w) = ℓ(w˜ s) + ℓ(˜ s). Thus (ii) and (20.14) hold under the assumption that ℓ(ws) = ℓ(w) − 1. ˜ . A compatible representation of w is a word Definition 20.17. Let w ∈ W f = s1 · · · sk ∈ MS\A such that w = s˜1 · · · s˜k and ℓ(w) =

k X

ℓ(˜ si ).

i=1

˜ has a compatible representation. Lemma 20.18. Every element of W ˜ . We proceed by induction with respect to ℓ(w). If Proof. Let w ∈ W ℓ(w) = 0, then w = 1 and the empty word is a compatible representation of the element w. Suppose that ℓ(w) > 0 and choose s ∈ J − (w). By 20.13(iii), s ∈ S\A. Hence 20.13(ii) implies that ℓ(w˜ s) = ℓ(w) − ℓ(˜ s). Thus w˜ s has a compatible representation t1 · · · tk (by induction) and t1 · · · tk s is a compatible representation of w. ˜ and let f = s1 · · · sk ∈ MS\A be a compatible Remark 20.19. Let w ∈ W ˜ represented by a proper representation of w. If w1 is an element of W subword sp · · · sq of f (where 1 ≤ p ≤ q ≤ k), then k X i=1

ℓ(˜ si ) >

q X

ℓ(˜ si )

i=p

and hence w1 6= w. ˜ and let f = s1 · · · sk be a compatible represenLemma 20.20. Let w ∈ W tation of w. Then the following hold: (i) s˜i 6= s˜i−1 for all i ∈ [2, k]. (ii) If J is a Θ-invariant subset of S containing A and w ∈ WJ , then si ∈ J for all i ∈ [1, k].

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157

(iii) For all p, q such that 1 ≤ p ≤ q ≤ k, the word sp · · · sq is a compatible representation of s˜p · · · s˜q . Proof. For each m ∈ [1, k], let gm ∈ MS be a reduced representation of s˜m . By 20.17, ℓ(g1 · · · gk ) = ℓ(w), so g1 · · · gk is a reduced representation of w. It follows, in particular, that s˜i−1 s˜i 6= 1 for all i ∈ [2, k]. Therefore (i) holds since each s˜i is an involution. If J is a Θ-invariant subset of S containing A and w ∈ WJ , then si ∈ J for all i ∈ [1, k] by 19.8(i). Thus (ii) holds. If 1 ≤ p ≤ q ≤ k, then gp · · · gq is a reduced representation of the image of gp · · · gq in W . Hence (iii) holds. ˜ , let J be a Θ-invariant subset of S conProposition 20.21. Let w ∈ W J ˜ and there taining A and let v = w as defined in 19.8(ii). Then v ∈ W exists a compatible representation f = s1 · · · sk of w such that s1 · · · sm is a compatible representation of v for some m ∈ [1, k]. Proof. We proceed by induction with respect to ℓ(w). If ℓ(w) = 0, then v = 1. Suppose that ℓ(w) > 0. By 20.13(iii), J − (w) ⊂ S\A. Suppose first that there exists s ∈ J − (w) ∩ J and put w1 = w˜ s. By 20.13(ii), we have ℓ(w1 ) = ℓ(w) − ℓ(˜ s) and by 20.20(ii), the J-residue containing w is the same as the J-residue containing w1 . By 19.8(ii), therefore, w1J = wJ = v. By induction, there exists a compatible representation f = s1 · · · sn of w1 such that s1 · · · sm is a compatible representation of v for some m ∈ [1, n]. It follows that s1 · · · sm s is a compatible representation of w with the desired property. Now suppose that J ⊂ J + (w). By 19.8(iii), we have v = w. Hence any compatible representation of w has the desired property. We are done, therefore, by 20.18. ˜ ∩WJ . Lemma 20.22. Let s, t ∈ S\A, let J = Θ(s)∪Θ(t)∪A and let w ∈ W Then any two compatible representations of w have the same length. Proof. Let f = s1 · · · sk be a compatible representation of w. By 20.5, we can assume that si ∈ {s, t} for all i ∈ [1, k] and by 20.20(i), the si ’s are alternately equal to s and to t (and k = 1 if s˜ = t˜). Thus if g is a second compatible representation of w whose length is greater than or less than the length of f , then one of these two words is a subword of the other. This is impossible, however, by 20.19. ˜ , we denote by ℓc (w) the minimal length Definition 20.23. For each w ∈ W of a compatible representation of w. ˜ Proposition 20.24. Every compatible representation of an element w of W has length ℓc (w). ˜ . We proceed by induction with respect to ℓc (w). If Proof. Choose w ∈ W ℓc (w) = 0, then the only compatible representation of w is the empty one. Suppose that ℓc (w) > 0 and let f = s1 · · · sk be a compatible representation of w with k = ℓc (w). Then f1 := s1 · · · sk−1 (which is empty if k = 1) is a

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compatible representation of w1 := s˜1 · · · s˜k−1 . Let g = t1 · · · tm be a second compatible representation of w, let s = sk and let t = tm . We want to show that m = k. Suppose that s˜ = t˜. Then w1 = wt˜ and hence t1 · · · tm−1 is a compatible representation of w1 . Hence k = m by induction. We can thus assume that s˜ 6= t˜. Let J = Θ(s) ∪ Θ(t) ∪ A, let u = wt˜ and let R be the unique Jresidue containing w. By 20.20(ii), u and w1 are also contained in R. Let ˜ and there exist words h, h′ , h1 , h′ in v = wJ = projR (1). By 20.21, v ∈ W 1 the monoid MS\A such that h and h1 are compatible representations of v, hh′ is a compatible representation of w1 and h1 h′1 is a compatible representation of u. By induction, we have (20.25)

ℓ(h) + ℓ(h′ ) = ℓc (w1 ) = k − 1

as well as (20.26)

ℓ(h1 ) + ℓ(h′1 ) = ℓc (u) = m − 1

and (20.27)

ℓ(h1 ) = ℓc (v) = ℓ(h).

Note, too, that hh′ s and h1 h′1 t are compatible representations of w and hence h′ s and h′1 t are compatible representations of v −1 w (by 20.19). Furthermore, ˜ ∩ WJ . By 20.22, therefore, ℓ(h′ s) = ℓ(h′1 t). Thus ℓ(h′ ) = ℓ(h′1 ). v −1 w ∈ W By (20.25), (20.26) and (20.27), it follows that m = k. ˜ and s, t ∈ S\A and let ℓc be as in 20.23. Then Lemma 20.28. Let w ∈ W the following hold: (i) ℓc (w˜ s) = ℓc (w)♯1 if and only if s ∈ J ♯ (w) for ♯ ∈ {+, −}. (ii) Suppose that s˜ 6= t˜ and ℓc (w˜ s) = ℓc (w) − 1 = ℓc (wt˜). Then the product ˜ s˜t has finite order and if x and x′ denote the two alternating products of the elements s˜ and t˜ of length m := |˜ st˜|, then x = x′ and ℓc (wx) = ℓc (w) − m. Proof. Suppose first that ℓ(ws) = ℓ(w) + 1. Then ℓ(w˜ s) = ℓ(w) + ℓ(˜ s) by 20.13(i). Thus if s1 · · · sk is a compatible representation of w, then s1 · · · sk s is a compatible representation of w˜ s. By 20.24, it follows that ℓc (w˜ s) = ℓc (w) + 1. Suppose next that ℓ(ws) = ℓ(w)−1. Then ℓ(w˜ s) = ℓ(w)−ℓ(˜ s) by 20.13(ii). Thus if t1 · · · tm is a compatible representation of w˜ s, then t1 · · · tm s is a compatible representation of w. By 20.24, it follows this time that ℓc (w˜ s) = ℓc (w) − 1. Thus (i) holds. Suppose now that s˜ 6= t˜ and that ℓc (w˜ s) = ℓc (w) − 1 = ℓc (wt˜). Let J = Θ(s) ∪ Θ(t) ∪ A. We have (20.29)

Θ(s) ∪ Θ(t) ⊂ J − (w)

by (i) and 20.11(ii)–(iii) and A ⊂ J + (w) by 20.13(iii).

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˜ , wA ] = 1 by 20.11(i) and hence v = wA w. Let v = wwA . Recall that [W We have ℓ(v) = ℓ(w) + ℓ(wA ) by 19.8(iii). Thus J − (w) ∪ A ⊂ J − (v) by 19.8(iv). By (20.29), therefore, J ⊂ J − (v). Let u = v J . By 19.8(v), J is a spherical subset of S, v = uwJ and ℓ(v) = ℓ(u) + ℓ(wJ ). Let w1 = w˜ s and w2 = wt˜. Then ℓc (w1 ) = ℓc (w) − 1 = ℓc (w2 ) by (i). Since w, v, w1 and w2 are all contained in the same J-residue, we have ˆ and there u = v J = w1J = w2J by 19.8(ii). By 20.21, therefore, u ∈ W exist words g1 , g2 , h1 , h2 such that g1 and g2 are compatible representations of u and gi hi is a compatible representation of wi for i = 1 and 2. Thus hi is a compatible representation of u−1 wi for i = 1 and 2. Since u−1 w1 and u−1 w2 lie in WJ , we can assume by 20.5 that h1 and h2 both lie in the monoid M{s,t} . The words h1 s and h2 t are both compatible representations of u−1 w. Hence by 20.20(i), they are both alternating products in s and t and by 20.24 they have the same length p. This means that the two alternating products in s˜ and t˜ of length p are both equal to u−1 w and hence equal to each other. This implies that (˜ st˜)p = 1 in W . Hence m := |˜ st˜| divides p. Suppose that m < p and hence 2m ≤ p. Let f be an alternating word in s and t of length 2m. Then f is a subword of a compatible representation of u−1 w and hence a compatible representation of its image in W . Its image in W is, however, 1. With this contradiction, we conclude that p = m. Hence ℓc (w) = ℓc (u) + m and u = wx, where x is either of the two alternating products of the elements s˜ and t˜ of length m. Thus (ii) holds. ˜ Note that ˜ , S) ˜ by ℓ. Notation 20.30. We denote the length function of (W ˜ ˜ ℓ(w) ≤ ℓc (w) for all W . Lemma 20.31. Let f = s1 · · · sk be an element of the monoid MS\A and ˜ . Then the following are equivalent: let w = s˜1 · · · s˜k ∈ W (i) f is a compatible representation of w. ˜ , S). ˜ (ii) s˜1 · · · s˜k is a reduced representation of w in (W Proof. Suppose that (ii) holds We proceed by induction with respect to k := ℓ(f ) to show that f is a compatible representation of w. If k = 0, then w = 1 and f = ∅ is a compatible representation of w. Let k > 0 and f1 = s1 · · · sk−1 . Then s˜1 · · · s˜k−1 is a reduced representation of w˜ sk ˜ , S), ˜ so by induction, f1 is a compatible representation of w˜ in (W sk . In particular, we have ℓc (w˜ sk ) = k − 1. Suppose that ℓ(w˜ sk sk ) = ℓ(w˜ sk ) − 1. Then ℓ(w) = ℓ(w˜ sk · s˜k ) = ℓ(w˜ sk ) − ℓ(˜ sk ) by 20.13(ii). Let t1 · · · tm be a compatible representation of w. As ℓ(w˜ sk ) = ℓ(w) + ℓ(˜ sk ), the word t1 · · · tm sk is a compatible representation of w˜ sk of length m + 1. By 20.24 and the conclusion of the previous paragraph, ˜ we have k − 1 = m + 1. Thus ℓ(w) = k > m = ℓc (w). This contradicts the inequality in 20.30. We conclude that ℓ(w˜ sk sk ) = ℓ(w˜ sk ) + 1. This implies that ℓ(w) = ℓ(w˜ sk ) + ℓ(˜ sk ) by 20.13(i). Since f1 is a compatible representation of w˜ sk , it follows that f is a compatible representation of w. Thus (ii) implies (i).

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Now suppose that (i) holds and let g = t1 · · · tm be an element of MS\A ˜ , S). ˜ Since (ii) such that t˜1 · · · t˜m is a reduced representation of w in (W implies (i), the word g is also a compatible representation of w. By 20.24, therefore, k = m. Thus (ii) holds. Here, finally, is the main result of this chapter. Theorem 20.32. Under the assumptions in 20.4, the following hold: ˜ , S) ˜ is a Coxeter system. (i) ℓc = ℓ˜ and (W ˜ s) = ℓ(w)♯1 ˜ ˜ , all (ii) ℓ(w˜ if and only if ℓ(ws) = ℓ(w)♯1 for all w ∈ W s ∈ S\A and ♯ ∈ {+, −}. ˜ ˜ . Both assertions follow, thereProof. By 20.31, ℓc (w) = ℓ(w) for all w ∈ W fore, from 19.3 and 20.28. ˜ and ℓ(vw) ˜ ˜ + ℓ(w). ˜ Remark 20.33. Suppose that v, w ∈ W = ℓ(v) Then the product of reduced representations for v and w is a reduced representation for vw. By 20.31, it follows that ℓ(vw) = ℓ(v) + ℓ(w). ˜ be the Coxeter diagram corresponding to the Definition 20.34. Let Π ˜ ˜ Coxeter system (W , S). We will call (W, S) the absolute Coxeter system and ˜ , S) ˜ the relative Coxeter system and W ˜ the relative Coxeter group of the (W Tits index T = (Π, Θ, A). We will call Π the absolute Coxeter diagram (or ˜ the relative Coxeter diagram (or relative type) of T and absolute type) and Π we will call the number of vertices of Π (i.e. |S|) the absolute rank and the ˜ (i.e. the number of orbits of Θ in S\A) the relative number of vertices of Π rank of T. We end this chapter with some observations about the case that (W, S) is spherical (in 20.35), irreducible (in 20.40) or affine (in 20.43). We continue with the hypotheses in 20.4. ˜ , S) ˜ be the absolute and relative Proposition 20.35. Let (W, S) and (W Coxeter systems of the Tits index T = (Π, Θ, A). Then the following hold: ˜ , S) ˜ is spherical. (i) (W, S) is spherical if and only if (W ˜ , S) ˜ are both spherical, then wS = w ˜ wA = wA w ˜ , (ii) If (W, S) and (W S S wS˜ = (wS )A and ℓ(wS ) = ℓ(wS˜ ) + ℓ(wA ), where wS , wS˜ and wA are as in 19.5 and (wS )A is as in 19.8(ii). ˜ is a subgroup of W , it is finite if W is finite. Suppose, Proof. Since W ˜ is finite. By 20.11(i), w ˜ wA = wA w ˜ , by 20.13(iii), conversely, that W S S + A ⊂ J (wS˜ ) and thus by 19.8(iii), (20.36)

ℓ(wA wS˜ ) = ℓ(wS˜ wA ) = ℓ(wS˜ ) + ℓ(wA ).

Since A ⊂ J − (wA ), it follows from 19.8(iv) that A ⊂ J − (wS˜ wA ).

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˜ ˜ s˜) = ℓ(w ˜ ˜ ) − 1 for ˜ , S), ˜ we have ℓ(w Since wS˜ is the longest element of (W S S all s ∈ S\A. By 20.32(ii), therefore, we have S\A ⊂ J − (wS˜ ). By 19.8(iv) and (20.36), it follows that S\A ⊂ J − (wS˜ wA ). Hence S = J − (wS˜ wA ). By 19.8(v), we conclude that (W, S) is spherical, so (i) holds, and (20.37)

wS˜ wA = wS .

Since A ⊂ J − (wS ), we have wS = (wS )A wA , again by 19.8(v). Hence (wS )A = wS wA . By (20.36) and (20.37), we conclude that (wS )A = wS˜ and ℓ(wS ) = ℓ(wS˜ ) + ℓ(wA ). Thus (ii) holds. Our next goal is to prove 20.40 below. Recall the notion of a direct summand introduced in 19.21. ˜ , S) ˜ be the absolute and relative Coxeter Lemma 20.38. Let (W, S) and (W systems of the Tits index T = (Π, Θ, A) and suppose that there exist s, t ∈ S such that S˜ = {˜ s, t˜}, s˜ 6= t˜ and s˜t˜ = t˜s˜. Let Ks = supp(˜ s), let Kt = supp(t˜) and let A1 = A\(Ks ∪ Kt ). Then Θ(s) ⊂ Ks , Θ(t) ⊂ Kt and {Ks , Kt , A1 }

(or {Ks , Kt } if A1 = ∅) is a partition of S into direct summands of (W, S).

˜ | = 4 and w ˜ = s˜t˜. By 20.35, it follows that W is Proof. We have |W S spherical and wS = s˜t˜wA . By 20.4(i) and 20.11(i), therefore, [wt , wS ] = 1 and wS wt = s˜. Note, too, that S\Jt = Θ(s) by 20.1(iii). We now apply 19.22 with J = Jt to conclude that Ks = supp(˜ s) = D(S\Jt ) = D(Θ(s)). By symmetry, we also have Kt = supp(t˜) = D(S\Js ) = D(Θ(t)). Since s˜ ∈ WJs , we have Ks ⊂ Js = Θ(s) ∪ A and hence Ks ∩ Θ(t) = ∅. Thus Kt \Ks is a direct summand of (W, S) containing Θ(t). By 19.21, it follows that Kt \Ks = Kt . In other words, Ks ∩ Kt = ∅. Therefore {Ks , Kt , A1 } is a partition of S into direct summands of (W, S).

˜ , S) ˜ be the absolute and relative Coxeter Lemma 20.39. Let (W, S) and (W systems of the Tits index T = (Π, Θ, A) and suppose that there exist nonempty subsets S1 and T1 of S\A such that S˜ = S˜1 ∪ T˜1 , S˜1 ∩ T˜1 = ∅ and [S˜1 , T˜1 ] = 1. Let Ks = supp(˜ s) for all s ∈ S1 ∪ T1 , let [ [ Ks and T¯ = Kt S¯ = t∈T1

s∈S1

¯ T¯ , A1 } (or {S, ¯ T¯} if A1 = ∅) is a partition and let A1 = A\(S¯ ∪ T¯). Then {S, of S into direct summands of (W, S). Proof. Choose s ∈ S1 and t ∈ T1 . We set J = Θ(s) ∪ Θ(t) ∪ A and let TJ = (ΠJ , ΘJ , A) as defined in 20.6. Since S˜1 ∩ T˜1 = ∅, we have s˜ 6= t˜. Thus (h˜ s, t˜i, {˜ s, t˜})

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is the relative Coxeter system of the Tits index TJ . Since [˜ s, t˜] ∈ [S˜1 , T˜1 ] = 1, it follows from 20.38 that Θ(s) ⊂ Ks , that Θ(t) ⊂ Kt and that {Ks , Kt , A\(Ks ∪ Kt )}

is a partition of J into direct summands of (WJ , J). Therefore S¯ ∩ T¯ = ∅, ¯ T¯] = 1 and [S¯ ∪ T¯ , u] = 1 for every u ∈ A not contained in S¯ ∪ T¯. Since [S, ˜ ¯ T¯, A1 } is a partition of S S = S˜1 ∪ T˜1 , we have S = S¯ ∪ T¯ ∪ A. Hence {S, into direct summands of (W, S). ˜ , S) ˜ be the absolute and relative CoxTheorem 20.40. Let (W, S) and (W eter systems of the Tits index T = (Π, Θ, A) and suppose that (W, S) is ˜ , S) ˜ is also irreducible. irreducible. Then (W Proof. This holds by 20.40. The converse of 20.40 is false. It does not hold, for instance, for the Tits index T = (A1 × A1 , Θ, ∅) if Θ 6= 1. We turn, finally, to affine Coxeter systems. Definition 20.41. The irreducible affine Coxeter diagrams are the diagrams ˜ℓ , . . . , G ˜ℓ A˜ℓ , B for suitable ℓ. See, for example, [65, Fig. 1.1]. We call a Coxeter system (W, S) affine if at least one connected component of the corresponding Coxeter diagram is affine and all the other components are affine or spherical. Suppose that (W, S) is an irreducible affine Coxeter system with Coxeter diagram Π, let ℓ + 1 be the rank of Π, let ΣΠ be as in 19.7 and let T be the subgroup of W consisting of all the elements of W which map every root of ΣΠ to a parallel root (as defined in [65, 1.2]). Then T ∼ = Zℓ by [65, 1.24] and |W/T | < ∞ by [65, 1.6 and 1.22]. Remark 20.42. In [26], it is shown that if (W, S) is an arbitrary irreducible Coxeter system, then W is spherical or affine if and only if W does not contain a free non-abelian subgroup. ˜ , S) ˜ be the absolute and relative Proposition 20.43. Let (W, S) and (W Coxeter systems of the Tits index T = (Π, Θ, A) and suppose that (W, S) is ˜ , S) ˜ is also affine. affine. Then (W ˜ , S) ˜ are irreducible. Let Proof. By 20.40, we can assume that (W, S) and (W ˜ ˜ . Then T be the subgroup of translations in W and let T = T ∩ W ˜ /T˜ ∼ ˜ /T ⊂ W/T. W = TW

˜ /T˜ is finite. Since T ∼ Hence W = Zℓ for some ℓ > 0, we have T˜ ∼ = Zm for ˜ ˜ some m. By 20.35(i), m > 0. By 20.42, it follows that (W , S) is affine.

We expect that it can be shown that the converse of 20.43 is valid under the additional assumptions that (W, S) is irreducible and the relative rank of T is at least 3, but we have not worked out all the details.

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Definition 20.44. We will call a Tits index spherical if its absolute type is spherical, affine if its absolute type is affine, irreducible if its absolute type is irreducible and exceptional if its absolute type is exceptional in the sense of 36.13 below. We observe that the only irreducible spherical or affine Coxeter diagrams whose automorphism group has order greater than 2 are A˜ℓ for ℓ ≥ 2, D4 , ˜ ℓ for ℓ ≥ 4 and E ˜6 . Thus if (Π, Θ, A) is a Tits index and Π is an irreducible D spherical or affine diagram not in this short list, then, of necessity, |Θ| ≤ 2.

Chapter Twenty One Parallel Residues In this chapter we examine the notion of parallel residues in a building; see 21.7 for the definition. Parallel residues were first introduced by Tits in [58]. They play an important role in the theory of buildings. Hypothesis 21.1. Throughout this chapter, we assume that ∆ is a building of type Π (as defined in [62, 7.1]). The building ∆ is arbitrary except in a few places where it is explicitly assumed to be spherical. We do not assume that ∆ is thick. Let S be the vertex set of Π, let (W, S) be the corresponding Coxeter system, let δ be the W -distance function on the set of ordered pairs of chambers of ∆ and let ℓ be the length function on (W, S). Thus dist(c, d) = ℓ(δ(c, d)) for all chambers c and d of ∆ (by [62, 7.8]). Notation 21.2. We denote by Typ(R) the type of a residue R. Thus Typ(R) is the set of colors (i.e. elements of S) that appear on edges in the residue R. A residue R of type J is a building of type ΠJ whose WJ -distance function is the restriction of δ to R × R (by [62, 7.20], for example). We denote by typ the natural homomorphism from Aut(∆) to Aut(Π) = Aut(W, S). An automorphism of ∆ is type-preserving if it is in the kernel of the map typ. Remark 21.3. We will say that a word f in the free monoid MS is reduced if it is a reduced representation of its image rf in W . By 19.3(ii), a word f is reduced if and only if it is p-reduced as defined in 19.1. Remark 21.4. Let x and y be two chambers of ∆ and let f be a reduced word in MS . By [62, 7.1], δ(x, y) = rf if and only if there is a galley of type f in ∆ from x to y, by [62, 7.3], δ(x, y) = δ(y, x)−1 and by [62, 7.7(ii)], a gallery from x to y is minimal if and only if its type is reduced. Lemma 21.5. Let c, d, e be chambers of ∆, let w = δ(c, d) and let v = δ(d, e). Then dist(c, e) = dist(c, d) + dist(d, e) if and only if ℓ(wv) = ℓ(w) + ℓ(v) and if either of these equalities holds, then δ(c, e) = δ(c, d) · δ(d, e).

Proof. Let γ be a minimal gallery from c to d, let γ ′ be a minimal gallery from d to e, let f and f ′ be the types of γ and γ ′ , respectively, and let γ ′′ denote the concatenation (γ, γ ′ ). Thus dist(c, e) = dist(c, d) + dist(d, e) ′′

holds if and only if γ is minimal and, as noted in 21.4, γ ′′ is minimal if and only if the word f f ′ is reduced. Also the equality ℓ(wv) = ℓ(w) + ℓ(v) holds

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if and only if f f ′ is reduced and if f f ′ is reduced, then δ(c, e) = rf f ′ = rf rf ′ , again by 21.4. Lemma 21.6. Let d = projR (c) for some residue R and some chamber c, let J = Typ(R), let e ∈ R and let v = δ(c, e). Then the following hold: (i) dist(c, e) = dist(c, d) + dist(d, e) and v = δ(c, d) · δ(d, e). (ii) projR (u) = d for each chamber u on a minimal gallery from c to d. (iii) projT (c) = projT (d) for each subresidue T of R. (iv) e = d if and only if J ⊂ J + (v). Proof. By 1.2, there is a minimal gallery γ from c to e that passes through d. By 21.5, therefore, (i) holds. Suppose u is a chamber contained in a minimal gallery from c to d and let z = projR (u). Then dist(z, c) ≤ dist(z, u) + dist(u, c) ≤ dist(d, u) + dist(u, c) = dist(d, c) and hence z = d. Thus (ii) holds. Let T be a subresidue of R. By (i) and (ii), the unique chamber of T nearest c is also the unique chamber of T nearest d. Thus (iii) holds. If e = d, then (i) implies that J ⊂ J + (v). Suppose, instead, that e 6= d and let i be the last letter in the type f of the gallery γ. Since residues are convex, i ∈ J. By 21.3, f is reduced and hence v = rf by 21.4. Thus i ∈ J ∩ J − (v). Thus (iv) holds. Here now is the central notion in this chapter. Definition 21.7. Residues R and T of a building will be called parallel if R = projR (T ) and T = projT (R). Parallelism is not, in general, a transitive relation on the set of residues of a building. See, however, 21.21. Pairs of parallel residues are ubiquitous, as the following basic result shows (compare [20, Prop. 3]). In part (v), “σ-isomorphism” means an isomorphism of graphs that sends s-panels to σ(s)-panels for all s (as in [62, 1.13]). Proposition 21.8. Let R be a J-residue, let T be a K-residue, let R′ = projR (T ) and let T ′ = projT (R). Then the following hold: (i) R′ and T ′ are parallel residues. (ii) The restriction of projR to T ′ and the restriction of projT to R′ are inverses of each other. (iii) w := δ(c, projT (c)) is independent of the choice of c ∈ R′ .

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(iv) The type of R′ is J ′ := {s ∈ J | w−1 sw = t for some t ∈ K} and the type of T ′ is K ′ := {t ∈ K | wtw−1 = s for some s ∈ J}, where w is as in (iii). (v) The restriction of projT to R′ is a σ-isomorphism from R′ to T ′ , where σ is the unique bijection from J ′ to K ′ such that σ(s) = w−1 sw for all s ∈ J ′. Proof. Let x ∈ T , let d = projR (x) and let u = projT (d). By 21.6(i), there exists a minimal gallery from x to d which passes through u. By 21.6(ii), therefore, d = projR (u). Thus (ii) holds. Now let d and e be arbitrary adjacent chambers of R, let u = projT (d), let v = projT (e), let k = dist(d, u), let m = dist(e, v) and let n = dist(v, u). By 21.6(i) again, m + n = dist(e, u) ≤ dist(d, u) + 1 = k + 1 and k + n = dist(d, v) ≤ dist(e, v) + 1 = m + 1. Hence n is bounded by both (k − m) + 1 and (m − k) + 1. Therefore n ≤ 1 and if n = 1, then k = m. Since R is connected, it follows that T ′ = projT (R) is connected. Similarly, R′ is connected. With all the notation in the previous paragraph, suppose now that d and e are in R′ . By (ii), it follows that u 6= v and hence n = 1 and k = m. Let s ∈ J be the type of the panel containing d and e, let t ∈ K be the type of the panel containing u and v and let f be the type of a minimal gallery γ from d to u. We choose an apartment Σ containing d and v. Since apartments are convex, the chambers e and u and the gallery γ are all contained in Σ. By 19.17, there is a gallery of type f from c to v in Σ and rf t = srf . By 21.4, therefore, δ(d, u) = rf = δ(e, v). Since R′ is connected, we conclude that (iii) holds with w = rf and that if e is an arbitrary chamber of R′ that is s-adjacent to d, then s is contained in the set J ′ defined in (iv). We can also observe that (v) will follow automatically once we finish verifying that (i) and (iv) hold. Now suppose that s is an arbitrary element of J ′ , let t be the unique element of K such that sw = wt and let e′ be a chamber of R which is s-adjacent to d. Then there exists a minimum gallery of type sf from e′ to u passing through d, where f is as in the previous paragraph. Thus sf is reduced. Since sw = wt, also f t is reduced. By 21.4, therefore, there exists a gallery γ = (e′ , . . . , v ′ , u) of type f t from e′ to u. Thus v ′ is t-adjacent to u, so v ′ ∈ T , and dist(e′ , v ′ ) = k, where v ′ is the penultimate chamber of γ. By 21.6(i) and (iii), it follows that e′ = projR (v ′ ) and hence e′ ∈ R′ . We conclude that a vertex of R that is s-adjacent to d for some s ∈ J is contained in R′ if and only if s ∈ J ′ . Similarly, a vertex of T that is t-adjacent to u for some t ∈ K is contained in T ′ if and only if t ∈ K ′ . Since R′ and T ′ are both connected, it follows that R′ is a J ′ -residue and T ′ is a K ′ -residue. Thus (iv) holds and, by (ii), also (i) holds. As observed above, it follows that (v) holds as well.

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Remark 21.9. Let R and T be two residues. It follows from 21.8(ii) that the following hold: (i) R and T are parallel to each other if and only if the restriction of projR to T and the restriction of projT to R are both injective. (ii) R and T are parallel to each other if and only if the restriction of projR to T is a bijection. It will be convenient to repeat the most important parts of 21.8: Proposition 21.10. Suppose that R and T are parallel residues of type J and K. Then the following hold: (i) The restriction of projR to T is an isomorphism1 from T to R, the restriction of projT to R is an isomorphism from R to T and these two isomorphisms are inverses of each other. (ii) w := δ(c, projT (c)) is independent of the choice of c ∈ R.

(iii) K = w−1 Jw and two chambers of R are s-adjacent for some s ∈ J if and only if their images under projT are w−1 sw-adjacent. Proof. This is a special case of 21.8. Notation 21.11. Let R and T be parallel residues. By 21.10(ii), the element δ(c, projT (c)) of W is independent of the choice of c ∈ R. We denote this element by δ(R, T ). Remark 21.12. Let R and T be parallel residues, let w = δ(T, R) and let R1 be a subresidue of R. Then by 21.10(i), T1 := projT (R1 ) is a residue parallel to R1 and by 21.11, δ(T1 , R1 ) = w. Proposition 21.13. Let w ∈ W , let J ⊂ J + (w−1 ), let K ⊂ J + (w) and suppose that K = w−1 Jw. Let c and d be chambers such that w = δ(c, d), let R be the J-residue containing c and let T be the K-residue containing d. Then R is parallel to T and δ(R, T ) = w. Proof. Let R′ = projR (T ) and let T ′ = projT (R). By 19.8(iii), d = projT (c) and c = projR (d). Thus, in particular, c ∈ R′ . By 21.8(iv), R′ is a J-residue and T ′ is a K-residue. Thus R′ = R and T ′ = T . ˆ be a residue Proposition 21.14. Let R and T be parallel residues, let R containing R and let R1 = projRˆ (T ). Then R1 is a residue parallel to both R and T , δ(T, R) = δ(T, R1 ) · δ(R1 , R) and ℓ(δ(T, R)) = ℓ(δ(T, R1 )) + ℓ(δ(R1 , R)). 1 Here “isomorphism” means (as in [62, 1.14]) “σ-isomorphism for some bijection σ from J to K.”

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Proof. By 21.6(iii), we have (21.15)

projR (c) = projR (projRˆ (c))

for all chambers c. Hence R = projR (T ) = projR (projR1 (T )) and the restriction of projR1 to T is injective. Since projR1 (T ) = projRˆ (T ) = R1 , the restriction π of projR1 to T is, in fact, a bijection. By (21.15), it follows that the restriction of projR to R1 is the composition of two bijections. By 21.9(ii), we conclude that T is parallel to both R and R1 . Let c ∈ T , let d = projR1 (c) and let e = projR (c). Then d = projRˆ (c), so e = projR (d) by (21.15). By 21.6(i) with e in place of d, we thus have δ(T, R) = δ(c, d) · δ(d, e) = δ(T, R1 ) · δ(R1 , R) and ℓ(δ(T, R)) = ℓ(δ(T, R1 )) + ℓ(δ(R1 , R)).

Remark 21.16. Let Σ be an apartment containing chambers of a residue R. By [62, 8.23], projR (x) = projΣ∩R (x) for every chamber x in Σ. Proposition 21.17. Let R and T be residues and let Σ be an apartment containing chambers of both R and T . Then R and T are parallel if and only if R ∩ Σ and T ∩ Σ are parallel residues of Σ. Proof. If R is parallel to T , then by 21.9(i) and 21.16, R ∩ Σ is parallel to T ∩ Σ. Suppose, conversely, that R ∩ Σ is parallel to T ∩ Σ. Let J = Typ(R), let K = Typ(T ), choose c ∈ R ∩ Σ and let w = δ(c, projR (c)). Then K = w−1 Jw by 21.10(iii). By 21.8(iv). therefore, projR (T ) is a J-residue of R and projT (R) is a K-residue of T . Thus R and T are parallel. Definition 21.18. Let R be a residue and let α be a root of some apartment Σ of ∆. We will say that α cuts R if R ∩ Σ contains chambers in α and others not in α. This notion is independent of the choice of the apartment Σ containing α. By [62, 8.13], in fact, a root α cuts a residue R if and only if α ∩ R is a root of R. By 19.15, a panel is cut by α if and only if it is in the wall of α. Proposition 21.19. Let R and T be residues, let Σ be an apartment containing chambers of both R and T , let ΩR be the set of roots of Σ cutting R and let ΩT be the set of roots of Σ cutting T . Then the following hold: (i) The residues R and T are parallel if and only if ΩR = ΩT . (ii) If R and T are parallel, then for each chamber c in T ∩ Σ, projR (c) is the unique chamber of R ∩ Σ contained in every root of ΩT containing the chamber c.

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Proof. We first observe that if α ∈ ΩR and c ∈ Σ, then by [65, 29.16] applied to α and to its opposite −α in Σ, we have (21.20)

c ∈ α if and only if projR (c) ∈ α.

Now suppose that R and T are parallel. By 21.17, R ∩ Σ and T ∩ Σ are parallel residues of Σ. Hence projR (T ∩ Σ) = R ∩ Σ (by 21.16). By (21.20), therefore, ΩR = ΩT . Suppose, conversely, that ΩR = ΩT . If x, y ∈ Σ ∩ T are distinct, then there exists a root of ΩT containing x but not y. By (21.20), it follows that the restriction of projR to Σ ∩ T is injective. By symmetry, the restriction of projT to Σ ∩ R is injective. By 21.9(i), therefore, Σ ∩ R and Σ ∩ T are parallel residues of Σ. Hence by 21.17, R and T are parallel residues. Thus (i) holds. We continue to assume that ΩR = ΩT . Let c ∈ Σ ∩ T . By (21.20) again, it follows that the set of roots in ΩT containing c is the same as the set of roots of ΩR containing projR (c). The chamber projR (c) is the only chamber of R contained in all these roots. Hence (ii) holds. Corollary 21.21. If ∆ is thin, then parallelism is an equivalence relation on the set of all residues of ∆. Proof. If ∆ is thin, it consists of a single apartment. The claim is thus an immediate consequence of 21.19(i). Proposition 21.22. If R and T are parallel residues, then either R = T or R ∩ T = ∅. Proof. Suppose that R and T are parallel residues and that c is a chamber in R ∩ T . Then projT (c) = c. By 21.11, it follows that δ(R, T ) = 1 and hence dist(d, projT (d)) = ℓ(1) = 0 for all d ∈ R. Therefore T = projT (R) = R. Proposition 21.23. Let R, T, T ′ be residues such that R is parallel to both T and T ′ and suppose that T ∩ T ′ 6= ∅. Then T = T ′ . Proof. Let J = Typ(R), let c be a chamber in T ∩ T ′ , let d = projR (c) and let w = δ(c, d). By 21.10(iii), Typ(T ) = wJw−1 = Typ(T ′ ). Since T and T ′ have the same type and both contain c, they are equal. We now want to make a number of observations (many of which can be found in [55]) concerning the notion of opposite residues defined in 19.25. We emphasize that only spherical buildings have opposite residues. Proposition 21.24. Opposite residues in a spherical building are parallel. Proof. Suppose that ∆ is spherical and let R and T be opposite residues, let Σ be an apartment containing chambers of both R and T and let opΣ be the map which sends each chamber of Σ to its unique opposite in Σ. By [62, 9.8], R ∩ Σ and T ∩ Σ are opposite residues of Σ. By [62, 5.3], therefore, opΣ (T ∩ Σ) = R ∩ Σ and opΣ (R ∩ Σ) = T ∩ Σ. By 19.25, finally, a chamber

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of Σ is contained in a given root of Σ if and only if its image under opΣ is not contained in this root. We conclude that a root of Σ cuts R ∩ Σ if and only if it cuts T ∩ Σ. By 21.19(i), therefore, R and T are parallel. Proposition 21.25. Suppose that ∆ is spherical, let R and T be opposite residues of ∆, let J = Typ(R) and let c and e be opposite chambers of R. Then c is opposite projT (e) in ∆ and δ(T, R) = wS wJ . Proof. Let d = projT (e) and let w = δ(d, e). By 21.24, R and T are parallel, so e = projR (d) by 21.10(i). We have δ(e, c) = wJ by hypothesis and w = δ(T, R) = wS wJ by [62, 9.11(iii)]. Therefore δ(d, c) = δ(d, e) · δ(e, c) = w · wJ = wS by 21.6(i). Proposition 21.26. Suppose that ∆ is spherical and let R and T be parallel residues. Then R and T are opposite in ∆ if and only if for some c ∈ R there exists a chamber of T opposite c in ∆. Proof. Let J = Typ(R), let c ∈ R and suppose that d is a chamber of T opposite c in ∆, so δ(d, c) = wS . Let e := projR (d). By 21.6(i), we have (21.27)

dist(d, x) = dist(d, e) + dist(e, x)

and (21.28)

δ(d, x) = δ(d, e) · δ(e, x)

for every chamber x of R. Since c maximizes dist(d, x), it follows from (21.27) that c and e are opposite in R. Thus δ(e, c) = wJ . Hence δ(d, e) = wS wJ by (21.28). By 21.10(iii), therefore, Typ(T ) = wS wJ JwJ wS = wS JwS . Thus by [62, 5.12], T has the type of a residue opposite R. By [62, 9.10], it follows that T is opposite R. The converse holds by [62, 9.9]. Proposition 21.29. Suppose that ∆ is spherical. Then two residues R and T of ∆ are opposite if and only if for each c ∈ R there exists a chamber in T opposite c and for each d ∈ T there exists a chamber in R opposite d. Proof. Suppose that for each c ∈ R there exists a chamber in T opposite c and for each d ∈ T there exists a chamber in R opposite d. Let R′ = projR (T ), let c ∈ R′ , let d ∈ T be opposite c, let e = projR (d), so e ∈ R′ , and choose z opposite e in R. By 21.6(i), we have dist(z, e) + dist(e, d) = dist(z, d) ≤ diam(∆)

= dist(c, d) = dist(c, e) + dist(e, d)

and hence diam(R) = dist(z, e) ≤ dist(c, e) ≤ diam(R′ ). By [62, 5.8], it follows that R′ = R. By symmetry, we have T = projT (R). By 21.26, therefore, R and T are opposite. The converse holds by [62, 9.9].

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Note that in the following, K ⊂ J + (w) holds by 21.6(iv). Proposition 21.30. Let R and T be parallel spherical residues of type J, respectively K, of a building ∆, let w = δ(R, T ) and suppose that J + (w) ⊂ K. Then ∆ is spherical, T is opposite R in ∆, wS JwS = K and w = wS wK = wJ wS . Proof. Choose c ∈ R, let d = projT (c), choose d′ opposite d in T and let c′ = projR (d′ ). By 21.10(i), we have d′ = projT (c′ ). Thus δ(c, d) = δ(c′ , d′ ) = w by 21.10(ii). In particular, dist(c, d) = dist(c′ , d′ ). By 21.6(i), we have (21.31) dist(c, d′ ) = dist(c, d) + dist(d, d′ ) = dist(c, c′ ) + dist(c′ , d′ ) and hence dist(c, c′ ) = dist(d, d′ ). By 21.10(i) again, R and T are isomorphic and thus have the same diameter. It follows that c is opposite to c′ in R. Thus δ(c, c′ ) = wJ and δ(d, d′ ) = wK . Let w0 = wwK . By 21.6(i) again, we thus have δ(c, d′ ) = wwK = wJ w and by 21.5 and (21.31), therefore, (21.32) ℓ(w0 ) = ℓ(w) + ℓ(wK ) and (21.33) ℓ(w0 ) = ℓ(wJ ) + ℓ(w). − Since K ⊂ J (wK ) and w0 = wwK , we have K ⊂ J − (w0 ) by 19.8(iv) and (21.32). By hypothesis, we have J + (w) ⊂ K. Hence S\K is contained in J − (w). By 19.8(iv) and (21.33) applied to w0 = wJ w, we conclude that S ⊂ J − (w0 ). By 19.8(v), therefore, ∆ is spherical and w0 = wS (since wS = 1 by 19.8(ii)). In particular, c is opposite d′ in ∆, so R and T are opposite by 21.25, and wS JwS = wS−1 JwS = w−1 wJ JwJ w = w−1 Jw. By 21.10(iii), therefore, wS JwS = K. (This last conclusion holds also by [62, 5.13 and 9.8].) In the following, recall that since R and T are opposite, they are parallel (by 21.24), so δ(R, T ) is defined. Proposition 21.34. Let R and T be opposite residues of a spherical building ∆, let w = δ(R, T ) and let K = Typ(T ). Then J + (w) = K. Proof. By 21.6(iv), K ⊂ J + (w). Now choose s ∈ J + (w). Let c ∈ R, let d = projT (c), let L = K ∪ {s} and let T1 be the unique L-residue containing d. By 21.6(iv) again, d = projT1 (w). Let e be a chamber opposite d in T and let e1 be a chamber opposite d in T1 . By 21.6(i), we have dist(c, d) + dist(d, e) = dist(c, e) and dist(c, d) + dist(d, e1 ) = dist(c, e1 ). By 21.25, c and e are opposite in ∆. Hence dist(c, e1 ) ≤ dist(c, e) and therefore dist(d, e1 ) ≤ dist(d, e), so diam(T1 ) ≤ diam(T ). Hence T = T1 (by [62, 5.8]). Thus s ∈ K. We conclude that J + (w) = K.

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173

The following observation is a consequence of [62, 9.7], but only under the assumption that ∆ is thick. It will be used in the proof of 21.55 and again in the proof of 24.5. Lemma 21.35. Suppose that ∆ is spherical. Let Σ be an apartment of ∆ and let g be an automorphism of ∆ acting trivially on every panel containing chambers of Σ. Then g = 1. Proof. Let θ denote the automorphism of the Coxeter diagram Π induced by conjugation by the longest element wS of (W, S) (as described in 19.6), let c be a chamber of Σ, let z be an arbitrary chamber of ∆, let γ be a minimal gallery from c to z and let v be the penultimate chamber in γ. We claim that g fixes z. By induction, we can assume that g fixes v. By [62, 8.17 and 9.4], there exists a map ω from ∆ to Σ such that for every chamber u of ∆, ω(u) is opposite u and ω(v) 6= ω(z). Let P be the unique panel containing v and z and let Q be the unique panel containing ω(v) and ω(z). By [62, 9.10], the panels P and Q are opposite each other. Since g stabilizes all the panels containing c, it is type-preserving. Since g fixes v, it therefore stabilizes P . Since Q contains chambers of Σ, g acts trivially on Q. By 21.24, the restriction of projP to Q is a bijection from Q to P . Since g commutes with this bijection, it must act trivially on P . Hence g fixes z. We revert now to the assumption that ∆ is an arbitrary building (that is to say, not necessarily spherical) and we continue to apply all the notation in 21.1. The following remark will be needed in 30.2 and 32.8. Remark 21.36. Let M be the wall of some root of ∆, let R be the set of roots with wall M and let Z be the union of all the panels in M . We set x ≈ y for chambers x, y ∈ Z whenever y = projP (x), where P is the unique panel on M containing y. By 21.19(i), the panels in M are pairwise parallel and thus by 21.10(i), the relation ≈ is symmetric. Let P1 , P2 , P3 be three panels in M and let x be a chamber in P1 . By [65, 29.33], there exists α ∈ R
containing x and by [65, 29.39], projP2 (x), projP3 (x) and projP3 projP2 (x) are all contained in α. By 19.15, every root in R contains exactly one chamber in each panel of M . It follows that
projP3 (x) = projP3 projP2 (x) .

We conclude that ≈ is an equivalence relation whose equivalence classes are the intersections Z ∩ α for all α ∈ R, and for each panel P in M , the map z 7→ z ∩ P is a bijection from the set of equivalence classes to P .

Proposition 21.37. Suppose that S is the disjoint union of subsets J and K such that [J, K] = 1. Let R be a J-residue of ∆ and let π = projR . Then the following hold: (i) For each K-residue T , the intersection R ∩ T consists of a single chamber c, π(T ) = {c} and π −1 (c) = T .

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(ii) If c and d are i-adjacent chambers of ∆, then π(c) and π(d) are also i-adjacent if i ∈ J and π(c) = π(d) if i ∈ K. (iii) π(T ) = R ∩ T and T = π −1 (R ∩ T ) for each residue T whose type contains K; and for each subset J0 of J, the map T 7→ R ∩ T is a bijection from the set of (J0 ∪K)-residues of ∆ to the set of J0 -residues of R. Proof. Let T be a K-residue. By [62, 7.33], ∆ is isomorphic to the direct product R × T as defined in [62, 7.32]. It follows, in particular, that R ∩ T consists of a single chamber c. By [62, 8.25], we have π(T ) = {c} and hence T ⊂ π −1 (c). Let d ∈ π −1 (c) and let T ′ be the unique K-residue T ′ containing d. Then π(T ′ ) is the unique chamber in R ∩ T ′ . Therefore c ∈ T ∩ T ′ and hence T ′ = T . We conclude that T = π −1 (c). Thus (i) holds. Let c, d and i be as in (ii) and let T and T ′ be the unique K-residues containing c and d, respectively. If i ∈ K, then T = T ′ and hence π(c) = π(d) by (i). Suppose that i ∈ J. From ∆ ∼ = R × T it follows that every chamber of T is i-adjacent to a chamber of T ′ . If a chamber is i-adjacent to π(c) ∈ R, then it is also in R. By (i), π(d) is the unique chamber of T ′ in R. Hence π(d) is i-adjacent to π(c). Thus (ii) holds. Let T be a residue of ∆ whose type contains K. Then T contains every K-residue containing a chamber of T . By (i), therefore, π(T ) = R ∩ T and T = π −1 (R ∩ T ). Since R ∩ T is non-empty, T is the unique Typ(T )-residue containing R ∩ T and by [62, 7.25], R ∩ T is a (Typ(T )\K)-residue. Hence if T0 is an arbitrary residue of R, then there exists a unique (K ∪ Typ(T0 ))residue of ∆ containing T0 and its image under π is T0 . Thus (iii) holds. Proposition 21.38. Suppose that S is the disjoint union of subsets J and K such that [J, K] = 1. Then the following hold: (i) If R and R1 are J-residues, then R and R1 are parallel and the restriction of projR to R1 is a type-preserving isomorphism from R1 to R. (ii) If R, R1 and R2 are J-residues, then projR and the composition projR ◦ projR1

have the same restriction to R2 .

(iii) If R and T are J- and K-residues and g ∈ Aut(∆) acts trivially on R ∪ T , then g = 1. Proof. Let R and R1 be two J-residues, let c be a chamber of R1 and let T be the unique K-residue of ∆ containing c. By 21.37(i), R ∩ T consists of a single chamber d, d = projR (c) and c = projR1 (d). Thus the restriction of projR1 ◦ projR to R1 is the identity map. By symmetry, also the restriction of projR ◦ projR1 to R is the identity map. Hence R and R1 are parallel. Hence by 21.10(i), the restriction of projR to R1 is an isomorphism and by 21.37(ii), this isomorphism is type-preserving. Thus (i) holds.

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Now suppose that R2 is a third J-residue and let c, d and T be as in the previous paragraph. Then R2 ∩T contains a unique chamber e, projR1 (e) = c and projR (e) = d. Thus (ii) holds. Suppose, finally, that g is a non-trivial element of Aut(∆) that acts trivially on a K-residue T and let R be an arbitrary J-residue. By (i), there exists a unique chamber c in R ∩ T . Since the permutation induced by g on S stabilizes K, it stabilizes J as well. Since g fixes c, it therefore stabilizes R. Thus g stabilizes every J-residue. In particular, it stabilizes the unique J-residue containing a chamber d not fixed by g. By (i), therefore, g does not fix the chamber projR (d). Thus (iii) holds. Proposition 21.39. Suppose that S is the disjoint union of subsets J and K such that [J, K] = 1, let R be a J-residue and let T and T1 be L-residues for some subset L of S containing K. Then the following holds: (i) T is parallel to T1 if and only if T ∩ R is parallel to T1 ∩ R. (ii) Suppose that ∆ is spherical. Then T is opposite T1 in ∆ if and only if T ∩ R is opposite T1 ∩ R in R. Proof. By [62, 8.25], projT ∩R and projT have the same restriction to R as do projT1 ∩R and projT1 . If T is parallel to T1 , it follows by 21.9 that T ∩ R is parallel to T1 ∩ R. Suppose, conversely, that T ∩ R is parallel to T1 ∩ R. Let w = δ(T1 ∩ R, T ∩ R), T ′ = projT (T1 ) and T1′ = projT1 (T ). By 21.8(i), T ′ and T1′ are residues. We have projT (T1 ∩ R) = projT ∩R (T1 ∩ R) = T ∩ R

and ′

projT1 (T ∩ R) = projT1 ∩R (T ∩ R) = T1 ∩ R,

so T ∩ R ⊂ T and T1 ∩ R ⊂ T1′ . Since T ′ ⊂ T and T1′ ⊂ T1 , it follows that T ∩ R = T ′ ∩ R and T1 ∩ R = T1′ ∩ R. We have w ∈ WJ , so [K, w] = 1. By 21.8(iv), therefore, K is contained in both Typ(T ′ ) and Typ(T1′ ). By 21.37(iii), it follows that T = T ′ and T1 = T1′ . Hence T is parallel to T1 . Thus (i) holds. Now suppose that ∆ is spherical and that T is opposite T1 in ∆ or T ∩ R is opposite T1 ∩ R in R. By (i) and 21.24, T is parallel to T1 and T ∩ R is parallel to T1 ∩ R. If x1 ∈ T1 ∩ R, then projT (x1 ) = projT ∩R (x1 ) ∈ T ∩ R, so w := δ(T1 , T ) = δ(x1 , x) = δ(T1 ∩ R, T ∩ R) ∈ WJ .

We have L = K ∪ M for M = Typ(T ∩ R) and K ⊂ J + (w). Thus M ⊂ J + (w) ∩ J if and only if L ⊂ J + (w). By 21.30 and 21.34, T ∩ R is opposite T1 ∩ R in R if and only if M ⊂ J + (w) ∩ J and T is opposite T1 in ∆ if and only if L ⊂ J + (w). Thus (ii) holds. Proposition 21.40. Let G = Aut(∆) and suppose that S is the disjoint union of subsets J and K such that [J, K] = 1. Let GJ = {g ∈ G | typ(g)(J) = J},

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where typ is as in 21.2, let R be a J-residue of ∆, let π = projR and let ξ be the map from GJ to the set of maps from R to itself s such that for all g ∈ GJ , cξ(g) = π(cg )

for all chambers c of R. Then ξ is a homomorphism from GJ to Aut(R). Proof. Let g and h be elements of GJ , let c be a chamber of R, let T1 be the unique K-residue containing cg and let T2 be the unique K-residue containing cgh . By 21.37(i), π(cg ) is the unique chamber in T1 ∩ R. In particular, π(cg ) ∈ T1 , so π(cg )h ∈ T1h . Since cgh ∈ T2 ∩T1h and typ(h)(K) = K, we have T1h = T2 . Hence π(cgh ) and π(π(cg )h ) are both equal to the unique chamber in T2 ∩ R. We conclude that ξ(g)ξ(h) = ξ(gh) for all g, h ∈ GJ . Since ξ(1) = 1, it follows that for each g ∈ GJ , ξ(g) is a permutation of the set of chambers of R. By 21.37(ii), it follows that ξ(g) ∈ Aut(R) for all g ∈ GJ . Definition 21.41. We call a set X of residues in a building coherent if the residues in X are pairwise parallel and
projR1 projR2 (c) = projR1 (c)

for all R0 , R1 , R2 ∈ X and for all c ∈ R0 . Thus 21.36 says that walls are coherent sets of panels and 21.38(i)–(ii) says that if S has a partition into two subsets J and K such that [J, K] = 1, then the set of J-residues of ∆ is coherent. See also 21.42 and 21.54 as well as 23.15 and 23.17 below. The following observation is a special case of 21.55. Proposition 21.42. Every set of pairwise parallel residues in a thin building is coherent. Proof. Suppose that ∆ is thin and let X be a set of pairwise parallel residues. Since ∆ is thin, it consists of a single apartment. By 21.19(i), every residue in X is cut by the same set of roots. The claim holds, therefore, by 21.19(ii). Lemma 21.43. Let R be a residue of type J, let T be a residue parallel to R and suppose that t := δ(R, T ) ∈ S. Then [J, t] = 1. Proof. Let s ∈ J, let d and e be s-adjacent chambers of R, let d1 = projT (d), let e1 = projT (e) and let γ = (d1 , d, e, e1 ). By 21.10(ii), γ is a gallery of type tst and by 21.10(i), the chambers e1 and d1 are adjacent. Thus γ is not a minimal gallery. Hence the type tst of γ is not a reduced word in MS . Therefore [s, t] = 1. Lemma 21.44. Let R and T be parallel residues, let J = Typ(R), let w = δ(T, R), let t ∈ J − (w) and let J1 be a spherical subset of J. Then the following hold: (i) J1 ∪ J − (w) is a spherical subset of S.

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(ii) K1 := {s ∈ J | [s, t] 6= 1} ⊂ J − (wt). In particular, K1 is a spherical subset of J. Proof. Choose c ∈ T , let d = projR (c) and let R1 be the J1 -residue containing d. Then d = projR1 (c). Let e be a chamber of R1 opposite d, let w1 = δ(c, e), let s ∈ J1 and let P be the s-panel containing e. Then projP (c) = projP (d) by 21.6(iii) and projP (d) 6= e by the choice of e. Thus projP (c) 6= e and hence s ∈ J − (w1 ). Thus J1 ⊂ J − (w1 ). Now let Q be the t-panel containing e and let e′ = projT (e). Then δ(e′ , e) = δ(T, R) = w, so projQ (e′ ) 6= e by 21.6(iv). By 21.6(i), there is a minimal gallery from e to c passing through e′ , so projQ (c) = projQ (e′ ). Therefore projQ (c) 6= e, so t ∈ J − (w1 ), again by 21.6(iv). Hence J − (w) ⊂ J − (w1 ). We have thus shown that (21.45)

J1 ∪ J − (w) ⊂ J − (w1 ).

By 19.8(v), it follows that J1 ∪ J − (w) is spherical. Thus (i) holds. Let v = wt. We now specialize to the case that J1 = {s} for some s ∈ J and assume that [s, t] 6= 1. Thus R1 = P , w1 = ws and, by 21.6(iv), s ∈ J + (w). Therefore ℓ(w1 ) = ℓ(w) + 1 and hence ℓ(w1 ) = ℓ(v) + 2 by the choice of t. Let u = sts. Since [s, t] 6= 1, we have ℓ(u) = 3. By (21.45), {s, t} ⊂ J − (w1 ). By 19.8(vi), therefore, ℓ(w1 u) = ℓ(w1 ) − 3 < ℓ(v). Since vs = wts = w1 sts = w1 u, we conclude that s ∈ J − (v). Therefore K1 = {s ∈ J | [s, t] 6= 1} ⊂ J − (v). By 19.8(v), it follows that K1 is spherical. Thus (ii) holds. Lemma 21.46. Let R and T be parallel residues, let J = Typ(R), let w = δ(T, R), let t ∈ J − (w), let v = wt, let K1 = {s ∈ J | [s, t] 6= 1}, let K = J\K1 , let X be a K-residue of R, let Y be the ({t} ∪ K)-residue containing X, let X ′ = projT (X) and let X ′′ = projY (X ′ ). Then the following hold: (i) X ′ and X ′′ are parallel, Typ(X ′′ ) = K and δ(X ′ , X ′′ ) = v. (ii) If K0 is a spherical subset of K, then the union K0 ∪ K1 is a spherical subset of J. Proof. By 21.12, X is parallel to X ′ and δ(X ′ , X) = w. By 21.14, X ′′ is parallel to both X and X ′ and (21.47)

w = δ(X ′ , X ′′ ) · δ(X ′′ , X).

Since [K, t] = 1 and δ(X ′′ , X) ∈ W{t}∪K , it follows from 21.10(iii) that Typ(X ′′ ) = K and from 21.6(iv) that δ(X ′′ , X) = t. Hence δ(X ′ , X ′′ ) = wt = v by (21.47). Thus (i) holds. Let K0 be a spherical subset of K. By 21.44(ii), K1 ⊂ J − (v). By (i), we can thus apply 21.44(i) with X ′′ and X ′ in place of R and T to conclude that K0 ∪ K1 is spherical. Thus (ii) holds. The notion of a component of a subset of S, which appears in the next result, was defined in 19.21.

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Proposition 21.48. Let R and T be parallel residues, let J = Typ(R), let w = δ(T, R) and suppose that J0 is a non-spherical component of J. Then [J0 , t] = 1 for all t ∈ supp(w). Proof. We proceed by induction with respect to ℓ(w). If w = t ∈ S, then [J, t] = 1 by 21.43. Suppose that ℓ(w) > 1 and choose t ∈ J − (w). Let K = {s ∈ J | [s, t] = 1}, K1 , X, X ′ and X ′′ be as in 21.46 and let K0 be the union of all the spherical components of K. By 21.46(ii), K0 ∪ K1 is spherical. Since J0 is non-spherical and J = K ∪ K1 , it follows that J0 ∩ K 6⊂ K0 . Thus J0 ∩ K2 6= ∅ for some non-spherical component K2 of K. We have [K2 , K\K2 ] = 1 by definition. By 21.46(i) and induction, [K2 , supp(v)] = 1. By 21.44(ii), K1 ⊂ J − (v) ⊂ supp(v), so also [K1 , K2 ] = 1. Hence [K2 , J\K2 ] = 1. Thus K2 is also a component of J, hence J0 = K2 ⊂ K and therefore [J0 , t] = 1. Since supp(w) = {t} ∪ supp(v), we conclude that [J0 , supp(w)] = 1. Lemma 21.49. Suppose that K is a direct summand of S and that J − (w) ⊂ K for some w ∈ W . Then supp(w) ⊂ K. Proof. Let L = S\K. There exist unique u ∈ WK and v ∈ WL such that w = uv. Furthermore, ℓ(w) = ℓ(u) + ℓ(v) and ℓ(ws) = ℓ(u) + ℓ(vs) for all s ∈ L. Since L ⊂ J + (w), it follows that L ⊂ J + (v) and therefore v = 1. Proposition 21.50. Let t ∈ S, let J = S\{t}, let K be the component of S containing t, let L = K ∩J, let R be a J-residue and suppose that there exists a residue T 6= R parallel to R. Then K is spherical and δ(T, R) = wK wL . Proof. Let w = δ(T, R). By 21.6(iv), J ⊂ J + (w) and by 21.22, w 6= 1. Thus J − (w) = {t}, so (21.51)

supp(w) ⊂ K

by 21.49 and (21.52)

J = J + (w).

Let J0 be a component of J contained in L. Then t 6∈ J0 , so J0 is a proper subset of K. Since K is irreducible, it follows that [J0 , t] 6= 1. By 21.48, therefore, J0 is spherical. Thus L is spherical. Since t ∈ J − (w), it follows by 21.44(i) that K is spherical. Let R1 be an L-residue of R, let R0 be the unique K-residue of R containing R0 and let T1 = projT (R1 ). By 21.12, R1 and T1 are parallel and δ(T1 , R1 ) = w. By 21.10(iii), Typ(T1 ) = wLw−1 . By (21.51) and [62, 4.8], wLw−1 ∈ WK ∩ S = K and hence T1 ⊂ R0 . Thus by 21.30 with R0 in place of ∆ and (21.52), we have w = wK wL . Corollary 21.53. Let R be a residue of rank |S| − 1 and suppose that ∆ is thin. Then there is at most one residue other than R that is parallel to R.

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PARALLEL RESIDUES

Proof. Let T 6= R be a residue parallel to R and let w = δ(T, R). For each chamber c, there is a unique chamber d such that δ(d, c) = w (by 19.7). Since w = δ(c, projT (c)) for all c ∈ R by 21.10(iii), it follows that T is uniquely determined by R and w. By 21.50, w is uniquely determined by R. Hence T is unique. Proposition 21.54. Suppose that K is a direct summand of S that is contained in the type of a residue C, let L = S\K and let T be a residue parallel to C. Then K is contained in the type of T and δ(C, T ) ∈ WL . Proof. Let w = δ(T, C). We have K ⊂ J + (w) by 21.6(iv). By 21.49, therefore, w ∈ WL . Thus K = wKw−1 and hence K ⊂ Typ(T ) by 21.10(iii). We conclude this chapter with a generalization of 21.42. Proposition 21.55. Let Σ be an apartment and let X be a set of pairwise parallel residues of ∆ each containing chambers of Σ. Then X is coherent. Proof. Let XΣ = {T ∩ Σ | T ∈ X}, choose R, R1 , R2 ∈ X and let 
 π(x) = projR projR2 projR1 (x)

for all x ∈ R. It follows from 21.10(i) that π is an automorphism of R and that it will suffice to show that π must be the identity in order to conclude that X is coherent. By 21.17, the residues in XΣ are pairwise parallel. Hence by 21.42, the set XΣ is coherent. By 21.16, therefore, π acts trivially on R ∩ Σ. Thus, in particular, π is type-preserving. Let P be a panel of R containing chambers of Σ. Then P is stabilized by π. Since the residues R, R1 and R2 are pairwise parallel, the panels P , P1 := projR1 (P ) and P2 := projR2 (P ) are also pairwise parallel (by 21.12) and 
 π(x) = projP projP2 projP1 (x)

for all x ∈ P . By 21.16, the panels P1 and P2 also contain chambers of Σ. By 21.19(i), therefore, there is a root of Σ whose wall contains P , P1 and P2 . Thus the set {P, P1 , P2 } is coherent by 21.36. Hence π acts trivially on P . By 21.35, it follows that π is the identity if R is spherical. Let J = Typ(R) and suppose that J has no spherical components. Let wi = δ(R, Ri ) for i = 1 and 2, let L = supp(w1 ) ∪ supp(w2 ),

ˆ be the M -residue containing R. By 21.48, [J, L] = 1. let M = L∪J and let R ˆ It follows that Typ(Ri ) = J for i = 1 and 2 by 21.10(iii), so R1 ∪ R2 ⊂ R, ˆ and [J, K] = 1 for K := M \J. By 21.38(ii) applied to R, we conclude that the set {R, R1 , R2 } is coherent. Thus π is trivial if J has no spherical components.

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Suppose, finally, that R is non-spherical but that the union J1 of all the spherical components of J is non-empty. Let c ∈ R ∩ Σ, let J2 = J\J1 and let Ti be a Ji -residue of R containing the chamber c for i = 1 and 2. Since π is type-preserving, it stabilizes both T1 and T2 . By 21.12, the residues Ti , Ti′ := projR1 (Ti ) and Ti′′ := projR2 (Ti ) are pairwise parallel and 
 π(x) = projTi projTi′′ projTi′ (x)

for both i = 1 and 2 and for all x ∈ Ti . Since T1 is spherical and J2 has no spherical components, it follows from our previous conclusions that the restriction of π to T1 ∪ T2 is trivial. By 21.38(iii), we conclude that π is trivial also in this last case.

Chapter Twenty Two Fixed Point Buildings In this chapter we prove what might be called the Fundamental Theorem of Descent in buildings: that if Γ is a descent group as defined in 22.19, the set of residues of a building ∆ that are stabilized by a subgroup Γ of Aut(∆) forms a thick building. The exact statement of our result is in 22.20. Hypothesis 22.1. Let Π be an arbitrary Coxeter diagram, let S be the vertex set of Π and let (W, S) be the corresponding Coxeter system. Throughout this chapter, we assume that ∆ is a building of type Π (not necessarily thick) and that Γ is a subgroup (not necessarily type-preserving) of Aut(∆) and we denote by Θ the subgroup of Aut(W, S) induced by Γ. Definition 22.2. A Γ-residue is a residue of ∆ stabilized by Γ. We denote by Fix(Γ) the set of all Γ-residues. The set Fix(Γ) is partially ordered by inclusion. A Γ-chamber is a minimal element of this partially ordered set. Proposition 22.3. Let R and T be Γ-residues and let R′ = projR (T ). Then the following hold: (i) R′ is a Γ-residue. (ii) If T is a Γ-chamber, then so is R′ , and R′ is parallel to T . (iii) If R and T are Γ-chambers, then they are parallel and δ(R, T ) ∈ Fix(Θ). Proof. By 21.8(i), R′ is a residue. Since R and T are stabilized by Γ, so is R′ . Thus (i) holds. Now suppose that T is a Γ-chamber. By (i), T ′ := projT (R′ ) is a Γ-residue contained in T . Hence T = T ′ . Thus T is parallel to R′ . By 21.10(i), the restriction of projT to R′ is an isomorphism from R′ to T . Since this isomorphism maps Γ-residues to Γ-residues and T is a Γ-chamber, so is R′ . Thus (ii) holds. Assertion (iii) follows immediately from (ii). Remark 22.4. Let R be a Γ-residue. Then Typ(R) is a Θ-invariant subset of S. A residue T containing R is uniquely determined by R and Typ(T ). Thus a residue containing R is a Γ-residue if and only if its type is Θ-invariant. Proposition 22.5. Let C be a Γ-chamber and suppose that each Γ-residue containing C properly contains at least two Γ-chambers. Let A = Typ(C) and suppose that A0 is a non-spherical component of A. Then A0 is a component of S.

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Proof. Let t ∈ S\A, let K = Θ(t) ∪ A and let R be the unique K-residue containing the Γ-chamber C. By 22.4, R is a Γ-residue. Since R contains C properly, it contains a second Γ-chamber T . By 22.3(iii), C and T are parallel and w := δ(T, C) ∈ Fix(Θ). By 21.22, w 6= 1, so we can choose t′ ∈ J − (w). By 21.6(iv), A ⊂ J + (w), so t′ ∈ Θ(t). Thus t ∈ J − (w) by 19.14. Hence t ∈ supp(w). By 21.48, we conclude that [A0 , t] = 1. Since t is arbitrary, it follows that A0 is a component of S. It will be useful for the reader to keep in mind from now on that if R and T are Γ-chambers, then by 22.3(iii), R and T are parallel and therefore δ(R, T ) is defined. Lemma 22.6. Suppose that K is a direct summand of S that is contained in the type of some Γ-chamber and let J = S\K. Then K is contained in the type of every Γ-chamber and δ(C, T ) ∈ WJ for all Γ-chambers C and T . Proof. This holds by 21.54. Proposition 22.7. Suppose that S is the disjoint union of Θ-invariant subsets J and K such that [J, K] = 1. Suppose, too, that the set K is contained in the type of some Γ-chamber. Let R be a J-residue of ∆ and let GJ and ξ : Aut(∆) → Aut(R) be as in 21.40. Then Γ ⊂ GJ , the map T 7→ R ∩ T is an inclusion-preserving bijection from the set of all Γ-residues of ∆ to the set of all ξ(Γ)-residues of R, Typ(T ) = Typ(R ∩ T ) ∪ K for all T ∈ Fix(Γ), δ(C, T ) = δ(R ∩ C, R ∩ T ) for all Γ-chambers C and T and (22.8)

typ(ξ(g)) = typ(g)|J

for all g ∈ Γ. In particular, typ(ξ(Γ)) equals the subgroup ΘJ of Aut(ΠJ ) defined in 20.6. Proof. Since K is stabilized by Θ, we have Γ ⊂ GJ . Let X be the set of all residues of ∆ whose type contains K and let π = projR . By 21.37(iii), π(T ) = R ∩ T , Typ(T ) = Typ(R ∩ T ) ∪ K for all T ∈ X and π induces a bijection from X to the set of all residues of R. Suppose that T is a Γ-residue. Then T ∈ X by 22.6 and thus cξ(g) = π(cg ) ∈ π(T ) = R ∩ T

for all c ∈ R ∩ T and all g ∈ Γ, so π(T ) is a ξ(Γ)-residue. Let i ∈ J, let c and d be i-adjacent chambers of R and let g ∈ Γ. Then cg and dg are j-adjacent for j = typ(g)(i) ∈ Θ(J) = J. By 21.37(ii), it follows that π(cg ) is j-adjacent to π(dg ). Thus (22.8) holds. Now suppose that T0 is an arbitrary ξ(Γ)-residue of R and let T = π −1 (T0 ). We claim that T is stabilized by Γ. By 21.37(iii) again, T is the unique (K∪Typ(T0 ))-residue of ∆ such that R∩T = T0 . Since T0 is ξ(Γ)-invariant, it follows from (22.8) that Typ(T0 ) is Θ-invariant. Since also K is Θ-invariant, we conclude that (22.9)

Typ(T ) = Typ(T h )

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FIXED POINT BUILDINGS

for all h ∈ Γ. Choose c ∈ T and g ∈ Γ. Then π(c) ∈ T0 , so
π π(c)g = π(c)ξ(g) ∈ T0

−1

and hence π(c)g ∈ π −1 (T0 ) = T . Therefore π(c) ∈ R ∩ T g . By (22.9), −1

K ⊂ Typ(T ) = Typ(T g ) and thus T g

−1

∈ X. By another application of 21.37(iii), therefore, we have Tg

−1

−1

= π −1 (R ∩ T g ).

−1

−1

Hence c ∈ T g . From c ∈ T and Typ(T ) = Typ(T g ), it follows that −1 T g = T . Thus T is stabilized by Γ, as claimed. We conclude that the map T 7→ R ∩ T is a bijection from the set of all Γ-residues of ∆ to the set of all ξ(Γ)-residues of R. Suppose, finally, that C and T are Γ-chambers. Thus R ∩ C and R ∩ T are ξ(Γ)-chambers. By 22.3(iii), C is parallel to T and R ∩ C is parallel to R ∩ T . Choose a chamber c ∈ R ∩ C and let d = projT (c). By 22.6, δ(c, d) ∈ WJ . Hence there exists a gallery from c to d whose type is a word in the alphabet J. Thus d ∈ R and hence d = projR∩T (c). Thus δ(C, T ) = δ(c, d) = δ(R ∩ C, R ∩ T ) by 21.10(ii). Proposition 22.10. Let R and T be distinct Γ-chambers, let J = Typ(R) and let K = Typ(T ). Suppose that T is spherical and that S\K is a single Θ-orbit. Then the following hold: (i) (W, S) is spherical, T is opposite R in ∆ and δ(R, T ) = wS wK . (ii) S\J is also a single Θ-orbit. (iii) If there exists a third Γ-chamber, then J = wS JwS and J is the type of every Γ-chamber. Proof. By 22.3(iii), R and T are parallel and w := δ(R, T ) is fixed by Θ. By 21.10(i), R is spherical. Since R 6= T , we have R ∩ T = ∅ by 21.22. Therefore w 6= 1 and hence J + (w) 6= S. By 21.6(iv), K ⊂ J + (w). It follows that s ∈ J − (w) for some s ∈ S\K. Thus Θ(s) ⊂ J − (w) by 20.11(iii). Since S = Θ(s)∪K by hypothesis, it follows that K = J + (w). By 21.30, therefore, (W, S) is spherical, R and T are opposite, (22.11)

wS JwS = K

and w = wS wK . Thus (i) holds. Since wS is fixed by Θ, conjugation by wS maps Θ-orbits to Θ-orbits. Hence (ii) holds. Suppose, finally, that T1 is a third Γ-chamber distinct from R and T and let K1 = Typ(T1 ). By (22.11) applied to the residues R, T and T1 pairwise, we have J = wS K1 wS = K and K1 = wS JwS = K. Thus (iii) holds.

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Definition 22.12. A Γ-panel is a Γ-residue P that contains a Γ-chamber R such that Typ(P )\Typ(R) is a single Θ-orbit. By 22.10(ii), this property is independent of the choice of the Γ-chamber in P , given that one of them is spherical. Corollary 22.13. Suppose that R is a spherical Γ-chamber such that R 6= ∆ and each Γ-panel containing R contains at least three Γ-chambers. Then T := (Π, Θ, Typ(R)) is a Tits index as defined in 20.1 and 20.3. Proof. This holds by 22.10. Proposition 22.14. Suppose that there exists a Γ-chamber C such that
T = Π, Θ, Typ(C)

˜ , S) ˜ be the relative Coxeter is a Tits index. Let A = Typ(C) and let (W system of T as defined in 20.4, 20.32(i) and 20.34. Then the following hold: (i) Typ(R) = A for all Γ-chambers R. ˜ for all Γ-chambers R and T . (ii) δ(R, T ) ∈ W ˜ δ) ˜ is a build(iii) If each Γ-panel contains at least two Γ-chambers, then (∆, ˜ ˜ ˜ ing of type (W , S), where ∆ is the set of Γ-chambers of ∆ and δ˜ is the ˜ ∆. ˜ This building restriction of the map δ defined in 21.11 to the set ∆× is thick if and only if every Γ-panel contains at least three Γ-chambers. Proof. We proceed in a series of steps. Claim 1: Let R be a Γ-chamber of type A, let s ∈ S\A, let P be the Γ-panel of type Θ(s) ∪ A containing R and let T ⊂ P be a Γ-chamber distinct from R. Then Typ(T ) = A and δ(R, T ) = s˜. Proof of Claim 1. Let ws , wA and s˜ = ws wA be as in 20.4 and let K = Typ(T ). By 22.10, R is opposite T in P , δ(R, T ) = ws wK and ws Kws = A. By 20.1, we have ws Aws = A. Hence K = A and thus δ(R, T ) = ws wA = s˜. Thus Claim 1 holds. Claim 2: Let R be an arbitrary Γ-chamber. Then the following hold: (a) Typ(R) = A.

˜. (b) w := δ(C, R) ∈ W (c) Let s ∈ S\A and let P be the Γ-panel of type Θ(s) ∪ A containing R. Then projP (C) = R if and only if s ∈ J + (w). Proof of Claim 2. We proceed by induction with respect to ℓ(w). Suppose that ℓ(w) = 0. Then C = R by 21.22. In particular, Typ(R) = Typ(C) = A. Thus (a), (b) and (c) hold.

185

FIXED POINT BUILDINGS

Now suppose that ℓ(w) > 0, so w 6= 1, and let J = Typ(R). Since projR (C) = R by 22.3(iii), we have J ⊂ J + (w) by 21.6(iv). Therefore we can choose t ∈ S\J such that (22.15)

t ∈ J − (w).

Let Q be the unique (Θ(t)∪J)-residue that contains R and let T = projQ (C). By 22.4, Q is a Γ-panel. By 22.3(iii) and 21.14, therefore, T is a Γ-chamber, (22.16)

ℓ(w) = ℓ(w′ ) + ℓ(δ(T, R))

and (22.17)

w = w′ · δ(T, R),

where w′ = δ(C, T ). Let c0 be a chamber in C, let c1 = projQ (c0 ) and let c2 be a chamber that is t-adjacent to c1 . Then δ(c0 , c1 ) = w′ and c2 ∈ Q. By 21.6(i), there is a minimal gallery from c0 to c2 that passes through c1 . It follows that δ(c0 , c2 ) = w′ t and t ∈ J + (w′ ). By (22.15), therefore, w′ 6= w. Hence R 6= T by (22.17) and thus ℓ(w′ ) < ℓ(w) by (22.16). By Claim 1 and ˜ and δ(T, R) = t˜. induction, it follows that A = Typ(R) = Typ(T ), w′ ∈ W ˜ . Thus (a) and (b) hold. By (22.17) again, also w ∈ W Now let s ∈ S\A and let P denote the unique (Θ(s)∪A)-residue containing R. By 22.4, P is a Γ-panel. If s ∈ J − (w), then we can repeat the argument in the previous paragraph with s in place of t to conclude that projP (C) 6= R. Suppose instead that s ∈ J + (w). By (b) and 20.13(iii), we have A ⊂ J + (w) and by 20.11(iii), we have Θ(s) ⊂ J + (w). Thus (22.18)

Θ(s) ∪ A ⊂ J + (w).

Let d0 be a chamber in C and let d1 = projR (d0 ). Thus d1 ∈ R ⊂ P and w = δ(C, R) = δ(d0 , d1 ). By (22.18), Typ(P ) ⊂ J + (w). By 21.6(iv), therefore, d1 = projP (d0 ). Let T1 = projP (C). By 22.3(ii), T1 is a Γ-chamber parallel to C. By Claim 1 and (a), Typ(T1 ) = A = Typ(R). Thus T1 and R are two residues of the same type containing d1 . Hence R = T1 . We conclude that s ∈ J + (w) if and only if projP (C) = R. Thus (c) holds. This concludes the proof of Claim 2. By Claim 2(a), assertion (i) holds and
T = Π, Θ, Typ(R)

for every Γ-chamber R. Hence Claim 2(b) holds with C replaced by an arbitrary Γ-chamber. Thus also (ii) holds. It remains only to verify (iii). Claim 3: Let R, T and T1 be Γ-chambers, let s ∈ S\A, let P denote the unique Γ-panel of type Θ(s) ∪ A containing R and let w = δ(T, R). Then the following hold: (I) w = 1 if and only if T = R.

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CHAPTER 22

(II) δ(R, T1 ) = s˜ if and only if R 6= T1 and T1 ⊂ P . (III) R = projP (T ) if and only if s ∈ J + (w). (IV) If δ(R, T1 ) = s˜, then δ(T, T1 ) ∈ {w, w˜ s}. (V) If R is not the only Γ-chamber contained in P , then there exists a Γchamber R1 contained in P such that δ(R, R1 ) = s˜ and δ(T, R1 ) = w˜ s. Proof of Claim 3. Assertion (I) holds by 21.22. By Claim 1, δ(R, T1 ) = s˜ if T1 6= R and T1 ⊂ P . Suppose, conversely, that δ(R, T1 ) = s˜. By (I), we have R 6= T1 . Let d0 be a chamber in R and let d′0 = projT1 (d0 ). Then δ(d0 , d′0 ) = δ(R, T1 ) = s˜ ∈ WΘ(s)∪A

and therefore d0 and d′0 are contained in the same (Θ(s) ∪ A)-residue. Since d0 ∈ R ⊂ P , it follows that d′0 ∈ P . By Claim 2(a), we have Typ(T1 ) = A. Since d′0 ∈ T1 ∩ P , we conclude that T1 ⊂ P . Thus (II) holds. Assertion (III) holds by Claim 2(c) (with T in place of C). Suppose again that δ(R, T1 ) = s˜ and let T ′ = projP (T ). By (II), we have T1 ⊂ P . If T ′ = R, then δ(T, T1 ) = w˜ s by (III) and 20.32(ii). Suppose that T ′ 6= R. By (II), δ(T ′ , R) = s˜ and thus by two applications of 21.14, we have w = δ(T, R) = δ(T, T ′ ) · s˜

and hence δ(T, T1 ) = δ(T, T ′ ) · δ(T ′ , T1 ) = w˜ s · δ(T ′ , T1 ). The residues T ′ and T1 are both contained in P . By (I) and (II), therefore, δ(T ′ , T1 ) ∈ {1, s˜}. We conclude that δ(T, T1 ) ∈ {w, w˜ s} and that δ(T, T1 ) = w˜ s ′

if and only if T1 = T . In particular, (IV) holds. By (II), we have δ(R, R1 ) = s˜ for all Γ-chambers R1 contained in P other than R. Recall that T ′ = projP (T ). By the conclusions of the previous paragraph, δ(T, R1 ) = w˜ s for all Γ-chambers R1 contained in P other than R if T ′ = R and if T ′ 6= R, then T ′ is a Γ-residue contained in P other than R and δ(T, T ′ ) = w˜ s. Thus (V) holds. This concludes the proof of Claim 3. Claim 4: Suppose that every Γ-panel contains at least two Γ-chambers. Let X and Y be Γ-chambers, let w = δ(X, Y ), let s ∈ S\A and let ℓ˜ be as in 20.30. Then the following hold: (1) w = 1 if and only if X = Y . (2) If Z is a Γ-chamber with δ(Y, Z) = s˜, then δ(X, Z) ∈ {w, w˜ s}; and if ˜ s) = ℓ(w) ˜ ℓ(w˜ + 1, then δ(X, Z) = w˜ s. (3) There exists a Γ-chamber Z such that δ(X, Z) = w˜ s and δ(Y, Z) = s˜.

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187

Proof of Claim 4. Assertion (1) holds by Claim 3(I) and the first statement ˜ s) = ℓ(w) ˜ + 1 and let P be the in (2) holds by Claim 3(IV). Suppose that ℓ(w˜ Γ-panel of type Θ(s) ∪ A containing Y . By 20.32(ii), we have s ∈ J + (w). Hence Y = projP (X) by Claim 3(III) and thus δ(X, Z) = w˜ s by 21.14. Thus (2) holds. Assertion (3) holds by Claim 3(V). This completes the proof of Claim 4. The assertions (1)–(3) of Claim 4 are the axioms for a building given in ˜ is a building of type (W ˜ , S). ˜ Thus (iii) holds and [58]. We conclude that ∆ the proof of 22.14 is finished. Definition 22.19. A descent group of ∆ is a subgroup Γ of Aut(∆) such that there exist Γ-panels and every Γ-panel contains at least three Γ-chambers. A descent group Γ is called spherical if there exist spherical Γ-chambers. Note that there exist Γ-panels (as defined in 22.12) if and only if there exist proper residues of ∆ stabilized by Γ. See 22.36 and 22.37 for characterizations of descent groups in the case that the building ∆ is spherical. We come now to the main result of this chapter. Theorem 22.20. Let ∆ be a building of type Π, let (W, S) be the corresponding Coxeter system, let Γ be a descent group of ∆ as defined in 22.19, let Θ = typ(Γ), let C0 be a Γ-chamber, let A = Typ(C0 ) and let K1 be the union of all the non-spherical components of A.1 Then K1 is a Θ-invariant direct summand of S. Let K ⊂ A be a Θ-invariant direct summand of S containing K1 , let J = S\K and let R be a J-residue of ∆. Then the following hold: (i) ΓR := ξ(Γ) is a spherical descent group of R, where ξ(Γ) is as in 22.7. (ii) T 7→ R∩T is an inclusion-preserving bijection from Fix(Γ) to Fix(ΓR ), Typ(T ) = Typ(R∩T )∪K for all T ∈ Fix(Γ), δ(C, T ) = δ(R∩C, R∩T ) for all Γ-chambers C, T and typ(ΓR ) equals the group ΘJ induced by Θ on ΠJ . (iii) If R′ is another J-residue and ΓR′ is as in (ii) applied to R′ , then the restriction of projR to R′ is a type-preserving isomorphism from R′ to R carrying ΓR′ to ΓR . (iv) TK := (ΠJ , ΘJ , A ∩ J) is a Tits index. (v) Every Γ-chamber has type A. ˜ for all Γ-chambers C and T , where (W ˜ , S) ˜ is the relative (vi) δ(C, T ) ∈ W Coxeter system of the Tits index TK in (iv). 1 Thus

K1 is empty if A is spherical.

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˜ , S) ˜ does not depend on the choice of the set (vii) The Coxeter system (W K ⊂ A. ˜ δ) ˜ is a thick building of type (W ˜ , S), ˜ where ∆ ˜ is the set of (viii) ∆Γ := (∆, Γ-chambers and ˜ δ(C, T ) = δ(C, T ) for all Γ-chambers C and T . Proof. By 22.5, K1 is a Θ-invariant direct summand of Π. By 22.7, (ii) holds. In particular, T 7→ R ∩ T is an inclusion-preserving bijection from Fix(Γ) to Fix(ΓR ). Since Γ is a descent group, it follows that ΓR is one too. Thus (i) holds. Assertion (iii) holds by 21.38(i)-(ii). Since K1 ⊂ K, the set A ∩ J is spherical. Thus (iv) holds by 22.13 applied to ΓR . By (iii), Typ(T ) = Typ(R ∩ T ) ∪ K and δ(C, T ) = δ(R ∩ C, R ∩ T ) for all Γ-chambers C, T . Thus (v) and (vi) hold by 22.14(i) and 22.14(ii) applied to ΓR , (vii) holds by 20.7 and (viii) holds by 22.14(iii) applied to ΓR . Remark 22.21. If the descent group Γ in 22.20 is spherical, then all Γchambers are spherical by 22.20(v) and thus K1 = ∅. Conversely, Γ is, of course, spherical if K1 = ∅. Definition 22.22. Suppose that Γ is a descent group of a building ∆ and let K, K1 and ∆Γ be as in 22.20. We call ∆Γ the fixed point building of Γ and we call TK for K = K1 the Tits index of Γ. Definition 22.23. Let Ψ and ∆ be two buildings. We will say that Ψ is a Γ-form of ∆ if it is isomorphic to ∆Γ for some descent group Γ of ∆. We will say that Ψ is a form of ∆ if it is a Γ-form of ∆ for some Γ. Remark 22.24. Let R, Γ and ΓR be as in 22.20. By 22.20(i), ΓR is a descent group, so we can apply 22.22 to ΓR , and 22.20(ii) says that the fixed point buildings RΓR and ∆Γ are canonically isomorphic. By 22.24, every fixed point building in the sense of 22.22 is automatically the fixed point building of a spherical descent group. Thus to study fixed point buildings, it suffices to consider spherical descent groups. For our applications, it will, in fact, be useful to have formulations of the following two special cases of 22.20. Theorem 22.25. Let ∆ be a building of type Π, let (W, S) be the corresponding Coxeter system, let Γ be a spherical descent group of ∆ as defined in 22.19, let Θ = typ(Γ) and let A be the type of a Γ-chamber. Then all Γ-chambers are of type A, T = (Π, Θ, A) ˜ is a thick building of type (W ˜ δ) ˜ , S), ˜ where is a Tits index and ∆ := (∆, ˜ and δ˜ are as in 22.14(iii) and (W ˜ , S) ˜ is the relative Coxeter system of T ∆ defined in 20.34. Γ

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Proof. This holds by 22.20 with K1 = K = ∅. Theorem 22.26. Let ∆ be a building of type Π, let (W, S) be the corresponding Coxeter system, let Γ be a spherical descent group of ∆ with Tits index T = (Π, Θ, A) as defined in 22.19 and suppose that S is the disjoint union of subsets J and K such that K ⊂ A, K is stabilized by Θ and [J, K] = 1. Let R be a J-residue of ∆ and let GJ and ξ : GJ → Aut(R) be as in 21.40. Then Γ ⊂ GJ , ξ(Γ) is a spherical descent group of R, the Tits index of ξ(Γ) is TJ as defined in 20.6 and the map T 7→ R ∩ T is an isomorphism of buildings from ∆Γ to Rξ(Γ) , where ∆Γ and Rξ(Γ) are as in 22.25. Proof. This holds by 22.20 with K1 = ∅. There are descent groups which are not spherical: Example 22.27. Suppose that S has a partition into subsets J and K such that [J, K] = 1 and K is non-spherical and let R and T be residues of type J and K, respectively. Suppose, too, that Γ1 is a descent group of R and that Γ2 is a subgroup of Aut(T ) stabilizing no proper residues of T . Then ∆∼ = R × T (by [62, 7.33]) and hence there is a natural action of Γ := Γ1 × Γ2 on ∆ extending the action of Γ1 on R and the action of Γ2 on T . If C is a Γ1 -chamber, then (C, T ) ⊂ ∆ is a Γ-chamber. By 22.6, it follows that Γ is a descent group of ∆ which does not stabilize any spherical residues. We also mention the following: Proposition 22.28. Every finite subgroup of Aut(∆) stabilizes spherical residues. In particular, every finite descent group is spherical. Proof. Suppose that Γ ⊂ Aut(∆) is finite. Then Γ fixes a point x in the Davis-complex of ∆ by [13, Cor. 11.9]. The point x lies in a unique open cell of the Davis complex, which, in turn, corresponds to a unique spherical residue R of ∆. Since Γ fixes x, it stabilizes R.

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Our next goal is to prove 22.36 and 22.37. Hypothesis 22.29. We assume now that ∆ is a spherical building of type Π, that Γ is a subgroup of Aut(∆) and that C is a Γ-chamber such that T = (Π, Θ, A) is a Tits index, where Θ = typ(Γ) and A = Typ(C).

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˜ , S) ˜ be the absolute and relative Coxeter systems of T Let (W, S) and (W and let ℓ and ℓ˜ denote their length functions. For each s ∈ S\A, let s˜ be the ˜ corresponding element of S. ˜ for all Γ-chambers By 22.14, we know that Typ(R) = A and δ(R, T ) ∈ W R and T . Lemma 22.30. Let R and R1 be Γ-chambers, let w = δ(R, R1 ) and let ˜ s) = ℓ(w) ˜ − 1. Then there exists a Γ-chamber R0 such s ∈ S\A such that ℓ(w˜ that δ(R, R0 ) = w˜ s and δ(R0 , R1 ) = s˜. Proof. Let T be the (Θ(s)∪A)-residue containing R1 and let R0 = projT (R). By 22.4, T is a Γ-residue. Thus by 22.3(ii), R0 is a Γ-chamber and by Claim 1 and Claim 2(c) in the proof of 22.14, δ(R0 , R1 ) = s˜. Thus δ(R, R0 ) = w˜ s by 21.14. ˜ ˜ + ℓ(v) ˜ Lemma 22.31. Suppose that w = uv and ℓ(w) = ℓ(u) for elements ˜ . Then the following hold: u, v, w ∈ W (i) If R and R1 are Γ-chambers such that δ(R, R1 ) = w, then there exists a Γ-chamber R0 such that δ(R, R0 ) = u. (ii) If R, R0 and R1 are Γ-chambers such that δ(R, R0 ) = u and δ(R0 , R1 ) = v, then δ(R, R1 ) = w. Proof. By 19.8(iv), (i) follows by induction from 22.30. By 20.31, there exists compatible representations of u and of v whose product is a compatible representation of w. Thus (22.32)

ℓ(w) = ℓ(u) + ℓ(v).

Let R, R0 and R1 be Γ-chambers such that δ(R, R0 ) = u and δ(R0 , R1 ) = v. Choose x ∈ R, let y = projR0 (x) and let z = projR1 (y). Then δ(x, y) = u and δ(y, z) = v. By 21.5 and (22.32), therefore, δ(x, z) = w. By 20.13 and 22.14(i), we have Typ(R1 ) = A ⊂ J + (w). Hence by 21.6(iii), z = projR1 (x). Therefore δ(R, R1 ) = δ(x, z) = w. Thus (ii) holds. ˜ , S) ˜ with the unique longest Remark 22.33. By 19.8(vi) applied to (W ˜ element wS˜ of W in place of w, we have ˜ −1 w ˜ ) + ℓ(w ˜ −1 ) ˜ ˜ ) = ℓ(w ℓ(w S S ˜ −1 w ˜ ) + ℓ(w) ˜ = ℓ(w S

˜. for all w ∈ W Proposition 22.34. Given 22.29 holds, suppose that there exist Γ-chambers C and C1 such that δ(C, C1 ) = wS˜ . Then the following holds: (i) For each Γ-chamber R there exists a Γ-chamber R1 such that δ(R, R1 ) = wS˜ .

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(ii) Every Γ-panel contains at least two Γ-chambers. Proof. By 22.31(i) and 22.33, there exists a Γ-chamber Cw such that δ(C, Cw ) = w ˜ . Now let R be a Γ-chamber, let v = δ(R, C) and let for each w ∈ W −1 w = wS˜ v . By 22.31(ii) and 22.33, we have δ(R, Cw ) = wS˜ . Thus (i) holds. By 22.30, it follows that every Γ-panel containing R contains at least one other Γ chamber. Thus (ii) holds. Lemma 22.35. Let ∆ be a spherical building such that for some chamber c, every panel containing c contains at least two other chambers. Then ∆ is thick. Proof. We call a chamber u thick if every panel containing u contains at least two other chambers. Since ∆ is connected, it will suffice to show that if x is adjacent to c, then x is thick. By 21.24, every chamber opposite a thick chamber is thick. Choose x adjacent to c and choose a chamber d opposite c. Then d is thick. If d is opposite x, then x is thick. Assume, therefore, that d is not opposite x. Thus x is on a minimal gallery from c to d and therefore x is contained in the unique apartment Σ containing c and d. Let y = opΣ (x). By 19.25, y is adjacent to d. Let P be the unique panel containing d and y and let Q be the unique panel containing c and x. By [62, 9.8], P and Q are opposite residues. Since x is not opposite d, we have d = projP (x) and since c is not opposite y, we have y = projP (c). Thus x is opposite every chamber in P except d and c is opposite every chamber in P except y. Since y is thick, we can choose z ∈ P \{d, y}. Then z is opposite both c and x. Hence x is thick. Corollary 22.36. Suppose that 22.29 holds, that there exist Γ-chambers C and C1 such that δ(C, C1 ) = wS˜ and a Γ-chamber C0 such that every Γ-panel containing C0 contains at least three chambers. Then Γ is a descent group. Proof. By 22.14(iii) and 22.34(ii), ∆Γ is a spherical building and by 22.35, ∆Γ is thick. Thus Γ is a descent group. Corollary 22.37. Suppose that ∆ is a spherical building of type Π and that Γ is a subgroup of Aut(∆) such that there exist a Γ-chamber C such that every Γ-panel containing C contains at least three chambers. Then the following hold: (i) T := (Π, Θ, A) is a Tits index, where Θ = typ(Γ) and A = Typ(C). (ii) If there exists a Γ-chamber C1 such that δ(C, C1 ) equals the longest element in the relative Coxeter system of T, then Γ is a descent group. Proof. This holds by 22.13 and 22.36.

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Hypothesis 22.38. For the rest of this chapter, we assume that Γ is a descent group of our building ∆ (see 22.1) and that ∆Γ is its fixed point building as defined in 22.22. Remark 22.39. Suppose that Γ is spherical with Tits index T = (Π, Θ, A). Let R be a Γ-residue and let J = Typ(R). Choose elements s1 , . . . , sk of S\A in distinct Θ-orbits such that J = Θ(s1 ) ∪ · · · ∪ Θ(sk ) ∪ A, let ΓR denote the subgroup of Aut(R) induced by Γ, let TJ = (ΠJ , ΘJ , A) be as in 20.6 and let J˜ = {˜ s1 , . . . , s˜k }, where s˜i for all i ∈ [1, k] is as in 20.4(i). Then ΓR is a descent group of ˜ := RΓR is a R, the Tits index of ΓR is TJ and the fixed point building R Γ ˜ ˜ J-residue of ∆ = ∆ . Remark 22.40. By 20.35(i), ∆ is spherical if and only if ∆Γ is spherical. If ∆ is spherical, then, of course, Γ is also spherical. Proposition 22.41. Suppose that ∆ is spherical and let R and T be two ˜ is opposite T˜ in ∆, ˜ Γ-residues. Then R is opposite T in ∆ if and only if R ˜ ˜ ˜ where R, T and ∆ are as in 22.39. Proof. By [62, 9.9 and 9.10], we know that if the residues R and T are opposite in ∆, then for every residue R1 contained in R there is a residue of T which is opposite R1 in ∆ and by 21.29, we know that if for some K ⊂ Typ(R), to every K-residue R1 in R there is a residue of T which is opposite R1 in ∆, then R and T are opposite in ∆. It follows from these two observations that it suffices to consider the case that R and T are Γchambers. Let T = (Π, Θ, A) be the Tits index of Γ. By 20.35(ii), we have (22.42)

wS wA = wS˜ and wS˜ wA = wS .

Suppose that R and T are Γ-chambers. Thus by 22.3(iii), R and T are parallel. Choose c ∈ R. If R and T are opposite in ∆, then ˜ R, ˜ T˜) = δ(R, T ) δ( = δ(c, projT (c)) = wS wA = wS˜

by 22.14(iii) by 21.11 by 21.25 by (22.42)

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˜ and T˜ are opposite in ∆. ˜ Suppose, conversely, that and hence R ˜ R, ˜ T˜) = w ˜ . δ( S Let x ∈ R, let x′ = projT (x) and let y ∈ T be a chamber opposite to x′ in T . By 21.6(i) and (22.42), we have δ(x, y) = δ(x, x′ ) · δ(x′ , y) = wS˜ wA = wS . Hence y is opposite x in ∆. By 21.26, we conclude that R is opposite T in ∆. Lemma 22.43. Let R be a spherical residue containing two residues T and Z that are opposite residues of R and are both stabilized by Γ. Then R is also stabilized by Γ. Proof. Let J = Typ(R), let K = Typ(Z) and let w = δ(T, Z). Then w and wK are both in Fix(Θ) and by 21.25, w = wJ wK , from which it follows that also wJ ∈ Fix(Θ). Since J = {i ∈ S | i ∈ J − (wJ )}, we conclude that J is stabilized by Θ. Hence R is stabilized by Γ. Proposition 22.44. Suppose that Γ is spherical, let R be a Γ-residue and ˜ be as in 22.39. Then the following hold: let R ˜ is a spherical residue of ∆. ˜ (i) R is spherical if and only if R (ii) If R is spherical and T and Z are Γ-residues contained in R, then T ˜ and Z are opposite in R if and only if T˜ and Z˜ are opposite in R. Proof. Assertion (i) holds by 22.40 and (ii) is just a restatement of 22.41. Definition 22.45. Two distinct residues R and T of ∆ are locally opposite if there is a spherical residue of ∆ containing R and T in which R and T are opposite. Corollary 22.46. Two Γ-residues T and Z are locally opposite in ∆ if and ˜ only if T˜ and Z˜ are locally opposite in ∆. Proof. This holds by 22.43 and 22.44. Proposition 22.47. Let Γ1 be a normal subgroup of Γ, suppose that Γ1 is ¯ of Aut(∆Γ1 ) induced by Γ is a descent group of ∆ and that the subgroup Γ Γ1 a descent group of ∆ . Then the following hold: (i) Γ is a descent group of ∆ and there is a canonical isomorphism from ¯ ∆Γ to (∆Γ1 )Γ . ¯ are spherical, then so is Γ. (ii) If Γ1 and Γ Proof. A residue of ∆ is stabilized by Γ if and only if it is stabilized by Γ1 ¯ acts) and by Γ. ¯ The map R 7→ R (which makes it a residue of ∆Γ1 on which Γ ¯ ¯ Γ Γ1 Γ is thus a canonical isomorphism from ∆ to (∆ ) . Since (∆Γ1 )Γ is a thick

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building, it follows that every Γ-panel contains at least three Γ-chambers. Thus (i) holds. ¯ are spherical and let R be a Γ-chamber. Then Now suppose that Γ1 and Γ ¯ R is a Γ-chamber of the fixed point building ∆Γ1 . Therefore R is a spherical residue of ∆Γ1 . By 22.44(i), therefore, R is a spherical residue of ∆. Thus (ii) holds.

Chapter Twenty Three Subbuildings In this chapter we give a generalization of the usual notion of a subbuilding; see 23.1 and 23.4. Our main result is 23.19. We do not assume that buildings are thick in this chapter. Definition 23.1. Let ∆ be a building of type Π, let T = (Π, Θ, A) ˜ , S) ˜ be a Tits index of absolute type Π as defined in 20.1 and 20.3 and let (W be the relative Coxeter system of T as defined in 20.34. A subbuilding of ˜ of A-residues of ∆ such that the following type T of ∆ is a non-empty set ∆ hold: ˜ are parallel and δ(R, T ) ∈ W ˜. (i) Any two elements R, T of ∆

˜ δ) ˜ is a building of type (W ˜ , S), ˜ where δ(R, ˜ (ii) The pair (∆, T ) = δ(R, T ) ˜ for all R, T ∈ ∆. Notation 23.2. Let ∆ be a building of type Π and let T = (Π, Θ, A) be a Tits index of absolute type Π. A T-group of ∆ is a subgroup Γ of Aut(∆) inducing Θ on Π for which there exists a Γ-chamber of type A. If Γ is a T-group of ∆ such that every Γ-panel contains at least two Γ-chambers, ˜ is a subbuilding of type T of ∆. In ˜ δ) then by 22.3(iii) and 22.14(iii), (∆, ˜ ˜ is a subbuilding of type T. particular, if Γ, T and ∆ are as in 22.25, then ∆ Notation 23.3. We will sometimes refer to a subbuilding of type T as a T-building of ∆. Notation 23.4. Let ∆ be a building of type Π and let TΠ denote the split Tits index of absolute type Π as defined in 20.10. A subbuilding of split type is a subbuilding of type TΠ . Thus a subbuilding in the sense of 23.1 is of split type if and only if its chambers are chambers of ∆ and its panels are contained in panels of ∆. This is often what is meant by the term “subbuilding.” In [62, 7.18], the term “subbuilding” is used to mean a subbuilding of split type of a residue. Convention 23.5. If we refer to a subbuilding without reference to a Tits index, we will always mean a subbuilding of split type as defined in 23.4. Note that the apartments of a building are subbuildings in this sense. We will make frequent use in this chapter of the following observation about apartments.

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Proposition 23.6. Let X be a subset of a building (∆, δ). Then there exists an apartment containing X if and only if δ(x, z) = δ(x, y) · δ(y, z)

(23.7) for all x, y, z ∈ X.

Proof. Let (W, S) be the type of ∆. Suppose that (23.7) holds, choose x ∈ X, let Y = {δ(x, y) | y ∈ X} and let ξ : X → Y be given by y ξ = δ(x, y) for all y ∈ X. By (23.7), ξ is injective. Let π be its inverse. Then δ(x, y)π = y for all y ∈ X and hence by (23.7), δ δ(x, y)π , δ(x, z)π ) = δ(y, z)

= δ(x, y)−1 · δ(x, z) for all y, z ∈ X. Thus π is an isometry from Y to ∆ as defined in [62, 8.3]. By [62, 8.5], therefore, X = π(Y ) is contained in an apartment of ∆. Suppose, conversely, that X is contained in an apartment Σ of ∆. By [62, 8.3], there exists a bijection π from W to Σ such that δ(uπ , v π ) = u−1 v for all u, v ∈ W . Choose x, y, z ∈ X and let u, v, w be their pre-images under π. Then δ(x, y) = u−1 v, δ(y, z) = v −1 w and δ(x, z) = u−1 w. Hence (23.7) holds. Remark 23.8. If x, y, z is an ordered triple satisfying condition (23.7), then every rearrangement of the elements x, y, z also satisfies this condition. Proposition 23.9. Let (∆′ , δ ′ ) be a subbuilding of (∆, δ) as defined in 23.5 and let R and T be residues of ∆ containing chambers of ∆′ . Then R′ := R ∩ ∆′ and T ′ := T ∩ ∆′ are residues of ∆′ and (23.10)

∆′ ∩ projR (T ) = projR′ (T ′ ).

Furthermore, the residues R and T are parallel in ∆ if and only if the residues R′ and T ′ are parallel in ∆′ , and if R and T are parallel, then δ(R, T ) = δ ′ (R′ , T ′ ). Proof. By [62, 7.18], ∆′ is convex. It follows that R′ and T ′ are residues of ∆′ (of the same type as R and T , respectively). Choose chambers x ∈ R′ and y ∈ T ′ . Suppose c ∈ ∆′ ∩ projR (T ). Thus c = projR (d) for some d ∈ T . By 21.8(ii), c = projR (d′ ) for d′ = projT (c). By 21.6(i), there is a minimal gallery from y to c passing through d′ . Since y and c are in ∆′ so is d′ . Thus (23.11)

c = projR (d′ ) = projR′ (d′ ) ∈ projR′ (T ′ ).

Therefore ∆′ ∩ projR (T ) ⊂ projR′ (T ′ ). Now suppose that u = projR′ (v) for some v ∈ T ′ and let w = projR (v). By 21.6(i) again, there is a minimal gallery from v to u passing through w. Since v and u are in ∆′ , so is w. Therefore w = u. Thus (23.10) holds. Since ∆′ has the same rank as ∆, there is no proper subresidue R1 of R such that R1 ∩ ∆′ = R′ . Thus by (23.10), projR′ (T ′ ) = R′ if and only if

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projR (T ) = R. By symmetry, projT ′ (R′ ) = T ′ if and only if projT (R) = T . Therefore R is parallel to T if and only if R′ is parallel to T ′ . Suppose, finally, that R and T are parallel. Then R′ and T ′ are parallel as well. We saw in (23.11) that there exists d′ ∈ T ′ such that projR (d′ ) = projR′ (d′ ). Hence δ(R, T ) = δ ′ (R′ , T ′ ) by 21.11. Proposition 23.12. Let (∆, δ) be a building of type Π, let T = (Π, Θ, A) ˜ be a T-building of (∆, δ) and ˜ δ) be a Tits index of absolute type Π, let (∆, let (∆′ , δ ′ ) be a subbuilding of ∆ as defined in 23.5 such that every chamber ˜ contains chambers of ∆′ . Let of ∆ ˜ ′ := {R ∩ ∆′ | R is a chamber of ∆}. ˜ ∆ ′ ˜′ ′ ˜ Then (∆ , δ ) is a T-building of ∆ , where δ˜′ (R′ , T ′ ) = δ ′ (R′ , T ′ ) ˜ ′ in the sense of 21.11 and the map R 7→ R ∩ ∆′ is an isometry for R′ , T ′ ∈ ∆ ˜ to ∆ ˜ ′. from ∆ Proof. This holds by 23.1 and 23.9. Definition 23.13. Let ∆ be a building of type Π. A T-apartment of ∆ is a thin T-building of ∆ for some Tits index T = (Π, Θ, A) of absolute type Π. As was observed in 19.7, the only thin buildings are the Coxeter chamber systems. Thus by 23.1(ii), a T-apartment is isomorphic to the Coxeter chamber system Σ( W ˜ ,S) ˜ ˜ , S) ˜ of T. associated with the relative Coxeter system (W Proposition 23.14. Let (∆, δ) be a building of type Π, let T = (Π, Θ, A) ˜ , S) ˜ be the relative Coxeter system be a Tits index of absolute type Π, let (W ˜ be a set of residues of ∆. Then Σ ˜ is a T-apartment of ∆ if of T and let Σ and only if the following conditions are satisfied: ˜ for all R, T ∈ Σ. ˜ (a) R and T are parallel and δ(R, T ) ∈ W

˜ the map X 7→ δ(R, X) is a bijection from Σ ˜ to W ˜. (b) For all R ∈ Σ, ˜ (c) δ(R, Z) = δ(R, T ) · δ(T, Z) for all R, T, Z ∈ Σ.

˜ is a T-apartment. By 23.1(i), (a) holds and by 23.13, Proof. Suppose that Σ ˜ is isomorphic to Σ ˜ ˜ . Thus (b) and (c) hold. Suppose, conversely, that Σ (W ,S) ˜ ˜ (a), (b) and (c) hold. By (a), we can set δ(R, T ) = δ(R, T ) for all R, T ∈ Σ. ˜ ˜ Choose R ∈ Σ and let ϕ(X) = δ(R, X) for every X ∈ Σ. By (b), ϕ is a ˜ to W ˜ . Let T, Z ∈ Σ. ˜ Then δ(T, Z) = δ(R, T )−1 · δ(R, Z) bijection from Σ −1 by (c). By [62, 2.5], δ(R, T ) · δ(R, Z) is the Weyl-distance from ϕ(T ) to ˜ ˜ ϕ(Z) in Σ(W ˜ ,S) ˜ . Thus ϕ is an isomorphism from (Σ, δ) to Σ(W ˜ ,S) ˜ . Therefore ˜ Σ is a T-apartment.

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Proposition 23.15. Let ∆ be a building of type Π, let T = (Π, Θ, A) be a ˜ be a T-apartment of ∆ and let Σ be an Tits index of absolute type Π, let Σ ˜ Then apartment of ∆ such that R ∩ Σ 6= ∅ for all chambers R of Σ. ˜ {R ∩ Σ | R is a chamber of Σ} is a T-apartment of Σ.

Proof. This is a special case of 23.12. Our next goal is to prove 23.19, in some sense a converse of 23.15. Lemma 23.16. Let (∆, δ) be a building of type Π, let T = (Π, Θ, A) be a ˜ , S) ˜ be the relative Coxeter system of T, Tits index of absolute type Π, let (W ˜ ˜ Then let Σ be a T-apartment of ∆ and let T, Z be adjacent chambers of Σ.
projZ (c) = projZ projT (c) ˜ for all chambers c of ∆ contained in some chamber of Σ.

˜ let c be a chamber of R and let w = δ(R, T ). Proof. Let R be a chamber of Σ, ˜ . Since T and Z are adjacent in Σ, ˜ we have s˜ := δ(T, Z) ∈ S. ˜ Then w ∈ W By 23.14(c), therefore, δ(R, Z) = w˜ s. Hence (by 21.11) δ(c, projT (c)) = δ(R, T ) = w as well as δ(c, projZ (c)) = δ(R, Z) = w˜ s and


δ projT (c), projZ (projT (c)) = δ(T, Z) = s˜.

˜ s) = ℓ(w) ˜ Assume first that ℓ(w˜ + 1. By 20.33, it follows that ℓ(w˜ s) = ℓ(w) + ℓ(˜ s). Hence


dist c, projZ (projT (c)) ≤ dist(c, projT (c))


+ dist projT (c), projZ (projT (c)) = ℓ(w˜ s) = dist(c, projZ (c))

and therefore projZ (projT (c)) = projZ (c). ˜ s) = ℓ(w) ˜ Now suppose that ℓ(w˜ − 1. Let w′ = w˜ s. Then ′ ′ ˜ ˜ ℓ(w s˜) = ℓ(w ) + 1, δ(R, Z) = w′ and δ(R, T ) = w′ s˜. By the conclusion of the previous paragraph, therefore, projT (projZ (c)) = projT (c) and hence
projZ projT (projZ (c)) = projZ (projT (c)).

Since T and Z are parallel, we have


projZ projT (projZ (c)) = projZ (c)

by 21.10(i). Thus projZ (c) = projZ (projT (c)) also in this case.

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Proposition 23.17. Let ∆ be a building of type Π, let T = (Π, Θ, A) be a ˜ be a T-apartment of ∆ and let R, T, Z Tits index of absolute type Π, let Σ ˜ Then be chambers of Σ.
projZ (c) = projZ projT (c) ˜ is a for all chambers c of R. In other words, the set of chambers of Σ coherent set of residues of ∆ as defined in 21.41. ˜ , S) ˜ be the relative Coxeter system of T, let v = δ(R, T ), let Proof. Let (W w = δ(T, Z) and choose c ∈ R. We proceed by induction with respect to ˜ k := ℓ(w). If k = 0, then T = Z and there is nothing to prove. Suppose ˜ s) = ℓ(w) ˜ that k > 0 and choose an element s˜ of S˜ such that ℓ(w˜ − 1. By ′ ˜ 23.13, there exists a unique chamber Z of Σ such that δ(T, Z ′ ) = w˜ s and δ(Z, Z ′ ) = s˜. By induction, we have
(23.18) projZ ′ (c) = projZ ′ projT (c) . By two applications of 23.16, we have

and


projZ projZ ′ (c) = projZ (c)


projZ projZ ′ (projT (c)) = projZ (projT (c)).

By (23.18), it follows that


projZ (c) = projZ projT (c) .

We will apply the following result in the proof of 27.9. Proposition 23.19. Let ∆ be a building of type Π, let T = (Π, Θ, A) be ˜ be a T-apartment of ∆, let R be a Tits index of absolute type Π, let Σ ˜ and let ΣR be an apartment of R. Then there exists an a chamber of Σ apartment Σ of ∆ such that R ∩ Σ = ΣR and T ∩ Σ 6= ∅ for all chambers T ˜ of Σ. Proof. Let c be a chamber of ΣR , let cT = projT (c) for each chamber T of ˜ and let Y = {cT | T ∈ Σ}. ˜ By 23.17, we have Σ
(23.20) δ(cT , cZ ) = δ projT (c), projZ (projT (c)) = δ(T, Z) ˜ By 23.14(c), it follows that for all T, Z ∈ Σ.

δ(x, z) = δ(x, y) · δ(y, z) for all x, y, z ∈ Y . ˜ Since T is parallel to R, we have proj (cT ) = c by Now choose T ∈ Σ. R 21.10(i) and thus (23.21)

δ(cT , x) = δ(cT , c) · δ(c, x)

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for all x ∈ R by 21.6(i). It follows that

δ(x, cT ) · δ(cT , y) = δ(x, c) · δ(c, y)

for all x, y ∈ R. Therefore

δ(x, cT ) · δ(cT , y) = δ(x, y)

for all x, y ∈ ΣR by 23.6. ˜ Since (23.21) holds also with Z in place of T , we have Next let T, Z ∈ Σ. and hence

δ(cT , x) · δ(x, cZ ) = δ(cT , c) · δ(c, cZ )

δ(cT , x) · δ(x, cZ ) = δ(cT , c) · δ(c, cZ ) = δ(T, R) · δ(R, Z) by 21.11 = δ(T, Z) = δ(cT , cZ )

by 23.14(c) by (23.20)

for all x ∈ R. Finally, we have (23.22)

δ(x, z) = δ(x, y) · δ(y, z)

for all x, y, z ∈ ΣR by 23.6. Combining these equations (and applying 23.8), we conclude that (23.22) holds for all x, y, z ∈ Y ∪ ΣR . By 23.6 again, we conclude that there exists an apartment Σ of ∆ containing Y ∪ ΣR . Remark 23.23. The apartment Σ in 23.19 is not, in general, unique. Sup˜ is spherical (by 20.35(i)). Let T pose, however, that ∆ is spherical. Then Σ ˜ be the unique chamber of Σ opposite R, let c ∈ ΣR , let c′ = projT (c) and let e ∈ T be a chamber opposite to c′ in T . By 21.6(i), we have δ(c, e) = δ(c, c′ ) · δ(c′ , e) = wS˜ wA

and by 20.35(ii), wS˜ wA = wS . Hence e is opposite c in ∆ and thus by 21.26, R is opposite T in ∆ (compare 22.41). Let d be the unique chamber opposite c in ΣR . By 21.25, c is opposite projT (d) in ∆. Every apartment containing both d and some chamber of T must contain projT (d) as well. Thus Σ is the unique apartment of ∆ containing both ΣR and some chamber of T . ∗

∗

∗

We now focus on thin T-buildings. Notation 23.24. For the rest of this chapter, we fix a Tits index T = (Π, Θ, A). ˜ ˜ Let (W, S) and (W , S) be the absolute and relative Coxeter systems of T and let Σ = ΣΠ and δ be as in 19.7. Thus (Σ, δ) is the unique thin building of type (W, S), its chambers are the elements of W and (23.25)

δ(v, w) = v −1 w

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for all v, u ∈ W . Let G = Aut(Σ) and let G◦ denote the group of typepreserving elements of G. To each w ∈ W we associate the type-preserving automorphism λw of Σ, where λw (x) = wx for all x ∈ W . By [62, 2.8], we have G◦ = {λw | w ∈ W }. Proposition 23.26. Suppose that H := {δ(c, x) | x ∈ X} is a subgroup of W for some subset X of W and some c ∈ X and let U = {λg | g ∈ cHc−1 }. Then U is the unique subgroup U of G◦ such that X is an orbit of U . Proof. By (23.25), H = c−1 X. It follows that U is a subgroup of G◦ such that U (c) = X. Since G◦ acts sharply transitively on W , the subgroup U is unique. The following result will be applied in the proof of 27.12. ˜ is a T-apartment of Σ. Then there Proposition 23.27. Suppose that Σ ◦ ˜ is an orbit of U in the set of all exists a subgroup U of G such that Σ ˜ is A-residues of Σ. Furthermore, every type-preserving automorphism of Σ ˜ the restriction to Σ of some element of U . ˜ let c ∈ R, let cT = projT (c) for each Proof. Let R be a chamber of Σ, ˜ and let X = {cT | T ∈ Σ}. ˜ Then δ(R, T ) = δ(c, cT ) for each chamber T ∈ Σ ˜ By 23.14(b), we have T ∈ Σ. ˜ = {δ(R, T ) | T ∈ Σ} ˜ = {δ(c, x) | x ∈ X}. W By 23.26, there exists a subgroup U of G◦ such that X is a U -orbit. Since ˜ to X, also Σ ˜ is an orbit of U in the the map T 7→ cT is a bijection from Σ ˜ ˜ induced by set of all A-residues of Σ. Let U denote the subgroup of Aut(Σ) ◦ ˜ U . Since U ⊂ G , the subgroup U is type-preserving. Since the group of ˜ acts sharply transitively, we conclude type-preserving automorphisms of Σ ˜ ˜ that U is the group of all type-preserving elements of Σ.

˜ for chambers x, y ∈ Σ and let Lemma 23.28. Suppose that δ(x, y) ∈ W R and T be the unique A-residues containing x and y. Then R and T are parallel, y = projT (x) and δ(R, T ) = δ(x, y). Proof. Let w = δ(x, y). By 20.13(iii), A ⊂ J + (w) ∪ J + (w−1 ). By 21.6(iii), therefore, y = projT (x) and x = projR (y). By 20.8, we have wAw−1 = A = w−1 Aw. Thus 21.8(iv), projT (R) and projR (T ) are both A-residues. Hence projT (R) = T and projR (T ) = R. Proposition 23.29. Let R be an A-residue of Σ. Then the following hold: ˜ there exists an A-residue Rw of Σ parallel to R such (i) For each w ∈ W that δ(R, Rw ) = w. ˜ , Rw is the only A-residue of Σ parallel to R such (ii) For given w ∈ W that δ(R, Rw ) = w. ˜. (iii) Rv is parallel to Rw and δ(Rv , Rw ) = v −1 w for all v, w ∈ W

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˜ . Let cw be the unique chamber of Σ Proof. Fix c ∈ R and choose w ∈ W such that δ(c, cw ) = w and let Rw be the unique A-residue containing cw . By 23.28, δ(R, Rw ) = w and Rw is parallel to R. Thus (i) holds. Suppose that T is a second A-residue parallel to R such that δ(R, T ) = w. Then δ(c, projT (c)) = w = δ(c, cw ) and hence projT (c) = cw by (23.25). Therefore T = Rw . Thus (ii) holds. ˜ and let cv and Rv be defined as in the previous Now choose v ∈ W ˜ by 23.6. paragraph. Then δ(cv , cw ) = δ(c, cv )−1 · δ(c, cw ) = v −1 w ∈ W Hence by 23.28 again, Rv and Rw are parallel and δ(Rv , Rw ) = v −1 w. Thus (iii) holds. Proposition 23.30. Let R be an A-residue of Σ. Then there exists a T˜ R of Σ one of whose chambers is R and Σ ˜ R is the only T-building apartment Σ of Σ one of whose chambers is R. In particular, T-buildings of Σ are Tapartments. ˜ be as in 23.29(i) and let Proof. Let Rw for each w ∈ W ˜ R = {Rw | w ∈ W ˜ }. Σ

˜ R satisfies the conditions (a)–(c) in 23.29 and is thus a TBy 23.14, Σ ˜ is a T-building of Σ one of whose chambers apartment of Σ. Suppose that ∆ ˜ then T is an A-residue of Σ which is is R. If T is another chamber of ∆, ˜ ˜ ⊂Σ ˜ R. parallel to R and w := δ(R, T ) ∈ W , so T = Rw by 23.29(ii). Thus ∆ ˜ ˜ Since the map T 7→ δ(R, T ) from ∆ to W is surjective, it follows that, in ˜ =Σ ˜ R. fact, ∆ Proposition 23.31. Suppose that Γ is a T-group of Aut(Σ) as defined in ˜ R be as in 23.30. Then an 23.2, let R be Γ-chamber of type A and let Σ ˜ A-residue of Σ is in ΣR if and only if it is a Γ-chamber. ˜ , let g ∈ Γ and let θ be the element of Θ induced by Proof. Let w ∈ W g g. Since g stabilizes the residue R, the residue Rw is parallel to R and g g g δ(R, Rw ) = w = w by 20.11(ii). By 23.29(ii), therefore, Rw = Rw . Thus Γ ˜ R. acts trivially on Σ Suppose, conversely, that T is a Γ-chamber. Since R is also a Γ-chamber, ˜ by 22.3(iii). If Rw is as in 23.29(i), T is parallel to R and w := δ(R, T ) ∈ W ˜ R. then T = Rw by 23.29(ii) and hence T ∈ Σ The following result will be used in the proof of 27.10. ˜ be a T-apartment of Σ. Then there exists a Proposition 23.32. Let Σ ˜ T-group Γ of Σ such that Σ is the set of Γ-chambers. ˜ and let c ∈ R. By [65, 29.29], there exists a unique Proof. Let R ∈ Σ ˆ of G = Aut(Σ) such that Θ ˆ fixes c and induces Θ on Π. Since A subgroup Θ ˆ It follows that Θ ˆ normalizes is Θ-invariant, the residue R is stabilized by Θ. the stabilizer G◦R . Since G◦R does not fix any proper residue of R, it follows ˆ · G◦ is a T-group and R is a Γ-chamber. By 23.30, Σ ˜ =Σ ˜ R. that Γ := Θ R ˜ Thus by 23.31, Σ is the set of Γ-chambers.

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We close this chapter with another application of 23.29. We will apply this result in the proof of 27.9. Proposition 23.33. Let (∆, δ) be a building and let Γ be a spherical descent ˜ are apartments of ∆ group of ∆ with Tits index T. Suppose that Σ and Σ ˜ := ∆Γ such that every chamber of Σ ˜ contains chambers of Σ. Then and ∆ ˜ every Γ-residue of ∆ containing chambers of Σ contains chambers of Σ. Proof. Let R be a Γ-residue of ∆ containing chambers of Σ. Choose a Γ˜ and let R0 = projR (T ). By hypothesis, T contains chamber T contained in Σ chambers of Σ. By 21.16, therefore, R0 contains chambers of Σ. By 22.3(ii), ˜. R0 is a Γ-chamber parallel to T and by 22.14(ii), w ˜ := δ(R0 , T ) ∈ W ˜ By 22.3 and 22.14(i)–(ii), the set of chambers of Σ forms a T-apartment ˜ are parallel (as as defined in 23.13. In particular, every two chambers of Σ ˜ such that residues of ∆). By 23.14(b), there is a unique chamber R0′ of Σ ′ ′ δ(R0 , T ) = w. ˜ By 21.17, the residues R0 ∩ Σ and R0 ∩ Σ of Σ are parallel to the residue T ∩ Σ. By 22.14(i), all three of these residues have type A. By 23.29(ii), we conclude that R0′ ∩ Σ = R0 ∩ Σ and hence R0′ = R0 .

Chapter Twenty Four Moufang Structures Our main goal in this chapter is to show (in 24.31) that if ∆ is a spherical building satisfying the Moufang condition and Γ is a descent group of ∆ as defined in 22.19, then the fixed point building ∆Γ defined in 22.22 also satisfies the Moufang condition. We do this via the notion of a Moufang structure defined in 24.6. Notation 24.1. Throughout this chapter, we suppose that ∆ is a spherical building. Let (W, S) denote the type of ∆, let G = Aut(∆) and let G◦ denote the group of type-preserving elements of G. We do not assume that ∆ is thick (until 24.31). Definition 24.2. An element g of G is unipotent if the following hold: (i) g is contained in the stabilizer G◦c for some chamber c. (ii) If g fixes two adjacent chambers x and y, it acts trivially on the unique panel containing x and y. A subgroup U of G is unipotent if it fixes a chamber and all its elements are unipotent. Remark 24.3. A subgroup of a unipotent group is unipotent. If a unipotent group U stabilizes a residue R, then U fixes projR (c) for each fixed chamber c of U and hence its restriction to R is a unipotent group in Aut(R). Remark 24.4. If a unipotent element fixes two chambers x and y, then it fixes every minimal gallery from x to y (by [62, 7.7(iii)]) and hence acts trivially on every panel containing two chambers in such a gallery. Proposition 24.5. The identity is the only unipotent element fixing two opposite chambers of ∆. Proof. Let g be a unipotent element fixing opposite chambers c and d and let Σ be the unique apartment containing c and d. Then g stabilizes Σ. Since unipotent elements are type-preserving, g must, in fact, act trivially on Σ. By 24.2, therefore, g acts trivially on every panel containing a chamber of Σ. By 21.35, we conclude that g = 1. Definition 24.6. A Moufang structure on ∆ is a collection of subgroups Uc of the group G◦ defined in 24.1, one for each chamber c of ∆, such that the following hold:

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(i) For each chamber c, Uc fixes c and acts transitively on the set of all chambers of ∆ opposite c. (ii) For each chamber c, for each element g ∈ Uc and each panel P , if g fixes at least 2 chambers of P , then g ∈ Ud for all chambers d in P . (iii) For all chambers c, d and each g ∈ Uc , gUd g −1 = Ug(d) . Note that we are composing elements of Aut(∆) from left to right. Remark 24.7. If ∆ is thick and of type A1 , so that ∆ consists of a single panel containing at least three chambers, then a Moufang structure on ∆ is the same thing as a Moufang set as defined in 1.5. If ∆ satisfies the Moufang condition defined in 1.1, then we will show in 24.13 and 24.14 that it has a unique Moufang structure. Recall that we have set G = Aut(∆) throughout this chapter. Proposition 24.8. Let {Uc | c ∈ ∆} be a Moufang structure on ∆. Then the following hold for every x ∈ ∆: (i) Ux is unipotent. (ii) Ux ∩ Gy ⊂ Uy for all y ∈ ∆. (iii) Ux acts sharply transitively on the set of all chambers of ∆ opposite x. Proof. Choose x ∈ ∆. It follows from 24.6(i)–(ii) that every element of Ux is unipotent. Hence (i) holds. Let y ∈ ∆ and let k = dist(x, y). We will show that Ux ∩ Gy ⊂ Uy by induction with respect to k. The claim holds by 24.6(i) for k = 0. Suppose that k > 0 and let P be a panel containing both y and a chamber z such that dist(x, z) = k − 1. By (i) and 24.4, Ux ∩ Gy ⊂ Gz and by induction, Ux ∩ Gz ⊂ Uz . Since Uz ∩ Gy ⊂ Uy by 24.6(ii), it follows that Ux ∩ Gy ⊂ Uy . Thus (ii) holds. The assertion (iii) holds by 24.5 and 24.6(i). Proposition 24.9. Suppose that ∆ is thick, irreducible and of rank at least 2 and that ∆ has a Moufang structure {Uc | c ∈ ∆}. Let α be a root of ∆ and let Uα be the corresponding root group as defined in [62, 11.1]. Then the following hold: (i) The root group Uα acts transitively on the set of all apartments containing α. (ii) If v and c are two chambers in α at distance diam(∆) − 1 from each other, then Uα = Uc ∩ Gv . (iii) Uα ⊂ Ux for all chambers x in α.

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Proof. By [65, 29.33], we can choose an apartment Σ of ∆ containing α. Let {c, d} be an edge of Σ with c ∈ α and d 6∈ α and let v be the unique chamber of Σ opposite d. Then dist(c, v) = diam(∆) − 1 and by [62, 5.2], no root contains two opposite chambers. Thus v ∈ α. We conclude that α contains two chambers at distance diam(∆) − 1 from each other. Now suppose that c and v are an arbitrary pair of chambers in α at distance diam(∆) − 1 from each other and let z be the unique chamber of Σ opposite c. Then z 6∈ α and by [62, 5.4], v lies on a minimal gallery from c to z. Hence z is adjacent to v. Let A = Uc ∩ Gv . By [65, 29.46], a chamber of ∆ is on a minimal gallery from c to v if and only if it is contained in α. By 24.4, therefore, A acts trivially on α and by 24.8(i), A acts trivially on every panel containing two chambers of α. Thus A ⊂ Uα

(24.10)

(by [62, 11.1]). Let P be the unique panel of ∆ containing v and z and let z ′ ∈ P \{v}. Since z is opposite c, we must have v = projP (c) and hence z ′ is also opposite c. By 24.6(i), therefore, there exists g ∈ Uc mapping z to z ′ . Since Uc ⊂ G◦ , the element g must stabilize P . Hence g ∈ A. We conclude that A acts transitively on P \{v}. By [62, 11.4], Uα acts sharply transitively on P \{v}. By (24.10), therefore, Uα = A ⊂ Gc and by [62, 9.3], it follows that Uα acts transitively on the set of apartments containing α. Thus (i) and (ii) hold. By 24.8(ii), we have Uα ⊂ Uc ∩ Gx ⊂ Ux for all x ∈ α. Thus (iii) holds. Recall that by definition (see 1.1), a building can only satisfy the Moufang condition if it is thick, spherical, irreducible and has rank at least 2. Proposition 24.11. Suppose that ∆ satisfies the Moufang condition. For each chamber c, let Uc be the subgroup generated by the root groups Uα for all roots α of ∆ containing c. Then the following hold: (i) M := {Uc | c ∈ ∆} is a Moufang structure on ∆. (ii) For each apartment Σ and each residue R containing chambers of Σ, \ Ux = hUα | α is a root of Σ containing R ∩ Σi. x∈R

Proof. Let Σ be an apartment and let α be a root of Σ. By [62, 4.12], there exists a chamber u in α such that every chamber of Σ adjacent to u is contained in α. By [62, 3.14], Uα acts trivially on α and by [62, 11.1], Uα acts trivially on the set of all chambers adjacent to u. Hence, in particular, Uα ⊂ G◦ . From these observations, we conclude that Uc ⊂ G◦ and Uc ⊂ Gc for every chamber c. By [62, 11.11(ii)], Uc acts transitively on the set of chambers of ∆ opposite c. Thus 24.6(i) holds. Now let c and x be distinct chambers and let Σ be an apartment containing c and x. By [62, 5.4], we can choose a minimal gallery γ = (x0 , x1 , . . . , xm )

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in Σ from x0 = c to xm = opΣ (c) such that x = xk for some k ∈ [1, m]. For each i ∈ [1, m], let βi be the unique root of Σ containing xi−1 but not xi , let Ui = Uβi and let Mi = Ui · · · Um . We also set Mm+1 = 1. Thus Mi fixes (x0 , . . . , xi−1 ) for all i ∈ [1, m + 1]. By [62, 11.11(ii)-(iii)], every element of Uc can be written uniquely as a product g1 · · · gm with gi ∈ Ui for all i ∈ [1, m]. For each j ∈ [1, m], we have Uj ∩ Gxj = 1 and by [62, 7.7(iii)], the group Uc ∩ Gx fixes (x0 , . . . , xk ). Thus no non-trivial element of U1 · · · Uk fixes x. It follows that Uc ∩ Gx = Mk+1 . By [62, 5.3], either k < m and βk+1 , . . . , βm are the only roots of Σ containing c and x or k = m and there are no roots of Σ containing c and x. We thus have (24.12)

Uc ∩ Gx = hUα | α is a root of Σ containing c and x} ⊂ Ux

for all values of k. Now suppose that an element g ∈ Uc fixes two chambers x, y in some panel P . Interchanging x and y if necessary, we can assume that dist(c, y) ≤ dist(c, x). Let Σ be an apartment containing c and x and let u = projP (c). If α is a root of Σ containing c and x, then by [62, 3.19] and the convexity of Σ, u ∈ α and by [62, 11.1], therefore, Uα acts trivially on P . By (24.12), it follows that Uc ∩ Gx acts trivially on P and thus Uc ∩ Gx ⊂ Ud for every d ∈ P . Hence 24.6(ii) holds. Finally, we observe that 24.6(iii) holds simply because every element of G maps apartments to apartments (by [62, 9.2]) and therefore maps roots to roots. Thus (i) holds. Now let Σ be an apartment, let R be a residue containing a chamber c of Σ and let α be a root of Σ containing R∩Σ. If P is a panel of R containing c, then |P ∩ Σ| = 2 and hence Uα acts trivially on P . By [62, 9.7], we conclude that Uα acts trivially on R. By (i) and 24.8(ii), therefore, \ Uα ⊂ Ux . x∈R

Now choose an element g such that g ∈ Ux for all x ∈ R and let d be the unique chamber opposite c in Σ ∩ R. Then g ∈ Uc ∩ Gd . By [62, 4.10 and 5.2], a root of Σ that contains c and d must contain all of R ∩ Σ. It follows that g ∈ Uc ∩ Gd ⊂ hUα | α is a root of Σ containing R ∩ Σi

by (24.12). Thus (ii) holds.

Definition 24.13. Suppose that ∆ satisfies the the Moufang condition as defined in 1.1 and let M be as in 24.11. We call M the standard Moufang structure on ∆. Proposition 24.14. Suppose that ∆ satisfies the Moufang condition. Then the standard Moufang condition on ∆ is the only Moufang structure on ∆. Proof. Let M = {Uc | c ∈ ∆} be the standard Moufang structure on ∆ and ¨ = {U ¨c | c ∈ ∆} be a second Moufang structure on ∆. By 24.9(iii) and let M ¨c for each c ∈ ∆. It follows from 24.8(iii), therefore, that 24.11(i), Uc ⊂ U ¨ Uc = Uc for all c ∈ ∆.

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Definition 24.15. Let M = {Uc | c ∈ ∆} be a Moufang structure on ∆ and let G+ M = hUc | c ∈ ∆i.

An M-automorphism of ∆ is an automorphism g such that gUc g −1 = Ug(c) for all c ∈ ∆. The set of all M-automorphisms forms a subgroup of G which we denote by GM . By 24.6(iii), G+ M ⊂ GM . By 24.14, G = GM if ∆ is Moufang. Definition 24.16. Let M = {Uc | c ∈ ∆} be a Moufang structure on ∆ and let R be a residue of ∆. The unipotent radical UR of R (with respect to M) is the intersection of the subgroups Uc for all c ∈ R. Thus UR is a normal subgroup of the stabilizer GM,R of R in GM . For each element g ∈ GM,R , we denote by gR the automorphism of R induced by g and let πR denote the homomorphism g 7→ gR from GM,R to Aut(R). Note that the elements of πR (G◦M,R ) are type-preserving. Remark 24.17. Suppose that ∆ satisfies the Moufang condition and let M be the standard Moufang structure on ∆. Then by 24.11(ii), UR = hUα | α is a root of Σ containing Σ ∩ Ri for every residue R and every apartment Σ such that R ∩ Σ 6= ∅. Hypothesis 24.18. For the rest of this chapter, we fix a Moufang structure M = {Uc | c ∈ ∆} on ∆. Proposition 24.19. Let R be a residue of ∆, let g ∈ Uc for some chamber c and suppose that πR (g) = 1, where πR is as in 24.16. Then g ∈ UR . Proof. This holds by 24.8(ii). Proposition 24.20. Let R, T, T ′ be residues of ∆ and let Z be a residue containing T and T ′ . Suppose that R is parallel to both T and T ′ , that T and T ′ are opposite each other in Z and that g is an element of UR stabilizing T and T ′ . Then g ∈ UZ . Proof. By 22.43, g stabilizes Z. Since g stabilizes T and T ′ and acts trivially on R, it follows from 21.10(i) that g acts trivially on both T and T ′ . By 24.5, therefore, g acts trivially on Z. Hence g ∈ UZ by 24.19. Proposition 24.21. For each residue R, the unipotent radical UR acts sharply transitively on the set of all residues of ∆ that are opposite R. Proof. Let R be a residue, let c ∈ R, let d be a chamber opposite c in R, let T be a residue opposite R in ∆ and let e = projT (d). By 21.24, R and T are parallel, so d = projR (e) by 21.10(i). By 21.25, e is opposite c in ∆. Let g be an element of UR stabilizing T . The group UR acts trivially on R. Hence g fixes e = projT (d). By 24.8(iii), therefore, g = 1. We conclude that UR acts freely on the set of all residues of ∆ opposite R.

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Now let T ′ be a second residue opposite R in ∆ (and hence also parallel to R) and let e′ = projT ′ (d). By 21.10(i), d = projR (e′ ) and by 21.25 again, c is opposite e′ in ∆. By 24.6(i), there exists g ∈ Uc ⊂ Gc mapping e to e′ . It follows from d = projR (e) = projR (e′ ) that g ∈ Gd . By 24.5, g acts trivially on R and hence by 24.19, g ∈ UR . Since g maps e ∈ T to e′ ∈ T ′ and is type-preserving, g also maps T to T ′ . Proposition 24.22. Let c be a chamber of a residue R of ∆. Then Uc acts transitively on the set of chambers opposite c in R. Proof. Let d, d′ be two chambers both opposite c in R, let T be a residue opposite R in ∆, let e = projT (d) and e′ = projT (d′ ). By 21.24, T is parallel to R, so d = projR (e) and d′ = projR (e′ ) by 21.10(i). By 21.25, e and e′ are both opposite c in ∆. Thus by 24.6(i), there exists g ∈ Uc mapping e to e′ . From d = projR (e) and d′ = projR (e′ ) it follows that g maps d to d′ . ¯c = Proposition 24.23. Let R be a residue of ∆ and for each c ∈ R, let U ¯ πR (Uc ), where πR is as in 24.16. Then MR := {Uc | c ∈ R} is a Moufang structure on R. Proof. Let c ∈ R. Since the elements of Uc are type-preserving (by 24.6), so ¯c and since the elements of Uc fix c, so do the elements are the elements of U ¯ of Uc . By 24.22, therefore, MR satisfies 24.6(i). Let g ∈ Uc and suppose that πR (g) fixes two chambers of a panel P of R. Then g fixes these two chambers and hence, by 24.6(ii) for M, g ∈ Ud for ¯d for all d ∈ P . Thus MR satisfies 24.6(ii). each d ∈ P . Therefore πR (g) ∈ U Now let c, d ∈ R and g ∈ Uc . Then g(d) = πR (g)(d) and by 24.6(iii) for M, ¯d πR (g)−1 = U ¯π (g)(d) . Thus MR satisfies gUd g −1 = Ug(d) . Therefore πR (g)U R 24.6(iii). Remark 24.24. Suppose that ∆ is thick and has a Moufang structure M and let G+ M be as in 24.15. Let P be a panel. Then |P | ≥ 3. By 24.23, therefore, the stabilizer of P in G+ M acts transitively (in fact 2-transitively) on P . Since ∆ is connected, it follows that G+ M acts transitively on the set of chambers of ∆. We now introduce a T-group (as defined in 23.2) into the picture. Hypothesis 24.25. Let ∆ and M be as in 24.18. Let Π be the Coxeter diagram of ∆ and let (W, S) be the corresponding Coxeter system. For the rest of this chapter, we assume that T = (Π, Θ, A) is a Tits index, that ˜ , S) ˜ is its relative Coxeter system and that Γ is a T-group of ∆. We (W ˜ the set of Γ-chambers and suppose that every Γ-panel contains denote by ∆ ˜ is a T-building of ∆ as defined ˜ By 23.2, (∆, ˜ δ) at least two elements of ∆. ˜ ˜ = CG (Γ) and let πΓ denote the in 23.3, where δ is as in 23.1(ii). We set G ˜ ˜ natural homomorphism from G to Aut(∆).

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Remark 24.26. If g is a type-preserving element of G, then δ(R, T ) = ˜ δ(Rg , T g ) for all pairs of parallel residues R and T . Thus if g also lies in G, ˜ then its image under πΓ is a type-preserving automorphism of ∆. Lemma 24.27. Suppose G is a group acting on a set X and that N is a sharply transitive normal subgroup of G. Then for every subgroup H of G, CN (H) acts sharply transitively on FixX (H). Proof. Choose a subgroup H of G, let x, y ∈ FixX (H) and let g be the unique element of N mapping x to y. Then xhg = xg = y = y h = xgh and hence [g, h] = 1 for all h ∈ H (since the commutator [g, h] lies in N and N acts sharply transitively on X). Thus CN (H) acts transitively on FixX (H). Since CN (H) ⊂ N , this action is sharply transitive. ˜c˜ = πΓ (Uc˜ ∩ G) ˜ for each Γ-chamber c˜ ∈ ∆, ˜ let Proposition 24.28. Let U ˜c˜ | c˜ ∈ ∆} ˜ MΓ = {U

and suppose that Γ is contained in the subgroup GM defined in 24.15. Then ˜ MΓ is a Moufang structure on ∆. ˜ By 24.26, the group U ˜c˜ is type-preserving and since Uc˜ Proof. Let c˜ ∈ ∆. ˜c˜. Let X denote the set of residues of ∆ opposite c˜. By fixes c˜, so does U 24.6(iii), Uc˜ is a normal subgroup of the stabilizer GM,˜c . By 24.21, Uc˜ acts sharply transitively on X and by hypothesis, Γ ⊂ GM,˜c . By 24.27, therefore, ˜ = CUc˜ (Γ) acts sharply transitively on FixX (Γ) and by 22.41, FixX (Γ) Uc˜ ∩ G ˜ opposite c˜. It follows that Uc˜ ∩ G ˜ acts faithfully is the set of chambers of ∆ ˜ so we can identify Uc˜ ∩ G ˜ with U ˜c˜ from now on, and we conclude that on ∆, MΓ satisfies 24.6(i). ˜ d˜′ ˜ and suppose that g ∈ U ˜c˜ fixes two chambers d, Let P˜ be a panel of ∆ ′ in P˜ . By 22.10(i), d˜ and d˜ are opposite residues of P˜ and by 22.3(iii), the ˜ d˜′ are pairwise parallel. By 24.20, therefore, g ∈ U ˜ . Therefore residues c˜, d, P ˜=U ˜x˜ for every x g ∈ Ux˜ ∩ G ˜ ∈ P˜ . We conclude that MΓ satisfies 24.6(ii). ˜ and let g ∈ U ˜c˜. By 24.15, we have g ∈ Uc˜ ⊂ GM . Hence Finally, let d˜ ∈ ∆ −1 ˜ −1 ˜ ˜g = g(U ˜ ∩ G)g gU d

d

˜ = gUd˜g −1 ∩ G ˜=U ˜ =U ˜ ∩G g(d)

˜ g(d)

˜ Thus MΓ satisfies 24.6(iii). since g ∈ G.

Remark 24.29. By 24.14, the condition Γ ⊂ GM in 24.28 is superfluous if ∆ satisfies the Moufang condition. Proposition 24.30. Let R be a Γ-residue, let J = Typ(R) and suppose that Γ ⊂ GM . Let MR be the Moufang structure on R defined in 24.23, let MΓ be ˜ defined in 24.28, let R ˜ be the building and ΓR the Moufang structure on ∆ ¯ ˜ the TJ -group of R defined in 22.39, let GR denote the subgroup of Aut(R) induced by the stabilizer GR and let M1 := (MΓ )R˜

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˜ obtained by applying 24.23 to MΓ and the be the Moufang structure on R ˜ of ∆. ˜ Then ΓR ⊂ G ¯ R,MR and if residue R M2 := (MR )ΓR ˜ obtained by applying 24.28 to MR and denotes the Moufang structure on R ΓR , then M1 = M2 .

Proof. Let πR and πΓ be as in 24.16 and 24.25, let πR˜ and πΓR be as in 24.16 ˜ and to ΓR . Since Γ ⊂ GR is contained in GM , the and 24.26 applied to ∆ ¯ R,MR . The centralizer of ΓR in Aut(R) contains group ΓR is contained in G ˜ = CG (Γ). Thus the subgroup of Aut(R) induced by the stabilizer of R in G πR˜ (πΓ (h)) = πΓR (πR (h)) ˜ R . Let for every h in the stabilizer G (i)

˜ Mi = {Uc˜ | c˜ ∈ R}

˜ and let g ∈ U (1) . Then for i = 1 and 2. Choose c˜ ∈ R c˜

(2)

g = πR˜ (πΓ (h)) = πΓR (πR (h)) ∈ Uc˜

˜ Therefore U (1) ⊂ U (2) . Since U (1) and U (2) both act for some h ∈ Uc˜ ∩ G. c˜ c˜ c˜ c˜ ˜ opposite c˜, the two groups sharply transitively on the set of chambers of R are equal. We come now to the main results of this chapter. Theorem 24.31. Suppose that ∆ satisfies the Moufang condition as defined in 1.1 and that Γ is a descent group of ∆ as defined in 22.19 and let ∆Γ be as in 22.22. Let M be the standard Moufang structure on ∆ as defined in 24.13, let MΓ be the Moufang structure on ∆Γ obtained by applying 24.28 to M and let k be the rank of ∆Γ . Then the following hold: (i) If k ≥ 2, then ∆Γ also satisfies the Moufang condition and MΓ is its standard Moufang structure. (ii) If k = 1, then the pair (∆Γ , MΓ ) is a Moufang set as defined in 1.5. Proof. By 20.35(i), ∆Γ is spherical. By 1.1(i), ∆ is irreducible. By 20.40, therefore, also ∆Γ is irreducible. By 24.9(i), ∆Γ satisfies the Moufang condition if k ≥ 2. By 24.14, therefore, MΓ is the standard Moufang structure on ∆Γ . Thus (i) holds. Suppose that k = 1. By 22.10(i), the chambers of ∆Γ are pairwise opposite. By 24.6(i) and 24.6(iii), therefore, the pair (∆Γ , MΓ ) satisfies 1.5(i)-(ii). (Note that in 1.5 we are composing from left to right in order to be able to quote verbatim results from [60] in 3.9 where this convention also holds. In the present chapter, we are composing from right to left.) Thus (ii) holds. Note for the following results that 23.19 assures that for a given apartment ˜ of ∆Γ , apartments of ∆ containing chambers of every Γ-chamber in Σ ˜ Σ always exist.

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˜ := ∆Γ and k be as in 24.31, let Σ ˜ be an Theorem 24.32. Let ∆, Γ, ∆ ˜ and let Σ be an apartment of ∆ containing chambers of apartment of ∆ ˜ Let R1 and R2 be adjacent Γ-chambers in Σ, ˜ let β˜ every Γ-chamber in Σ. ˜ containing R1 but not R2 , let M be the set of all be the unique root of Σ roots of Σ that contain R1 ∩ Σ but are disjoint from R2 ∩ Σ and let V be the subgroup hUα | α ∈ M i ˜ and the subgroup of Aut(∆). Then the centralizer CV (Γ) acts faithfully on ∆ ˜ of Aut(∆) it induces is equal to the root group Uβ˜, where “root group” is to be understood in the sense of 1.5 if k = 1. Proof. Let T be the unique Γ-panel that contains R1 and R2 and let T ′ be ˜ By 22.41, T ′ and the unique Γ-panel that is opposite T in the apartment Σ. T are opposite residues of ∆. Hence by 21.24, T and T ′ are parallel. Let Ri′ = projT ′ (Ri ) for i = 1 and 2. By 22.3(ii), R1′ and R2′ are Γ-chambers and by 22.10(i), R1 and R2 are opposite residues of T . By 21.10(i), therefore, R1′ and R2′ are opposite residues of T ′ . By 21.25, R2 and R1′ are opposite in ∆ as are R1 and R2′ . Hence, in particular, R2 and R1′ are opposite chambers of ˜ (by 22.41). Since β˜ does not contain the chamber R2 of ∆, ˜ it does contain ∆ ′ ˜ the chamber R1 (by [62, 5.2]). Since R1 and R2 are adjacent chambers of ∆ ′ ˜ and both are contained in Σ, the chambers R1 and R1 are at distance n − 1 ˜ where n is the diameter of ∆. ˜ from each other in ∆, Let M be the standard Moufang structure on ∆. Applying 24.23 with MΓ in place of M and T ′ in place of R, we conclude that the group CUR′ (Γ) acts 1 transitively on the set of Γ-chambers in T ′ different from R1′ . By 24.21, UR1 ˜ By 24.9(ii) and 24.28, the root group U ˜ equals the acts faithfully on ∆. β ˜ subgroup of Aut(∆) induced by the centralizer CUR1 ∩GR′ (Γ). 1 Let α be a root of Σ containing R1′ ∩ Σ. Since R1′ is opposite R2 in ∆, α is disjoint from R2 ∩ Σ (by [62, 5.2 and 9.8]). Suppose that α contains a chamber of R2′ . Since R2′ is opposite R1′ in T ′ , it follows that T ′ ∩ Σ ⊂ α. By 24.17, therefore, Uα ⊂ UT ′ and hence α acts trivially on T ′ . Suppose next that α ∩ R2′ = ∅. Since R2′ is opposite R1 in ∆, it follows that R1 ∩ Σ ⊂ α. Thus Uα ⊂ UR1 ∩ GR′1 and Uα ⊂ V ⊂ UR1 . By the observations in the ˜ is a previous paragraph, we conclude that the group induced by CV (Γ) on ∆ subgroup of Uβ˜ acting transitively on the set of Γ-chambers opposite R1′ in ˜ By [62, 11.4] if k ≥ 2 and 1.5(ii) if T ′ and that CV (Γ) acts faithfully on ∆. k = 1, the root group Uβ˜ acts sharply transitively on the set of Γ-chambers contained in T ′ different from R1′ . We conclude that the image of CV (Γ) in ˜ equals U ˜ . Aut(∆) β ∗

∗

∗

We close this chapter with several more results about Moufang structures in which we continue to assume, as everywhere in this chapter, that ∆ is a spherical building.

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Proposition 24.33. Suppose that ∆ has a Moufang structure M, that S is the disjoint union of subset J and K such that [J, K] = 1 and that ∆ is thick. Let R and R1 be two J-residues. Then there exists ζ ∈ GM such that projR (x1 ) = ζ(x1 ) for all x1 ∈ R1 . Proof. Choose chambers c ∈ R and c1 ∈ R1 . Since ∆ is thick, there exists a chamber c2 that is opposite both c and c1 (by [65, 29.50]). Let R2 be the unique J-residue containing c2 . By 21.38(i), R and R1 are both parallel to R2 . Hence by 21.26, R and R1 are both opposite R2 . By 24.21, therefore, there exists a unique element ζ in the unipotent radical UR2 (and thus, in particular, in GM ) mapping R1 to R. Choose x1 ∈ R1 and let x = ζ(x1 ). Since ζ acts trivially on R2 , we have projR2 (x) = projR2 (x1 ). Let x2 = projR2 (x1 ). Then x1 = projR1 (x2 ) and x = projR (x2 ) by 21.10(i) and hence projR (x1 ) = projR (projR1 (x2 )) = projR (x2 ) = x = ζ(x1 ) by 21.38(ii). Proposition 24.34. Suppose that there exist Γ, T = (Π, Θ, A), J, K, S, R and ξ as in 22.26 and that ∆ is thick. Let M be a Moufang structure on ∆ such that Γ ⊂ GM and let MR and MΓ be as in 24.23 and 24.28. Then ξ(Γ) is a TJ -group of R contained in Aut(R)MR and for each chamber c˜ of ∆Γ , ˜c˜ the map ξ : Aut(∆) → Aut(R) induces an isomorphism from the group U Γ ξ(Γ) ˜ in M to the group UprojR (˜c) in (MR ) . In other words, the isomorphism from ∆Γ to Rξ(Γ) induced by projR carries the Moufang structure MΓ on ∆Γ to the Moufang structure (MR )ξ(Γ) on Rξ(Γ) . Proof. Let τ ∈ Γ, let c ∈ R, let d = cτ , let e = projR (d) = cξ(τ ) and let ζ be as in 24.33 with Rτ in place of R1 . Then ξ(τ ) is just the restriction of ζ ◦ τ to R and hence ξ(τ )Uc ξ(τ )−1 = ζUd ζ −1 = Ue . Thus ξ(Γ) is contained in Aut(R)MR . Let c˜ be a Γ-chamber of ∆. By 22.26, ξ(Γ) is a TJ -group of R and R ∩ c˜ = projR (˜ c) is a ξ(Γ)-chamber of R. In particular, R∩˜ c 6= ∅. Since the unipotent radical Uc˜ of c˜ acts trivially on c˜ and the elements of Uc˜ are type-preserving (by 24.6 and 24.16), the group Uc˜ stabilizes R. It follows that for each g ∈ Uc˜, ξ(g) is just the restriction of g to R. By 21.38(iii), Uc˜ acts faithfully on R (since c˜ contains a K-residue) and by 21.40, ξ is a homomorphism. Thus ξ induces an isomorphism from CUc˜ (Γ) to Cξ(Uc˜) (ξ(Γ)) and ξ(Uc˜) is a subgroup of the unipotent radical of R ∩ c˜ with respect to MR . The type of c˜ is A. There exist Γ-chambers opposite c˜ in ∆Γ . By 22.44(ii), these Γ-chambers are opposite c˜ in ∆. By 22.20(v), therefore, there exist residues opposite c˜ of type A. By [62, 9.8], therefore, every residue opposite c˜ is of type A. By 24.21, Uc˜ acts transitively on the set of residues opposite c˜ in ∆. By 21.37(iii) and 21.39(ii), it follows that ξ(Uc˜) acts transitively on the set of residues opposite c˜ ∩ R in R. Since ξ(Uc˜) is a subgroup of the unipotent radical of R ∩ c˜ with respect to MR and by 24.21, this unipotent radical acts sharply transitively on the set of residues of R opposite R ∩ c˜ in R, we conclude that ξ(Uc˜) equals the unipotent radical of R ∩ c˜ in R. By the

MOUFANG STRUCTURES

215

conclusion of the previous paragraph, therefore, ξ induces an isomorphism ˜c˜ in MΓ to the group U ˜proj (˜c) in (MR )ξ(Γ) . from the group U R Lemma 24.35. Let ∆ be a Moufang building of type (W, S), let s, t ∈ S, let J = S\{s} and let K = S\{s, t}. Suppose that |st| = 3, that [s, K] = 1 and that the {s, t}-residues of ∆ are isomorphic to A2 (F ) for some commutative field F . Let R be a J-residue, let T be an ({s} ∪ K)-residue such that R ∩ T is a K-residue and let X be the set of all ordered pairs (Q, R′ ), where Q is a residue of ∆ opposite R and R′ is a residue which is different from R, contains a K-residue of T and is opposite Q. Then the pointwise stabilizer of R in Aut(∆) acts sharply transitively on X. Proof. Let A be the pointwise stabilizer of R in Aut(∆). Since |J| = |S| − 1, the group A is type-preserving. By 24.21, A acts transitively on the set of residues opposite R. Let Q be one of these residues, let c be a chamber of R ∩ T , let P be the s-panel containing c, let d be a chamber of Q opposite c and let Σ be the unique apartment containing c and d. By 21.10(i) and 21.24, the stabilizer of Q in A acts trivially on Q and hence trivially on Σ (because A is type-preserving). By 21.37(i) and 21.38(i), every K-residue of T is parallel to R ∩ T and contains a unique chamber in the panel P . Let e be the unique chamber of Σ in the panel P other than c. The residue R is the unique residue opposite Q that contains chambers of Σ. Thus the unique J-residue containing e is not opposite Q. By 21.6(i), therefore, all the chambers in P \{e} are opposite d. By [62, 9.10], it follows that each J-residue containing a chamber of P \{e} is opposite Q. Let D denote the pointwise stabilizer in Aut(∆) of the union of Σ with all the panels of R containing c. Then D fixes d = opΣ (c) and is type-preserving, so D stabilizes Q. By [62, 9.7], D acts trivially on R and by 3.16 and 3.19, D acts sharply transitively on P \{c, e}. We conclude that A acts sharply transitively on the set X. Proposition 24.36. Let ∆, R, T , etc., be as in 24.35 and let ΓR be a subgroup of Aut(R) stabilizing R ∩ T . Then there is a subgroup Γ of Aut(∆) which acts faithfully on R and whose restriction to R is ΓR . Proof. Let X be as in 24.35 and let (Q, R′ ) ∈ X. By 3.19 and 24.35, every element of ΓR has a unique extension to an automorphism of ∆ fixing (Q, R′ ). The set of these extensions forms a group acting faithfully on R.

Chapter Twenty Five Fixed Apartments In this short chapter, we examine conditions which force a group of order 2 acting on a building to stabilize various kinds of substructures. In our principal result (25.15), we show that a group of order 2 acting on a spherical building must fix an apartment. Our results will be applied in Chapter 31. Hypothesis 25.1. We assume that (∆, δ) is a building of type (W, S) and that τ is an automorphism of ∆ of order 2. We denote by θ the automorphism of (W, S) induced by τ and we set Γ = hτ i. Proposition 25.2. There exists a spherical residue R stabilized by Γ such that dist(x, xτ ) = diam(R) for all x ∈ R. Proof. Choose a chamber c such that m = dist(c, cτ ) is minimal, let (25.3)

w = δ(c, cτ ),

let J = J − (w) and let R be the unique J-residue containing c. If m = 0, then R = {c} and dist(c, cτ ) = 0 = diam(R). Suppose that m > 0, so J 6= ∅. Choose s ∈ J, let P denote the s-panel containing cτ , let (25.4)

d = projP (c)

and let Q denote the θ(s)-panel containing c. By 21.6(iv), we have d 6= cτ , so dist(c, d) = m − 1 and δ(c, d) = ws. By the choice of c, we have δ(d, dτ ) ≥ m.

Since dτ ∈ P τ = Q, it follows that projQ (d) = c. Applying τ , both to this equation and to (25.4), we obtain projP (dτ ) = cτ and projQ (cτ ) = dτ . Thus |projP (Q)| > 1 and |projQ (P )| > 1. By 21.8(i), therefore, P and Q are parallel panels and ws = δ(c, d) = δ(Q, P ) (by 21.11). By 21.10(iii), therefore, (ws)−1 θ(s)ws = s. We conclude that (25.5)

w = θ(s)ws

for all s ∈ J. We have θ(w) = θ(δ(c, cτ )) = δ(cτ , c) and thus θ(w) = w−1 by 21.4. It follows that (25.6) by 19.14.

J = J − (w) = J − (θ(w−1 )) = θ(J − (w−1 ))

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By 19.8(v), J is spherical, w = wJ wJ

(25.7) and

ℓ(w) = ℓ(wJ ) + ℓ(wJ ).

(25.8)

We claim that w = wJ . Suppose that this is false. By (25.7), we can choose t ∈ S such that ℓ(twJ ) < ℓ(wJ ) and thus ℓ(tw) = ℓ(twJ · wJ ) ≤ ℓ(twJ ) + ℓ(wJ ) < ℓ(wJ ) + ℓ(wJ ).

Hence ℓ(tw) < ℓ(w) by (25.8). Therefore t ∈ J − (w−1 ) and hence θ(t) ∈ J by (25.6). Let s = θ(t). Then s ∈ J and (25.9)

ℓ(θ(s)wJ ) = ℓ(twJ ) < ℓ(wJ ).

Thus ℓ(w) = ℓ(θ(s)ws) J

by (25.5)

= ℓ(θ(s)w · wJ s)

by (25.7)

< ℓ(wJ ) + ℓ(wJ )

by (25.9) and 19.18.

J

≤ ℓ(θ(s)w ) + ℓ(wJ s)

By (25.8), this is impossible. We conclude that w = wJ as claimed. It follows that w = w−1 , so θ(J) = J by (25.6), and it follows that J = supp(w) by (19.19). Thus by 21.4 and (25.3), there is a J-gallery from c to cτ and hence cτ ∈ R. We conclude that Rτ = R. By the choice of c, therefore, dist(x, xτ ) = diam(R)

for all x ∈ R. Corollary 25.10. Suppose that ∆ is spherical and that Σ is an apartment of ∆ stabilized by τ . Then either τ stabilizes two opposite residues of Σ or τ maps every chamber of Σ to its opposite in Σ. Proof. By 25.2, there exists a residue R of Σ stabilized by τ such that τ maps each chamber of R to its unique opposite in R. If R is a proper residue of Σ, then τ also stabilizes opΣ (R). Notation 25.11. Let Opp(τ ) denote the set of chambers c such that cτ is opposite c in ∆. If ∆ is non-spherical, then, of course, Opp(τ ) is empty. Proposition 25.12. Suppose that ∆ is spherical and x ∈ Opp(τ ) for some chambers x of ∆ but not for all. Then τ stabilizes a root or there exist two opposite residues stabilized by τ . Proof. By connectivity, we can choose adjacent chambers c and d such that c ∈ Opp(τ ) but d 6∈ Opp(τ ). Let m = diam(∆) and let Σ be the unique apartment containing c and cτ . Then Στ = Σ and either dist(d, dτ ) = m − 1 or dist(d, dτ ) = m − 2. If dist(d, dτ ) = m − 1, then by [65, 29.46], there is a

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FIXED APARTMENTS

unique root of ∆ containing d and dτ and this root is, of course, stabilized by τ . Suppose, instead, that dist(d, dτ ) = m − 2. Then there is a minimal gallery from c to cτ that passes through both d and dτ and hence d and dτ are both contained in Σ. By 25.10, it follows that Σ contains two opposite residues stabilized by τ . Proposition 25.13. Suppose that ∆ is spherical but that Opp(τ ) is empty. Then τ stabilizes an apartment Σ and a root of Σ. Proof. Since ∆ is spherical, we can choose a chamber c such that m := dist(c, cτ ) is maximal. Let w = δ(c, cτ ), let J = J + (w), let R be the unique Jresidue containing cτ and let T = Rτ . By 19.8(iii), cτ = projR (c). By 19.8(v), S ⊂ J − (w) would imply that w = wS . Since Opp(τ ) = ∅, we thus have J 6= ∅. Let s ∈ J and let d be a chamber s-adjacent to cτ . Then dist(c, d) = m + 1. Since dτ is adjacent to c, we have dist(d, dτ ) ≥ m. By the choice of c, we must, in fact, have dist(d, dτ ) = m. Thus (25.14)

dist(c, cτ ) = dist(c, d) − 1 = dist(d, dτ ).

Let Σ0 be an apartment containing c and d. There is a minimal gallery from c to d passing through dτ and another passing through cτ . It follows that P := {c, dτ } and Q := {cτ , d} are parallel residues of Σ0 with projQ (c) = cτ . By 21.10(ii), therefore, δ(dτ , d) = δ(c, cτ ) = w. By 19.8(iii) again, it follows that d = projR (dτ ). Since R is connected, we conclude that projR (eτ ) = e for all e ∈ R. Applying τ to both sides of this equation, we deduce that projT (e) = eτ for all e ∈ R. Thus the restriction of τ to T coincides with the restriction of projR to T , and the restriction of τ to R coincides with the restriction of projT to R. Hence R and T are parallel. By 21.30, finally, we conclude that R and T are opposite residues of ∆. Let c1 be a chamber opposite c in T1 . By 21.25, c is opposite cτ1 in ∆. Let Σ be the unique apartment containing c and cτ1 . By 21.6(i), there exists a minimal gallery from c to cτ1 that passes through c1 and another that passes through cτ . Therefore c1 and cτ both lie in Σ. Applying τ to c and cτ1 , we deduce that cτ and c1 are also opposite in ∆. Thus Σ is also the only apartment containing cτ and c1 . It follows that Σ = Στ . Let d be a chamber adjacent to cτ in R ∩ Σ and let α be the unique root of Σ that contains cτ but not d. By (25.14), dist(c, cτ ) < dist(c, d) and dist(dτ , d) < dist(dτ , cτ ). Hence c ∈ α and dτ 6∈ α, so α is also the unique root of Σ containing c but not dτ . Thus ατ = α. Corollary 25.15. If ∆ is spherical then the group Γ = hτ i stabilizes an apartment of ∆.

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Proof. By 25.13, we can assume that Opp(τ ) 6= ∅. If c ∈ Opp(τ ), then τ stabilizes the unique apartment containing c and cτ . See [24, 4.9(3) and 6.4] for results similar to 25.1 and 25.15. Corollary 25.16. Suppose that ∆ is spherical and that τ stabilizes a proper residue R of ∆. Then Opp(τ ) 6= ∆ and there exist two opposite residues or a root (or both a root and two opposite residues) stabilized by τ . Proof. By [62, 5.8], the diameter of R is less than the diameter of ∆. Thus if c is a chamber of R, then c 6∈ Opp(τ ). The claim holds now by 25.12 and 25.13. Corollary 25.17. The group Γ = hτ i does not stabilize any proper residues of ∆ if and only if ∆ is spherical and Opp(τ ) = ∆. Proof. If Γ does not stabilize any proper residue of ∆, then by 25.2, ∆ is spherical and Opp(τ ) = ∆. The converse holds by 25.16.

Chapter Twenty Six The Standard Metric Our next main goal is to prove 27.29. In this chapter we prove some results about groups generated by reflections and the standard metric on a BruhatTits building which will be needed in the proof of 27.29. The standard metric on a Bruhat-Tits building is the metric on its geometric realization introduced in [7, §2.5]; see 26.35. Affine spaces. Definition 26.1. A subset Y of a real vector space X is called an affine subspace if it is either empty or a coset of a linear subspace U . (The subspace U is, of course, uniquely determined by Y .) If Y = x+U for some x ∈ X and some linear space U , we define the dimension of Y to be the dimension of U , and we set the dimension of the empty set to be −1. An affine hyperplane is an affine subspace of co-dimension 1. The intersection of affine subspaces is again an affine subspace. The affine span hAiaff of a subset A of X is the intersection of all affine subspaces containing A. Definition 26.2. Let X and X ′ be two real vector spaces. An affine map from X to X ′ is a map of the form x 7→ φ(x) + u′ for some u′ ∈ X ′ and some linear map φ from X to X ′ . An affine transformation of X is a bijective affine map from X to itself. We fix a real vector space X. Notation 26.3. For x, y ∈ X, the segment [x, y] between them is the set {(1 − t)x + ty | t ∈ [0, 1]}. A subset a of X is called convex if [x, y] ⊂ a for all x, y ∈ a. The convex closure of a subset a of X is the intersection of all convex sets containing a; we will denote it by |a|. Remark 26.4. Suppose that a is a non-empty finite subset of X. Let k + 1 be the cardinality of a and let x0 , . . . , xk be the points in a. Then |a| is the set of sums (26.5)

k X

ti xi

i=0

for all t0 , . . . , tk in [0, 1] whose sum is 1. If, in addition, a is not contained in any affine subspace of dimension k − 1 (a vacuous condition if k = 0), then

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each element of |a| has a unique representation of the form (26.5). In this case, we denote by a◦ the set of all points of |a| that are not contained in |b| for any proper subset b of a. Thus a sum of the form (26.5) is contained in a◦ precisely when ti > 0 for all i ∈ [0, k] (and if k = 0, then |a| = a◦ = a). Remark 26.6. We have |a| ⊂ haiaff and if α is an affine map from X to a second real vector space X ′ , then α(|a|) = |α(a)| for all subsets a of X. Definition 26.7. A translation of X is a map of the form x 7→ x + u

for some u ∈ X.

Notation 26.8. We denote by AGL(X) the group of all affine transformations of X and by Trans(X) the set of all translations of X. The set Trans(X) is a normal subgroup of AGL(X) and AGL(X) = GL(X) · Trans(X).

We set Trans(H) = Trans(X) ∩ H for all subgroups H of AGL(X). Remark 26.9. The fixed point set of a subgroup Γ of AGL(X) is an affine subspace of X. We denote this subspace by X Γ . Euclidean spaces. Notation 26.10. Let M = (U, d) and let M ′ = (U ′ , d′ ) be two metric spaces and let µ be a positive real number. A µ-similarity from M to M ′ is a bijection α : U → U ′ such that d′ (α(x), α(y)) = µ · d(x, y) for all x, y ∈ U . A similarity from M to M ′ is a µ-similarity from some µ > 0 and an isometry is a 1-similarity. We denote by Isom(M ) the group of all isometries from M to itself. Hypothesis 26.11. From now on, we will assume that our real vector space X is of finite dimension n and that d is a Euclidean metric on X. We let E denote the Euclidean metric space (X, d). By [6, Thm. 2.24(1)], we have Trans(X) ⊂ Isom(E) ⊂ AGL(X). Definition 26.12. A reflection of E is an isometry of E whose fixed point set is an affine hyperplane. A reflection is uniquely determined by its fixed point set and has order 2; see [5, Chapter 2, §3]. Let Refl(E) denote the set of all reflections of E. We set Refl(G) = Refl(E) ∩ G

for all subgroups G of Isom(E).

Definition 26.13. A subset A of X is discrete if |A ∩ Br (x)| < ∞ for all positive real numbers r and all x ∈ X, where Br (x) denotes the ball of radius r with center x. A subgroup G of Isom(E) is called discrete if each G-orbit is a discrete subset of X.

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Definition 26.14. A lattice in E is a discrete subgroup of Trans(X) isomorphic to Zn . Remark 26.15. If T is a subgroup of Trans(X) isomorphic to Zn , then by [41, Chapter 1, 4.2], T is a lattice if and only if there exists a basis u1 , . . . , un of X such that {x 7→ x + ui | i ∈ [1, n]} is a basis of T (as a Z-module). The following is essentially [41, Chapter 1, 4.3]. Proposition 26.16. Let T be a subgroup of Trans(X) isomorphic to Zm for some m and suppose that [ X= Br (τ (x)) τ ∈T

for some x ∈ X and some positive r. Then m ≥ n and if m = n, then T is a lattice.

Proof. We choose u1 , . . . , um ∈ X such that {x 7→ x + ui | i ∈ [1, m]} is a basis of T and let A denote the linear subspace hu1 , . . . , um i of X. Choose v ∈ X. For each p ∈ N, there exists xp ∈ Br (x) and ap ∈ A such that pv = xp + ap . Since Br (x) is bounded, we have lim p−1 · xp = 0. p→∞

Thus the sequence {p−1 ·ap }p∈N converges to v. Since A is closed, we conclude that v ∈ A. Thus A = V . Hence m ≥ n and by 26.15, T is a lattice if m = n. Definition 26.17. Let a = {x1 , . . . , xk } be a finite set of k distinct points in X. The barycenter of a is the point a0 := (x1 + · · · + xk )/k. Definition 26.18. A half-space of X is a subset of the form {x | α(x) ≤ m} for some non-zero linear functional α : X → R and some m ∈ R and the wall of the half-space {x | α(x) ≤ m} is the affine hyperplane {x | α(x) = m}. An open half-space is a half-space with its wall removed. We will refer to the wall of a half-space also as the wall of the corresponding open half-space. Thus to every affine hyperplane H, there are exactly two half-spaces and two open half-spaces with wall H. Note that we are defining a half-space to be closed, but we will sometimes refer to closed half-spaces for emphasis. Remark 26.19. The convex closure |a| of a finite subset a of X equals the intersection of all the half-spaces containing a. Proposition 26.20. Let Γ be a finite subgroup of Isom(E), let a = {x1 , . . . , xk } ⊂ X be a Γ-orbit and let δ be an affine half-space containing the barycenter a0 of a. Then there exists an affine half-space δ1 containing a such that δ1 ∩ X Γ = δ ∩ X Γ , where X Γ is as in 26.9.

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Proof. The group Γ fixes the barycenter a0 of a. Replacing Γ by a conjugate, we can assume that a0 = 0 and that Γ, therefore, is linear. Let M be the linear subspace hx1 , . . . , xk i. We claim that Γ has no non-zero fixed points in M . To show this, we suppose that u is a non-zero element of M fixed by Γ. By Maschke’s Theorem, there exists a Γ-invariant complement M1 to hui in M . If x1 ∈ M1 , then a ⊂ M1 , but this would imply that M = hai ⊂ M1 . Hence x1 = y + z1 where y = tu for some non-zero t ∈ R and some z1 ∈ M1 . Applying Γ, we deduce that y − xi ∈ M1 for each i ∈ [1, k]. This implies, however, that
y = (y − x1 ) + · · · + (y − xk ) /k ∈ M1 since a0 = 0. Hence y = 0. Since t 6= 0, it follows that u = 0. With this contradiction, we conclude that M Γ = 0 as claimed. By Maschke’s Theorem again, we can choose a Γ-invariant complement Z of M in X. Since X = M ⊕ Z is a Γ-invariant decomposition, we have X Γ = M Γ ⊕ Z Γ and hence (26.21)

XΓ = ZΓ

by the conclusion of the previous paragraph. We have δ = {x | α(x) ≤ m} for some non-zero linear functional α and some constant m. Since δ contains a0 = 0, we have m ≥ 0. Choose p ≥ m such that α(xi ) ≤ p for each i ∈ [1, k] and let δ ′ = {x | α(x) ≤ p}. Then δ ⊂ δ ′ . If Z Γ ⊂ δ, then δ′ ∩ X Γ = δ′ ∩ Z Γ = Z Γ = δ ∩ Z Γ = δ ∩ X Γ

by (26.21) and hence we can set δ1 = δ ′ . Suppose, instead, that Z Γ 6⊂ δ. In this case, we let β be the linear functional which agrees with α on Z and sends M to 0. Since m ≥ 0 and Z 6⊂ δ, the linear functional β is non-zero. Thus δ1 := {x | β(x) ≤ m} is a half-space. We have a ⊂ δ1 , again because m ≥ 0, and δ1 ∩ X Γ = δ1 ∩ Z Γ = δ ∩ Z Γ = δ ∩ X Γ

by (26.21). Proposition 26.22. Let a be subset of X of cardinality k + 1 that is not contained in any affine subspace of dimension less than k, let b0 , . . . , bs be non-empty subsets forming a partition of a and let ui be the barycenter of bi for each i ∈ [0, s]. Then there is no (s − 1)-dimensional affine subspace containing {u0 , . . . , us }. Proof. Let ei be the cardinality of bi for each i ∈ [0, s]. Suppose that {u0 , . . . , us } is contained in an (s − 1)-dimensional affine subspace. This means that there exist t0 , . . . , ts ∈ R not all zero such that t0 u 0 + · · · + ts u s = 0

and t0 + · · · + ts = 0.

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THE STANDARD METRIC

Thus s X X

(ti /ei )x = 0

i=0 x∈bi

and the sum of the k +1 = e1 +· · ·+es coefficients in this sum is 0. Therefore a is contained in an affine subspace of dimension k − 1. This contradicts the hypothesis that a is not contained in such an affine subspace. Notation 26.23. Let a = {x0 , . . . , xk } be a subset of X of cardinality k + 1 that is not contained in any affine subspace of dimension less than k. If x ∈ a, then (as already observed in 26.4) there are unique t0 , . . . , tk in [0, 1] whose sum is 1 such that k X x= ti xi i=0

and b := {xi | ti 6= 0} is the unique subset of a such that x ∈ b◦ (where b◦ is as in 26.4). Thus a is partitioned by the sets b◦ for all subsets b of a containing at least two elements and the barycenter of a (as defined in 26.17) is contained in a◦ . Euclidean representations. For the rest of this chapter we assume the following: Hypothesis 26.24. Let E = (X, d) and n be as in 26.11 (so n ≥ 1), let Π be an irreducible affine Coxeter diagram of rank n + 1, let (W, S) be the corresponding Coxeter system and let ΣΠ be the corresponding Coxeter chamber system as in 19.7 and let c0 be a chamber of ΣΠ . We identify W with the group Aut◦ (Σ) of type-preserving automorphisms of Σ so that S is both the vertex set of Π and the set of elements of W mapping c0 to an adjacent chamber. Notation 26.25. Following the recipe in [65, Chapter 2], we use the canonical embedding of the appropriate root system into X to identify S with a subset of Refl(E) and W with the subgroup of Isom(E) generated by S. We denote by XΠ the Euclidean space E = (X, d) endowed with this set S of isometries and proceed with a series of observations about XΠ : (i) The set H of reflection hyperplanes corresponding to the elements of W conjugate to an element of S is locally finite (in the sense of 27.18 below). An alcove of XΠ is a connected component of the space X minus the union of all the reflection hyperplanes in H. For each alcove a, there exists a unique set Va consisting of n + 1 points of X such that the convex closure |Va | of Va equals the topological closure of a. Alcoves are not contained in any affine hyperplane of X. (ii) By [65, 2.28], we can identify the set of chambers of ΣΠ with the set of alcoves so that pairs of adjacent chambers correspond to pairs of alcoves a and a′ such that |Va ∩ Va′ | = n.

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(iii) There is a distinguished alcove a0 such that s(a0 ) is adjacent to a0 for all s ∈ S. Let Va0 = {v0 , . . . , vn }. Then for each i ∈ [0, n], there exists a unique si ∈ S fixing vj for all j 6= i in [0, n], and S = {s0 , . . . , sn }. (iv) A vertex of XΠ is a point of X contained in Va for some alcove a. For each vertex v of XΠ , the W -orbit W (v) intersects the set Va0 = {v0 , . . . , vn } in a unique element. We say that a vertex v is of type si if W (v) ∩ Va0 = {vi }. (v) A simplex of XΠ is a subset b of Va for some alcove a. The type of a simplex is the set of types of all the vertices in the simplex and the dimension of a simplex is its cardinality minus one. Thus a vertex and a simplex of dimension 0 are essentially the same thing. A facet is a set of the form b◦ for some simplex b, where b◦ is as defined in 26.4. Each point of XΠ is contained in a unique facet. (vi) There exists a unique W -equivariant bijection λ from the set of all residues of ΣΠ to the set of all simplices of XΠ that maps the chamber c0 in 26.24 to the simplex Va0 = {v0 , . . . , vn }, inverts inclusion and maps residues of type J ⊂ S to simplices of type S\J. In particular, the maximal residues of ΣΠ correspond under λ to the vertices of XΠ and the chambers of ΣΠ correspond to the maximal simplices of XΠ (and hence to the alcoves of XΠ ). (vii) By [65, 1.24], Trans(W ) ∼ = Zn . There is a minimal distance separating the vertices, W is discrete and Trans(W ) is a lattice in E. (viii) It follows from [65, 1.30] that Trans(W ) has finite index in W (since W acts sharply transitively on alcoves). (ix) Let T denote the group consisting of all translations of XΠ that normalize the set Refl(W ). Then T acts on the set of maximal simplices and hence we can identify T with a subgroup of Aut(ΣΠ ). By [65, 1.28], Aut(ΣΠ ) = T W . Hence we can think of Aut(ΣΠ ) as a subgroup of Isom(E). (x) By [65, 1.30 and 2.42], there exists i ∈ [0, n] such that Trans(W ) acts transitively on the set of vertices of type si . (xi) For each i ∈ [0, n], the group Wi := hS\{si }i fixes vi but does not fix any simplex of positive dimension containing vi . It follows that vi is the unique fixed point of the subgroup Wi in X. Notation 26.26. The space XΠ is endowed with reflection hyperplanes (i.e. the fixed point sets of the conjugates of elements of S in W ). We call the two half-spaces determined by one of these reflection hyperplanes (as in 26.18) the reflection half-spaces of X.

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THE STANDARD METRIC

Proposition 26.27. Let XΠ = (X, d, S) and W = hSi be as in 26.25, let E ′ = (X ′ , d′ ) be a second Euclidean metric space and let ρ be an injective homomorphism from W into Isom(E ′ ). Suppose that ρ(S) is contained in Refl(E ′ ) and that there exists a subgroup T1 of finite index in Trans(W ) such that ρ(T1 ) is a lattice in E ′ . Then there exists a unique affine isomorphism π : X → X ′ such that ρ(w) = πwπ −1 for all w ∈ W , and this isomorphism π is a similarity from E = (X, d) to E ′ . Proof. Let (W ′ , S ′ ) be the image of (W, S) under ρ. By 26.25(vii), T1 ∼ = Zn . ′ ′ Since ρ(T1 ) is a lattice in E , we have dim X = n (by 26.14). Since ρ(T1 ) has finite index in W ′ , it follows that W ′ satisfies condition (D′ 2) at the beginning of §3 in [5, Chapter V]. By [7, 1.3.2], Trans(W ) ∩ Z(W ) = 1. Hence ρ(T1 ) ∩ Z(W ′ ) = 1. Since ρ(T1 ) is a lattice in E ′ and of finite index in Trans(W ′ ), the group Trans(W ′ ) is also a lattice in E ′ . Let m be the index of ρ(T1 ) in Trans(W ′ ). If t ∈ Trans(W ′ ) ∩ Z(W ′ ), then tm ∈ ρ(T1 ) ∩ Z(W ′ ) and hence t = 1. Thus Trans(W ′ ) ∩ Z(W ′ ) = 1.

(26.28)

Since W ′ is generated by elements of Refl(E ′ ), we have W ′ ⊂ AGL(X ′ ). Let W ′′ denote the image of W ′ in AGL(X ′ )/Trans(X ′ ) and let B = {u1 , . . . , un } be a basis of X ′ obtained by applying 26.15 to Trans(W ′ ). The group W ′′ is finite and each element z in W ′′ is represented with respect to the basis B by a matrix Az with integer entries. By (26.28), the system of equations ′′

(Az − I)x = 0

for all z ∈ W has no non-zero rational solution. Therefore this system has no non-zero real solution. We conclude that W ′ does not centralize any nontrivial element of Trans(E ′ ). In other words, W ′ is essential as defined in [5, Chapter V, §3.7]. Since (W ′ , S ′ ) is an irreducible Coxeter system, it follows from [5, Chapter V, §3.7, Corollary] that W ′ is an irreducible reflection subgroup of Isom(E ′ ) as defined in [5, Chapter V, §3.7]. The claim holds, therefore, by [5, Chapter V, §4, Prop. 11]. Definition 26.29. A Euclidean representation of (W, S) is a pair (α, E ′ ), where E ′ = (X ′ , d′ ) is a Euclidean metric space and α is a bijection from X to X ′ (but not necessarily an affine map) such that the following hold: (a) The metric dα := d′ ◦ α on X is W -invariant. (b) There exists a subgroup T1 of finite index in Trans(W ) such that αT1 α−1 ⊂ Trans(X ′ ). (c) α (1 − t)x + ty) = (1 − t)α(x) + tα(y) for all alcoves a, all x, y ∈ Va and all t ∈ [0, 1]. Lemma 26.30. Let (α, E ′ ) be a Euclidean representation of (W, S), let d′ ˆ = αHα−1 for all H ⊂ W . Then the following and T1 be as in 26.29 and let H hold:

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(i) If a is a simplex of dimension k, then dimhα(a)iaff = k. (ii) Tˆ1 is a lattice in E ′ . (iii) Sˆ ⊂ Refl(E ′ ). Proof. Let vˆ = α(v) for all v ∈ X, let Aˆ = {ˆ v | v ∈ A} for all A ⊂ X and let w ˆ = α · w · α−1 for all w ∈ W . Thus the map v 7→ vˆ is an isometry from Eα := (X, dα ) to E ′ (but not necessarily an affine map) and the map w 7→ w ˆ ˆ . Hence (W ˆ , S) ˆ is an irreducible Coxeter is an isomorphism from W to W system of rank n + 1. We prove (i) by induction with respect to k. Let a = {x0 , . . . , xk } be a simplex of dimension k. Since W acts transitively on the set of alcoves of XΠ , we can assume that a is a subset of the alcove Va0 , where a0 is as in 26.25(iii). The claim is clear for k = 0. Suppose that k > 0 and let b denote the simplex {x1 , . . . , xk }. Then dimhˆbiaff = k −1 by induction. By 26.25(iii), ˆ there exists an element w ∈ W fixing x1 , . . . , xk but not x0 . Therefore W fixes xˆ1 , . . . , x ˆk but not x ˆ0 . Since w ˆ ∈ Isom(E ′ ), it fixes hˆbiaff pointwise. Therefore x ˆ0 6∈ hˆbiaff . Hence dimhˆ aiaff = 1 + dimhˆbiaff = k.

Thus (i) holds. It follows, in particular, that dimhˆ ciaff = n for every alcove c in X and hence dim X ′ ≥ n.

(26.31)

Let c = Va0 = {v0 , . . . , vn } be as in 26.25(iii). By 26.29(b), the elements of Tˆ1 are translations of X ′ . By 26.25(x), we can choose i ∈ [0, n] such that T acts transitively on the set of all vertices of type si . We reorder the vertices in c so that i = 0. Thus the orbits T (v0 ) and W (v0 ) are equal. Next, we choose R > max{dα (v0 , vi ) | i ∈ [1, n]}. By 26.29(c), we have α(|c|) = |ˆ c| and by 26.29(a) we have cˆ ⊂ BR (ˆ v0 ), where BR (ˆ v0 ) denotes the ball in X ′ of radius R with center vˆ0 with respect to the metric d′ . Hence |ˆ c| ⊂ BR (ˆ v0 ) since BR (ˆ v0 ) is convex. Thus α(w(|c|)) = w(α(|c|)) ˆ ⊂ w(B ˆ R (ˆ v0 ))

ˆ -invariant (by 26.29(a)), we have w(B for all w ∈ W . Since d′ is W ˆ R (ˆ v0 )) = BR (w(ˆ ˆ v0 )) for all w ∈ W . Since W acts transitively on the set of alcoves of ˆ (ˆ XΠ and the orbits Tˆ(ˆ v0 ) and W v0 ) are equal, we thus have ! [ [ [ X ′ = α(X) = α w(|c|) ⊂ BR (w(ˆ ˆ v0 )) = BR (ˆ τ (ˆ v0 )). w∈W

w∈W

τ ∈T

Since T1 is of finite index in T , it follows that there exists R1 ≥ R such that [ X′ = BR1 (ˆ τ (ˆ v0 )). τ ∈T1

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By 26.16, therefore, dim X ′ ≤ n. Thus

dim X ′ = n by (26.31). Hence by 26.16 again, Tˆ1 is a lattice in E ′ . Thus (ii) holds. Let i ∈ [0, n] be arbitrary and let s = si . By 26.25(iii), s acts trivially on the simplex c\{vi }. Therefore sˆ acts trivially on the set cˆ\{ˆ vi }. By (i) and (26.32), this set is contained in a unique affine hyperplane of X ′ . The element sˆ must act trivially on this hyperplane. By 26.12, therefore, sˆ is a reflection of E ′ . Thus (iii) holds.

(26.32)

Proposition 26.33. Let (α, E ′ ) be a Euclidean representation of (W, S). Then α is both an affine map and a similarity from E to E ′ . Proof. Let the maps w 7→ w ˆ and v 7→ vˆ be as in the proof of 26.30. By 26.27 and assertions (ii) and (iii) of 26.30, there exists a unique bijection π from X to X ′ that is both an affine map and a similarity from E to E ′ such that w ˆ = πwπ −1 for all w ∈ W . We only need to show that α = π. Let Wi be as in 26.25(xi) for some i ∈ [0, n]. Since vi is the unique fixed point of the subgroup Wi , vˆi is the unique fixed point of ˆ i := hˆ W sj | j 6= ii = πWi π −1 . Therefore π(vi ) = α(vi ) for all i ∈ [0, n]. Since π is an affine map, it follows from 26.29(c) that π and α coincide on |c|, where c = Va0 = {v0 , . . . , vn } is as in 26.25(iii). Since [ X= w(|c|) w∈W

and

α(w(x)) = w(α(x)) ˆ = w(π(x)) ˆ = π(w(x)) for all x ∈ |c| and all w ∈ W , we conclude that α = π. The standard metric. We continue with all the hypotheses and notation in 26.24 and 26.25. In particular, Π is an irreducible affine Coxeter diagram. Theorem 26.34. Let Ξ be a thick building of type Π and let G = Aut(Ξ). Then there exists: (a) a metric space (D, ∂) and a subset V of D; (b) a bijection π from the set of maximal residues of Ξ to V ; (c) a set A of convex subsets AΣ of D, one for each apartment Σ of Ξ, such that D is the union of the subsets in A; (d) an action of G on (D, ∂) by isometries such that V is G-invariant, the map π is G-equivariant and g(AΣ ) = Ag(Σ) for each apartment Σ and each g ∈ G;

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(e) an isometry ξΣ from X to AΣ mapping the set of vertices of X to V ∩AΣ , one for each apartment Σ of Ξ, such that for all apartments Σ and Σ′ , −1 −1 ξΣ′ ◦ ξΣ acts trivially on AΣ ∩ AΣ′ and ξΣ (AΣ ∩ AΣ′ ) is an intersection of reflection half-spaces of X; and (f) an injective inclusion-reversing G-invariant map β from the set of all proper residues of Ξ to the set of subsets of V extending the map π such that for each proper residue R and for each apartment Σ containing chambers of R, there is a simplex a of X such that β(R) = ξΣ (a). If (D′ , ∂ ′ ), . . . , β ′ is another set of data satisfying (a)–(f), then there exists a unique similarity ψ from (D, ∂) to (D′ , ∂ ′ ) such that β ′ = ψ ◦ β. Proof. The space set D endowed with the subsets |β(R)| for all proper residues R of Ξ can be identified with the set J endowed with its facets constructed in [7, 2.1.1]. With this identification, the subsets AΣ of D correspond to the apartments of J as defined in [7, 2.2.2]. The existence and uniqueness (up to similarity) of the metric ∂ holds, therefore, by [7, 2.5.1]. Definition 26.35. Let Ξ, (D, ∂) and β be as in 26.34. We identify Ξ (viewed as a simplicial complex) with its image under β. With this identification, we are endowing Ξ with the additional structure of a metric space. We call ∂, which is unique up to similarity, the standard metric on Ξ. We call (D, ∂) endowed with the simplicial complex β(Ξ) the metric realization of Ξ. It follows from [1, 11.53] (and the thickness of Ξ) that β(Ξ) can be reconstructed from the metric space (D, ∂). This justifies referring to (D, ∂) alone as the metric realization of Ξ. Once we have made the identification of Ξ with β(Ξ) ⊂ D, we think of residues as simplices, chambers as maximal simplices and maximal residues as vertices in D and we think of an apartment Σ of Ξ as the subspace AΣ of D. In fact, we will also start to use letters like ˜ etc., rather than AΣ , A ˜ or Σ, Σ ˜ to denote apartments. A, A, Σ Remark 26.36. Each apartment A of D is endowed with reflection hyperplanes and reflection half-spaces via the isometry in 26.34(e). Remark 26.37. Let x be a point of D. By 26.34(c), there exist apartments containing x. In each of these apartments, there exists a unique simplex a such that x is contained in the facet a◦ . It follows from 26.34(e) that the same facet a◦ is contained in all the apartments of D containing x. Thus a◦ is the unique facet of D containing x. Remark 26.38. Let a be a simplex of D and let A be an apartment containing a. In A we can describe the elements of |a| and the barycenter of a as linear combinations of the elements of a as in 26.5. It follows from 26.34(e) that these descriptions are independent of the choice of the apartment A containing a. We conclude this chapter with one more observation. It will be applied in the proof of 27.14.

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Proposition 26.39. Let ∂ ′ be a metric on D whose restriction to A is similar to the restriction of ∂ to A for each apartment A of Ξ. Then (D, ∂ ′ ) is similar to (D, ∂). Proof. Let A1 and A2 be two apartments and let c1 and c2 be chambers of A1 and A2 (which we think of now as simplices of D of dimension n). By [62, 8.6], there exists an apartment A3 containing both c1 and c2 . By hypothesis, there exists a constant µi > 0 such that ∂ ′ coincides with µi · ∂ on Ai for each i ∈ [1, 3]. If v and v ′ are two vertices in c1 , then µ1 ·∂(v, v ′ ) = ∂(v, v ′ ) = µ3 · ∂(v, v ′ ) and hence µ1 = µ3 . Similarly, µ2 = µ3 . Hence µ1 = µ2 .

Chapter Twenty Seven Affine Fixed Point Buildings The main goal of this chapter, which we achieve in 27.29, is to show that if Ξ is an affine building and Γ is a finite descent group of Ξ, then Γ is a descent group of Ξ∞ and (ΞΓ )∞ ∼ = (Ξ∞ )Γ . By 26.34, ΞΓ and Ξ can be viewed as metric spaces. The key intermediate step in the proof of 27.29 is 27.14 in which we show that the first of these metric spaces is similar to the subspace of the second consisting of the fixed points of Γ. Hypothesis 27.1. We continue with all the assumptions, notation and identifications in 26.24 and 26.25. In particular, Π is an irreducible affine Coxeter diagram. Let Ξ be a thick building of type Π, let Γ be a finite descent group of Ξ as defined in 22.19 (see 22.28) and let T = (Π, Θ, A), ˜ and (W ˜ , S) ˜ be as in 22.25 applied to Ξ and Γ. By 20.43 and 20.40, Ξ =Ξ ˜ , S) ˜ is irreducible and affine. This means that the the Coxeter system (W ˜ is also irreducible and affine. In particular, the rank |S| ˜ of Ξ ˜ is building Ξ at least 2. Let (D, ∂) be the metric realization of Ξ as described in 26.34 and 26.35. By 26.34(d), there is an action of G in (D, ∂) by isometries. We denote by DΓ the set of fixed points of Γ ⊂ G in D. Next we identify Ξ with (D, ∂) via the G-equivariant map β in 26.34(f); see also 26.35. Finally, we ˜ be the geometric realization of ∆ ˜ ˜ ∂) ˜ and we identify Ξ ˜ with (D, ˜ ∂) let (D, ˜ ˜ via the map β obtained by applying 26.34(d) to ∆. Γ

Notation 27.2. We call a simplex of Ξ a Γ-simplex if it is stabilized by ˜ is the building whose simplices are the Γ-simplices of Ξ. Let G ˜ Γ. Thus Ξ denote the centralizer of Γ in G := Aut(Ξ). ˜ means that we ˜ with (D, ˜ ∂) Comment 27.3. Identifying Ξ with (D, ∂) and Ξ ˜ simultaneously as simplices of D, ˜ Γ-simplices are thinking of residues of Ξ ˜ of D and Γ-residues of Ξ. Thus, in particular, to each simplex a ˜ of D, there is a corresponding Γ-simplex a of D and a corresponding Γ-residue R of Ξ. These are all different ways of looking at the same object. Note, ˜ are now certain subsets too, that the apartments of Ξ (respectively, of Ξ) ˜ of D (respectively, of D) and we refer to these subsets as apartments of D ˜ (respectively, of D).

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˜ and let a be the corresponding Notation 27.4. Let a ˜ be a simplex of D Γ-simplex of D in the sense of 27.3. By 26.38, we can let b be the set of barycenters (as defined in 26.17) of all the Γ-orbits in a. Thus b ⊂ DΓ . We ˜ to the set of subsets of denote the map a ˜ 7→ b from the set of simplices of D ˜ DΓ by κ. Note that κ is G-equivariant and that |κ(˜ a)| = 1 for all vertices a ˜ ˜ Thus κ restricts to a map from the set of vertices of D ˜ to DΓ itself. of D. Notation 27.5. We define a simplex of DΓ to be the image of a simplex of ˜ under the map κ defined in 27.4. Let x˜ ∈ D. ˜ By 26.37, there is a unique D ˜ simplex a ˜ = {˜ v0 , . . . , v˜k } of D such that x ˜ is contained in the facet a ˜◦ and by 26.4 (and 26.38), there exist unique ti ∈ [0, 1] for all i ∈ [0, k] such that x ˜ = t0 v˜0 + · · · + tk v˜k .

We set κ(˜ x) = t0 κ(˜ v0 ) + · · · + tk κ(˜ vk ). ˜ ˜ This extends κ to G-equivariant map from D to DΓ mapping simplices to ˜ is as in 27.2. simplices, where G Proposition 27.6. The map κ in 27.4 is an inclusion-preserving bijection ˜ to the set of simplices of DΓ . from the set of simplices of D Proof. This map is surjective and inclusion-preserving by 27.5. Suppose that ˜ have the same barycenter x. By 26.37, there is a two simplices a ˜ and ˜b of D ˜ unique simplex c˜ of D such that x ∈ c˜◦ . By 26.23, we must have a ˜ = c˜ = ˜b. Γ ˜ Therefore κ maps distinct vertices of D to distinct points of D . It follows ˜ to distinct simplices of DΓ . that κ maps distinct simplices of D By 27.6, the simplices of DΓ have the structure of a simplicial complex ˜ # (see 1.31). From now on, we isomorphic via κ to the simplicial complex Ξ Γ will think of D as being endowed with its simplices. Proposition 27.7. Let a be a Γ-simplex of D and let a ˜ be the corresponding ˜ in the sense of 27.3. Then the following hold: simplex of D (i) The restriction of κ to the convex hull |˜ a| is a bijective map from |˜ a| to |κ(˜ a)|. (ii) |κ(˜ a)| = |a| ∩ DΓ . Proof. Assertion (i) follows from 26.4 and 26.22 and assertion (ii) from 26.19 and 26.20. ˜ → DΓ is a bijection. Proposition 27.8. The map κ : D Proof. Let x ∈ DΓ . By 26.37, there exists a unique simplex a of D such that x is contained in the facet a◦ . Hence x ∈ |a| ∩ DΓ . Since a is unique and ˜ in the Γ fixes x, a is a Γ-simplex. Let a ˜ be the corresponding simplex of D ˜ sense of 27.3. Then x ∈ |κ(˜ a)| ⊂ κ(D) by 27.7(ii). Thus κ is surjective. Now ˜ By 26.37 applied to D, ˜ suppose that κ(˜ x) = κ(˜ x′ ) for two points x ˜, x ˜′ ∈ D.

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˜ such that x there exist unique simplices a ˜ and a ˜′ of D ˜∈a ˜◦ and x ˜′ ∈ (˜ a′ )◦ . ′ ′ Let a and a be the Γ-simplices of D corresponding to a ˜ and a ˜ in the sense of 27.3. By 27.5, κ(˜ x) is a linear combination of all the vertices of a with every coefficient in the half-closed interval (0, 1] and hence κ(˜ x) ∈ a◦ . Similarly, ′ ′ ◦ ′ κ(˜ x ) ∈ (a ) . Since κ(˜ x) = κ(˜ x ), we conclude that a = a′ by 26.37. Hence ′ a ˜=a ˜ . By 27.7(i), therefore, x ˜ = x˜′ . Thus κ is injective. In the next result, AΓ = A ∩ DΓ is the set of fixed points of Γ in A; we are not implying with this notation that A is, as a whole, stabilized by Γ. Recall, too, the observation about apartments in 27.3. ˜ Then there exists an apartProposition 27.9. Let A˜ be an apartment of Ξ. Γ ˜ ment A of Ξ such that κ(A) = A . Proof. By 23.19, there exists an apartment A of D that contains the Γsimplex a corresponding to a ˜ (in the sense of 27.3) for every simplex a ˜ ˜ Let a be an arbitrary Γ-simplex in A and let a contained in A. ˜ be the ˜ corresponding to a (again, in the sense of 27.3). By 23.33, simplex of D ˜ a ˜ ⊂ A. Let x ∈ AΓ be arbitrary. Let a be the unique simplex of D such that x is contained in the facet a◦ . Then a is a Γ-simplex. By 26.37, a ⊂ A. Thus a is the Γ-simplex corresponding to a simplex a ˜ of A˜ by the conclusion of the previous paragraph. Therefore every point of AΓ is in |a| ∩ AΓ for ˜ By 27.7(ii), therefore, some Γ-simplex a corresponding to a simplex a ˜ of A. Γ ˜ κ(A) = A . Proposition 27.10. Let A˜ and A be as in 27.9. Then AΓ is an affine subspace of A. Proof. By 23.32, there exists a T-group Ω of A such that maximal Ωsimplices are precisely the maximal Γ-simplices in A. Note that by 26.25(ix), Ω ⊂ Isom(A). Let x ∈ A and let a be the unique simplex of A such that x ∈ a◦ . Then Ω stabilizes a if and only if Γ does. Suppose Ω and Γ both stabilize a. Since Ω and Γ both induce the same group Θ on the Coxeter diagram Π, they induce the same group on the vertices of a. By 26.4, it follows that Ω and Γ have the same fixed points in the convex closure |a|. We conclude that Ω fixes x if and only if Γ does. By 26.9 and 26.12, therefore, AΓ = AΩ is an affine subspace of A. Remark 27.11. By 27.9, there is a unique surjective map α from A˜ to AΓ ˜ such that α(x) = κ(x) for all x ∈ A. Proposition 27.12. Let A˜ and A be as in 27.9 and let α be as in 27.11. ˜ , S) ˜ as defined in Then the pair (α, AΓ ) is a Euclidean representation of (W 26.29. Proof. By 27.10, AΓ is a Euclidean space with metric d equal to the restriction of ∂ and by 27.8, α is injective. Thus α is bijective.

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By 26.34(e), the group W acts on A. This action is by isometries and ˜ we the elements of Trans(W ) act by translations. By 26.34(e) applied to Ξ, ˜ on A. ˜ The automorphisms x 7→ w(x) of A and also have an action of W ˜ and every x ˜ 7→ w(˜ ˜ x) of A˜ are type-preserving for all w ∈ W , and all w ˜∈W ˜ is of this form. Let type-preserving automorphism of A (respectively, A) ˜ . By 23.27, every type-preserving automorphism of A˜ extends to a w ˜ ∈W ˜ there type-preserving automorphism of A. Thus to every element w ˜ of W ˜ exists an element w of W whose action on A extends the action of w ˜ on A. Let w, ˜ w be such a pair. If a is a Γ-simplex in A, the Γ-orbits in a depend only on the group Θ. Since w is type-preserving, it maps Γ-orbits in a to Γ-orbits in w(a). By 27.4 (and 27.5), therefore, we have (27.13) w(α(u)) = α(w(u)) ˜ ˜ (In particular, w stabilizes AΓ .) Hence for all u ∈ A.

d α(w(x)), ˜ α(w(x ˜ ′ )) = d w(α(x)), w(α(x′ )) = d(α(x, x′ )) ˜ Thus 26.29(a) holds. By 27.5, also 26.29(c) holds. It for all x, x′ ∈ A. remains only to show that 26.29(b) holds. By 26.25(viii), the index m of T := Trans(W ) in W is finite and T˜ := ˜)∼ Trans(W = Zk for some k. Let T˜1 = T˜ m . Then T˜1 is of finite index in T˜. Choose w ˜ ∈ T˜ and let w be an element of W whose action on A extends the ˜ By (27.13), αwα action of w ˜ on A. ˜ −1 is the restriction of w to AΓ . Hence m −1 m αw˜ α is the restriction of w to AΓ . Since wm ∈ Trans(W ) ⊂ Trans(A), we conclude that αw ˜m α−1 ∈ Trans(AΓ ). Therefore αT˜1 α−1 ⊂ Trans(AΓ ). Thus 26.29(b) holds. ˜ be as in 27.1 and let κ be as in ˜ ∂) Theorem 27.14. Let (D, ∂) and (D, ˜ ˜ 27.5. Then κ is a similarity from (D, ∂) to (DΓ , ∂ Γ ), where ∂ Γ denotes the restriction of ∂ to DΓ . Proof. By 27.12, we can apply 26.33 to each apartment. The claim holds, therefore, by 26.39. The metric ∂˜ is unique only up to similarity. Thus we can adjust ∂˜ so that ˜ to (DΓ , ∂ Γ ). This allows us to identify ˜ ∂) κ is, in fact, an isometry from (D, these two metric spaces via κ. Notation 27.15. The assertions in 27.9 and 27.10 can now be rephrased ˜ is contained in an apartment of D to say simply that every apartment of D ˜ then A˜ is an and if an apartment A of D contains an apartment A˜ of D, ˜ affine subspace of A equal to A ∩ D. Remark 27.16. As noted in 26.36, all the apartments of D are endowed with reflection hyperplanes and reflection half-spaces as are the apartments ˜ The reflection hyperplanes are, in fact, hyperplanes of the respective of D. ˜ such that Euclidean spaces. Thus if A and A˜ be apartments of D and D ˜ ˜ ˜ A ⊂ A and H and H are reflection hyperplanes such that H ⊂ H, then ˜ either A ⊂ H or H ∩ A = H.

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Proposition 27.17. Let A be an apartment of D containing an apartment A˜ ˜ and let H ˜ be a reflection hyperplane of A. ˜ Then there exists a reflection of D ˜ hyperplane H of A such that H ∩ A˜ = H. ˜ , S) ˜ whose Proof. We identify A with XΠ . Let t˜ be the reflection of (W ˜ By [5, Chapter V, §3.3, Prop. 2], there exist reflection hyperplane is H. reflections t1 , . . . , tm of (W, S) whose reflection hyperplanes H1 , . . . , Hm all ˜ such that t˜ = t1 · · · tm . By 27.16, we know that for each i ∈ [1, m], contain H ˜ or A˜ ⊂ Hi . It follows that Hi ∩ A˜ = H ˜ for some i. either Hi ∩ A˜ = H Definition 27.18. Let V be a real vector space. A set U of affine subspaces of V is locally finite if for every v ∈ V , there is an open ball B around v which contains points of only finitely many elements of U. Note that if U is a locally finite set of affine hyperplanes of V and M is an arbitrary affine subspace, then the set of affine hyperplanes of M which are the intersection with M of an element of U is locally finite. Lemma 27.19. Let V be a non-trivial finite-dimensional real vector space and let U be a locally finite set of affine hyperplanes. Then V is not the union of the subspaces in U. Proof. We use induction with respect to n := dim V . Let U be the union of all the subspaces in U. If n = 1, then U is a discrete set of points and hence U 6= V . Suppose that n > 1. We can assume that U contains at least two elements M1 and M2 . By induction and the observation in 27.18, we can choose points x1 ∈ M1 and x2 ∈ M2 such that M1 is the only element of U containing x1 and M2 is the only element of U containing x2 . Thus the unique affine line L containing x1 and x2 is not contained in an element of U. It follows (again by induction and the observation in 27.18) that the set of points in L contained in an element of U is a discrete set of points. Therefore L 6⊂ U . Proposition 27.20. Let A be an apartment of D containing an apartment ˜ and let H be a reflection hyperplane of A containing points in A. ˜ A˜ of D ˜ ˜ ˜ Then either A ⊂ H or H ∩ A is a reflection hyperplane of A. Proof. We identify A with XΠ and let t be the reflection of (W, S) whose reflection hyperplane is H. Suppose that H ∩ A˜ is not a reflection hyperplane ˜ By 26.25(i), the set of reflection hyperplanes of A˜ is locally finite. of A. By 27.18 and 27.19, it follows that H ∩ A˜ contains a point x which is not ˜ Let a contained in any reflection hyperplane of A. ˜ be the unique simplex of ◦ ˜ A such that x ∈ a ˜ and let a be the unique simplex of A such that x ∈ a◦ . Since Γ and t both fix x, they both stabilize a. Since t is type-preserving, it acts trivially on a and hence a ⊂ H. By 27.5, the vertices of a ˜ are the barycenters of the Γ-orbits of vertices in a. Therefore a ˜ ⊂ H. Since x is ˜ the simplex a not contained in any reflection hyperplane of A, ˜ is a maximal ˜ simplex of A. In particular, there is no proper affine subspace of A˜ containing a ˜. Therefore A˜ ⊂ H.

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Definition 27.21. Let A be an apartment of D. We will call an affine hyperplane of A a wall and we will say that a wall of A is true if it is parallel to a reflection hyperplane. A true half-space is a half-space whose wall is true and true open half-space is a true half-space with its wall removed. We ˜ analogously. define a wall and a true wall of an apartment of D Proposition 27.22. Let A be an apartment of D containing an apartment ˜ let H be the set of true walls of A whose intersection with A˜ is a A˜ of D, non-empty proper subset of A˜ and let K be the set of true open half-spaces of A whose wall is in H. Then H 7→ H ∩ A˜ is a surjection from H to the set of true walls of A˜ and γ 7→ γ ∩ A˜ is a surjection from K to the set of true ˜ open half-spaces of A. Proof. Suppose that H is a hyperplane of A. If A ⊂ H, then H ′ ∩ A˜ = ∅ for all hyperplanes H ′ parallel to H. Thus if H ∩ A˜ is a non-empty proper subset ˜ then H ′ ∩ A˜ is a non-empty proper subset of A˜ for every hyperplane of A, parallel to H. It follows that if H is H, then there is a reflection hyperplane H ′ in H that is parallel to H. The first claim is a consequence, therefore, of 27.17 and 27.20 and the second claim follows from the first. Definition 27.23. Let x be a point of an apartment A of D and let Hx be the set of all true walls of A containing x. The set Hx defines a decomposition of A\{x} into conical cells: Two points y and z of A\{x} belong to the same conical cell in this decomposition if and only if for each H ∈ Hx , either y and z are both contained in H or they are both contained in the same open half-space with wall H. As in [1, 11.44] we call these cells conical cells based at x. (In [5, Chapter V, §1.2] they are called facets.) A conical cell of D is a conical cell C of some apartment based at some point of that apartment which we refer to as the cone point of C. We define conical cells ˜ analogously. A Γ-cone is a conical cell of D that is stabilized by Γ. of D Remark 27.24. Let A be an apartment, let C be a conical cell in A with cone point x, let Hx be as in 27.23 and let A′ be a second apartment containing C. By 26.34(e), there is an isometry from A to A′ acting trivially on A ∩ A′ and mapping Hx to the set Hx′ of true walls of A′ containing x. It follows that C is also a conical cell in A′ . Thus the notion of a conical cell is independent of the choice of the apartment in which the conical cell is contained. Definition 27.25. A ray is the image of an isometry from the interval [0, ∞) of R into D. The point γ(0) is called the initial point of the ray. By [1, 11.53], every ray is contained in an apartment. Let r be a ray with initial point x and let A be an apartment containing r. The conical closure (in A) of r is the unique conical cell C of A with cone point x that contains r\{x}. Let A′ be a second apartment containing r and let C ′ be the conical closure of r in A′ . By 26.34(e), A ∩ A′ is an intersection of closed reflection half-spaces of A. A closed reflection half-space of A contains r if and only if it contains C.

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Thus C ⊂ A′ . By 27.24, it follows that C ′ = C. Thus the conical closure of r is independent of the choice of the apartment in which the ray r is contained. Proposition 27.26. Let C be a Γ-cone. Then the following hold: (i) C ∩ A˜ 6= ∅. (ii) There exists an apartment A of D containing both C and some apart˜ ment of D. Proof. Let x be the cone point of C. Since Γ is finite and C is convex, we can choose a fixed point y of Γ in C. Thus (i) holds. Let r be the ray with initial point x containing y. Then r is contained in C, so C is the conical ˜ By [1, 11.53], closure of r, and Γ fixes every point on r. Thus r is a ray of D. ˜ ˜ we can choose an apartment A of D containing r. By 27.15, we can choose ˜ By 27.25, the apartment A contains the an apartment A of D containing A. conical closure C of r. Thus (ii) holds. Proposition 27.27. Let A be an apartment of D containing an apartment ˜ let x ∈ A, ˜ let C be the set of all Γ-cones in A with cone point x, let A˜ of D, ˜ C be the set of all conical cells of A˜ with cone point x and let π(C) = C ∩ A˜

for every C ∈ C. Then π is a bijective map from C to C˜ and for each C ∈ C, C is contained in every apartment of D which contains π(C). Proof. Let Hx be the set of true walls of A containing x (as in 27.23) and ˜ Choose C ∈ C. Then let Hx′ be the set of walls in Hx that do not contain A. ′ ˜ ˜ C ∩ A is the intersection of H ∩ A for all H in Hx containing C with γ ∩ A˜ for all open half-spaces γ containing C whose wall is in Hx′ . By 27.22 and ˜ 27.26(i), it follows that π(C) := C ∩ A˜ is an element of C. ˜ ˜ ˜ For each C ∈ C, let C := ψ(C) be the intersection of all the affine hyper˜ with all the planes H in Hx containing C˜ (including those that contain A) ˜ open half-spaces γ of A containing C whose wall is in Hx . The cone C is stabilized by Γ since C˜ is stabilized by Γ. By 27.22 again, ψ is a map from C˜ to C and is, in fact, the inverse of π. Thus π is a bijection. Now let C ∈ C and let A′ be a second apartment containing π(C). Then C is contained in the intersection of all the closed reflection half-spaces of A containing π(C). By 26.34(e), A ∩ A′ is bounded by closed reflection half-spaces of A. Hence C ⊂ A′ . ˜ with cone point x. Then Proposition 27.28. Let C˜ be a conical cell of D ˜ = C. ˜ there exists a unique Γ-cone C with cone point x such that C ∩ D ˜ containing C. ˜ By 27.15, there exists Proof. Let A˜ be an apartment of D ˜ an apartment A of D containing A. By 27.27, there exists a Γ-cone C in ˜ = C˜ and C is contained in every A with cone point x such that C ∩ D ˜ apartment containing C. Suppose that C ′ is a second Γ-cone with cone ˜ = C. ˜ By 27.26(ii), we can choose an apartment point x such that C ′ ∩ D

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˜ Then C ⊂ A′ (since A′ of D containing both C ′ and some apartment of D. ′ ˜ and by 27.24, C is, in fact, a conical cell A is an apartment containing C) ˜ = C ′ ∩ D. ˜ By 27.27 applied to A′ , it follows that in A′ . We have C ∩ D ′ C =C . We come now to the main result of this chapter. Theorem 27.29. Let Ξ and Γ be as in 27.1. Then Γ is a descent group of ∆ := Ξ∞ and there is a canonical isomorphism from the fixed point building ∆Γ to (ΞΓ )∞ . ˜ let C denote the set of all Γ-cones Proof. Let x be an arbitrary point of D, ˜ ˜ of D with cone point x and let C denote the set of all conical cells of D ′ with cone point x. We denote by C the set obtained from C by replacing each conical cell in C by its topological closure and we denote by C˜′ the set ˜ obtained analogously from C. ˜ Let ∆ denote the simplicial complex consisting of the Γ-simplices of ∆ ˜ = ∆Γ ). By [1, 11.75], there are canonical containment-preserving (so ∆ bijections ˜ and from C ′ to ∆

C 7→ C ∞ C˜ 7→ C˜ ∞

˜ ∞ , i.e. to (ΞΓ )∞ . By 27.28, we conclude from C˜′ to the set of simplices of D ˜ that the map C 7→ C ∩ D from C ′ to C˜′ induces an isomorphism ξ from ˜ = ∆Γ to D ˜ ∞ = (ΞΓ )∞ . ∆ ˜ ˜ ∞ is also thick (by [65, 11.3]). Since D is a thick building, the building D ˜ to D ˜ ∞ , it follows that Since ξ is a bijection of simplicial complexes from ∆ Γ ˜ D = D is also a thick building. Hence Γ is a descent group of ∆. Remark 27.30. It follows from the assumption that Ξ is thick that the construction of Ξ∞ in [1, §11.8] (which we used in the proof of 27.29) and the construction of Ξ∞ in [65, Chapter 8] yield the same building. Remark 27.31. Let Ξ, Γ and ∆ be as in 27.29. Let n + 1 denote the rank of Ξ, let k + 1 denote the rank of ΞΓ . Thus n is the rank of ∆ and k is the rank of ∆Γ . By [65, 8.25], ∆ is irreducible and by [65, 11.3], ∆ is thick. Suppose that Ξ is a Bruhat-Tits as defined in 1.20 (which, by 1.4, is automatic if n ≥ 3 since ∆ is thick and irreducible). Thus n ≥ 2 by 1.1. If k ≥ 2, then ∆Γ is Moufang by 24.31(i) and hence (by 27.29) ΞΓ is also a Bruhat-Tits building. If instead k = 1, then ΞΓ is a tree, ∆Γ is its set of ends and by 24.31(ii), the pair (∆Γ , MΓ ) is a Moufang set, where M is the standard Moufang structure on ∆.

PART 4

Galois Involutions

Chapter Twenty Eight Pseudo-Split Buildings Our goal of this chapter is to make precise the notion of the field of definition of a spherical building satisfying the Moufang condition (in 28.8), and to introduce a class of Moufang spherical buildings we call pseudo-split (in 28.16). In 28.22 we characterize pseudo-split buildings as the spherical buildings which can be embedded in a split building (see 28.10) of the same type. As in Chapter 18, we rely heavily in this chapter on the summary of the classification of Moufang spherical buildings in terms of root group sequences contained in Appendix B of [65]. Notation 28.1. Let ∆ be an irreducible spherical building satisfying the Moufang condition as defined in 1.1, and let ℓ denote its rank (so ℓ ≥ 2 by definition). Then by [60, 17.1 and 17.7], [62, 5.17 and 11.8] and [65, 30.14–30.15], one of the following holds: (i) ∆ ∼ = Xℓ (Λ) for X = A, D, E, F or G, X (ii) ∆ ∼ = BX ℓ (Λ) = Cℓ (Λ) for X = Q, I, P, D, E or F ,

where Λ is a suitable parameter system as described in [65, 30.14], ℓ = 4 if X = F and ℓ = 2 if X = G in (i) and ℓ = 2 if X = D, E or F in (ii), or (iii) ∆ is isomorphic to a Moufang octagon O(Λ) for some octagonal set Λ as defined in [60, 10.11] and ℓ = 2. Notation 28.2. We will say that ∆ is unitary if it is as in case (iv)–(vii) of [65, 30.14]. In cases (iv), (vi) and (vii), it follows from [60, 35.7 and 35.10] that K is an invariant of ∆ and the involution σ is an invariant up to inner automorphisms. Thus the set F := Z(K)σ of fixed points of σ in the center Z(K) of K is also an invariant of ∆. Suppose that ∆ and (K, F, σ) is as in case (v) of [65, 3.14]. Then (K, F ) is a composition algebra, σ is its standard involution, F = Z(K)σ and QI (K, F, σ) = QQ (F, K, q), where q is the norm of this composition algebra. By [60, 35.8], therefore, the quadratic space (K, F, q) is an invariant of ∆ up to similarity. By [60, 20.28], it follows that the composition algebra (K, F ) is an invariant of ∆. We conclude that F = Z(K)σ is an invariant of ∆ also in this case. Definition 28.3. The building ∆ is indifferent if it is isomorphic to one of the following:

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(i) BD 2 (Λ) for Λ = (K, A, B) an indifferent set such that B 6= K and if A = K, then B = 6 K 2. (ii) F4 (Λ) for Λ = (L, K) an inseparable composition algebra as defined in [65, 30.17]. (iii) G2 (Λ) for Λ = (L/K)◦ an inseparable hexagonal system as defined in [65, 30.23]. The building ∆ is mixed if it is either indifferent or isomorphic to BF 2 (Λ) for Λ = (K, L, q) a quadratic space of type F4 . Notation 28.4. In case (i) of 28.3, we set F0 = K and F1 = L1/2 if B 6= K 2 , where L denotes the subfield of K generated by B; if B = K 2 , we set F0 = K 2 and F1 = K. In cases (ii) and (iii), we set F0 = K and F1 = L. If ∆ is isomorphic to BF 2 (Λ) for Λ = (K, L, q) a quadratic space of type F4 , we let F0 be the subfield of K called F in [60, 14.3] (which is a proper subfield of K by [60, 14.8(iii)]) and set F1 = K. Then in every case, F1 /F0 is a proper inseparable extension such that F1p ⊂ F0 , where p = char(F0 ). Notation 28.5. Let F1 /F0 be a field extension in characteristic p > 0 such that F1p ⊂ F0 . Then F1p is a subfield of F0 canonically isomorphic to F1 that contains F0p . It is thus ambiguous which of the two fields F0 or F1 is a subfield of which, and we can write F1 /F0 or F0 /F1 depending on which of the two fields we want to think of as the larger. Since an automorphism of the smaller field extends uniquely to an automorphism of the larger field, there is a canonical isomorphism between Aut(F1 /F0 ) (i.e. the subgroup of Aut(F1 ) consisting of those elements that map F0 to itself) and Aut(F0 /F1 ) and we will use this isomorphism to simply think of Aut(F1 /F0 ) and Aut(F0 /F1 ) as the same group which we will denote by Aut(F0 , F1 ). Remark 28.6. Suppose that ∆ is mixed as defined in 28.3 and let F1 /F0 be the field extension defined in 28.4. By [60, 35.9, 35.12 and 35.13], the pair of field extensions {F1 /F0 , F0 /F1 } is an invariant of ∆. Hence also the group Aut(F1 , F0 ) defined in 28.5 is an invariant of ∆. Notation 28.7. Suppose that ∆ is neither unitary or mixed. We let K be as [60, 16.9] if ∆ is a Moufang octagon and we let K be as in [65, 30.14] in every other case. By [65, 30.29] and [60, 35.14], F := Z(K) is an invariant of ∆. Definition 28.8. In the non-mixed cases, we define the field of definition (or the defining field) of ∆ to be the field F described in 28.2 in the unitary case and in 28.7 in the non-unitary cases. In each of these cases, the field F is an invariant of ∆. In the mixed cases, we define the defining extensions of ∆ to be the pair {F1 /F0 , F0 /F1 } described in 28.6 and we will refer to F0 and F1 as the two fields of definition of ∆. (Note that we can think of the invariant {F1 /F0 , F0 /F1 } as a pair of fields F0 and F1 together with embeddings of π : F0 → F1 and ρ : F1 → F0 such that ρ ◦ π is the Frobenius map of F0 and π ◦ ρ is the Frobenius map of F1 .)

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If ∆ is the spherical building associated via a BN-pair to the group of k-rational points of an absolutely simple group G, then ∆ is neither mixed nor a Moufang octagon and k is isomorphic to the field of definition F of ∆ as defined in 28.8. Definition 28.9. Let ∆ be a thick irreducible spherical building of rank ℓ at least 2. Then ∆ is split if for some commutative field F , ∆ is isomorphic to one of the following: (i) Aℓ (F ), Dℓ (F ) or Eℓ (F ). 2 (ii) BQ ℓ (Λ), where Λ is the anisotropic quadratic space (F, F, x 7→ x ).

(iii) BIℓ (Λ), where Λ is the involutory set (F, F, id). (iv) F4 (Λ), where ℓ = 4 and Λ is the composition algebra (F, F ). (v) G2 (Λ), where ℓ = 2 and Λ is the quadratic Jordan division algebra (E/F )◦ as defined in [60, 15.20] with E = F . Remark 28.10. Note that the buildings in 28.9 are not mixed and in each case, F is the defining field of ∆. The split buildings are precisely the spherical buildings associated with the F -points of split absolutely simple algebraic groups of F -rank at least 2 (for some field F ). Roughly speaking, a building is split if all its root groups are 1-dimensional over the field of definition (but note that we have not actually defined the dimension of a root group). Remark 28.11. Let F be a field of characteristic 2. Then we have the following isomorphisms between root group sequences (where the notation is as in 1.14): QQ (F, F, x 7→ x2 ) ∼ = QQ (F 2 , F, x 7→ x2 )op ∼ = QI (F, F, id) ∼ = QI (F, F 2 , id)op ∼ = QD (F, F, F )op ∼ = QD (F, F, F 2 ).

Q 2 2 2 It follows that the buildings BQ 2 (F, F, x 7→ x ) and B2 (F , F, x 7→ x ) as well I I 2 D 2 D as B2 (F, F, id), B2 (F, F , id), B2 (F, F, F ) and B2 (F, F, F ) are all isomorphic to each other. We observe, in particular, that the conditions “B 6= F ” and “if A = F , then B 6= F 2 ” in 28.3(i) are there precisely to eliminate those D buildings of the form BD 2 (F, A, B) which are split. (These split B2 (F, A, B)’s exist for arbitrary fields F of characteristic 2; as observed in 28.4, all the others require a purely inseparable extension F1 /F0 .)

Remark 28.12. Let Λ = (F, A, B) be an indifferent set and let L be the subfield of F generated by B. If A = F , then (F, B 1/2 , x 7→ x2 ) is an anisotropic quadratic space, (F, B, id) is an involutory set and QD (Λ) ∼ = QQ (F, B 1/2 , x 7→ x2 ) ∼ = QI (F, B, id)op

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and if B = L, then (L, A, x 7→ x2 ) is an anisotropic quadratic space, (L, A2 , id) is an involutory set and ∼ QQ (L, A, x 7→ x2 )op ∼ QD (Λ) = = QI (L, A2 , id).

If neither A = F nor B = L, then (F, A, B) is proper as defined in [60, 38.8] and there is no thick spherical building of rank greater than 2 having a residue isomorphic to BD 2 (Λ). Remark 28.13. Let Λ = (F, L, q) be an anisotropic quadratic space such 1 that BQ 2 (Λ) is indifferent and 1 ∈ q(L). Then ∂q = 0, the map u 7→ q(u) 2 is additive, F ⊂ q(L), (F, F, q(L)) is an indifferent set, (F, q(L), id) is an involutory set and ∼ QD (F, F, q(L)) QQ (Λ) = ∼ = QI (F, q(L), id)op . Thus, in particular,

∼ I BQ ℓ (Λ) = Cℓ (F, q(L), id) for all ℓ ≥ 2. Remark 28.14. Let Λ = (F, B, σ) be an involutory set such that BI2 (Λ) is indifferent. Then σ = id, char(F ) = 2, (F, F, B) is an indifferent set, B is a vector space over F 2 and QI (Λ)op ∼ = QD (F, F, B) ∼ = QQ (F 2 , B, x 7→ x2 ) ∼ = QQ (F, B 1/2 , x 7→ x2 ).

Thus, in particular, for all ℓ ≥ 2.

1/2 CIℓ (Λ) ∼ , x 7→ x2 ) = BQ ℓ (F, B

Remark 28.15. Let L/F be a field extension in characteristic 2 such that L2 ⊂ K. Then (L, F, id) is an involutory set and hence CI (L, F, id) ∼ = BQ (L2 , F, x 7→ x2 ) ℓ

ℓ

for all ℓ ≥ 3. This is just a special case of 28.14.

We come now to the main definition of this chapter. Definition 28.16. Let ∆ be an irreducible spherical building of rank ℓ ≥ 2. Then ∆ is pseudo-split if it is Moufang and all of its irreducible rank 2 residues are split as defined in 28.9 or indifferent as defined in 28.3, i.e. split ◦ or isomorphic to BD 2 (Λ) for some indifferent set Λ or to G2 ((E/F ) ) for some 3 field extension E/F in characteristic 3 such that E ⊂ F , where (E/F )◦ is as defined in [60, 15.20]. Thus every split building is pseudo-split and the 1 By

[60, 35.17], this is not a real restriction.

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only mixed buildings that are not pseudo-split are the Moufang quadrangles of type F4 . By 28.12, 28.13 and 28.14, every pseudo-split building is either split or mixed or isomorphic to 1/2 (28.17) BIℓ (Λ) ∼ , x 7→ x2 ) = BQ ℓ (F, B for some ℓ ≥ 3 and for some involutory set Λ = (F, B, id) with char(F ) = 2 and F 2 $ B $ F . Remark 28.18. The building Aℓ (K) is pseudo-split if and only if it is split if and only if K is a commutative field. The buildings Dℓ (K), E6 (K), E7 (K) and E8 (K) are all split. Suppose ∆ is a building of type Bℓ or F4 and let R be an irreducible rank 2 residue whose Coxeter diagram is B2 . Then ∆ is pseudo-split (respectively, split) if and only if R is pseudo-split (respectively, split). Moufang octagons are neither split nor pseudo-split. Remark 28.19. Let F be a field of characteristic p. If p = 2, then the 2 2 isomorphisms in 28.11 extend to isomorphisms from BQ ℓ (F , F, x 7→ x ) to I 2 the split building Cℓ (F, F, id) for all ℓ and from F4 (F, F ) to the split building F4 (F, F ). If p = 3, then there is, analogously, an isomorphism from G2 ((F/F 3 )◦ ) to the split building G2 ((F/F )◦ ), where (F/F 3 )◦ and (F/F )◦ 2 2 2 are as in [60, 15.20]. Thus the buildings BQ ℓ (F , F, x 7→ x ) and F4 (F, F ) 3 ◦ (with p = 2) and G2 ((F/F ) ) (for p = 3) are split even though at first glance they might seem to be only pseudo-split. In particular, the buildings in [65, Table 28.4] are split rather than mixed and belong in [65, Table 28.5], in the lines with “affine index” Cℓ (but with the condition char(K) 6= 2 removed), F4 and G2 . Our last goal in this chapter is to prove the characterization of pseudo-split buildings in 28.22. We recall the definition of a subbuilding in 23.5. Proposition 28.20. Every pseudo-split building is a subbuilding of a split building. Proof. Let ∆ be a pseudo-split building that is not split, let J be the set consisting of the two vertices of the Coxeter diagram Π connected by an edge with label n ≥ 4 and let Ω be the root group sequence of an arbitrary Jresidue of ∆. To proceed, we use the observations in 1.17. If ∆ is of type G2 , then Ω ∼ = H((L/F )◦ ) for some field extension L/F in characteristic 3 such 3 that L ⊂ F , thus Ω is a subsequence of H(Λ) for Λ = (L/L)◦ and therefore ∆ is isomorphic to a subbuilding of the split building G2 (Λ). If ∆ is of type F4 , then ∆ ∼ = F4 (C, F ) for some field extension C/F in characteristic 2 such that C 2 ⊂ F , (C, C, F ) is an indifferent set, Ω (or Ωop ) is isomorphic to QD (C, C, F ) and ∆ is isomorphic to a subbuilding of the split building F4 (C, C). If ∆ is of type Bℓ for some ℓ ≥ 3, then ∆ ∼ = BIℓ (Λ) for Λ = (F, B, id) as in (28.17) and ∆ is isomorphic to a subbuilding of the split building BIℓ (F, F, id). Suppose, finally, that ∆ is of type B2 . Then ∆ ∼ = BD 2 (Λ) for some indifferent set (F, A, B) and hence ∆ is isomorphic to a subbuilding of the split building BD 2 (F, F, F ) which, as we observed in 28.11, goes by many names.

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Proposition 28.21. Let ∆ be an irreducible spherical building and suppose that ∆ is pseudo-split as defined in 28.16. Let ∆′ be a thick subbuilding of ∆. Then ∆′ is also pseudo-split. Proof. Let S be the vertex set of the Coxeter diagram of ∆ and let ℓ = |S| denote the rank of ∆. By [65, 29.61(iii)], ∆′ also satisfies the Moufang condition. This means that ∆′ is linked to a root group labeling that embeds in the root group labeling with which ∆ is linked as described in [65, 30.13 and 30.16]. Suppose first that Π is simply laced. Then ∆ is isomorphic to Xℓ (F ) for X = A, D or E and for some (commutative) field F . By [65, 30.14], we know that ∆′ = Xℓ (L) for some alternative division ring L. The embedding of ∆′ in ∆ gives rise to an embedding of the root group sequence T (L) into the root group sequence T (F ). The first paragraph of the proof of [60, 35.23] now yields the conclusion that L is isomorphic to a subfield of F . Thus L is commutative and hence ∆′ is split. Suppose next that Π is not simply laced, let J denote the unique 2-element subset of S such that the subdiagram ΠJ is not simply laced and let n be the label on the unique edge of ΠJ . To see that ∆′ is pseudo-split, it suffices to show that the J-residues of ∆′ are pseudo-split. Let R′ be a J-residue of ∆′ , let Σ be an apartment of R′ and let R be the unique J-residue of ∆ containing R′ . The residues R and R′ are Moufang n-gons and Σ is also an apartment of R. We number the chambers of Σ and let Ω = (U+ , U1 , . . . , Un ) and ′ Ω′ = (U+ , U1′ , . . . , Un′ )

be the corresponding root group sequences of R and R′ (as in [60, 5.1]). ′ Then U+ is a subgroup of U+ and Ui′ is a subgroup of Ui for all i ∈ [1, 4]. Since ∆ is pseudo-split, we can then observe from the commutator relations in [60, 16.2–16.8] that the following conditions hold: (i) If n = 4, then [U1 , U3 ] = 1 and either [U2 , U4 ] = 1 or [a, b] 6= 1 for all a ∈ U2∗ and all b ∈ U4∗ ; and (ii) if n = 6, then either [U1 , U3 ] = 1 or [a, b] 6= 1 for all a ∈ U1∗ and all b ∈ U3∗ . We claim that, in fact, the converse also holds: These same two conditions imply that ∆ is pseudo-split. Suppose that (i) and (ii) hold. By 28.18, it will suffice to show that R is pseudo-split. Suppose that Ω ∼ = QI (Λ) for some involutory set Λ = (D, D0 , τ ) (as in [60, 16.2]). If [U2 , U4 ] = 1, then by [60, 38.1], R is of indifferent type and hence ∆ is pseudo-split. Suppose that [a, b] 6= 1 for all a ∈ U2∗ and all b ∈ U4∗ . Then xτ + x 6= 0 for all x ∈ K ∗ . Hence char(K) 6= 2 since 1τ + 1 = 2. Since xτ + x = 0 for all x equal to a − aτ for some a ∈ K, it follows that τ = 1. Hence D0 = D by [60, 11.1(i)] and D is commutative (since τ is an anti-automorphism). Therefore R is split. If Ω ∼ = QQ (Λ) for some anisotropic quadratic space (F, V, q) (as in [60,

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249

16.3]), then (i) implies that either ∂q is identically zero or dimF V = 1. In the first case R is pseudo-split and in the second R is split. By (i), Ω is not as [60, 16.5–16.7]. Suppose, finally, that Ω = R ∼ = H(Λ) for some hexagonal system Λ = (J, F, #) (as in [60, 16.8]). If [U1 , U3 ] = 1, then the trace T of Λ is identically zero and hence R is pseudo-split by [60, 30.5]. If [a, b] 6= 1 for all a ∈ U1∗ and all b ∈ U3∗ , then by [60, 30.6], we have dimF J = 1 and hence by [60, 30.3], J is isomorphic to (F/F )◦ . Thus R is split. This concludes the proof of the claim. Since ∆ satisfies conditions (i) and (ii), also ∆′ satisfies conditions (i) and (ii) with Ui′ in place of Ui for all i ∈ [1, n]. Thus ∆′ is pseudo-split by the claim in the previous paragraph. Theorem 28.22. Let ∆ be a thick irreducible spherical building. Then ∆ is a pseudo-split if and only if it is a subbuilding of a split building. Proof. This holds by 28.20 and 28.21. Remark 28.23. It follows from [59, Cours 1991-1992, Prop. 5A] and 28.22 that the group of linear automorphisms (as defined in 29.19 below) of a pseudo-split building is the group of rational points of a pseudo-split group as long as at least one of the defining extensions F1 /F0 or F0 /F1 is finite. (See also [59, Cours 1991-1992, §5.3].) Thus our notion of a pseudo-split building is compatible with the notion of a pseudo-split group given in [11, Definition 2.3.1] (see also [12]). Note, however, we are not assuming that F1 /F0 or F0 /F1 is finite in 28.16.

Chapter Twenty Nine Linear Automorphisms In this chapter we make precise the notion of a linear automorphism of an arbitrary spherical building satisfying the Moufang property. We develop the definition in a series of steps culminating in 29.19. Notation 29.1. Let Ω = (U+ , U1 , . . . , Un ) be the root group sequence and x1 , . . . , xn the isomorphisms obtained by applying the recipe in [60, 16.x] for x = 1, 2, 3, . . . or 9 to a parameter system Λ of the suitable type (and for suitable n) and let ∆ be the corresponding Moufang n-gon. Thus Ω and Λ are as in 1.14(i)–(ix) and if x = 1, then n = 3 and Λ is an alternative division ring K, if x = 2, then n = 4 and Λ is an involutory set (K, K0 , σ), etc. As was observed in 4.2, we can assume without loss of generality that in case (ii), the involutory set Λ is proper as defined in [60, 35.3] and in case (v), the anisotropic pseudoquadratic space Λ is proper as defined in [60, 35.5]. Let F (or the pair {F1 /F0 , F0 /F1 }) be as in 28.8 applied to ∆. Notation 29.2. Suppose that K is either a field, a skew field or an octonion division algebra with center F . If K is octonion with norm N , we let ΦK denote the set of all linear automorphisms φ of K as a vector space over F such that N (φ(a)) = N (a) and φ(1) = 1. If K is associative, we let ΦK denote the set containing only the identity map on K. If φ ∈ ΦK , then φ is injective (because the norm of an octonion division algebra is anisotropic) and hence invertible and its inverse is also in ΦK . Notation 29.3. Let HΩ be the automorphism group of Ω and let h ∈ HΩ . By [60, 33.5], H acts faithfully on U1 × Un . The group HΩ is described in terms of the parameter system Λ in [60, 37.12, 37.29, 37.30, 37.32, 37.41 and 37.46] for x ∈ {1, 2, 3, 4, 8, 9}. In particular, we have the following: (i) If x = 1, then F = Z(K) and there exists an automorphism ρ of K, elements s, w ∈ K ∗ , and an element φ of the set ΦK defined in 29.2 such that x1 (t)h = x1 (s · φ(tρ )) and x3 (t)h = x3 (φ(tρ )w · s) for all t ∈ K. (ii) If x = 2 and the involutory set Λ is proper, then F = FixZ(K) (σ) and there exists an automorphism ρ of K, an element s ∈ K0∗ and an

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element w ∈ K ∗ such that x1 (t)h = x1 (stρ ) and x4 (u)h = x4 (tρ w) for all t ∈ K0 and all u ∈ K. (iii) If x = 3 and ∂q is not identically zero, then F = K and there exist an automorphism ρ of K, a ρ-linear automorphism ξ of L and an element w ∈ K ∗ such that x1 (t)h = x1 (wtρ ) and x4 (u)h = x4 (ξ(u)) for all t ∈ K and all u ∈ L. (iv) If x = 4, then (after interchanging F0 and F1 if necessary) F0 = hL0 i, F1 = K and there exists an element ρ of the group Aut(F0 , F1 ) defined in 28.5 and elements s ∈ K0 and w ∈ L0 such that x1 (t)h = x1 (stρ ) and x4 (u)h = x4 (wuρ ) for all t ∈ K0 and all u ∈ L0 . (v) If x = 8, then F = K (or K = F0 or F1 if the hexagonal system (J, F, #) is inseparable as defined in [65, 30.23], in which case ∆ is mixed) and there exists an automorphism ρ of K, a ρ-linear isotopy ξ of J and an element w ∈ K ∗ such that x1 (u)h = x1 (ξ(u)) and x6 (t)h = x6 (wtρ ) for all u ∈ J and all t ∈ K. (vi) If x = 9, then F = K and there exists an automorphism ρ of K and s, w ∈ K ∗ such that x1 (t)h = x1 (stρ ) and x8 (t, v) = (wσ+1 tρ , wv ρ ) for all t, v ∈ K. Furthermore: (a) In the case x = 5, F is as in (ii) and Λ is a proper anisotropic pseudoquadratic space defined over an involutory set (K, K0 , σ). Let U1′ = x1 (0, K0 ), U3′ = x3 (0, K0 ) and W = U1′ U2 U3′ U4 ⊂ U+ . Then Ω′ := (W, U1′ , U2 , U3′ , U4 )

is the root group sequence QI (K, K0 , σ). By [60, 38.2], either (K, K0 , σ) is proper or K0 = F , (K, F ) is a composition algebra and QI (K, K0 , σ) = QQ (F, K, q), where q is the norm of (K, F ). By [60, 38.10], Ui′ = Z(Ui ) and hence h normalizes Ui′ for i = 1 and 3. We define ρ to be the automorphism of K obtained by applying (ii) to the restriction of h to Ω′ if (K, K0 , σ) is proper or (iii) to the restriction of h to Ω′ if (K, K0 , σ) is not proper. (b) In the case x = 6, F = K and Λ is a quadratic space (K, L, q) of type E6 , E7 or E8 . Then Ω′ := (W, U1′ , U2 , U3′ , U4 ) for U1′ = x1 (0, K), U3′ = x3 (0, K) and W = U1′ U2 U3′ U4 ⊂ U+ is the root group sequence QQ (Λ). By [60, 38.10], Ui′ = Z(Ui ) and hence h normalizes Ui′ for i = 1 and 3. We define ρ to be the automorphism of F obtained by applying (iii) to the restriction of h to Ω′ .

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(c) In the case x = 7 there is, after interchanging F0 and F1 if necessary, a subsequence Ω′ := (W, U1′ , U2′ , U3′ , U4′ ) of Ω with Ui′ = xi (0, F0 ) for i = 1 and 3 and Ui′ = xi (0, F1 ) for i = 2 and 4, where (by 28.8) F0 and F1 are the fields called K and F in [60, 16.6]. This subsequence is the root group sequence QD (F0 , F0 , F1 ). We have CUi (U4−i ) = Ui′ for i = 1 and 3. Hence h normalizes U1′ and U3′ . We define ρ to be the element of Aut(F0 , F1 ) obtained by applying (iv) to the restriction of h to Ω′ . Remark 29.4. Suppose that we are in case (i) of 29.3. Then x1 (s) = x1 (1)h , so s is uniquely determined by h, and x1 (stρ ) = x1 (t · 1)h for all t ∈ F . Thus the restriction of ρ to F is uniquely determined by h. By a similar argument, the same conclusion holds in all other cases of 29.3. Notation 29.5. In each case of 29.3, we set λΩ (h) equal to the element ρ of Aut(F0 , F1 ) or to the restriction of ρ to F , whichever applies. By 29.4, the map λΩ is well defined. In every case, λΩ is a homomorphism from HΩ to Aut(F0 , F1 ) or Aut(F ). Notation 29.6. Suppose that Ω = T (K) and Ω′ = Ω or T (K op ) for some alternative division ring K and that h is an isomorphism from Ω to Ω′ . Let F = Z(K). It follows from [60, 35.23, 37.10 and 37.12] that there exist F -linear maps ψ1 and ψ3 from K to K and an automorphism (if Ω = Ω′ ) or anti-automorphism (if Ω 6= Ω′ ) ρ of K whose restriction to F is unique such that h maps xi (t) to xi (ψi (tρ )) for all t ∈ K and for both i = 1 and 3. Let κΩ denote the restriction of ρ to F . Thus if Ω′ = Ω, then κΩ = λΩ . Let ξ denote the unique isomorphism from T (K op ) to T (K)op that extends the maps xi (t) 7→ xi (t) for i = 1 and 3. If we identify T (K op ) with T (K)op via ξ, then the union of HΩ and the set of isomorphisms from Ω to T (K op ) can be thought of as the group of all automorphisms and anti-automorphisms of Ω. When we do this, κΩ becomes a homomorphism from this group to Aut(F ). Notation 29.7. Let ∆ be the spherical Moufang building with Coxeter diagram Π, let Σ be an apartment of ∆, let c be a chamber of Σ, let G = Aut(∆), let G† be the subgroup of G generated by all the root groups of ∆, let HΣ denote the pointwise stabilizer of Σ in G and let F or {F1 /F0 , F0 /F1 } be as in 28.8 applied to ∆. Notation 29.8. If Π is simply laced, let Mc,Σ denote the stabilizer of the pair (c, Σ) in G and let G♭ = G. If Π is not simply laced, let Mc,Σ = HΣ and let G♭ denote the set of type-preserving elements of G. Notation 29.9. For each edge e = {s, t} of Π, let Re be the unique eresidue of ∆ containing the chamber c and let Ωst and Ωts be the root group sequences of the residue Re (opposite to each other) defined with respect

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to the pair (c, Re ∩ Σ) as in [65, 30.1] labeled so that the first term of Ωst acts non-trivially on the s-panel of ∆ containing c (and thus trivially on the t-panel containing c). Observation 29.10. For each edge {s, t} of Π, the root group sequence Ωst is isomorphic to one of the root group sequences in 1.14 or to the opposite of one of these root group sequences. Suppose that {s, t} and {s, r} are two edges of Π sharing a vertex s. Without loss of generality, we can assume that the label on the edge {s, t} is 3 and thus Ω ∼ = T (K) for some alternative division ring K (and either F = Z(K) or ∆ is unitary with involution σ as in 28.2 and F = Z(K)σ ). By [60, 40.25], one of the following holds: (i) K is associative and Ωsr ∼ = T (K op ).

(ii) K is commutative and Ωsr ∼ = QQ (Λ) for some anisotropic quadratic space over K. (iii) Ωsr ∼ = QQ (F, C, q)op for some composition algebra (C, F ) with norm q and C = K. (iv) Ωsr ∼ = QI (Λ)op for some proper involutory set Λ = (D, D0 , σ) with D = K op . (v) Ωsr ∼ = QP (Λ)op for some proper anisotropic pseudo-quadratic space Λ over an involutory set (D, D0 , σ) with D = K op . If C is a field of characteristic 2 and F is a subfield containing C 2 in case (iii), then as we observed in 28.11, QQ (F, C, x 7→ x2 )op = QQ (C 2 , F, x 7→ x2 ).

We redefine case (iii) to exclude this case. With this modification, it follows from [60, 38.9] that exactly one of the five cases (i)–(v) holds. Notation 29.11. If Π = F4 , then by [60, 40.49], (ii) or (iii) holds for some composition algebra (C, F ) with norm q with Λ = (F, C, q) in case (ii). If there is a third vertex r′ adjacent to s different from t and r, then by [60, 40.47], K is commutative and (i) holds. There is an isomorphism from T (K op ) = T (K)op mapping xi (t) to x4−i (t) for i = 1 and 3 and x2 (t) to x2 (−t) for all t ∈ K. It follows from these observations that we can define a root group labeling of Π as follows: Suppose first that the rank of ∆ is 2 and let {s, t} be the unique edge of Π. Interchanging s and t if necessary, we can choose a root group sequence ζ(s, t) in 1.14 isomorphic to Ωst and we set ζ(t, s) = ζ(s, t)op . Suppose that the rank of ∆ is at least 3. If {s, t} is an edge with label 3, we set ζ(s, t) = T (K), where K is the unique alternative division ring such that Ωst ∼ = T (K) and let ζ(t, s) = T (K op ). Suppose, finally, that {s, r} is an edge with label 4, that s is adjacent to another vertex t of Π and ζ(s, t) = T (K) for some alternative division ring K. In this case Ωsr is in exactly one of the four cases (ii)–(v) of 29.10. If it is in case (i), we set ζ(s, t) = QQ (Λ), in case (ii), we set QQ (F, C, q)op , etc., and in each case we set ζ(t, s) = ζ(s, t)op .

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Notation 29.12. Let ζ be as in 29.11. By [60, 40.22], there exists a family (πe ) of isomorphisms πe : Ωst → ζ(s, t), one for each edge e = {s, t} of Π, such that whenever e = {s, t} and f = {s, r} are two edges sharing a vertex s, in which case the first term of Ωst is canonically isomorphic to the first term of Ωsr and hence the restriction of the map πf ◦ πe−1 to the first term of ζ(s, t) makes sense, this restriction is the map sending the element x1 (u) in the first term of ζ(s, t) to the element x1 (u) in the first term of ζ(s, r) for every u ∈ K. We call such a family (πe ) a linking of ∆ to the root group labeling ζ based at (Σ, c). Let ζ and (πe ) be as in 29.12 and suppose that h ∈ HΣ . For each edge e = {s, t} of Π, we denote by he the automorphism of Ωst induced by h and thus πe he πe−1 is an automorphism of ζ(s, t). We set (29.13)

σe = λζ(s,t) (πe he πe−1 ),

where λζ(s,t) is as in 29.5. Observation 29.14. Now suppose that e = {s, t} and f = {s, r} are two edges of Π sharing a vertex s. If we use πe and πf to identify the first term of Ωst with the first term of Ωsr as described in the previous paragraph, then the automorphisms πe hπe−1 of ζ(s, t) and πf hπf−1 of ζ(s, r) agree on these two first terms. It follows that σe = σf . For example, if ζ(s, r) is in case (i) of 29.10, then there exist s, s′ ∈ K ∗ and ρ, ρ′ ∈ Ant(K) such that ′ sxρ = xρ s′ for all x ∈ K, from which it follows that ρ and ρ′ have the same restriction to the center of K. Notation 29.15. Since Π is irreducible, it follows from the conclusion of the previous paragraph that the automorphism σe given by (29.13) is independent of the edge e. In this fashion, we obtain a homomorphism λ from HΣ to Aut(F ) such that λ(h) = λζ(s,t) (πe he πe−1 ) for every edge e = {s, t} of Π. Definition 29.16. Let Mc,Σ be as in 29.8 and suppose that Mc,Σ 6= HΣ . We now extend the homomorphism λ : HΣ → Aut(F ) defined in 29.15 to a homomorphism from Mc,Σ to Aut(F ) as follows. Then Π is simply laced, there exists an alternative division ring K such that for each directed edge (s, t) of Π, ζ(s, t) equals T (K) or T (K op ) and F = Z(K). Let h ∈ Mc,Σ and let θ be the automorphism of Π induced by h. If e = {s, t} is an edge of Π, then h induces an isomorphism he from Ωst to Ωs′ t′ , where s′ = sθ , t′ = tθ and hence e′ := {s′ , t′ } = eθ . We set σe = κζ(s,t) (πe′ he πe−1 ),

where κζ(s,t) is as in 29.6. Just as in 29.14, we see that σe is independent of the choice of e. We denote this restriction by σh and let κ be the map

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h 7→ σh from Mc,Σ to Aut(F ). Then κ is a homomorphism extending the homomorphism λ and κ(h) = κζ(s,t) (πe′ he πe−1 )

(29.17) for all h ∈ Mc,Σ .

♮ Notation 29.18. If Π is simply laced, let Mc,Σ denote the kernel of the ♮ map κ defined in 29.16 and let HΣ♮ = Mc,Σ ∩ HΣ . If Π is not simply laced,

♮ let HΣ♮ = Mc,Σ denote the kernel of the map λ defined in 29.15 and let κ = λ. Thus in both cases, κ is a homomorphism from Mc,Σ to Aut(F ) or ♮ Aut(F0 , F1 ) with kernel Mc,Σ . ♮ Notation 29.19. Let G† be as in 29.7 and let M = Mc,Σ , M ♮ = Mc,Σ and ♮ ♮ H = HΣ be as in 29.18. By the formulas in [60, 33.10-33.16], we have

(29.20) and by [62, 11.12], (29.21)

M ∩ G† ⊂ H ♮ ⊂ M ♮ G♭ = M G† ,

where G♭ is as in 29.8. Thus for each g ∈ G♭ , we can choose an element g † ∈ G† such that gg † ∈ M . Let F and {F1 /F0 , F1 /F0 } be as in 28.8 and let κ be the homomorphism from M to Aut(F ) or Aut(F0 , F1 ) defined in 29.18. We define a map β from G♭ to Aut(F ) or Aut(F0 , F1 ) by setting (29.22)

β(g) = κ(gg † )

for g † ∈ G† such that gg † ∈ M . By (29.20), β(g) is independent of the choice of g † . Since G† is normal in G♭ , the map β is a homomorphism. Let G♮ denote the kernel of β. Observation 29.23. The homomorphism β defined in 29.19 depends on the choice of the pair (c, Σ) and the linking (πe ). Let (ˆ πe ) be another linking from ∆ to ζ based at (Σ, c) as defined in 29.12 and let βˆ be the homomorphism obtained by applying 29.19 with (ˆ πe ) in place of (πe ). Then there exists a unique element h of HΣ such that he = πe−1 π ˆe for each edge e of Π. Choose an edge e = {s, t} of Π and an element g ∈ G♭ . Let θ = typ(g) ∈ Aut(Π) and let s′ = sθ , t′ = tθ and e′ = eθ . If θ 6= 1, then by (29.17), β(g) = κ(gg † ) = κζ(s,t) (πe′ · gg † · πe−1 ) = κζ(s,t) (ˆ πe′ he′ · gg † · h−1 ˆe−1 ) e π

−1 = κζ(s,t) (ˆ πe′ he′ π ˆe−1 πe′ · gg † · π ˆe−1 ) · κζ(s,t) (πe h−1 ′ ) · κζ(s,t) (ˆ e πe ) ˆ β(g) ˆ β(h) ˆ −1 = β(h)

ˆ β(g) ˆ β(h) ˆ −1 holds by (29.13) and the by (29.17). If θ = 1, then β(g) = β(h) same calculation but with κζ(s,t) replaced by λζ(s,t) and e′ replaced by e. Now let Σ′ be a second apartment and let c′ be a chamber of Σ′ . By [62, 11.12], there exists an element g0 ∈ G† mapping (c, Σ) to (c′ , Σ′ ). Let g ∈ G♭ and choose g † ∈ G† such that gg † ∈ Mc,Σ . Then g0 · gg † · g0−1 = g · (g −1 g0 g) · g † g0−1 ∈ gG†

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257

and g0 · gg † · g0−1 fixes (c′ , Σ′ ), For each edge {u, v} of Π, let Ω′st and Ω′ts be as in 29.9 but with (c′ , Σ′ ) in place of (c, Σ). Conjugation by g0−1 gives rise to an isomorphism ψ from Ωst to Ω′st for each edge {s, t} of Π. Let (πe ) be a linking of ∆ to ζ based at (Σ, c) used to define β. Then (πe · g0−1 ) is a linking of ∆ to ζ based at (Σ′ , c′ ) with which we can define homomorphisms β ′ and κ′ as in 29.18 and (29.22). Choose an edge e = {s, t} of Π and an element g ∈ G♭ and let θ, s′ , t′ and e′ be as in the previous paragraph. If θ 6= 1, then β ′ (g) = κ′ (g0 · gg † · g0−1 )

= κζ(s,t) (πe′ g0−1 · g0 · gg † · g0−1 · (πe g0−1 )−1 )

= κζ(s,t) (πe′ · gg † · πe−1 )

= κ(gg † ) = β(g)

by (29.17). If θ = 1, then β(g) = β(g) holds by (29.13) and the same calculation but with κζ(s,t) replaced by λζ(s,t) and e′ replaced by e. We conclude that β depends on the choice of (c, Σ) and (πe ), but only up to an inner automorphism of Aut(F ) or Aut(F0 , F1 ). In particular, the kernel G♮ of β is independent of these choices. Remark 29.24. For each edge {s, t} of Π, the algebraic structure defining the root group sequence Ωst is uniquely determined by Ωst only up to a suitable notion of similarity as defined in [60, 35.6–35.14]. Thus the map β appears to depend on the choice of an algebraic structure in the various similarity classes associated to each edge of Π. By [62, 29.40 and 35.16– 35.19], however, the map β is independent of these choices up to an inner automorphism of Aut(F ) or Aut(F0 , F1 ). Definition 29.25. Let ∆ and G be as in 29.7, let G♭ be as in 29.8 and let β be the homomorphism from G♭ to Aut(F ) or Aut(F0 , F1 ) defined in (29.22). A Galois map of ∆ is a map from G♭ to Aut(F ) of the form ι ◦ β for some inner automorphism ι of Aut(F ) or Aut(F0 , F1 ). We call an element of G linear if it is contained in the kernel G♮ of a Galois map of ∆ (and hence in the kernel of all Galois maps of ∆). Definition 29.26. Let Γ be a subgroup of Aut(F ) if ∆ has no mixed rank 2 residues; otherwise let Γ be a subgroup of Aut(F0 , F1 ). A Galois embedding of Γ in G is a homomorphism φ from Γ to G♭ such that γ∆ ◦ φ is the identity map in Aut(F ) or Aut(F0 , F1 ) for some Galois map γ∆ of ∆. A Galois action of Γ on ∆ is the action of Γ on ∆ which results from a Galois embedding of Γ in G. Definition 29.27. We will call an element of G semi-linear if it is contained in G♭ and strictly semi-linear if, in addition, it is not in G♮ . If Π is not simply laced, a non-type-preserving element of G (if there is one) is not in G♭ (by 29.8) and hence not semi-linear. We will call a subgroup Γ of G♭ strictly semi-linear if Γ ∩ G♮ = 1. If Γ and φ are as in 29.26, then φ(Γ) is a strictly semi-linear subgroup of G♭ .

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Remark 29.28. Suppose that R is an irreducible residue of rank at least 2 of ∆ and Γ is a subgroup of the stabilizer (G♭ )R . Then R is Moufang, so we can apply the results of this chapter to R. It follows from 29.12 and 29.16 that Γ is strictly semi-linear if and only if its restriction to R is a strictly semi-linear automorphism of R.

Chapter Thirty Strictly Semi-linear Automorphisms In this chapter we make some observations about the action of a strictly semi-linear automorphism fixing a root on the corresponding root group. We restrict our attention to the spherical buildings which play a role in Chapter 36. Hypothesis 30.1. Let ∆ be a spherical Moufang building and let Π be the Coxeter diagram of ∆. We fix an apartment Σ of ∆ and a root α of Σ. Let M denote the wall of α, let G, G† and H := HΣ be as in 29.7 and let H † = H ∩ G† . In addition, we assume that either (i) ∆ is pseudo-split as defined in 28.16 or (ii) ∆ is isomorphic to F4 (C, F ) or to a residue of F4 (C, F ) for some composition algebra (C, F ) of type (iii), (iv) or (v) as defined in [65, 30.17]. This means that either C/F is a separable quadratic extension or C is a quaternion or octonion division algebra with center F . We let η be the non-trivial element of Gal(C/F ) if C is commutative and if C is quaternion or octonion, we let η denote its standard involution. Notation 30.2. Let R, Z and ≈ be as in 21.36 applied to the wall M of the root α, let X = Z/ ≈ and let [β] denote the element β ∩ Z of X for each β ∈ R. By [65, 29.47], each chamber in Z lies in a unique root in R (since we are now assuming that ∆ is spherical) and hence the map β 7→ [β] is a bijection from R to X. For each β ∈ R, the root group Uβ acts faithfully ¯β denote the permutation group induced on on X (by [62, 11.11(ii)]). Let U X by Uβ for each β ∈ R and let M∆,M denote the pair ¯β | β ∈ R}). (X, {U

Then M∆,M is a Moufang set and for each panel P in M , the map x 7→ x∩P is an isomorphism from M∆,M to the Moufang set M∆,P defined in 1.19. Notation 30.3. Let R and X = {[β] | β ∈ R} be as in 30.2, let P be a panel in the wall M of the root α, let −α denote the root of the apartment Σ opposite α, let c be the unique chamber in α ∩ P , let Π be the Coxeter diagram of ∆, let ζ be the root group labeling in 29.11 and let (πe ) be a linking of ∆ to ζ based at (Σ, c) as defined in 29.12. Let e = {s, t} be an edge of Π such that s is the type of the panel P , let Ωst be as in 29.9 and let ξ denote the restriction of πe : Ωst → ζ(s, t) to the first term of Ωst . The first term of Ωst is canonically isomorphic to Uα and thus we can think of ξ as an

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isomorphism from Uα to the first term of ζ(s, t). Let x1 be the isomorphism from E to the first term of ζ(s, t) in 29.1, where either ∆ is as in 30.1(i) and E is a field or E is an additive subgroup of a field F of characteristic 2 containing 1 such that F 2 E ⊂ E or ∆, F and C are as in 30.1(ii) and E = C or F . Thus x := ξ −1 ◦ x1 is an isomorphism from E to Uα . Let π0 be the bijection from E to X\{[α]} obtained by composing the map t 7→ x(t)(−α) ∈ R with the map β → [β]. Notation 30.4. Let π0 and E be as in 30.3, let τ denote the permutation x 7→ −1/x of E ∗ and let M(E, τ ) be as in (1.13). We denote by π the bijection from the set of points E ∪ {∞} of the Moufang set M(E, τ ) to the set of points X of the Moufang set M∆,M defined in 30.2 that sends ∞ to [α] and agrees with π0 on E. Proposition 30.5. The bijection π defined in 30.4 is an isomorphism of Moufang sets from M(E, τ ) to M∆,M . Proof. Let α, −α, Ωst , ζ(s, t), πe and x be as in 30.3 and let U = Uα and ˆ = U−α . As in [60, pp. 353–354], we parametrize U ˆ by setting U (30.6)

xˆ(a) = x(a)µ(x(1))

for all a ∈ E, where µ = µΣ is as in 1.18. Let ρ be the unique permutation of U ∗ such that (30.7)

αxˆ(a) = (−α)x(a)

ρ

for all a ∈ E ∗ and let ω denote the unique permutation of E ∗ such that (30.8)

x(aω ) = x(a)ρ

for all a ∈ E ∗ . The image of M∆,M under π −1 is a Moufang set with point set E ∪ {∞}. Applying 1.11 to this Moufang set (with a = 1), we conclude by (30.7) and (30.8) that π is an isomorphism of Moufang sets from M(E, ω) to M∆,M . Let R be the unique {s, t}-residue containing the panel P and choose a ∈ E ∗ . Thus Ωst is one of the two root group sequences of R based at (Σ ∩ R, c) and the isomorphism πe : Ωst → ζ(s, t) maps x(a) in the first term of Ωst to x1 (a) in the first term of ζ(s, t). By [62, 11.10], we can think of µ(x(a)) as the element µ1 (x1 (a)), where µ1 is as in 3.6 with R in place of ∆. By 28.16 and 30.1, the root group sequence ζ(s, t) is one of the following: • T (E). • QD (F, E, B) for some indifferent set (F, E, B). • QQ (E, V, q) for some anisotropic quadratic space (E, V, q) with ∂q identically zero. • QI (E, F, η), where E = C and η are as in 30.1(ii).

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• H(Λ), where Λ = (E/F )◦ for some field F as defined in [60, 15.20]. By [60, 32.5, 32.8, 32.9 and 32.12], we conclude that (30.9)

µ(x(a)) = x(1/a)µ(x(1)) x(a)x(1/a)µ(x(1))

for all a ∈ E ∗ . By (30.7) and (30.8), we have αx((−a)

ω µ(x(1))

)

ρ
µ(x(1)) = (−α)x(−a)
µ(x(1)) = αxˆ(−a) = (−α)x(−a)

for all a ∈ E ∗ . Thus for each a ∈ E ∗ , the product (30.10)

x((−a)ω )µ(x(1)) x(a)x(−(aω ))µ(x(1))

ˆ x(a)Uˆ that interchanges α and −α. As observed in the is an element of U ˆ such that the proof of [62, 11.22], the elements λ(x(a)) and κ(x(a)) in U product κ(x(a))x(a)λ(x(a)) interchanges α and −α and hence equals µ(x(a)) are unique. By (30.9) and (30.10), it follows that (−a)ω = 1/a for all a ∈ E ∗ and hence ω = τ . In the following, the Moufang sets A1 (C), BI1 (C, F, η) and BQ 1 (F, C, q) are as defined in 3.8. See 36.3 for a similar result. Recall (from 1.8) that ≈ means “weakly isomorphic.” Proposition 30.11. Let (C, F ) and η be as in 30.1(ii) and let q be the norm of (C, F ) as defined in [65, 30.17]. Then A1 (C) ≈ BQ 1 (F, C, q). Proof. Let ∆ = BI2 (C, F, η), let Ω be a root group sequence of ∆ isomorphic to QI (C, F, η) and let Q be the panel of ∆ obtained by applying 3.7 with i = 4. Then MA2 (C),P ≈ M∆,Q by [60, 33.10 and 33.13] and M∆,Q ∼ = BQ 1 (F, C, q) by [60, 38.2]. Lemma 30.12. Let E be an octonion division algebra with center F and norm N and let D be a quaternion subalgebra of E. Suppose that h is an additive automorphism of E such that h(1) = 1, h(ta) = h(t)h(a) for all t ∈ F and all a ∈ E and h(N (a)) = N (h(a)) for all a ∈ E. Then h(D) is a quaternion subalgebra of E and the restriction of h to D is either an isomorphism or an anti-isomorphism from D to h(D). Proof. The proof of [60, 20.28] can be applied virtually verbatim. See also [50, 1.7.1].

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Notation 30.13. Let E be as in 30.3. We define a group Ant(E) as follows. If E is either a field, a quaternion division algebra or an octonion division algebra, we set Ant(E) equal to the group consisting of all automorphisms and anti-automorphisms of E. In the remaining case, E is an additive subgroup of a field F of characteristic 2 containing 1 such that F 2 E ⊂ E. Let K be the subring of F generated by E. Since F 2 ⊂ E, the subring K is, in fact, a subfield. We will denote by Ant(E) the group Aut(K). This group depends, of course, not on E alone but on E as a subset of F . It will be convenient, though, to have one name for the group we have in mind for all the cases and we do not think calling this group Ant(E) also in this last case will cause any confusion. We also extend the definition of ΦE in 29.2 by declaring that ΦE is the set containing just the identity map on F in this last case. Proposition 30.14. Let M = M(E, τ ) be one of the Moufang sets in 30.4, let X = E ∪ {∞}, let Ant(E) and ΦE be as in 30.13 and let g ∈ Sym(X) be an automorphism of M fixing ∞. Then there exists a ∈ E ∗ , b ∈ E, σ ∈ Ant(E) and φ ∈ ΦE such that g(x) = aφ(xσ ) + b

for all x ∈ E ⊂ X. Proof. Let µ = µxy be as in 1.6 applied to M with x = ∞ and y = 0. If E is a field or a quaternion division algebra, the claim holds by [55, 8.12.3]. Suppose that E is an additive group of a field F of characteristic 2 containing 1 such that F 2 E ⊂ E. The root group U∞ of M is {ρu | u ∈ E}, where ρu (x) = x + u for all x, u ∈ E. We can thus assume that g(0) = 0. Since g normalizes U∞ , it follows from g(0) = 0 that g is additive. It also follows from g(0) = 0, by [17, Prop. 4.3.1(4)], that
(30.15) g cµ(a)µ(b) = g(c)µ(g(a))µ(g(b)) for all a, b, c ∈ E ∗ . In M, we have cµ(a) Therefore

−1

µ(b)

= a−2 b2 c for all a, b, c ∈ E ∗ ,

g(a)−2 g(b)2 g(c) = g(a−2 b2 c)

(30.16)

for all a, b, c ∈ E ∗ . Setting a = c = 1, it follows that g(b)2 g(1)−1 = g(b2 )

for all b ∈ E ∗ . Let h be the map from E to F given by h(x) = g(x)g(1)−1 for all x ∈ E (so, in particular, h(0) = 0). Then h(a2 b2 ) = g(a2 b2 )g(1)−1

= g(a−1 )−2 g(b)2 = h(a

−1 −2

)

by (30.16)

2

h(b)

∗

for all a, b ∈ E . Hence (30.17)

h(b2 ) = h(12 · b2 ) = h(b)2

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263

and h(a2 ) = h(a2 · 12 ) = h(a−1 )−2

for all a, b ∈ E ∗ since h(1) = 1. Thus (30.18)

h(a2 b2 ) = h(a−1 )−2 h(b)2 = h(a2 )h(b)2 = h(a)2 h(b)2

for all a, b ∈ E. As in 30.13, we let K denote the subring of F generated by E. Then K 2 ⊂ F 2 ⊂ E. It follows from (30.18) that h(K 2 ) ⊂ K 2 and hence that we can extend h to an endomorphism σ of K by setting xσ = h(x2 )1/2 for all x ∈ K. Let a = g(1). Then g(x)2 = a2 h(x)2 = a2 h(x2 ) = (axσ )2 by (30.17) and thus g(x) = axσ for all x ∈ E. We have a2 ∈ F 2 ⊂ E = g(E), so a2 = g(z) for some z ∈ E. Therefore a = g(z)a−1 = h(z) = h(z 2 )1/2 = z σ ∈ E σ

by (30.17). From g(E) = E it follows that E = (zE)σ ⊂ K σ and thus K σ = K. Hence σ ∈ Aut(K). It remains only to consider the case that E is an octonion division algebra. Let N be the norm of E, let T be its trace and let F be its center. Thus N is a quadratic form over F and (30.19)

T (x) = ∂N (x, 1)

for all x ∈ E. As in the previous case, we can assume that g(0) = 0, from which it follows that g is additive and that (30.15) holds for all a, b, c ∈ E ∗ . For each a ∈ E ∗ , the map x 7→ ax is an automorphism of M fixing ∞. Hence we can assume that g(1) = 1. By 6.10, 30.11 and (30.15), the map g is semi-linear over F and g(N (x)) = N (g(x)) for all x ∈ E. Thus also g(T (x)) = T (g(x)) for all x ∈ E by (30.19). Let D be an arbitrary quaternion subalgebra of E and let e be a non-zero element in D⊥ , where D⊥ = {x ∈ E | T (xD) = 0}. Let D′ = g(D) and e′ = g(e). By 30.12, D′ is a quaternion subalgebra of E and the restriction of g to D is either an isomorphism or an anti-isomorphism from D to D′ . Replacing g by its product with the standard involution of E, we can assume that the restriction of g to D is an isomorphism from D to D′ . We have D⊥ = eD and (D′ )⊥ = e′ D′ . There thus exists a unique bijective additive map h from D to D′ such that g(ex) = e′ h(g(x)) for all x ∈ D. Let

φ(a + eb) = a + e′ h(b)

for all a, b ∈ D. Since g is semi-linear over F , the map h is linear over F . Since g(N (x)) = N (g(x)) for all x ∈ E, we have h(N (x)) = N (h(x)) for all x ∈ D. Hence φ is a linear map from E to itself and φ(N (x)) = N (φ(x)) for all x ∈ E. Thus φ is contained in the set ΦE (as defined in 29.2) and φ−1 (g(a + eb)) = φ−1 (g(a) + g(eb)) = g(a) + e′ g(b)

for all a, b ∈ D. Since g is an isomorphism from D to D′ , it follows that the composition φ−1 g is multiplicative and hence an automorphism of E.

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The next few results will be needed in the proof of 30.23. See 21.18 for the notion of a root cutting a residue. Proposition 30.20. Suppose that ∆ is as in 30.1(ii) and that the root α has panels of more than one type in its wall. Let Z0 be the set of roots β 6= ±α of Σ such that there is a reducible rank 2 residue cut by both α and β and let J0 = hµΣ (Uβ∗ ) | β ∈ Z0 i,

where µΣ is as in 1.18. Then J0 is contained in the stabilizer G†Σ,α and acts transitively on the set of residues of Σ of type C2 containing panels in the wall of α. Proof. Let Ω be the graph whose vertex set is the set of panels in the wall M of α, where two panels in M are contained in an edge e of Ω if there exists a rank 2 residue Re of Σ which contains them both. It follows from the convexity of residues that for each edge e of Ω, the residue Re is unique. Let Q be the set of edges e of Ω such that |Re | = 4. If e ∈ Q, there is a unique pair of opposite roots of Σ contained in the set Z0 . We can thus choose a root βe ∈ Z0 cutting Re for each e ∈ Q. If e ∈ Q, then the elements of µΣ (Uβ∗e ) stabilize Σ and interchange the two chambers in Re ∩ α and the two panels in e. By [62, 3.15], it follows that the elements of µΣ (Uβ∗e ) stabilize α for each e ∈ Q. Thus J0 ⊂ G†α,Σ . Suppose that ∆ is of type F4 . Numbering the vertices of the diagram F4 from left to right or from right to left, we can assume the panels in the wall of α are of type 1 and 2 (by [65, 29.27]). We declare two vertices of the graph Ω to be equivalent if they are contained in the same {1, 2, 3}-residue of Σ. Suppose that e is an edge of the graph Ω joining a vertex in one equivalence class Ω0 to a vertex of a second equivalence class Ω1 . Then the residue Re is either a {1, 4}- or a {2, 4}-residue. In either case, we have |Re | = 4. Thus the elements of µΣ (Uβ∗e ) interchange the two panels in e and hence interchange Ω0 with Ω1 as well. By [65, 29.23], the graph Ω is connected. It follows that the group J0 acts transitively on the set of equivalence classes. Let R0 be the unique {1, 2, 3}-residue containing the equivalence class Ω0 . Then R0 is a building of type C3 satisfying 30.1, Σ ∩ R0 is an apartment of R0 and α ∩ R0 is a root of this apartment. To complete the proof, it thus suffices to show that the claim holds when ∆ is of type C3 . Suppose that ∆ is of type C3 . In this case, there are just two residues R1 and R2 of type C2 containing panels in the wall of α, the set Q consists of just two edges e1 and e2 , the roots βe1 and βe2 are either equal or opposite in Σ and for both β = βe1 and β = βe2 , the elements of µ(Uβ∗ ) interchange R1 and R2 .

Corollary 30.21. Suppose that ∆ is as in 30.1(ii) and suppose that the root α has panels of more than one type in its wall. Then the group CG† (Uα ) Σ,α

acts transitively on the set of all residues of Σ of type C2 containing panels in the wall of α.

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265

Proof. By [62, 11.28(iii)], we have [Uα , µ(Uβ∗ )] = 1 for each root β 6= ±α of Σ such that there is a reducible rank 2 residue cut by both α and β. The claim holds, therefore, by 30.20. Proposition 30.22. Let P be a panel in the wall of the root α and let D be the set of elements in the stabilizer G†Σ,α that either centralize Uα or invert every element Uα . If ∆ is as in 30.1(ii), suppose, too, that the root α is long as defined in [40, 3.14]. Then D acts transitively on the set of panels in the wall of α of the same type as P . Proof. Let R be the arctic region of α, let P be the set of panels in the wall of α having the same type as P , let Z be the set of all roots β of Σ cutting R and let J = hµ(Uβ∗ ) | β ∈ Zi.

For every root β, the elements of µ(Uβ∗ ) are type-preserving and stabilize the apartment Σ as well as every panel in the wall of β. Thus the elements of µ(Uβ∗ ) stabilize both R and Σ for all β ∈ Z. By [40, 2.13], it follows that J is a subgroup of G†Σ,α . Choose mβ ∈ µ(Uβ∗ ) for each β ∈ Z. By 19.16, the group hmβ | β ∈ Zi acts transitively on P. It thus remains only to show that µ(Uβ∗ ) ∩ D 6= ∅ for each β ∈ Z. Let β be a root of Σ different from ±α, let n denote the gonality of the pair α, β and let φ be the angle between α and β as defined in [40, 2.14]. By [40, 2.14], β ∈ Z if and only if φ = π/2. Suppose that β ∈ Z. If n = 2, then ˆ [Uα , µ(U ∗ )] = 1 and hence µ(U ∗ ) ⊂ D. If by [62, 11.28(iii)] applied to ∆, β β ∆ is as in 30.1(ii), then [Uα , µ(Uβ∗ )] = 1 and hence µ(Uβ∗ ) ⊂ D by [40, 3.14] and the hypothesis that α is long. We can thus assume that n > 2 and that ∆ is as in 30.1(i). Let T be a residue of rank 2 cut by both α and β. Then |T ∩ Σ| = 2n > 4 and thus T is irreducible. By [40, 3.16] and the formulas in [60, 32.6–32.8 and 32.12], we can choose an element mβ in µ(Uβ∗ ) which acts trivially on Uα or inverts every element of Uα modulo the action of Uα on R. Since Uα acts faithfully on R, we conclude that mβ ∈ D. Proposition 30.23. Let G♭ be as in 29.8, let α and Σ be as in 30.1, let M = M∆,M be as in 30.2, let E be as in 30.3, let π : M(E, τ ) → M∆,M be the map defined in 30.4, which is an isomorphism by 30.5, let M∆,M be identified with M(E, τ ) via π and let g be an element of the stabilizer G♭Σ,α . Then there exist an element a of E ∗ , an element σ of the group Ant(E) defined in 30.13 and an element φ in the set ΦE defined in 29.2 and 30.13 such that (30.24)

g(x) = aφ(xσ )

for all x ∈ E. If ∆ and (C, F ) are as in 30.1(ii) and E = C, let λ denote the restriction of σ to F . In all other cases, let λ = σ. Then the following hold:

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(i) The element λ ∈ Ant(E) depends on the choice of the panel P , the vertex t of Π and the linking (πe ) in 30.3, but only up to conjugation by an element of Aut(E) (respectively, the restriction of an element of Aut(E) to Z(E) if E is quaternion or octonion). (ii) λ = γ∆ (g) for some Galois map γ∆ of ∆ as defined in 29.25. Proof. The existence of a, φ and σ such that (30.24) holds for all x ∈ E follows from 30.14. Given g, the element λ is uniquely determined by the isomorphism π. If we make different choices of P , t and (πe ) in 30.3, we obtain another isomorphism π ′ from M(E, τ ) to M∆,M . Replacing π by π ′ has the effect of replacing the automorphism π −1 gπ of M(E, τ ) by the automorphism κ−1 (π −1 gπ)κ, where κ denotes the automorphism π −1 π ′ of M(E, τ ). By 30.14 applied to κ, it follows that λ depends on the choice of P , t and (πe ), but only up to conjugation by an element of Ant(E) (respectively, the restriction of an element of Aut(E) to Z(E) if E is quaternion or octonion). If E is a quaternion or octonion division algebra, the standard involution is in the center of Ant(E). Therefore λ is, in fact, unique up to conjugation by the restriction of an element in Aut(E) to Z(E). Thus (i) holds. Suppose that g stabilizes a panel P in the wall M of α and let c be the unique chamber in P ∩ α. Then g fixes c and hence acts trivially on the apartment Σ. Thus g is contained in the group Mc,Σ defined in 29.8. By 29.15, (29.17) and 30.3, there is a Galois map γ∆ of ∆ such that λ = γ∆ (g). Thus (ii) holds under the assumption that g stabilizes a panel in the wall M. Next we suppose that there is a panel P in M such that P g has the same type as P (which is automatic if the Coxeter diagram Π of ∆ is not simply laced by 29.8). If ∆ and (C, F ) are as in 30.1(ii), we assume as well that α is a long root. By 30.22, there exists h ∈ G†α,Σ such that the product gh stabilizes P and h either centralizes Uα or inverts every element of Uα . Since h ∈ G† , we have γ∆ (gh) = γ∆ (g) for every Galois map γ∆ of ∆. Since (ii) holds for gh by the conclusion of the previous paragraph, it therefore holds for g as well. Now suppose that there is no panel P in M such that P g has the same type as P . Then g is not type-preserving. Hence Π is simply laced, thus M contains panels of every type and therefore the automorphism θ of Π induced by g has no fixed vertices. The only possibility is that Π = Aℓ for some even ℓ. Let {s, t} be the unique edge of Π stabilized by θ, let P be an s-panel in M , let c be the unique chamber in P ∩ α and let R be the unique {s, t}-residue of ∆ containing the chamber P . Then R ∼ = A2 (E), E is commutative (by 30.1) and α ∩ R is a root of the apartment Σ ∩ R of R. Let d be the unique chamber opposite c in the apartment Σ ∩ R and let Q be the unique t-panel containing d. Then Q is contained in the wall of α ∩ R and hence in M . By 30.22, we can assume that g maps P to Q. Thus g stabilizes R. Let e be the unique chamber in the apartment Σ ∩ R of R which is contained in α but not in P ∪ Q, let α1 be the unique root of Σ containing e but not c and let α3 be the unique root of Σ containing e but

STRICTLY SEMI-LINEAR AUTOMORPHISMS

267

not Q ∩ α. We identify Uα , Uα1 and Uα3 with the subgroups of Aut(R) they R induce and let U+ = Uα1 Uα Uα3 . Then R (U+ , Uα1 , Uα , Uα3 )

is a root group sequence of R and hence isomorphic to the root group sequence T (E) = (U+ , U1 , U2 , U3 ). There is an anti-automorphism of T (E) which interchanges x1 (u) with x3 (u) for all u ∈ E and maps each element of U2 to its inverse. By 3.17, this anti-automorphism extends to an automorphism of the root group labeling ζ in 30.3. There thus exists an element h of G♭α,Σ that interchanges P and Q and inverts every element of Uα . By 29.16, γ∆ (h) = 1 and hence γ∆ (gh) = γ∆ (g) for every Galois map γ∆ of ∆. Since gh stabilizes the panel P , we know that (ii) holds for gh. Hence it holds for g as well. Suppose, finally, that ∆, (C, F ) and η are as in 30.1(ii) and that α is short. Let R be a residue of type C2 containing a panel P in M , let s = Typ(P ) and let {s, t} = Typ(R). Then R ∼ = CI2 (E, F, η), E = C and by 30.21, we can assume that g stabilizes R. Let c be the unique chamber in P ∩ α, let d be the unique chamber opposite c in the apartment Σ ∩ R of R and let Q be the unique panel containing d such that Typ(Q) = Typ(P ). Then Q is the other panel contained in the wall of the root α ∩ R. Since g stabilizes α and R, it either stabilizes P and Q or interchanges them. Since we know that (ii) holds for g if g stabilizes P , we assume that g interchanges P and Q. Let (c0 , c1 , . . . , c8 ) be a labeling of the chambers in the apartment Σ ∩ R of R with subscripts in Z8 such that ci is adjacent to ci−1 for each i, c1 = d and c5 = c, so c2 , c3 , c4 , c5 are the chambers in Σ∩R. Let αi be the unique root of Σ that contains ci but not ci−1 for all i ∈ Z8 . Thus α2 = α. We identify each R Uαi with the subgroup of Aut(R) it induces and let U+ = Uα1 Uα2 Uα3 Uα4 . Since α is short, we can identify the root group sequence R Ω := (U+ , Uα1 , Uα2 , Uα3 , Uα4 )

with the root group sequence QI (E, F, η) and by (i), we can assume that the isomorphism x2 from E to Uα2 = Uα is the isomorphism x in 21.21. The elements of µΣ (Uα∗4 ) are contained in G†α,Σ , interchange P and Q and stabilize Σ, α and M . By [60, 32.9], we can choose m ∈ µΣ (Uα∗1 ) so that x2 (u)m = x2 (−uη ) for all u ∈ E. Since m interchanges P and Q, the product mg stabilizes P . The product ση has the same restriction to F as σ. Since m ∈ G† , we have γ∆ (m) = 1 and hence γ∆ (mg) = γ∆ (g) for every Galois map γ∆ of ∆. Since mg stabilizes P , we know that (ii) holds for mg. Hence it holds for g as well. Thus (ii) holds in every case. We close this chapter with three versions of what is really one result about fixed points of non-linear automorphisms of the Moufang sets in 30.5. Proposition 30.25. Let E be either a field or a quaternion division algebra, let F be the center of E and let ρ be the permutation of E given by ρ(x) = axσ + b

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for all x ∈ E, where a ∈ E ∗ , b ∈ E and σ is an element of Ant(E) that does not act trivially on F . Suppose, too, that ρ2 is the identity map. Then ρ has at least two fixed points in E. Proof. It follows from ρ2 = id that abσ + b = 0

(30.26) 2

2

and aaσ xσ = x or axσ aσ = x for all x ∈ E. Setting x = 1, we conclude that aaσ = 1

(30.27) and (30.28)

2

xσ = x for all x ∈ F.

Since σ acts non-trivially on F , it follows that there exist u, v ∈ F such that c := u + auσ 6= 0 and v + v σ 6= 0. By (30.27) and (30.28), we have (30.29)

acσ = a(uσ + aσ u) = c.

Let r = v + v σ , so rσ = r by (30.28) and thus (30.30)

s + sσ = 1

for s := vr−1 . Since s ∈ F , it follows from (30.26) and (30.30) that and hence also

ρ(sb) = sσ abσ + b = (1 − s)(−b) + b = sb ρ(sb + c) = sb + acσ = sb + c

by (30.29). Proposition 30.31. Let F be a field of characteristic 2, let E be an additive subgroup of F containing 1 such that F 2 E ⊂ E and let ρ be the permutation of F given by ρ(x) = axσ + b for all x ∈ F , where a ∈ E ∗ , b ∈ E and σ is a non-trivial automorphism of F . Suppose, too, that ρ(E) ⊂ E and that ρ2 acts trivially on E. Then ρ has at least two fixed points in E. Proof. The proof of 30.25 can be applied almost verbatim. It is only necessary to observe that since σ acts non-trivially on F , the automorphism σ also acts non-trivially on F 2 , and hence the elements u and v such that u + auσ 6= 0 and v + v σ 6= 0 can be chosen to lie in F 2 . Proposition 30.32. Let E be an octonion division algebra, let F be the center of E, let φ be an F -linear bijection from E to itself such that φ(1) = 1 and let ρ be the permutation of E given by ρ(x) = aφ(xσ ) + b for all x ∈ E, where a ∈ E ∗ , b ∈ E and σ is an element of Ant(E) that does not act trivially on F . Suppose, too, that ρ2 is the identity map. Then ρ has at least two fixed points in E.

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269

Proof. We again only have to make minor changes in the proof of 30.25, but this time we give all the details: Since φ(1) = 1 and φ is F -linear, we have 2

φ(x) = x for all x ∈ F.

It thus follows from ρ = id that aφ(bσ ) + b = 0

(30.33) as well as

aφ(aσ ) = 1

(30.34) and (30.35)

2

xσ = x for all x ∈ F.

Since σ acts non-trivially on F , there thus exist u, v ∈ F such that c := u + auσ 6= 0 and v + v σ 6= 0. By (30.34), (30.35) and the F -linearity of φ, we have (30.36)

aφ(cσ ) = a(uσ + φ(aσ )u) = c.

Let r = v + v σ , so rσ = r by (30.35) and thus (30.37)

s + sσ = 1

for s := vr−1 . Since s ∈ F , it follows from (30.33) and (30.37) that ρ(sb) = sσ aφ(bσ ) + b = (1 − s)(−b) + b = sb

and hence also ρ(sb + c) = sb + aφ(cσ ) = sb + c by (30.36).

Chapter Thirty One Galois Involutions In this short chapter we examine the fixed points of a strictly semi-linear automorphism of order 2 of a spherical building which satisfies the conditions in 30.1. Our main result is 31.10. Definition 31.1. Let ∆ be a spherical building satisfying the Moufang condition. A Galois involution of ∆ is an automorphism of ∆ of order 2 that is strictly semi-linear as defined in 29.27. We recall that by 29.19, ∆ can have a non-type-preserving semi-linear automorphism only if its Coxeter diagram is simply laced. Hypothesis 31.2. We assume for the rest of this chapter that our building ∆ is as in 30.1 (so, in particular, all root groups are abelian) and that τ is a Galois involution of ∆. Let Π be the Coxeter diagram of ∆, let S be the vertex set of Π, let Γ = hτ i and let Θ denote the subgroup of Aut(Π) induced by Γ. We let G, G† and H := HΣ be as in 29.7 and we let G♭ be as in 29.8. By 29.27, τ ∈ G♭ . Lemma 31.3. Let α be a root and suppose that there exists an element h of CG† (Uα ) such that the product τ h stabilizes α. Then the following hold: (i) Conjugation by τ h does not invert every element of Uα . (ii) If h = 1, then τ stabilizes two opposite residues. Proof. Let M be the wall of α and let M(E, τ ), M∆,M , x : E → Uα and π : M(E, τ ) → M∆,M be as in 30.3–30.5. Let F = Z(E) if E is a quaternion or octonion division algebra; otherwise let F = E. Since τ h stabilizes M , it acts on M∆,M . Let τˆ denote the automorphism of M(E, τ ) induced by τ h via π. Note that T := {x 7→ x + c | c ∈ E} is the automorphism group of M(E, τ ) induced by Uα via π. We can choose g ∈ Uα such that τ hg ∈ G♭Σ,α for some apartment Σ containing α. By 30.23 applied to the product τ hg, there exist a in E ∗ , b in E, σ ∈ Ant(E) and φ in the set ΦE defined in 29.2 such that (31.4) τˆ(x) = aφ(xσ ) + b for all x ∈ E and the restriction of σ to F is the image of τ h under some Galois map γ∆ of ∆. Since h ∈ G† and τ is semi-linear, the restriction of σ to F is non-trivial. Let ξc (x) = x + c for all x, c ∈ E. Then ξc ∈ T and τˆξc τˆ−1 (x) = x + aφ(cσ )

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for all c ∈ E. Suppose now that aφ(cσ ) = −c for all c ∈ E. The restriction of φ to F is the identity. Setting c = 1, we deduce that a = −1 and c = φ(c) = cσ for all c ∈ F . With this contradiction, we conclude that conjugation by τˆ does not invert every element of the subgroup T . Thus (i) holds. Now suppose that h = 1. Thus τ itself stabilizes the root α and τˆ2 = 1. Hence by 30.25, 30.31 and 30.32 applied to (31.4), we can choose a fixed point s of τˆ in E. There exists a unique root β such that π(s) = [β], where [β] is as in 30.2. Since [α] = π(∞), we have α 6= β. Since τ stabilizes both α and β, it also stabilizes the unique apartment of ∆ containing both α and β. By 25.10, therefore, τ stabilizes two opposite residues. Thus (ii) holds. Notation 31.5. The polar region of a root α of ∆ is the unique residue of ∆ containing the arctic region of α defined in [40, 2.3]; see 19.15. Proposition 31.6. Let R be a proper residue of ∆ and let UR be the unipotent radical UR as described in 24.16 and 24.17. Then there is a root α which is long as defined in [40, 3.14] such that Uα ⊂ Z(UR ).

Proof. Let c be a chamber of R, let Σ be an apartment of ∆ containing c and let α be a long root of Σ whose polar region contains c. By 19.27, R ∩ Σ ⊂ α. Thus Uα ⊂ UR . Let β be a root of Σ containing R ∩ Σ. By [40, 2.18], the angle between α and β (as defined in [40, 2.14]) is at most π/2. By [40, 3.14], [Z(Uα ), Uβ ] = 1. It follows that Z(Uα ) is contained in the center of UR . By 31.2, Uα is abelian. Proposition 31.7. Let α be a long root of ∆. Then CG† (Uα ) acts transitively on the set of long roots having the same polar region as α. Proof. Let R be the polar region of α, let Σ be an apartment containing α and let Z be the group generated by the root groups Uβ for all roots β of Σ containing chambers of R. By [40, 2.18 and 3.14], Z ⊂ CG† (Uα ) (just as in the previous proof). Let α′ be another long root with polar region R and let Σ′ be an apartment containing α′ . We want to show that α′ is in the same CG† (Uα )-orbit as α. By [62, 11.12], Z acts transitively on the set of all apartments of R. We can thus assume that Σ′ ∩ R = Σ ∩ R. Let T be the unique residue such that T ∩ Σ is opposite R ∩ Σ in Σ and let T ′ be the unique residue such that T ′ ∩ Σ′ is opposite R ∩ Σ in Σ′ . By 21.16, projT (x) = projT ∩Σ (x) for all x ∈ Σ. By 21.24, therefore, projT (R ∩ Σ) = T ∩ Σ. It follows that Σ is the convex hull of R ∩ Σ and projT (R ∩ Σ). Similarly, Σ′ is the convex hull of R ∩ Σ and projT ′ (R ∩ Σ). By [62, 9.8], T and T ′ are both opposite R. By 24.21, UR ⊂ Z acts transitively on the set of residues opposite R. We can thus assume that T ′ = T . Hence projT (R ∩ Σ) = projT ′ (R ∩ Σ), and so Σ′ = Σ. By [40, 2.13], α is the unique root of Σ whose polar region is Σ. Thus α′ = α. We conclude that Z acts transitively on the set of roots whose polar region is R. Proposition 31.8. Let R be a proper Γ-residue of ∆ and let UR be the unipotent radical of R. Then CZ(UR ) (Γ) 6= 1.

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Proof. For each u ∈ Z(UR ), the product τ (u)u is contained in CZ(Uα ) (Γ). Suppose that τ (u)u = 1 for all u ∈ Z(UR ). By 31.6, Uα ⊂ Z(UR ) for some long root α. Thus τ (u)u = 1 for all u ∈ Uα . In particular, Uα = Z(Uα ) is normalized by τ . By [40, 3.19], it follows that τ stabilizes the polar region of α. By 31.7, there exists h ∈ CG† (Uα ) such that the product τ h stabilizes α. The product τ h inverts every element of Uα . This is impossible, however, by 31.3(i). We conclude that there exists u ∈ Z(UR ) such that the product τ (u)u is non-trivial. Proposition 31.9. Suppose that R and T are two Γ-chambers of ∆ which are opposite and that S\Typ(R) is a single Θ-orbit, where Γ, S and Θ are as in 31.2. Then there is at least one further Γ-chamber in addition to R and T and all Γ-chambers have the same type. Proof. By 31.8, we can choose a non-trivial element u in CUR (Γ). Since T is a Γ-chamber, so is T u . By 24.21, T 6= T u . Since Ru = R and T 6= R, we have T u 6= R. Thus T u is a Γ-chamber distinct from T and R. By 22.10(iii), the three residues R, T and T u have the same type. Theorem 31.10. Suppose that ∆, S, Γ and Θ are as in 31.2 and that there exists a Γ-chamber R such that S\Typ(R) is a single Θ-orbit. Then there are at least three Γ-chambers and they all have the same type. Proof. We claim that there exist two opposite Γ-residues T and T ′ . This holds by 31.3(ii) if Γ stabilizes a root and by 25.16 if it does not. Let T1 = projT (R). By 22.3(ii), T1 is a Γ-chamber. By 22.10(ii), therefore, also S\Typ(T1 ) is a single Θ-orbit. Since T is a proper residue, it follows that T1 = T . Therefore T is a Γ-chamber such that S\Typ(T ) is a single Θ-orbit. Similarly, T ′ is a Γ-chamber. By 31.9, therefore, there is at least one further Γ-chamber in addition to T and T ′ and all Γ-chambers have the same type. We will apply 31.10 in the proof of 32.16 and 32.27.

Chapter Thirty Two Unramified Galois Involutions In this chapter we examine the fixed point building of an automorphism of a Bruhat-Tits building Ξ which induces a Galois involution on the building at infinity Ξ∞ . Our main result is 32.16. Notation 32.1. Throughout this chapter, Ξ is a Bruhat-Tits building, let ∆ = Ξ∞ be the building at infinity of Ξ (with respect to its complete system of apartments) and let G = Aut(∆). As observed in 1.27, we can identify G with Aut(Ξ). Let G† and G♭ be as in 29.7 and 29.8 and let F and/or {F1 /F0 , F0 /F1 } be as in 28.8 applied to ∆. Definition 32.2. We will say that Ξ is split (respectively, pseudo-split) if ∆ is split (respectively, pseudo-split) as defined in 28.9 and 28.16. We will say that Ξ is residually split (respectively, residually pseudo-split) if all its proper irreducible residues of rank at least 2 are split (respectively, pseudo-split). If Ξ is split or pseudo-split, then by the results in Chapter 18, Ξ is also residually split or pseudo-split. It is important to observe, however, that the converse assertions are false. If L/K is a separable quadratic extension ¯ = K, ¯ for of fields complete with respect to a discrete valuation such that L ˜ example, then the affine building F4 (L, K) (see 1.22) is residually split but neither split nor pseudo-split. Definition 32.3. Since the discrete valuation of a complete field is unique, there is a canonical homomorphism π from Aut(F ) to Aut(F¯ ) and also from Aut(F0 , F1 ) (defined as in 28.5) to Aut(F¯0 , F¯1 ). Let γ∆ be a Galois map of ∆ as defined in 29.25. The map γ∆ is unique up to an inner automorphism of Aut(F ) or Aut(F0 , F1 ). We will say that a subgroup Γ of G is unramified if Γ ⊂ G♭ and the restriction of the canonical homomorphism π to γ∆ (Γ) is injective. This notion is independent of the choice of γ∆ . Definition 32.4. We will say that an element of G (for example, a Galois involution of ∆) is unramified if the subgroup of G it generates is unramified as defined in 32.3. Definition 32.5. Let Γ be a subgroup of Aut(F ) or Aut(F0 , F1 ) and let π be as in 32.3. A Galois embedding ξ of Γ in G as defined in 29.26 is unramified if ξ(Γ) is unramified as defined in 32.3. Thus every Galois embedding of Γ in G is unramified if and only if the restriction of π to Γ is injective.

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Notation 32.6. Let A be an apartment of Ξ,1 let α be a root of A and let z = α∞ . We denote by Uα the stabilizer of α in the root group Uz of ∆. Proposition 32.7. Let A be an apartment of Ξ, let c be a chamber of A, let R be a proper residue of Ξ containing c, let H † be the pointwise stabilizer of A in G† and let Q be the set of roots of A containing chambers of R. Then the following hold: (i) (G† )R = hUα | α ∈ Qi · H † , where (G† )R denotes the stabilizer of the residue R in G† and Uα for α ∈ Q is as in 32.6. (ii) The stabilizer (G† )c of c in G† acts transitively on the set of all apartments of Ξ containing c. Proof. Let M = hUα | α ∈ Qi · H † and M1 = hUα | α ∈ Qc i · H † , where Qc is the set of roots of A containing c. Then M ⊂ (G† )R and by [65, 14.47(i)], (G† )c = M1 . By [65, 13.5], the stabilizer of an arbitrary panel P of R in M acts transitively on P . Therefore M acts transitively on R. Hence for each g ∈ (G† )R , there exists h ∈ M such that gh ∈ (G† )c = M1 ⊂ M.

Thus (i) holds. The group G† acts transitively on the set of all apartments of ∆. By [65, 8.27], it follows that G† acts transitively on the set of all apartments of Ξ. Since the stabilizer (G† )A contains all reflections of A, it acts transitively on the set of chambers of A. Therefore (ii) holds. The following should be compared to 30.2, where the building is assumed to be Moufang (and thus spherical) rather than Bruhat-Tits. Notation 32.8. Let M be the wall of some root of Ξ and let R, Z and ≈ be as in 21.36 (where ∆ is arbitrary) and let X = Z/ ≈. For each β ∈ R, let [β] denote the corresponding element β ∩ Z of X. Let i be the type of a panel in the wall M . By [65, 18.20], we can choose a gem R containing a panel P in M . Choose x ∈ X. Let β be a root in R such that [β] = x. By [65, 18.17(iii)], R is Moufang and the subgroup of Aut(R) induced by Uβ is the root group of R corresponding to the root β ∩ R of R. By [62, 11.4], therefore, the subgroup of Sym(X) induced by Uβ (which we denote by Ux ) acts sharply transitively on X\x. As was observed in 1.19, an arbitrary root group of R fixing the unique chamber in P ∩ x but not all of P induces the same subgroup of Sym(P ) as Uβ . It follows that the group Ux is independent of the choice of the root β fixing z. Hence the pair (X, {Ux | x ∈ X}) is a Moufang set. We denote this Moufang set by MΞ,M . 1 This

use of the letter A should not be confused with the use of the letter A in 20.1.

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Notation 32.9. We will call a panel a gem-residue if it is contained in a gem of Ξ. Suppose that P is a gem-panel. Then there exists an irreducible residue R of rank 2 contained in a gem such that P ⊂ R. Let M denote the Moufang set MR,P defined in 1.19 with R in place of ∆. If M is any wall containing P , then M is canonically isomorphic to the Moufang set MΞ,M defined in 32.8. In particular, M is independent of the choice of R. We will denote the Moufang set M by MΞ,P . If Π = C˜2 and the type of P is the vertex in the middle of Π, then P is a gem-panel and a torus of the Moufang set MΞ,P is precisely the joint torus of Ξ as defined in 10.19. Remark 32.10. By [65, 14.47(i)], the group G† acts transitively on the set of panels of Ξ of any given type. Thus if P and Q are two gem-panels of the same type, the Moufang sets MΞ,P and MΞ,Q are isomorphic. Notation 32.11. Let Ξ and ∆ = Ξ∞ be as in 32.1 and suppose that ∆ is as in 30.1. Let A be an apartment of Ξ, let α be a root of A, let −α be the root opposite α in A, let M be the wall of α, let z be the root α∞ of ∆, let −z = (−α)∞ , let m be the wall of z and let π and M(E, τ ) be as in 30.3 and 30.4 applied to m and z (in place of M and α). Thus π maps ∞ to [z] and 0 to [−z], where [z] and [−z] are as in 30.2 applied to m, and by 30.5, π is an isomorphism of Moufang sets from M(E, τ ) to M∆,m . Notation 32.12. We continue with the assumptions and notation in 32.11. In addition, let ν be the discrete valuation on E (or on the field F in 30.3 in the case that E is not closed under multiplication), let ΦE be as in 29.2, let Ant(E) be as in 30.13 and let h be an automorphism of M(E, τ ) fixing ∞ and 0. By 30.14, there exists a ∈ E ∗ , σ ∈ Ant(E), and φ ∈ ΦE such that h(x) = aφ(xσ ) for all x ∈ E or B. If φ is not the identity, then E is an octonion division algebra, N (φ(x)) = N (x) for all x ∈ E, where N is the norm of E, and for δ = 1 or 2, we have ν(x) = ν(N (x))/δ for all x ∈ E. It follows that ν is φ-invariant in every case. Since ν is unique (by [65, 23.15]), it is also σ-invariant. There are thus unique maps φ¯ and σ ¯ from E¯ to itself ¯ x) = φ(x) and σ ¯ such that φ(¯ ¯ (¯ x) = σ(x) for all x ∈ OE . We have σ ¯ ∈ Ant(E) and the map φ¯ is F¯ -linear, where F is the center of E. The map φ¯ is also bijective; this follows from the fact that the inverse of φ is also in ΦE . Proposition 32.13. Let Ξ and ∆ = Ξ∞ be as in 32.1 and suppose that ∆ is as in 30.1. Let A, α, −α, M , π, E and τ be as in 32.11, let [α] and [−α] ¯ ∗ and let M(E, ¯ τ¯) be be as in 32.8, let τ¯ be the permutation x ¯ 7→ −¯ x−1 of E as in (1.13). There exist a gem R containing panels in the wall M and an ¯ τ¯) to MΞ,M mapping ∞ and 0 to [α] and [−α] isomorphism π ¯ from M(E, such that the following hold: (i) R satisfies 30.1.

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¯ τ¯) → MR,MR obtained by applying (ii) There is an isomorphism ξ2 : M(E, 30.3–30.5 with the R in place of ∆ and the root α ∩ R of R in place of α such that π ¯ = ξ1 ◦ ξ2 , where MR is the wall of α ∩ R and ξ1 is the canonical isomorphism from MR,MR to MΞ,M . (iii) For each h ∈ G♭A,α , if x 7→ aφ(xσ ) denotes the permutation of E in× duced by h via π, then a ∈ OE and h induces the permutation ¯ σ¯ ) (32.14) x 7→ a ¯φ(x ¯ via π on E ¯ , where φ¯ and σ ¯ are as in 32.12.

Proof. As was observed in 32.8, we can choose a gem R containing a panel P in the wall M . Since ∆ satisfies 28.16, it follows from the results summarized in Chapter 18 that R does too. Thus (i) holds. Let ν be as in 32.12 and let z = α∞ . By [65, 23.27(iii) and 26.39], G acts transitively on the set of gems of Ξ. We can thus assume that R is the gem called R0 in [65, 18.1]. Let X denote the set of roots of the apartment A∞ of ∆ and let φ = φR = {φa | a ∈ X} be as in [65, 13.8]. By [65, 16.14, 20.2, 23.3, 25.31 and 26.11], φz (x(u)) = ν(u)/δ for all u ∈ E ∗ and δ = 1 or 2 (not depending on u), where x is the isomorphism from the additive group of E to Uz in 30.3, and if δ = 2, then E is not closed under multiplication, ν is a valuation of the field F in 30.3 and ν(E ∗ ) = 2Z. By [65, 18.20], therefore, the group UαR induced by Uα on R is isomorphic to {x(u) | ν(u) ≥ 0}/{x(u) | ν(u) > 0} ¯ There is thus an isomorphism x and hence to the additive group of E. ¯ from ¯ to U R such that for all u ∈ OE , x E ¯ (¯ u ) is the element of Aut(R) induced by α the element x(u) of Uα . Let e = {s, t}, πe and Ωst be as in 30.3 applied to R and α ∩ R. By [65, 18.17], UαR is the first term of Ωst and by [65, 18.27], the ¯ → U R is as in the linking (πe ) in 30.3 can be chosen so that the map x ¯: E α ¯ ∪ {∞} to the last display in 30.3. Let π ¯ denote the bijection from the set E point set of MΞ,M which sends ∞ to [α] and u ¯ to x¯(¯ u)([−α]) for all u ∈ OE . ¯ τ¯) to MΞ,M . Thus (ii) holds. By 30.5, π ¯ is an isomorphism from M(E, ¯ τ¯) with MΞ,M via π We identify M(E, τ ) with M∆,m via π and M(E, ¯. Suppose that x 7→ aφ(xσ ) is the permutation of E induced by an element h of G♭A,α . The element h normalizes Uα . By [65, 18.20], it also normalizes the sets {x(u) | ν(u) = i} for each i ∈ Z. Therefore ν(aφ(uσ )) = ν(u) for all × ¯ u ∈ E ∗ . It follows that a ∈ OE and hence h induces the map (32.14) on E. Thus (iii) holds. Lemma 32.15. Let ∆ be a thick building of type (W, S) and suppose that S is the disjoint union of subsets J and K such that [J, K] = 1. Suppose, too, that θ is an automorphism of (W, S) induced by an automorphism τ of ∆ of order 2 and that θ interchanges J and K. Then for each s ∈ J, τ stabilizes at least three residues of type S\{s, sθ }. Proof. Choose s ∈ J, let L = S\{s} and let R be an arbitrary L-residue. By 21.37(i), R ∩ Rτ 6= ∅ and thus by [62, 7.25], R ∩ Rτ is an S\{s, sθ }-residue.

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Since τ is of order 2, it stabilizes this intersection. Thus every L-residue contains an S\{s, sθ }-residue stabilized by τ . Let P be an s-panel. No two chambers of P can lie in the same L-residue. Since ∆ is thick, we conclude that there are at least three L-residues. We come now to the main result of this chapter. Theorem 32.16. Let Ξ be a Bruhat-Tits building such that ∆ = Ξ∞ is as in 30.1 and suppose that τ is an unramified Galois involution of ∆ as defined in 31.1. Then Γ := hτ i is a descent group of both ∆ and Ξ. Furthermore, ΞΓ is a Bruhat-Tits building and there is a canonical isomorphism from ∆Γ to (ΞΓ )∞ , where ΞΓ and ∆Γ are the fixed point buildings obtained by applying 22.25 to the pairs (Ξ, Γ) and (∆, Γ). Proof. To prove 32.16, we first prove 32.17–32.20 in which we assume that Γ and Ξ satisfy all the hypotheses in 32.16. Lemma 32.17. The group Γ stabilizes infinitely many residues of Ξ. Proof. We identify Ξ with the metric space D as in 26.35. Let u be an arbitrary point of D and let x be the midpoint of the closed segment from u to τ (u). Then x is fixed by Γ. Let R be a maximal Γ-residue of ∆, let C denote the set of all conical cells of D with cone point x and let C ′ denote the set obtained from C by replacing each conical cell in C by its topological closure. By [1, 11.75], there is a canonical containment-preserving bijection from C ′ to the set of simplices of ∆. Let C be the element of C ′ corresponding to the residue R viewed as a simplex of ∆. Since |Γ| = 2, R is either a maximal residue of Σ or the intersection of two maximal residues. It follows that the intersection of C with the unit sphere B1 (x) in D centered at x is either a single point or a segment of a great circle of B1 (x). Since this intersection is stabilized by Γ, it contains a fixed point y of Γ. We conclude that Γ fixes every point on the unique ray with initial point x passing through y. By 26.37, every point on this ray is contained in a simplex of D that is stabilized by Γ. Since the simplices of D have bounded size, we conclude that Γ stabilizes infinitely many simplices. Hence Γ stabilizes infinitely many residues of Ξ. Let (W, S) be the Coxeter system of ∆ and let Θ be the subgroup of Aut(W, S) induced by Γ. Lemma 32.18. Γ-chambers are not maximal in ΞΓ . In particular, there exist Γ-panels. Proof. Let R be a Γ-chamber that is maximal in ΞΓ and let J = Typ(R) ⊂ S. The set S\J must be a single Θ-orbit. By 32.17, we can choose Γ-residues T distinct from R. Let T1 = projT (R). By 22.3(ii), T1 is a Γ-chamber parallel to R. Let w = δ(R, T1 ). By 21.10(iii), we have Typ(T1 ) = w−1 Jw. By 22.3(iii), w is fixed by Θ. Thus the subset wTyp(T )w−1 of S is also fixed by

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Θ and contains J. Hence J = wTyp(T )w−1 by the choice of R. It follows that T = T1 and S\K is also a single Θ-orbit, where K = Typ(T ). By 21.22 and the choice of T , we have w 6= 1. By 21.6(i), we have K ⊂ J + (w). Suppose that t ∈ J + (w) for some t ∈ S\K. Then Θ(t) ⊂ J + (w) by 19.14 and hence S = J + (w). This implies, however, that w = 1. With this contradiction, we conclude that K = J + (w), but then 21.30 implies that Ξ is spherical. Let Π denote the Coxeter diagram associated with (W, S). Recall that for each subset J of S, ΠJ denotes the subdiagram of Π spanned by J. Let θ denote the automorphism of (W, S) induced by the Galois involution τ (so Θ = hθi). Remark 32.19. We call attention to the proof of 32.27 below. As we prove 32.20 under the hypotheses in 32.16, we will point out in five footnotes the few modifications needed to prove 32.20 under the hypotheses in 32.27. Lemma 32.20. Let R be a Γ-panel, let T be a Γ-chamber contained in R, let J = Typ(R), let K = Typ(T ), let s ∈ J\K, let L be the unique subset of J containing s such that ΠL is a connected component of ΠJ and let L1 = L ∪ θ(L), so L1 = θ(L1 ) ⊂ J, [L0 , L1 ] = 1 for L0 = J\L1 (which might be empty) and either L1 = L or the Coxeter diagram ΠL1 has two connected components. Let R1 be an L1 -residue of R containing chambers of T , let ξ be the homomorphism from Aut(R) to Aut(R1 ) obtained by applying 21.40 with R and R1 in place of ∆ and R and let λ ∈ Aut(R1 ) be the image under ξ of the restriction τ |R of τ to R. Then the following hold: (i) If L = L1 , then λ is a Galois involution of R1 if |L| > 1 and λ fixes at least three but not all chambers in R1 if |L| = 1. (ii) R contains at least three Γ-chambers. Proof. Let T1 = R1 ∩ T . By [62, 7.25], T1 is an (L1 ∩ K)-residue. Since L1 is Θ-invariant and contains s, we have L1 ∩ K = L1 \Θ(s). Let π be the restriction of projR1 to R. Thus cλ = π(cτ ) for each chamber c of R1 . Since ξ is a homomorphism, we have λ2 = 1. Furthermore, T1λ = π(T1τ ) = π(R1τ ∩ T τ ) = π(R1τ ∩ T ). By 21.6(i) and the convexity of residues, we have π(R1τ ∩ T ) ⊂ T . Hence 2 T1λ ⊂ T1 . Therefore T1 = T1λ ⊂ T1λ and thus T1 is stabilized by λ. Suppose that T1′ is an arbitrary (L1 \Θ(s))-residue of R1 stabilized by λ. Thus (T1′ )τ is also an (L1 \Θ(s))-residue and (32.21)

π((T1′ )τ ) = (T1′ )λ = T1′ .

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Let κ be the restriction of projRτ1 to R, let T ′ = π −1 (T1′ ) and let T ′′ = κ−1 ((T1′ )τ ). By (32.21), we have (32.22) (T1′ )τ ⊂ T ′ . ′ By 21.37(iii), T is the unique K-residue of R such that T1′ ⊂ T ′ ∩ R1 and T ′′ is the unique K-residue of R such that (T1′ )τ ⊂ T ′′ ∩ R1τ . Thus T ′′ = (T ′ )τ and by (32.22), T ′ = T ′′ . Hence τ stabilizes T ′ . We conclude that distinct (L1 \Θ(s))-residues of R1 stabilized by λ give rise to distinct K-residues of R stabilized by τ . By 22.3(ii), a K-residue of R stabilized by τ is a Γ-chamber. Thus to prove (ii), it will suffice to show that there exist at least three L1 \Θ(s)-residues of R1 stabilized by λ. This holds by 32.15 if L1 6= L. We can thus assume that L1 = L. This means that R1 is irreducible. We define a relation ∼ on the residue R by setting x ∼ y whenever projV (y) = x, where V is the unique L-residue containing x. By 21.38(i)– (ii), ∼ is an equivalence relation on R. Let H denote the kernel of the action of the stabilizer G†R on R/ ∼. Let x be a chamber of T1τ ⊂ R1τ , let A be an apartment of Ξ containing both x and xτ and let Σ = A∞ . For i = 0 and 1, let αi be a root of A such that the wall of the root αi ∩ R of R contains panels whose type lies in Li . By [65, 29.27], the type of every panel in the wall of αi ∩ R lies in Li . By [65, 29.24], there is a residue R∗ of rank 2 of R containing a panel Pi in the wall of αi ∩ R for both i = 0 and i = 1. Since [L0 , L1 ] = 1, the residue R∗ ∩ A of A is a circuit containing just four chambers.2 Let u0 be an element of Uα0 acting non-trivially on P0 , let u1 be an element of Uα1 acting non-trivially on P1 and let mi = µΣ (ui ) for i = 0 and 1. By [65, 13.28], the elements m0 and m1 each induce a reflection on the circuit R∗ ∩ A. Hence (m0 m1 )2 acts trivially on R∗ ∩ A. It follows that (m0 m1 )2 acts trivially on A and thus ∞ also on Σ. This implies that the roots z0 := α∞ 0 and z1 := α1 of Σ are orthogonal (as defined in [40, 2.14]). Therefore either [Uz0 , Uz1 ] = 1 or there is a root z∗ of Σ at angle π/4 to both z0 and z1 such that [Uz0 , Uz1 ] ⊂ Uz∗ . Suppose that we are in the second case and that α∗ is a root of A such that α∞ ∗ = z∗ . Then R ∩ α∗ cannot be a root of R, since if it were, its wall would contain panels whose type is in L0 or L1 , which would imply, as we just saw, that z∗ is orthogonal to z0 or z1 . Thus if α∗ ∩ R contains a chamber of R, it equals R ∩ A. We conclude that if α0 ∩ α1 contains a chamber of R, then the commutator group [Uα0 , Uα1 ] acts trivially on R ∩ A. Now suppose that {c, d} is a pair of i-adjacent chambers of A ∩ R for some i ∈ L0 = J\L and let α1 be the unique root of A containing c but not d. Let R2 be the unique L-residue of Ξ containing c, let j ∈ L, let e be the unique chamber of A that is j-adjacent to c, let P be the j-panel containing c and e and let α0 be the unique root of A containing c but not e. We have c = projR2 (d) and hence every chamber of R2 ∩ A is nearer to c than to d. Thus R2 ∩ A ⊂ α1 . In particular, Uα1 fixes the chamber e. Since c ∈ α0 ∩ α1 , 2 For the proof of 32.27: By [60, 40.9(iii)], |R∗ ∩ A| = 4 implies that [U α0 , Uα1 ] = 1 and the rest of this paragraph can be ignored.

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the commutator group [Uα0 , Uα1 ] acts trivially on R ∩ A (by the conclusion of the previous paragraph). Thus the subgroup Uα1 fixes eg for all g ∈ Uα0 . Since Uα0 acts transitively on P \{c}, it follows that Uα1 acts trivially on P . Since j is an arbitrary element of L, we conclude by [62, 9.7] that Uα1 acts trivially on R2 . Therefore Uα1 acts trivially on R/ ∼. In other words, Uα1 is contained in the subgroup H defined above. Let x ∈ T1τ ⊂ R1τ be as above and let γ be a minimal gallery from x to projR1 (x). Since x and xτ are contained in A, all of γ is contained in A. Since [L0 , L1 ] = 1, the type of γ is a word in the free monoid on L0 . Suppose Q is an arbitrary panel containing successive chambers of γ and let α and −α denote the two roots of A whose wall contains Q. By [65, 13.28],3 the product U−α Uα U−α contains an element interchanging α and −α. By the conclusion of the previous paragraph, both Uα and U−α are contained in H. We conclude that H contains elements interchanging the two chambers in Q ∩ A. Since Q is arbitrary, it follows that we can choose an element of H mapping x ∈ R1τ to projR1 (x). Let ρ denote its inverse. Thus ρ ∈ (G† )R , ρ maps R1 to R1τ and ξ(ρ|R ) = 1. Continuing to let x ∈ T1τ be as above, we set d = xτ , so d ∈ T1 , and let P be a panel of R1 containing d. Let α be the unique root of the apartment A containing d but not P ∩A, let i = Typ(P ) and let j = θ(i). (Thus if |L| = 1, then P = R1 and j = i.) We have j ∈ θ(L) = L = typ(R1 ). By 21.38(i), the restriction of π to R1τ is a type-preserving isomorphism from R1τ to R1 . Hence the automorphism λ of R1 is type-preserving if and only if θ acts trivially on L. If j 6= i, then τ induces a non-type-preserving involution on ∆, thus the Coxeter diagram of ∆ is simply laced and hence also the Coxeter diagram Π of Ξ is simply laced. By [65, 29.52], therefore, we can choose a panel Q in the wall of α ∩ R1 of type j. If j = i, we choose Q = P . Let c be the unique chamber in Q ∩ α. By 32.7(i), there exists an element g ∈ (G† )R1 such that the product gτ ρ stabilizes R1 and maps d to c and hence P to Q, where ρ is as at the end of the previous paragraph. By 32.7(ii)4 applied to c, we can assume that gτ ρ stabilizes the apartment A. Since α is both the unique root of A containing d but not P ∩ A and the unique root of A containing c but not Q ∩ A, we conclude that gτ ρ stabilizes α. Thus gτ ρ ∈ G♭A,α . Let M denote the wall of α, let m = M ∞ and let π, E, a, φ and σ be as in 30.4 and 30.23 applied to ∆ and m with α∞ in place of α and h in place of g and let M(E, x 7→ −1/x) be identified with M∆,m via π. Modulo this identification, the product gτ ρ fixes ∞ and induces the map (32.23) x 7→ aφ(xσ ) ¯ on E. Now let M(E, x 7→ −1/x) be identified with MΞ,M via the map called × π ¯ in 32.13(ii). By 32.13(iii), a ∈ OE and the product gτ ρ fixes ∞ and induces the map ¯ σ¯ ) (32.24) x 7→ a ¯φ(x 3 Or 4 Or

[62, 11.22] for the proof of 32.27. [62, 11.12] for the proof of 32.27.

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¯ modulo this identification. There is a canonical identification of MΞ,M on E with MR1 ,M1 , where M1 denotes the wall of the root α ∩ R1 of R1 . Let g1 denote the restriction of the product gτ ρ to R1 . By 30.23(ii), we have σ = γ∆ (gτ ρ) for some Galois map γ∆ of ∆ and, if the rank of |L| of R1 is at least 2, then (32.25)

σ ¯ = γR1 (g1 )

for some Galois map γR1 of R1 . Since ρ and g both lie in G† , we have σ = γ∆ (τ ) and hence (32.26) 5

σ ¯ 6= 1

since τ is unramified. From now on, we replace the elements τ , ρ and g of GR with their restrictions to R. Suppose that |L| = 1, so that P = R1 , the points of the Moufang set MR1 ,M1 are simply the chambers of R1 and the type of T1 is empty. Hence T1 = {d} = {c} and λ = ξ(τ ) fixes d (since T1 is stabilized by λ). Since g and gτ ρ both stabilize R1 , we have ξ(g) = g|R1 and ξ(gτ ρ) = g1 . Since ξ is a homomorphism and ξ(ρ) = 1, it follows that g stabilizes d. Since g ∈ G† , we conclude that λ = ξ(τ ) induces the map ¯ σ¯ ) + ¯b x 7→ e¯φ(x

¯ for some e ∈ O× and some ¯b ∈ OE .6 Thus λ acts non-trivially on R1 on E E and by 30.25, 30.31 or 30.32, λ fixes at least three chambers in R1 . Suppose, finally, that |L| > 1. As in the previous paragraph, we have ξ(ρ) = 1, g ∈ G† , ξ(g) = g|R1 and ξ(gτ ρ) = g1 . By (32.25) and (32.26), therefore, λ = ξ(τ ) is strictly semi-linear. Thus (i) holds and by 31.10, λ stabilizes at least three (L\Θ(s))-residues of R1 . This concludes the proof of (ii). We can now finish the proof of 32.16. By 32.18 and 32.20(ii), there exist Γ-panels and each of them contains at least three Γ-chambers. Thus Γ is a descent group of Ξ as defined in 22.19. By 27.29 and 27.31, we conclude that Γ is also a descent group of ∆, that ΞΓ is a Bruhat-Tits building and that (ΞΓ )∞ ∼ = ∆Γ . Theorem 32.27. Let ∆ be a Moufang spherical building satisfying the conditions in 30.1. Suppose that τ is a Galois involution of ∆ as defined in 31.1, let Γ = hτ i and suppose that there exist proper residues stabilized by Γ. Then Γ is a descent group of ∆. 5 For the proof of 32.27, we set m = M . By 30.23, we can identify M(E, x 7→ −1/x) with M∆,M1 so that the product gτ ρ fixes infinity and induces the map in (32.23) on E for some a ∈ E ∗ and some φ ∈ ΦE and σ = γ∆ (gτ ρ) = γ∆ (τ ) 6= 1 for some Galois map γ∆ of ∆. 6 For the proof of 32.27, we conclude that λ = ξ(τ ) induces the map x 7→ eφ(xσ ) + b on E for some e ∈ E ∗ and some b, where φ and σ are as in the previous footnote.

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Proof. By assumption, there exist Γ-panels. As was observed in 32.19, the result 32.20 holds verbatim under our present hypotheses. By 32.20(ii), every Γ-panel contains at least three Γ-chambers.

PART 5

Exceptional Tits Indices

Chapter Thirty Three Residually Pseudo-Split Buildings In the remainder of this monograph we apply the results of Parts 2, 3 and 4 ˜ 2 is to show that every exceptional Bruhat-Tits building of type other than G the fixed point building of a strictly semi-linear descent group of a residually pseudo-split Bruhat-Tits building and to determine the corresponding Tits index. Let Ξ be an exceptional Bruhat-Tits building. In this chapter, we identify in each case a residually pseudo-split Bruhat-Tits building ΞL (in 33.20– 33.25). In the next chapter, we will show that ΞL contain Ξ as the fixed point building (in the sense of 22.22) of a finite strictly semi-linear descent ˜ 2 ; see 34.15 and group Γ (with some caveats in the case that Ξ is of type G 36.12). In the final chapter, we examine the corresponding Tits indices. ˜ E (Λ) (in the notation described in 1.22) We begin with the case that Ξ = B 2 for some quadratic space Λ = (K, V, q) of type E6 , E7 or E8 . Our goal in 33.1–33.16 will be to identify an unramified extension L/K (as defined in ¯ is a pseudo-splitting field (as defined in 7.7) such that the residue field L 1 2.33) of both q¯0 and q¯1 . Proposition 33.1. Let Λ = (K, V, q) be a quadratic space of type E6 , E7 or E8 , let E be a norm splitting field of q as defined in 2.23, let f = ∂q and let X, 1 ∈ V and π : X → V be as in 8.13 applied to Λ. Then there exists b ∈ X ∗ such that E is the splitting field of the polynomial x2 − f (π(b), 1)x + q(π(b)) over K. Proof. Let (a, x) 7→ ax and θ be as in 8.13. By [60, 12.13], we can choose a norm splitting T of q (as defined in [60, 12.14]) such that E is the splitting field of the minimal polynomial of T over K. By [60, 13.10], we can assume that T is linked to the map (a, x) 7→ ax at an element b ∈ X ∗ as defined in [60, 13.2] with 1 ∈ V in place of ǫ. Let θb denote the map v 7→ θ(b, v). By [60, 13.61], θb is a norm splitting map and there exist r ∈ K ∗ and s ∈ K such that T (v) = rθ(b, v) + sv for all v ∈ V . Thus E is also the splitting field of the minimal polynomial of θb over K. By [60, 13.42 and 13.56(iii)], therefore, this minimal polynomial is x2 − f (π(b), 1)x + q(π(b)). 1 In the three cases of 14.3 where q ¯0 and q¯1 are already both totally singular, we make an additional demand on L; see 33.8.

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Proposition 33.2. Let (K, V, q), R0 , R1 and δΨ be as in 11.1 and suppose ¯ : K] ¯ = δΨ for every norm that R0 and R1 are not both indifferent. Then [E splitting field E of q. Proof. Let E be a norm splitting field of q, let N be the norm of the extension E/K and let b ∈ X ∗ be as in 33.1, so there exists a root γ of the polynomial x2 − f (π(b), 1)x + q(π(b)) in E and q(π(b) + t) = N (γ) + f (π(b), 1)t + t2 = N (γ − t) ∈ N (E) for all t ∈ K. Therefore ν(N (E)) = 2Z if and only if ν(q(π(b) + t)) is even for all t ∈ K. Thus if δΨ = 2, then ν(N (E)) = 2Z (by 10.4 and [65, 21.24]) ¯ 6= K. ¯ and hence E Suppose that δΨ = 1, let S and Z be as in 8.12 and let mi for all i be as in 8.29. By 11.2 and 13.12, we have m0 = 0. By 8.32, therefore, mi = 0 for all even i. Hence by 8.29, (33.3)

Si = Si+1 + Zi

for all even i, where Sj and Zj are as in 8.17 for all j. Suppose that ν(q(π(b) + t)) is even for all t ∈ K. Choose t ∈ K and let i = ν(q(b) + t). It follows from (33.3) that (b, t) ∈ Sj + Zi for all j ≥ i. By 8.21, however, this implies that (b, t) ∈ Z and thus b = 0. This contradicts the choice of b. Hence ν(q(π(b) + t)) is odd for some t ∈ K. Therefore ν(N (E)) = Z and hence ¯ = K. ¯ E We now work our way through the various cases in 14.3. The references in 14.2 should be helpful. Proposition 33.4. Let (K, V, q), R0 , R1 and δΨ be as in 11.1. Suppose that δΨ = 2, that R0 and R1 are not both indifferent and that Ξ is in one of the generic cases of 14.3 as defined in 14.1. Then there exists an unramified quadratic extension L/K such that qL is hyperbolic. Proof. By 11.4(i) in cases (1.i.a), (2.i.a) and (3.i.a) of 14.3, by 12.7(i) in case (1.ii.a) of 14.3, by 12.8(i) in case (2.ii.a) and by 12.10 in cases (2.iii.a) ¯ such that M is a and (3.ii.a), there is a separable quadratic extension M/K splitting field of both q¯0 and q¯1 . Let L/K be the unique unramified extension ¯ = M (which exists by 7.26). By 7.44, L is a splitting field of such that L the quadratic form q. an In the following, qL denotes the anisotropic part of qL (as defined in 2.32). an Thus qL is hyperbolic if and only if qL = 0. We say that a quaternion or octonion division algebra D is wild if its norm is totally wild as defined in 7.6. By [65, 26.15], this is equivalent to the condition that T¯ is identically zero, where T is the trace of D.

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289

Proposition 33.5. Let (K, V, q), ℓ, f¯0 , f¯1 , R0 , R1 and δΨ be as in 11.1. Suppose that δΨ = 2 and that f¯0 and f¯1 are not both identically zero but that Ξ is not in one of the generic cases of 14.3 as defined in 14.1. Then R0 and R1 are not both indifferent and there exists an unramified quadratic extension L/K with the following properties: an (i) The quadratic form qL is similar to the norm of a wild unramified quadratic extension with ground field L in cases (1.i.b) and (1.ii.b) of 14.3. an (ii) The quadratic form qL is similar to the norm of a wild quaternion division algebra which is unramified in cases (2.i.b) and (2.ii.b) and ramified in case (2.iii.b). an (iii) The quadratic form qL is similar to the norm of a wild octonion division algebra which is unramified in cases (3.i.b) and (3.ii.c) and ramified in case (3.ii.b). an In particular, dimL (qL ) = 2ℓ−5 .

Proof. Since f¯0 and f¯1 are not both identically zero, the residues R0 and R1 are not both indifferent by 14.5. Since δΨ = 2, Ξ is in an unramified or semi-ramified case of 14.3 as defined in 11.3 and identified in 14.2. In each of the non-generic ramified or semi-ramified cases of 14.3 with f¯0 and f¯1 not both identically zero, we observe that for j = 0 or 1 (but not both) there exists a subspace Z of V¯j such that the restriction of q¯j to Z is similar to the norm of a quaternion division algebra. Let ϕ denote the restriction of q¯j to Z, let M be a norm splitting field of ϕ, let L/K be the unique unramified ¯ = M , let {e1 , . . . , e4 } be a subset of Vj whose image extension such that L ¯ let W be the subspace of V spanned {¯ e1 , . . . , e¯4 } in V¯j is a basis of Z over K, by the set {e1 , . . . , e4 }, let ψ be the restriction of q to W and let p be as in 7.1. It follows from 7.4 that ψ/pj is unramified and ψ¯j = ϕ. By 7.36, it follows that ψ is similar to the norm of a quaternion division algebra D, L is a splitting field of ψ and ξ := ψ/pj is the unique unramified quadratic form over K such that ξ¯0 ∼ = φ. Let Q denote the restriction of q to W ⊥ . Let ℓ = 6, so we are in case (1.i.b) or (1.ii.b) of 14.3 and j = 0, so ψ is unramified. By 8.10, Q is similar to the norm of the discriminant extension E/K of q. Therefore E is a splitting field of ψ (by 8.4). Since ψ is unramified, ¯ K ¯ is quadratic. Hence either Q or Q/p is it follows that the extension E/ unramified. The quadratic form q is unramified and the co-dimension of ¯ is 4 in case (1.i.b) and q¯0 = φ and f¯1 = 0 in case (1.ii.b). Rad(f¯0 ) over K ¯ 0 is totally singular in case (1.i.b) and from 7.18 It follows from 7.23 that Q ¯ 1 is totally singular in case (1.ii.b). After replacing Q by Q/p in that Q ¯ 0 is totally singular in both cases. Thus case (1.ii.b), we conclude that Q ¯ 0 )M is anisotropic. By 7.37, therefore, QL is anisotropic, so by 2.29, (Q an ∼ qL = QL , and ¯ 0 )M , (QL )0 ∼ = (Q

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so (QL )0 is totally singular. By 7.43, QL is unramified. Since Q is similar to the norm of the separable quadratic extension E/K, QL is similar to the norm of the separable quadratic extension F/L, where F denotes the composite field EL. Since (QL )0 is totally singular, the quadratic extension F/L is wild. We suppose now that ℓ = 7 or 8. By 8.10, Q is similar to the norm of a quaternion division algebra if ℓ = 7 and Q is of type E7 with Hasse invariant [D] ∈ Br(K) if ℓ = 8. The quadratic form q is unramified and the ¯ is 4 in cases (2.i.b) and (3.i.b). It follows co-dimension of Rad(f¯0 ) over K ¯ 0 is totally singular in these two cases. from 7.23 that Q is unramified and Q Suppose we are in case (2.ii.b) or (3.ii.c). Then j = 1, dimK¯ (¯ q1 ) = 4 ¯ 1 is totally and f¯0 is identically zero. By 7.18, Q/p is unramified and Q singular. Suppose, finally, that we are in one of the remaining two cases, ¯ 1 ) ≤ 2 in case (2.iii.b), dimK¯ (Q ¯ 1 ) ≤ 4 in (2.iii.b) or (3.ii.b), Then dimK¯ (Q case (3.ii.b) and j = 0 in both cases. By 7.4, we have ¯ 0 ) = 4 − dimK¯ (Q ¯ 1) ≥ 2 (33.6) dimK¯ (Q in case (2.iii.b) and (33.7)

¯ 0 ) = 8 − dimK¯ (Q ¯ 1) ≥ 4 dimK¯ (Q

in case (3.ii.b). Therefore (7.24) holds with ψ and Q in place of q ′ and q ′′ . ¯ is 4 in these two cases, it follows Since the co-dimension of Rad(f¯0 ) over K ¯ from 7.23 that Q0 is totally singular and ¯ 0 ) = (dimK Q)/2. dimK¯ (Q By (33.6) and (33.7), therefore, ¯ 1 ) = (dimK Q)/2 = dimK¯ (¯ dimK¯ (Q q1 ). ¯ 1 is totally singular. Since f¯1 = 0 in both cases, we conclude that also Q ¯ By 2.29, (Qi )M is anisotropic for i = 0 and 1 (in all six cases). By 7.37, an therefore, QL is anisotropic, so qL = QL , and ¯ i )M , (QL )i ∼ = (Q so (QL )i is totally singular for i = 0 and 1. Since Q has a norm splitting, so does QL . Since L is a splitting field of D, QL has trivial Hasse invariant if ℓ = 8. We conclude that QL is similar to the norm of a wild quaternion division algebra if ℓ = 7 and (by [60, 12.28]) that QL is similar to the norm of a wild octonion division algebra if ℓ = 8. By 7.43, QL is ramified if and only if Q is. Thus QL is ramified in cases (2.iii.b) and (3.ii.b) and unramified in the other cases. Proposition 33.8. Let (K, V, q), ℓ, f¯0 , f¯1 , R0 and R1 be as in 11.1. Suppose that f¯0 and f¯1 are both identically zero but that R0 and R1 are not both indifferent, so that ℓ = 7, and let D be the quaternion division algebra such that [D] is the Hasse invariant of q. Then there exists an unramified quadratic an extension L/K such that L is a splitting field of D and qL = qL is the norm of a wild octonion division algebra which is unramified in case (2.i.c) of 14.3 and ramified in cases (2.ii.c) and (2.iv.c).

RESIDUALLY PSEUDO-SPLIT BUILDINGS

291

Proof. We are in case (2.i.c), (2.ii.c) or (2.iv.c) of 14.3. Let T be the trace of ¯ is either a D. By 15.6, T¯ is not identically zero. By [65, 26.15], therefore, D ¯ K ¯ is a separable quadratic extension. Let quaternion division algebra or D/ ¯ K ¯ is a separable quadratic extension. L be as in 7.45 applied to D. Thus L/ Hence by 2.29, the quadratic forms (¯ q0 )L¯ and (¯ q1 )L¯ are anisotropic. By 7.37, therefore, qL is anisotropic and totally wild. Since L splits D, the Hasse invariant of qL is trivial. Since qL has a norm splitting, it must therefore be similar to the norm of an octonion division algebra. By 7.43, qL is ramified if and only if q is. Proposition 33.9. Let (K, V, q), f¯0 , R0 , R1 and δΨ be as in 11.1. Suppose that δΨ = 1 and that f¯0 6= 0. Then R0 and R1 are not both indifferent and there exists an unramified extension L/K with the following properties: an (i) L/K is quadratic and qL is similar to the norm of a ramified separable quadratic extension in case (1.iii.a) of 14.3. an (ii) L/K is biquadratic and qL = 0 in cases (2.iv.a) and (3.iii.a); furthermore, there exists a subfield L1 of L containing K such that L1 /K is an quadratic, qL is similar to the norm of a ramified but not wild quater1 an nion division algebra in case (2.iv.a) and qL is similar to the norm of 1 a ramified but not wild octonion division algebra in case (3.iii.a). an (iii) L/K is quadratic and qL is similar to the norm of a wild ramified quaternion division algebra in case (2.iv.b). an (iv) L/K is quadratic and qL is similar to the norm of a wild ramified octonion division algebra in case (3.iii.b).

Proof. Since f¯0 = 6 0, we can choose u, v ∈ V such that (33.10)

ν(q(u)) = ν(q(v)) = ν(f (u, v)) = 0,

where f = ∂q. By 8.1(iii), we can choose a norm splitting field E of q. By 33.2, the extension E/K is ramified. By 8.9, there exist subspaces W1 , . . . , We of V for e = ℓ − 5 (where ℓ is as in 14.3) that are pairwise orthogonal with respect to ∂q such that the restriction Qi of q to Wi is similar to the norm of a quaternion division algebra Di over K containing E for each i ∈ [1, e] and u, v ∈ W1 . If ℓ = 7, let D3 be the unique quaternion division algebra such that [D3 ] is the Hasse invariant of q. Then E is a subfield of D3 and by 8.5, (33.11)

[D1 ] + [D2 ] + [D3 ] = 0

in Br(K) for both ℓ = 7 and 8. Since the quadratic extension E/K is ramified, the quaternion division algebra D1 is also ramified as are D2 and ¯ i /K ¯ is a quadratic extension for each i. By D3 if ℓ ≥ 7. In other words, D ¯ 1 /K ¯ is separable (since u, v ∈ W1 ). By 8.4 and 8.10(i), (33.10), D (33.12)

q∼ = Q1 ⊕ αN

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for some α ∈ K ∗ if ℓ = 6, where N is the norm of the extension E/K. If ℓ ≥ 7, then q = Q1 ⊕ · · · ⊕ Qe and by 7.23, we have (33.13)

q¯0 ∼ = (Q1 )0 ⊕ (Q2 )0

if ℓ = 7 and (33.14)

q¯0 ∼ = (Q1 )0 ⊕ (Q2 + Q3 )0

¯1 = D ¯ 1. if ℓ = 8. Let L1 /K be the unique unramified extension such that L Then L1 is a splitting field of Q1 (by [43, Lemma 3]). By 14.5, the residues R0 and R1 are not both indifferent. Since δΨ = 1, we are in one of the ramified cases of 14.3 as defined in 11.3. Suppose first that ℓ = 6. Hence we are in case (1.iii.a). Let F denote the composite field EL1 . Then F/L1 is a ramified separable quadratic extension and by (33.12), an qL is similar to the norm of this extension. Thus (i) holds with L = L1 . 1 Now let ℓ ≥ 7. If L1 were also a splitting field of Q2 , it would follow from (33.11) that L1 is also a splitting field of Q3 and hence of q. This is impossible, however, since every norm splitting field of q is ramified over K (by 33.2). Thus L1 is not a splitting field of Q2 . If ℓ = 8, we set ϕ = Q2 ⊕ Q3 and W = W2 + W3 ⊂ V and observe that by 8.10(iii), ϕ is a quadratic form of type E7 with Hasse invariant [D1 ], so we can set (33.15)

˜ E (K, W, ϕ). ˆ=B Ξ 2

Suppose that we are in case (2.iv.a). Then ℓ = 7 and f¯0 is non-degenerate. ¯ 2 /K ¯ is also separable. Let L2 /K be the unique By (33.13), it follows that D ¯2 = D ¯ 2 . Since L1 is not a splitting field unramified extension such that L of D2 , we have L1 6= L2 . Hence D2 ⊗K L1 is a quaternion division algebra an is similar to its norm. Let L denote the composite field L1 L2 . Then and qL 1 L/K is an unramified biquadratic extension, and L is a splitting field of both Q1 and Q2 . Hence L is a splitting field of q as well. Suppose next that we are in case (3.iii.a). Thus ℓ = 8 and f¯0 is nonˆ degenerate. By (33.14), ϕ¯0 is non-singular. Thus the Bruhat-Tits building Ξ defined in (33.15) is in case (2.iv.a) of 14.3. By the conclusions of the previous paragraph, therefore, there exists an unramified biquaternion extension L/K such that L is a splitting field of ϕ. By (33.11), it follows that L is also a ¯ is a splitting field splitting field of Q1 (and hence of q). By 7.37, therefore, L ¯ ¯ of (Q1 )0 . Thus L1 ⊂ L and hence L1 is K-isomorphic to a subfield of L. Since L1 is not a splitting field of q (by 33.2), ϕL1 is not hyperbolic. Since ϕL1 has a norm splitting and its Hasse invariant is trivial, it follows that an ϕL1 is similar to the norm of an octonion division algebra. Hence ϕL1 = qL 1 and thus (ii) holds. Suppose, finally, that we are in case (2.iv.b) or (3.iii.b). In both cases the defect of q¯0 has co-dimension 2. By (33.13) and (33.14), it follows that Q2 is an ∼ wild if ℓ = 7 and ϕ is wild if ℓ = 8. If ℓ = 7, we conclude that qL = (Q2 )L1 1 is the norm of a wild quaternion division algebra (since L1 is not a splitting field of Q2 ). Thus (iii) holds. Suppose that ℓ = 8. By 15.4, the Bruhat-Tits

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ˆ in (33.15) is in case (2.iv.c) of 14.3. By 33.8, therefore, there building Ξ exists an unramified quadratic extension L/K such that L is a splitting field ¯=D ¯1 = L ¯ 1 and therefore L = L1 ) and ϕL is similar to the norm of D1 (so L of an octonion division algebra. Since L is a splitting field of D1 , we have an ∼ qL = ϕL . Thus (iv) holds. Notation 33.16. Let (K, V, q), f¯0 , f¯1 , R0 and R1 be as in 11.1. If R0 and R1 are not both indifferent, we let L/K be as in 33.4, 33.5, 33.8 or 33.9 and an if R0 and R1 are both indifferent, we let L = K, so that qL = q. We identify four cases: an (A) qL = 0. We are thus in either one of the unramified or semi-ramified generic cases of 14.3 or one of the two ramified generic cases (2.iv.a) or (3.iii.a). In these two cases, let L1 /K be as in 33.9(ii). In case an (2.iv.a) let C be the quaternion division algebra over L1 such that qL 1 is similar to the norm of C and in case (3.iii.a) let C be the octonion an division algebra over L1 such that qL is similar to the norm of C. See 1 33.17. an (B) qL is similar to the norm of a separable quadratic extension C/L which is either wild or ramified, a wild quaternion division algebra C over L or a wild octonion division algebra C over L, but f¯0 and f¯1 are not both identically zero. an (C) qL is similar to the norm of a wild octonion division algebra C over L and f¯0 and f¯1 are both identically zero. In this case, we denote by σ the standard involution of C. an (D) qL = q.

Case (A) corresponds to 33.4 and 33.9(ii), case (B) corresponds to 33.5 and 33.9(i), (iii) and (iv), case (C) corresponds to 33.8 and case (D) occurs if and only if R0 and R1 are both indifferent. Notation 33.17. We will denote by (A1 ) the subcase (2.iv.a) or (3.iii.a) of case (A) and by (A0 ) the remaining subcases of case (A). Suppose that ¯ L ¯ 1 is a separable we are in subcase (A1 ) and let C be as in (A). Then C/ quadratic extension in case (2.iv.a) and C¯ is a quaternion division algebra ¯ 1 in case (3.iii.a). over L Next we turn briefly to the case that ˜ F (Λ) Ξ=B 2

for some quadratic space Λ of type F4 . Proposition 33.18. Let q, qˆ, K, F , R0 and R1 be as in 17.1 and suppose ˆ that R0 and R1 are not both indifferent. Then there exist fields L and L ˆ such that K ⊂ L, F ⊂ L, ˆ ⊂ L, L2 ⊂ L

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an ˆ L/K and L/F are both unramified quadratic extensions, qL is similar to the 2 an ˆ ˆ norm of the extension L/L , qˆLˆ is similar to the norm of the extension L/L, ˆ and the map ρ 7→ ρ| ˆ is an isomorphism the group Gal(L/K) stabilizes L L an an ˆ from Gal(L/K) to Gal(L/F ). In particular, qL and qˆL ˆ are both totally singular.

Proof. By 9.15 and 17.10, neither q nor qˆ is totally wild. By 9.13 and 9.14, we can assume that the extensions E/K and D/F in 9.1 and 9.3 are both ˆ = D and choose γ ∈ L\K. Then L = K(γ) unramified. Let L = E and L 2 ˆ ˆ and and L = F (γ ). Since F ⊂ K, it follows that Gal(L/K) stabilizes L ˆ the map ρ 7→ ρ|Lˆ is an isomorphism from Gal(L/K) to Gal(L/F ). By 2.29, ˆ 1/2 and q an is similar F 1/2 ⊗K L is isomorphic to F 1/2 L = F 1/2 (γ) = L L ˆ 1/2 /L. Similarly, K ⊗F L ˆ is isomorphic to to the norm of the extension L ˆ = K(γ 2 ) = L and qˆan is similar to the norm of the extension L/L. ˆ KL ˆ L ˆ be as in 33.18 and let Aut(L, L) ˆ be as Notation 33.19. Let K, F , L and L ˆ in 28.5. By 33.18(ii), we can think of Gal(L/K) as the subgroup of Aut(L, L) generated by the unique non-trivial element acting trivially on both K ⊂ L ˆ We will denote this subgroup by Gal(L/K, L/F ˆ ). and F ⊂ L. ∗

∗

∗

We now proceed to identify the residually split buildings ΞL that we will work with in Chapters 34 and 36. See 33.26. Notation 33.20. Let Λ = (K, V, q) be as in 11.1, let ˜ E (K, V, q) and ∆ = Ξ∞ = CE (K, V, q), Ξ=C 2 2 and let L/K, as well as L1 /K and C in case (A1 ), C in case (B) and C and σ in case (C), be as in 33.16. Since K is complete with respect to a discrete valuation, so are L and L1 . In case (A), we set ˜ ℓ (L) and ∆L = Eℓ (L), ΞL = E in subcase (A1 ), we also set ˜4 (C, L1 ) and ∆L = F4 (C, L1 ), ΞL1 = F 1 in case (B), we set ˜4 (C, L) and ∆L = F4 (C, L), ΞL = F in case (C), we set ˜ I (C, L, σ) and ∆L = BI (C, L, σ) = CI (C, L, σ) ΞL = B 3 3 3 if C is ramified and ˜ I (C, L, σ) and ∆L = BI (C, L, σ) = CI (C, L, σ) ΞL = C 3 3 3 if C is unramified and in case (D) we set ΞL = Ξ and ∆L = ∆. In all four cases, ∆L = Ξ∞ L , the building ΞL is residually pseudo-split and the extension L/K is unramified. In case (A1 ), we also have ∆L1 = Ξ∞ L1 , but the building ΞL1 is not residually pseudo-split.

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Notation 33.21. Let Λ = (K, L, q) and F be as in 17.1, let ˜ F (Λ) and ∆ = Ξ∞ = BF (Λ). Ξ=B 2

2

If q is not totally wild, we set ˜4 (L, L) ˆ and ∆ ˆ = F4 (L, L), ˆ ΞL,Lˆ = F L,L ˆ is as in 33.18. If q is totally wild, we set L = K, L ˆ = F, Ξ ˆ = Ξ where L/L L,L and ∆L,Lˆ = ∆. In both cases, ∆L,Lˆ = Ξ∞ , Ξ is residually pseudo-split ˆ ˆ L, L L,L ˆ (by 17.10 if q is totally wild) and the extensions L/K and L/F are both unramified. It will sometimes be convenient to write ∆L and ΞL when we really mean ∆L,Lˆ and ΞL,Lˆ . Notation 33.22. Let ˜ I (Λ) and ∆ = Ξ∞ = XI (Λ) Ξ=X 3 3 for some honorary involutory set Λ = (C, F, σ) as defined in [65, 30.22] such that F is complete with respect to a discrete valuation, where X = B if Λ is ramified and X = C if Λ is unramified. Suppose that C is not wild. By 7.45, we can choose an unramified quadratic extension L/F such that L is a splitting field of C. We set ˜ 7 (L) and ∆L = E7 (L). ΞL = E If C is wild, we set L = F , ΞL = Ξ ∆L = ∆. Thus in both cases, ∆L = Ξ∞ L , the building ΞL is residually pseudo-split and the extension L/F is unramified. Notation 33.23. Let ˜ 2 (C) and ∆ = Ξ∞ = A2 (C) Ξ=A for some octonion division algebra C complete with respect to a discrete valuation. Let F denote the center of C. Suppose C is not wild. By 7.45, we can choose an unramified quadratic extension L/F such that L is a splitting field of C. We set ˜ 6 (L) and ∆L = E6 (L). ΞL = E If C is wild, we set L = F , ΞL = Ξ and ∆L = ∆. Thus in both cases, ∆L = Ξ∞ L , the building ΞL is residually pseudo-split and the extension L/F is unramified. Notation 33.24. Let ˜4 (Λ) and ∆ = Ξ∞ = F4 (Λ) Ξ=F for some composition algebra Λ = (C, F ) such that F is complete with ¯ := (C, ¯ F¯ ) is a composition algebra respect to a discrete valuation. Thus Λ ¯ is not and hence of type (i)–(v) as described in [65, 30.17]. Suppose that Λ of type (i) or (ii). By 7.45, we can choose an unramified quadratic L/F such that L is a splitting field of C. We set ˜ m (L) and ∆L = Em (L), ΞL = E

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where m = 6 if (C, F ) is of type (iii), m = 7 if (C, F ) is of type (iv) and m = 8 if Λ is of type (v). If Λ is of type (i) or (ii), we set L = F , ΞL = Ξ and ∆L = ∆. Thus in every case, ∆L = Ξ∞ L , the building ΞL is residually pseudo-split and the extension L/F is unramified. Notation 33.25. Let ˜ 2 (Λ) and ∆ = Ξ∞ = G2 (Λ) Ξ=G for some hexagonal system Λ = (J, F, #) over a field F complete with respect ¯ be as in [65, 25.28]. By [60, 17.6], Λ ¯ is in to a discrete valuation and let Λ one of the six families of hexagonal systems in the table on page 148 of [60]. ¯ M (as defined in [60, 15.40]) is of type k/M for By [60, 15.39 and 15.41], Λ k = 1, 3, 9 or 27, where either M = F¯ or k ≥ 9 and M/F¯ is a separable quadratic extension. Suppose k > 1. By [60, 15.22 and 15.29], it follows ¯ K contains a subalgebra isomorphic to (T /M )+ for some separable that Λ cubic extension T /M . We let S be the normal closure of the extension T /F¯ . Thus S/F¯ is a separable extension of degree at most 12. We then let L be the field obtained by applying 7.26 to the extension S/F¯ . Thus L/F is an unramified Galois extension of degree dividing 12 and we set ˜ 4 (L) and ∆L = D4 (L) ΞL = D if dimF J = 3, ˜ 6 (L) and ∆L = E6 (L) ΞL = E if dimF J = 9 and ˜ 8 (L) and ∆L = E8 (L) ΞL = E if dimF J = 27. If k = 1, we set L = F , ΞL = Ξ and ∆L = ∆. Thus in every case, ∆L = Ξ∞ L , the building ΞL is residually pseudo-split and the extension L/K is unramified. Every exceptional Bruhat-Tits building is isomorphic to one of the buildings called Ξ in 33.20–33.25. We are thus finished making our list of residually pseudo-split buildings. 33.26. Existence. The existence of ΞL in every case of 33.20–33.25 holds by [65, 27.2]. Note that to apply [65, 27.2] in case (D) of 33.20, we need to cite 8.15 as well (but since we are assuming Ξ exists in the chapter and we are setting ΞL = Ξ in this case, 8.15 is not really required here). See 34.16. We will make use of the following remark in Chapter 36. Observation 33.27. Let ΞL be as in 33.20 or 33.21 and let Π be the Coxeter diagram of ΞL . Suppose that Π is not simply laced and let J be a subset of the vertex set S of Π such that the Coxeter diagram ΠJ is isomorphic to the Coxeter diagram Bn for n = 2 or 3 but Π has no subdiagram isomorphic ¯ = 2 and to Bn+1 and let R be a J-residue of ΞL . Then either char(L) Q ∼ R = Bn (Λ) for some anisotropic quadratic space Λ which is totally singular ¯ 6= 2, Π = F˜4 , ΞL is as in case (B) of 33.20, C/L is a (by 28.15) or char(L) ramified separable quadratic extension and by [65, 26.17(ii)], R ∼ = BQ 3 (Λ) for Λ an anisotropic quadratic space Λ of dimension 1.

Chapter Thirty Four Forms of Residually Pseudo-Split Buildings In this chapter we show that every exceptional Bruhat-Tits building of rank ˜ 2 is the fixed point building of an unramified at least 3 but not of type G group of order 2 or 4 acting on a residually pseudo-split Bruhat-Tits building. Our main result is 34.13. Let ∆ = Ξ∞ and ∆L = Ξ∞ L be as in 33.20, 33.21, 33.23, 33.22 or 33.24. Our proof of 34.13 rests on the fact that in every case, there is a Galois action (as defined in 29.261) of Γ := Gal(L/K) on ∆L whose fixed point building is isomorphic to ∆. The exact assertions are formulated in 34.3–34.8. We discuss the proofs of these assertions in 34.12. 34.1. Γ-forms. When we say that “∆ is a Γ-form of ∆L ” in 34.3–34.9, what we really mean is that “∆ is a φ(Γ)-form of ∆L for some Galois embedding φ of Γ in G♭L ,” where G♭L is the subgroup of GL := Aut(∆L ) defined in 29.8 and the terms form and Galois embedding are used as defined in 22.23 and 29.26. Conventions 34.2. From now on we will display Tits indices using the conventions in [53]. Thus to display a Tits index T = (Π, Θ, A), we draw the Coxeter diagram Π, bending edges where necessary so that vertices in the same Θ-orbit are conspicuously near to each other, and put a circle around the set of vertices in each orbit of Θ disjoint from A. Thus the circles correspond to the vertices of the relative Coxeter diagram of T. When |Θ| ≤ 2, the Tits index can be reconstructed unambiguously from the display; if |Θ| > 2, we can make the Tits index unambiguous only by supplementing the display with an explicit indication of the structure of the group Θ, as, for example, in the fifth and seventh lines in the affine column of Table 36.8 below, where the superscript indicates the order of Θ (which must be isomorphic to either A3 or S3 ). The absence of an explicit indication means that |Θ| ≤ 2 and the absence of any bent edges as, for example, in the fourth Tits index in the affine column of Table 36.8, means that Θ is trivial. Note that it can happen, as, for example, in the sixth Tits index in the affine column of Table 36.8, that all the Θ-orbits disjoint from A are trivial even though Θ is not trivial. We begin now with our description of the Galois action of Γ = Gal(L/K) of ∆L such that ∆ ∼ = ∆ΓL in all the different cases: 1 Note that by 32.27 (and 22.47 in case (A )), “Galois action” implies automatically 1 that the image of Γ in Aut(∆L ) is a descent group of ∆L in all the cases we are considering; see also 34.10. We will refer to the Tits index of the image of Γ in Aut(∆L ) as the Tits index of Γ.

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Proposition 34.3. Let ∆ = BE2 (K, V, q), ℓ, L/K and ∆L be as in 33.20, suppose that we are in case (A0 ), (B) or (C) as defined in 33.16 and 33.17 and let Γ = Gal(L/K). Then ∆ is a Γ-form of ∆L with Tits index • •

• •

•

•

in case (A) with ℓ = 6, • •

•

•

•

•

•

in case (A) with ℓ = 7, • •

•

•

•

•

•

•

in case (A) with ℓ = 8, •

•

•

•

in case (B) and •

•

•

in case (C). Proposition 34.4. Let ∆ = BE2 (K, V, q), ℓ, L/L1 , ∆L and ∆L1 be as in the subcase (A1 ) of 33.20, let Γ = Gal(L/K) and let Γ1 = Gal(L/L1 ). Then the following hold: (i) ∆L1 is a Γ1 -form of ∆L with Tits index • •

•

•

•

•

•

if ℓ = 7 and Tits index • •

•

•

•

•

•

•

if ℓ = 8. In both cases, ∆L1 is a building of type F4 . (ii) ∆ is both a Γ/Γ1 -form of ∆L1 with Tits index •

•

•

•

and, as already indicated in 34.3, a Γ-form of ∆L with Tits index • •

•

•

•

•

•

if ℓ = 7 and Tits index • •

if ℓ = 8.

•

•

•

•

•

•

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FORMS OF RESIDUALLY PSEUDO-SPLIT BUILDINGS

Γ/Γ1

Note that by 22.47, there is a canonical isomorphism from ∆ΓL to ∆L1 in 34.4. Proposition 34.5. Let ∆ = BF 2 (K, V, q),

ˆ and ∆L = ∆ ˆ be as in 33.21, suppose that q is not totally wild and L/L L,L ˆ ) be as in 33.19. Then ∆ is a Γ-form of ∆L with Tits let Γ = Gal(L/K, L/F index •

•

•

•

Proposition 34.6. Let ∆ = CI3 (C, F, σ), L/F and ∆L be as in 33.22, let Γ = Gal(L/F ) and suppose that the octonion division algebra C is not wild. Then ∆ is a Γ-form of ∆L with Tits index • •

•

•

•

•

•

Proposition 34.7. Let ∆ = A2 (C), L/F and ∆L be as in 33.23, let Γ = Gal(L/F ) and suppose that the octonion division algebra C is not wild. Then ∆ is a Γ-form of ∆L with Tits index • •

•

•

•

•

Proposition 34.8. Let ∆ = F4 (C, F ), ¯ F¯ ) is not of L/F and ∆L be as in 33.24, so the composition algebra (C, type (i) or (ii) as defined in [65, 30.17], and let Γ = Gal(L/F ). Then ∆ is a Γ-form of ∆L with Tits index •

• •

•

• •

if C is commutative, • •

•

•

•

•

•

if C is quaternion and • •

if C is octonion.

•

•

•

•

•

•

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Proposition 34.9. Let ∆ = G2 (Λ), k, M/F¯ , L/F and ∆L be as in 33.25, suppose that k > 1 and let Γ = Gal(L/F ). Then ∆ is a Γ-form of ∆L with Tits index • • •

•

if k = 3, • •

•

•

•

•

if k = 9 and M = F¯ , •

• •

•

if k = 9 and M 6= F¯ and

• •

• •

•

•

•

•

•

•

if k = 27 (whether or not M = F¯ ). Remark 34.10. Note that Γ ∼ = Z2 × Z2 in 34.4 and |Γ| = 3, 6 or 12 in 34.9, but that |Γ| = 2 in every other case. Observation 34.11. Suppose we are in case (A0 )—that is, case (A) but not case (A1 )—of 33.20. Thus ∆L = Eℓ (L) for ℓ = 6, 7 or 8. Let T be one of the first three indices in 34.3, the first if ℓ = 6, the second if ℓ = 7 and the third if ℓ = 8, and suppose that there is an embedding (but not necessarily a Galois embedding) of Γ in Aut(∆L ) with respect to which Γ is a descent group of ∆L with Tits index T such that ∆ΓL ∼ = ∆. It follows from [40, 2.9(iv)] that Γ stabilizes the polar region of a root α in each of the three cases ℓ = 6, 7 and 8. Since the Coxeter diagram of ∆L is simply laced, every root is long as defined in [40, 3.14]. By [40, 3.17], therefore, Γ normalizes the root group Uα . 34.12. Comments about the proofs of 34.3–34.9. The claims in 34.3– 34.9 can be proved by explicit calculations after defining the action of the group Γ in terms of coordinates for the root group datum defining the building ∆L . Proofs in which this strategy is carried out can be found in [25] for the E6 -case of 34.3, in [39, §7, Main Thm.] for 34.5 and in [33, Thm. 2] for 34.6. These proofs serve as models for proofs of the remaining five results, although admittedly the required calculations could be quite involved in some cases. For a proof of case (A0 ) of 34.3 that uses instead the theory of algebraic groups, we can apply [60, 42.6]. This proof exploits the fact that ∆L is the building associated to the group G(L) of L-rational points of an absolutely simple group G. (See [60, 42.3.6] for a description of ∆L in terms of the

FORMS OF RESIDUALLY PSEUDO-SPLIT BUILDINGS

301

parabolic subgroups of G(L).) The field K, having a discrete valuation, is infinite. As indicated in [60, 42.4.1], this means that the algebraic K-group G (in the sense of [60, 42.2.1]) can be thought of simply as the abstract group G(L) endowed with an action of the Galois group Γ via an embedding φ of Γ in Aut(∆L ). The results in [60, 42.6] imply that Γ is a descent group of ∆L whose Tits index is one of the first three indices in 34.3 and that ∆ΓL ∼ = ∆. By 34.11, therefore, Γ normalizes a root group Uα of ∆L . Let Σ be an apartment of ∆L containing α, let c be a chamber in α adjacent to a chamber of Σ not in α and let (πe ) be as in 29.9 and 29.15 applied to ∆L with respect to the pair (Σ, c). The linking (πe ) gives rise to an isomorphism ι from the root group Uα to the additive group of L. The action of Γ coming from φ is a twisted version of the action of Γ = Gal(L/K) on ∆L with respect to which ι(u)σ = ι(uσ ) for all u ∈ L; see, for example, [53] or [60, 42.4]. It follows from this that the subgroup φ(Γ) of Aut(∆) is strictly semi-linear in our sense and that φ is thus a Galois embedding. For a proof of 34.7 in the same spirit, see [55, 5.12]. In cases (B) and (C) of 34.3, in 34.4 and in 34.6–34.9, ∆L (respectively, ∆L1 in 34.4) is also the building associated to the group of L-rational points of an absolutely simple group G (although in most cases, G is not split over L). It should thus be possible to obtain proofs in all these cases along similar lines by applying the results on forms of algebraic groups in [53] and [60, 42.6]. We come now to the main result of this chapter. Note that the hypothesis that Ξ is as in 33.20, 33.21, 33.23, 33.22 or 33.24 is equivalent to the assumption that Ξ is, up to isomorphism, an arbitrary exceptional Bruhat˜2. Tits building of type other than G Theorem 34.13. Let Ξ, ∆ = Ξ∞ , ΞL and ∆L = Ξ∞ L be as in 33.20, 33.21, 33.23, 33.22, respectively, 33.24 and let Γ be as in 34.3, 34.4, 34.5, 34.7, 34.6, respectively, 34.8. Then ΞL is residually pseudo-split and there is a Galois embedding of Γ into Aut(ΞL ) with respect to which Γ is an unramified descent group of both ∆L and ΞL , ∆ is isomorphic to the fixed point building ∆ΓL and Ξ is isomorphic to the fixed point building ΞΓL . Proof. By 33.20, 33.21, 33.23, 33.22 and 33.24, ΞL is residually split and the extension L/K is unramified. By 34.3, 34.4, 34.5, 34.7, 34.6 and 34.8, there is a Galois embedding of Γ into Aut(∆L ) with respect to which Γ is a descent group of ∆L and ∆ is isomorphic to the fixed point building ∆ΓL . By 32.5, Γ is unramified. We now apply 32.16 (twice, along with 22.47, in the two cases described in 34.4) to conclude that Γ is a descent group of Ξ and that ΞΓ is a Bruhat-Tits building such that (ΞΓ )∞ ∼ = (∆L )Γ ∼ = ∆. L

Since Ξ is the unique Bruhat-Tits building whose building at infinity is ∆, it follows that ΞΓL is isomorphic to Ξ. Theorem 34.14. Every exceptional Bruhat-Tits building of type other than ˜ 2 is a form of a residually pseudo-split Bruhat-Tits building. G

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Proof. This holds by 22.23 and 34.13. ˜ 2 . We are confident that the results formulated in 34.13 34.15. The Case G hold also if we allow Ξ, ∆ = Ξ∞ , ΞL and ∆L = Ξ∞ L to be as in 33.25 and Γ to be as in 34.9, so that 34.14 holds for all exceptional Bruhat-Tits buildings. We cannot justify this claim, however, with the methods developed in this monograph, where we have exploited the assumption |Γ| = 2 in several places to keep the arguments as elementary as possible. 34.16. Existence. Suppose that Ξ and ΞL are as in one of the cases (A)– (C) but not (D) of 33.20. As we observed in 33.26, ΞL exists by [65, 27.2]. Thus 34.14 yields an existence proof for Ξ under the assumption that Ξ is not itself residually pseudo-split. For an alternate approach to the existence of Ξ under a similar restriction, see [46]. See also 10.3.

Chapter Thirty Five Orthogonal Buildings By orthogonal building, we mean a spherical building isomorphic to Dn (K) for some field K or BQ n (Λ) for some anisotropic quadratic space Λ. In this chapter, we prove a few results about certain forms of these groups which we will require in the next chapter. The main results of this chapter are 35.13 and 35.14 We fix a field K. Notation 35.1. Let V be a K-vector space of positive dimension. For each m ≥ 1, we denote by Vm (V ) the set of m-dimensional subspaces of V . Thus V1 (V ) is the set of points P(V ) of the projective space P(V ) defined in 2.35. Notation 35.2. Let (K, V, q) be a quadratic space of positive dimension and let V(q) and P(q) be as in 2.37. For each m ≥ 1, let Vm (q) = Vm (V ) ∩ V(q). Thus V1 (q) = P(q). Let Γ(q) be the graph with vertex set V(q) where two distinct subspaces U, W ∈ V(q) are joined by an edge whenever either U ⊂ W or W ⊂ U . Notation 35.3. Let (K, V, q) be a regular quadratic space of positive Witt index and Aut1 (Γ(q)) denote the stabilizer of P(q) in Aut(Γ(q)). (It follows from 35.5 and 35.6 below that Aut1 (Γ(q)) 6= Aut(Γ(q)) only when char(K) = 2 and q has Witt index 2 and dimension 5.) Let ψq denote the natural homomorphism from ΓO(q) to Aut1 (Γ(q)) and let ρ denote the natural homomorphism from Aut(P(q)) to Aut1 (Γ(q)). Then ψq = ρ ◦ ϕq , where ϕq is as in 2.37. Proposition 35.4. Suppose (K, V, q) is a regular quadratic space of positive Witt index. Then the homomorphism ρ in 35.3 is an isomorphism. Proof. For each vertex u of Γ = Γ(q) we denote by Γu the set of vertices adjacent to u in Γ. Let θ ∈ Aut1 (Γ(q)). Let R be the set of subsets of P(q) of the form Γu ∩P(q) for some vertex u of Γ not in P(q) together with all the singleton subsets of P(q). Then (P(q), R) is precisely the polar space P(q). Since θ stabilizes P(q) and R, there is a unique element of Aut(P(q)) whose restriction to P(q) coincides with the restriction of θ to P(q). It follows that ρ is bijective. Notation 35.5. Let n ≥ 2, let S be the vertex set of the Coxeter diagram Bn and let the elements of S be numbered from left to right relative to the way Bn is drawn in 35.12 below by the integers 1, 2, . . . , n. Let (K, V, q) be

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a regular quadratic space with Witt index n, let ξ : V(q) → S be the map which sends an arbitrary element of Vm (q) to the element m of S for each m and let F (q) be the set of subsets of V(q) whose elements are pairwise adjacent in the graph Γ(q). The simplicial complex (V(q), F (q)) is called the flag complex of P(q), the elements of F (q) are called flags, the map ξ is a numbering of the flag complex, the pair

B(q) := ((V(q), F (q)), ξ) is a building of type Bn (by [55, 7.4]), the group of all type-preserving automorphisms of C(q) is canonically isomorphic to Aut1 (Γ(q)) and by [55, 8.4.2], this building is thick unless q is hyperbolic. Proposition 35.6. Let (K, V, q) be a regular quadratic space with Witt index n at least 2 which is not hyperbolic, let (K, Z, Q) be the anisotropic part of (K, V, q) and let B(q) be as in 35.5. Then B(q) ∼ = BQ n (K, Z, Q). Proof. This holds by 3.20 if n = 2, so we can suppose that n ≥ 3. Let U be a totally isotropic subspace of dimension 2(n − 2) of V , let W be the subspace U ⊥ of V , let qU and qW denote the restrictions of q to U and to W and let c be a maximal flag of the building B(qU ). Then c is also a flag of B(q), the anisotropic part (K, Z, Q) of (K, V, q) is also the anisotropic part of (K, W, qW ) and the {n − 1, n}-residue of B(q) containing c is isomorphic to B(qW ). By 3.20 again, this residue is isomorphic to BQ 2 (K, Z, Q). By [60, 40.52], B(q) is uniquely determined by any one of its {n − 1, n}-residues. Hence B(q) ∼ = BQ n (K, Z, Q). Notation 35.7. Let (K, V, q) be a hyperbolic quadratic space of dimension 2n for some n ≥ 3 and let Ω(q) be the graph with vertex set

X := V(q)\Vn−1 (q), where U, W ∈ X are adjacent whenever U ⊂ W , W ⊂ U or dimK (U ∩ W ) = n − 1. Since n ≥ 3, we have P(q) ⊂ X. Let Aut1 (Ω(q)) denote the stabilizer of P(q) in Aut(Ω(q)). (It follows from 35.9 below that Aut1 (Ω(q)) 6= Aut(Ω(q)) only when n = 4.) Let πq denote the natural homomorphism from ΓO(q) to Aut1 (Ω(q)) and let ρ denote the natural homomorphism from Aut(P(q)) to Aut1 (Ω(q)). Then πq = ρ ◦ ϕq , where ϕq is as in 2.37. Proposition 35.8. Suppose (K, V, q) is a hyperbolic quadratic space of dimension 2n for some n ≥ 3. Then the homomorphism ρ in 35.7 is an isomorphism. Proof. For each vertex u of Ω = Ω(q) we denote by Ωu the set of vertices adjacent to u in Ω. Let θ ∈ Aut1 (Ω(q)). Let R be the set of subsets of P(q) of the form Ωu ∩ P(q) for all vertices u of Ω not in P(q) together with all intersections of pairs of such subsets and all the singleton subsets of P(q). Then (P(q), R) is precisely the polar space P(q). Since θ stabilizes P(q) and R, there is a unique element of Aut(P(q)) whose restriction to P(q) coincides with the restriction of θ to P(q). It follows that ρ is bijective.

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Notation 35.9. Let S be the vertex set of the Coxeter diagram Dn for some n ≥ 3, let the two rightmost elements of S (relative to the way Dn is drawn in 35.12 below) be labeled with the letters a and b and let the remaining elements of S be numbered from left to right by the integers 1, 2, . . . , n − 2. Let (K, V, q) be a hyperbolic quadratic space of dimension 2n, let X be the vertex set of Ω(q), let W ∈ Vn (q) and let ξ : X → S be the map which sends an arbitrary element of Vm (q) to the element m of S for each m ≤ n − 2 and an arbitrary element U of Vn (q) to a if dimK (U ∩ W ) is congruent to n modulo 2 and to b if it is not. Let F (q) be the set of subsets of X whose elements are pairwise adjacent in the graph Ω(q). The simplicial complex (X, F (q)) is called the oriflamme complex of q. The map ξ is a numbering of the oriflamme complex, the pair D(q) := ((X, F (q)), ξ) is a building isomorphic to Dn (K) (by [8, 7.8.4] and [55, 8.4.3]) and the group of all automorphisms of this simplicial complex stabilizing the set of vertices of type 1 is canonically isomorphic to Aut1 (Ω). Remark 35.10. Let (K, V, q) be a regular quadratic space of Witt index at least 2 and dimension at least 5. If q is not hyperbolic, let ∆ = B(q) as defined in 35.5; otherwise let ∆ = D(q) as defined in 35.9. Let G = Aut(∆), let G† be as in 1.18 and let G# denote the subgroup of G containing all automorphisms of ∆ that preserve the set of maximal simplices of co-type 1 (so G# = G unless either q has Witt index 2 and dimension 5 and char(K) = 2 or q has Witt index 0 and dimension 8). By 2.38, 35.4 and 35.8, we have a canonical identification of PΓO(q) := ΓO(q)/HT(V ) with both G# and Aut(P(q)) in both cases. Every root of ∆ contains a pair of 1-adjacent chambers. Hence every root group of ∆ acts trivially on some panel of type 1. Thus (35.11)

G† ⊂ PGO(q) := GO(q)/HT(V ).

Let P be a panel of type 1 of ∆, let β be a root whose wall M contains P , let c be the unique chamber in P ∩ β and let d be a second chamber in P . The panel P is a flag of co-type 1. There is a unique 2-dimensional subspace U which is either contained in this flag or is the intersection of two 3-dimensional spaces in this flag (when q is hyperbolic of dimension 6), and the chambers in P correspond to the 1-dimensional subspaces of U . There exists an identification of the Moufang set M∆,P as defined in 1.19 with the Moufang set M(K, τ ) as defined in 30.5(i) with respect to which for each g in PΓO(q) stabilizing c and d, there exists a unique element a ∈ K ∗ such that g induces x 7→ axσg

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on M∆,P , where σg is the image of g under the canonical homomorphism from PΓO(q) to Aut(K) with kernel PGO(q). It follows from 30.23 that there exists a Galois map κ of ∆ as defined in 29.25 such that κ(g) = σg for all g ∈ PΓO(q) stabilizing the c and d. For each g ∈ PΓO(q), there exists h ∈ G† such that the product gh stabilizes c and d. We conclude by (35.11) that the canonical homomorphism from PΓO(q) to Aut(K) with kernel PGO(q) is a Galois map of ∆. In particular, an element of G# = PΓO(q) is contained in P GO(q) if and only if it is linear in the sense of 29.25. Notation 35.12. We denote by Dn,ℓ the Tits index •

•

•

•

•

•

• •

of absolute type Dn for n ≥ 3 and 1 ≤ ℓ ≤ n − 2 or ℓ = n, by 2Dn,ℓ the Tits index •

•

•

•

•

•

• •

of absolute type Dn for n ≥ 3 and 1 ≤ ℓ ≤ n − 1 and by Bn,ℓ the Tits index •

•

•

•

•

•

•

of absolute type Bn for n ≥ 2 and 1 ≤ ℓ ≤ n, where n is the number of vertices of the absolute Coxeter diagram, ℓ is the number of circles, i.e. the number of vertices of the relative Coxeter diagram, and the Tits indices are drawn according to the conventions in 34.2. Proposition 35.13. Suppose that ∆ = BQ n (K, V, q) for some n ≥ 2 and some anisotropic quadratic space (K, V, q) or that ∆ = Dn (K) for some n ≥ 3 and some field K. Let G = Aut(∆), let G♭ ⊂ G be as in 29.8, let γ∆ : G♭ → Aut(K) be a Galois map of ∆ as defined in 29.25, let M be the standard Moufang structure on ∆, let Γ be a finite strictly semi-linear descent group of ∆, let F ⊂ K be the fixed field of γ∆ (Γ) and suppose that the Tits index T of Γ is Dn,ℓ for some ℓ ∈ [1, n − 2] or 2Dn,ℓ or Bn,ℓ for some ℓ ∈ [1, n − 1]. Then there exists an anisotropic quadratic space (F, M, Q) over F such that ∆Γ ∼ = BQ (F, M, Q) ℓ

if ℓ ≥ 2 and the Moufang set

(∆Γ , MΓ )

described in 24.28 and 24.31(ii) is isomorphic to the Moufang set BQ 1 (F, M, Q) defined in 3.8 if ℓ = 1 and for all ℓ the following hold:

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(i) If T = Dn,ℓ or 2Dn,ℓ , then dimF Q = 2(n − ℓ) and Q is non-singular. (ii) If T = Bn,ℓ , then dimF Q = 2(n − ℓ) + dimK q and dimF Def(Q) = dimK Def(q),

where Def(Q) and Def(q) are as defined in 2.31. ˆ where Proof. Suppose first that ℓ = 1. We think of ∆ as a residue of ∆, Q ˆ ∆ is Dn+1 (K) or Bn+1 (K, V, q). By 24.36, we can extend Γ to a descent ˆ of ∆ ˆ acting faithfully on ∆ and having Tits index Dn+1,2 , 2Dn+1,2 group Γ ˆ (by 29.28). or Bn+1,2 . Since Γ is finite and strictly semi-linear, so is Γ It will thus suffice to assume that ℓ ≥ 2. Let ϕ0 be a hyperbolic quadratic form over K of dimension 2n and let ϕ = ϕ0 ⊕ q be the orthogonal sum of ϕ and q if ∆ is of type Bn and let ϕ = ϕ0 otherwise. Let T denote the vector space over K on which ϕ is defined. By 35.6 and 35.9, ∆ is isomorphic to B(ϕ) or D(ϕ). Let G# be as in 35.10 with ϕ in place of q and, as in 35.10, we identify G# with P ΓO(ϕ). The group Γ stabilizes a 1-dimensional isotropic subspace A of T . Let π denote the natural homomorphism from ΓO(ϕ) to ˆ = π −1 (Γ), let v be a non-zero vector in A and let Γ1 denote P ΓO(ϕ), let Γ ˆ Since HT(T ) acts sharply transitively on A\{0}, the the stabilizer of v in Γ. restriction of π to Γ1 is an isomorphism from Γ1 to Γ. By 2.41 applied to Γ1 (and 35.10), there is a quadratic form ψ over F such that ψK is similar to ϕ and the map U 7→ hU iK is an inclusion- and dimension-preserving bijection π from the set of subspaces of the polar space P(ψ) to the set of subspaces of P(ϕ) that are stabilized by Γ. (Note that in the case that the Tits index of Γ is 2Dn,n−1 , a maximal flag of P(ϕ) stabilized by Γ consists of subspaces of dimension i for all i from 1 up to n − 1.) Thus, in particular, the Witt index of ψ is ℓ. Since ψK is similar to ϕ, we have dimF ψ = dimK ϕ = 2n + dimK q and dimF Def(ψ) = dimK Def(q) if ∆ is of type Bn , and dimF (ψ) = dimK (ϕ) = 2n and ψ is non-singular if ∆ is of type Dn . Let (F, M, Q) be the anisotropic part of ψ. Then dimF Def(Q) = dimF Def(ψ) and, since the Witt index of ψ is ℓ, dimF Q = 2(n − ℓ) + dimK q if ∆ is of type Bn and dimF Q = 2(n − ℓ) otherwise. By 35.6, we have B(ψ) ∼ = BQ (F, M, Q) ℓ

and the bijection π induces an isomorphism from B(ϕ) to ∆Γ .

Proposition 35.14. Let ∆, Γ, M, T, n and ℓ be as in 35.13. For each panel P of ∆Γ , let the Moufang set M∆Γ ,P be as defined in 1.19 if ℓ ≥ 2 and let M∆Γ ,P = (∆Γ , MΓ ) if ℓ = 1. Then the following hold:

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(i) The root groups of M∆Γ ,P are abelian for all panels P of ∆Γ . (ii) The tori of M∆Γ ,P are non-abelian when the type of P is the rightmost circle of T as it is drawn in 35.12 and T 6= 2Dn,n−1 . In all other cases, the tori of M∆Γ ,P are abelian. Proof. If P is a panel of A2 (F ), then the root groups are abelian and, since F is commutative, also the tori of MA2 (F ),P are abelian (by [60, 33.10]). By 4.5, the root groups of the Moufang set BQ 1 (F, W, Q) are abelian, but by 4.8(iii), the tori of this Moufang set are non-abelian unless dim W ≤ 2 or ∂Q = 0. If P is a panel of ∆Γ not contained in a residue isomorphic to A2 (F ), then M∆Γ ,P ∼ = BQ 1 (F, W, Q) by 35.13. Thus (i) and (ii) hold.

Chapter Thirty Six Indices for the Exceptional Bruhat-Tits Buildings In 34.13, we showed that for each exceptional Bruhat-Tits building Ξ of type ˜ 2 , there is a residually pseudo-split building ΞL and a Galois other than G embedding of a group Γ in Aut(ΞL ) such that Ξ is the fixed point building ΞΓL obtained by applying 22.25 to the pair (ΞL , Γ) and Ξ∞ is the fixed point Γ ∞ building (Ξ∞ L ) obtained by applying 22.25 to the pair (ΞL , Γ). These two applications of 22.25 also yield two Tits indices which we call the affine and spherical Tits indices of Γ. The spherical Tits index of Γ is given in 34.3– 34.8. Our goal in this final chapter is to determine the affine Tits index of Γ in each case. The results are displayed in a series of tables at the end of this chapter. See 36.12 and 36.16. We begin with a few small observations and some notation. Proposition 36.1. The relative type of the affine Tits indices in the Tables 36.1, 36.2, 36.3 and 36.4 is C˜2 . The relative type of the affine Tits indices ˜3 or C˜3 , as indicated. The relative type of the affine Tits in Table 36.5 is B ˜ 2 , respectively. The indices in the Tables 36.6, 36.7 and 36.8 is A˜2 , F˜4 and G relative type of the three Tits indices •

•

•

•

•

•

•

•

•

and • •

• •

• •

•

•

and • •

•

•

˜ 2 . Furthermore, these last three Tits indices are the only non-split affine is G Tits indices of relative rank ℓ ≥ 3 which are exceptional in the sense of 20.44 but do not appear in Tables 36.1–36.8. Proof. It can be checked using only 19.6, 20.1, 20.3 and 20.10 that there are no other non-split exceptional affine Tits indices. An algorithm for determining the relative Coxeter diagram of a Tits index is described in [60, 42.3.5]. 36.2. Orientation. There is a canonical correspondence between the circles in a Tits index and the vertices of its relative Coxeter diagram. For all the affine Tits indices in Tables 36.1–36.4, the relative Coxeter diagram

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is C˜2 . In every case, we have drawn the Tits index so that the leftmost and rightmost circles are the two circles which correspond to the two special vertices of C˜2 (i.e. the two vertices of this diagram that are not adjacent to each other). In Table 36.5, the relative Coxeter diagram for the first affine ˜3 . In both cases, the leftmost and Tits index is C˜3 , and for the second it is B rightmost circles correspond to the two special vertices of C˜3 , respectively, ˜3 . In Table 36.7, the relative Coxeter diagram is F˜4 and in Table 36.8, B ˜ 2 . We have drawn the affine Tits indices the relative Coxeter diagram is G in both of these tables so that the leftmost circle corresponds to the unique ˜ 2 . (There is no significance in the way special vertex of F˜4 , respectively, G we have oriented the spherical Tits indices in any of these tables.) In the following, the Moufang sets BI1 (C, F, η) and BQ 1 (F, C, q) are as defined in 3.8. See 30.11 for a similar result. Recall from 1.8 that ≈ denotes “weakly isomorphic.” Proposition 36.3. Let (D, D0 , τ ) be an involutory set with D a quaternion division algebra, let ∆ = BI2 (D, D0 , τ ), let Ω be a root group sequence of ∆ isomorphic to QI (D, D0 , τ ) and let P be the panel of ∆ obtained by applying 3.7 with i = 4. Then M∆,P ≈ BQ 1 (F, D, q), where F is the center of D and q is its norm.

Proof. Let η be the standard involution of D, let ∆′ = BI2 (D, F, η), let Ω′ be a root group sequence of ∆′ isomorphic to QI (D, F, η) and let Q be the panel of ∆′ obtained by applying 3.7 with i = 4. Then by [60, 33.13] and

M∆,P ≈ M∆′ ,Q M∆′ ,Q ∼ = BQ 1 (F, D, q)

by [60, 38.2]. Notation 36.4. Let M(k) denote the class of Moufang sets isomorphic to BQ 1 (Λ)

for some non-singular anisotropic quadratic space Λ of dimension k over F (for each k ≥ 1) and let D(k) denote the class of Moufang sets isomorphic to BQ 1 (Λ) for some singular anisotropic quadratic space Λ over F of arbitrary dimension whose defect is of co-dimension k. We will denote by D∗ (k) the union of D(k) and M(k + 1). Proposition 36.5. If k ≥ 3, then a Moufang set in M(k) cannot be weakly isomorphic to a Moufang set in M(m) for any m different from k, nor can it be isomorphic to a Moufang set in D(m) for any m. If k ≥ 2, then a Moufang set in D(k) cannot be weakly isomorphic to a Moufang set in D(m) for any m different from k.

INDICES FOR THE EXCEPTIONAL BRUHAT-TITS BUILDINGS

311

Proof. This holds by 6.10.

∗

∗

∗

Hypothesis 36.6. For the rest of this chapter we assume that Ξ, L, ΞL and Γ are as in 34.13. ˜ be the Notation 36.7. Let Π be the Coxeter diagram of ΞL and let Π relative Coxeter diagram of the Tits index of Γ. Thus Ξ∼ = ΞΓ L

˜ is the Coxeter diagram of Ξ. We identify Γ with its image in Aut(ΞL ). and Π ˆ and Ξ ˆ rather than L and (Note that in case 33.21, we really mean (L, L) L,L ΞL . See 33.19.) Let T = (Π, Θ, A) denote the Tits index of Γ ⊂ Aut(ΞL ). Thus Θ is the subgroup of Aut(Π) induced by Γ and A is a subset of the vertex set S of Π stabilized by Θ. We let ℓ denote the number of Θ-orbits in S which are disjoint from A. These orbits are indicated by circles in the representation of T described in 34.2. They correspond to the vertices of ˜ of Ξ and hence to the types of the panels of Ξ. Let the Coxeter diagram Π P be a gem-panel of Ξ and let (36.8)

MΞ,P

be the Moufang set defined in 32.9. The type of P is s˜ for some element s ∈ S\A. We recall that by 32.10, the Moufang set MΞ,P depends, up to isomorphism, only on the type s˜ and not on the choice of the panel P of type s˜. Let s1 and s2 be elements of S\A such that [˜ s1 , s˜2 ] 6= 1, where s˜1 and s˜2 are as in 20.4(i). Thus, in particular, Θ(s1 ) 6= Θ(s2 ). Let J = Θ(s1 ) ∪ Θ(s2 ) ∪ A, let R be a J-residue of ΞL stabilized by Γ and let ΓR denote the subgroup of Aut(R) induced by Γ. By 22.39, ΓR is a descent group of R with Tits s1 , s˜2 }-residue of Ξ = ΞΓL . In index TJ (as defined in 20.6) and RΓR is an {˜ ΓR ˜ particular, R is irreducible. We denote this residue by R. Our next goal is to show (in 36.9 and 36.10) how it is possible in many cases to determine properties of the Moufang set (36.8) by applying 35.13 and 35.14. ˜ is contained in a gem of Ξ and that either Observations 36.9. Suppose R TJ = Bn,2 for some n ≥ 2 or TJ = Dn,2 or 2Dn,2 for some n ≥ 3 (so, in particular, R is irreducible) and that s˜1 corresponds to the leftmost circle in TJ as it is drawn in 35.12. Let T be a Γ-panel of R of type Θ(si ) ∪ A, where i = 1 if TJ = 2D3,2 and i = 2 otherwise, then by 32.20(i) with T in place of R, the group ΓR is strictly semi-linear. Let M denote the standard

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˜ By 24.31(i), M ΓR is the Moufang structure on R and let P˜ be a panel of R. ˜ Thus if we assume that P˜ is the panel P standard Moufang structure on R. in 36.7, then the Moufang set (36.8) is (P˜ , (MΓR )P˜ ). Hence by 33.27, we can apply 35.13 and 35.14 to the pair (R, ΓR ) to obtain information about the Moufang sets (36.8) for panels P˜ of Ξ of type s˜1 and s˜2 . For example, we conclude by 35.14 that the torus of a Moufang set (36.8) is abelian for every panel P˜ of type s˜1 in the following four cases: (a) T is the last affine Tits index in Table 36.2 and s˜1 and s˜2 correspond (always reading from left to right) to the second and third circles, respectively. (b) T is the first affine Tits index in Table 36.3 and s˜1 and s˜2 correspond to the third and second circles, respectively. (c) T is the fifth affine Tits index in Table 36.7 and s˜1 and s˜2 correspond to the fourth and third circles, respectively. (d) T is the sixth affine Tits index in Table 36.7 and s˜1 and s˜2 correspond to the third and fourth circles, respectively. In case (b), we can also conclude by 35.13 that the Moufang set (36.8) is in M(12) (as defined in 36.4) for every panel P˜ of type s˜2 . Observations 36.10. We return to the assumptions and notation in 36.7 ˜ is contained in a gem of Ξ. This time we supand again we suppose that R pose, in addition, that there exists a Θ-invariant subset J1 of J such that the subdiagram ΠJ1 is a connected component of ΠJ and Θ(s1 ) ∪ Θ(s2 ) ⊂ J1 . Let ξ be the homomorphism obtained by applying 22.26 with R in place of ∆ and R1 in place of R and let Γ1 = ξ(ΓR ). Then Γ1 is a descent group of ˜ Since R1 is R1 whose fixed point building is canonically isomorphic to R. irreducible, it is Moufang (by [65, 18.18]). Let M denote the standard Moufang structure on R1 . By 24.31(i), MΓ1 is the standard Moufang structure ˜ Now let R2 be a Γ1 -chamber in R1 whose type J2 , we assume, on R1Γ1 = R. contains Θ(s1 ) and let Γ2 denote the restriction of Γ1 to R2 . We assume as well that the panel P in (36.8) is the panel R2Γ2 . Finally, we assume that there exists a Θ-invariant subset J3 of J2 such that ΠJ3 is a connected component of ΠJ2 , |J3 | > 1 and Θ(s1 ) ⊂ J3 . Let R2′ be the unique Γ-panel containing R2 , let J2′ = Typ(R2′ ) (so J2′ = J2 ∪ A) and let Γ′2 denote the restriction of Γ to R2′ . Then J1 ∩ J2′ = J2 and ΠJ3 is a connected component of ΠJ2′ . Let ξ ′ be the homomorphism obtained by applying 22.26 with R2′ in place of ∆ and R3 in place of R. By 32.20, Γ3 := ξ ′ (Γ′2 ) ⊂ Aut(R3 ) is strictly semi-linear. Next let ξ ′′ be the homomorphism obtained by applying 22.26 with R2 in place of ∆, R3 in place of R and Γ2 in place of Γ and let Γ′3 = ξ ′′ (Γ2 ). Then Γ′3 is a descent group of R3 and by 24.34, the canonical ′ ˜ 2 := RΓ2 to R ˜ 3 := RΓ3 carries the Moufang structure isomorphism from R 2 3

INDICES FOR THE EXCEPTIONAL BRUHAT-TITS BUILDINGS

313

′

MΓR22 to the Moufang structure (MR3 )Γ3 . By 24.30, (MR2 )Γ2 = (MΓ1 )R˜ 2 . Hence the Moufang set (36.8) is ˜2 , (MR2 )Γ2 ), (R which is, in turn, isomorphic to ˜3 , (MR3 )Γ′3 ). (R We claim that Γ3 = Γ′3 . Let τ ∈ Γ, let τ3 be the image under ξ ′ of the restriction of τ to R2′ , let τ1 be the image under ξ of the restriction of τ to R and let τ3′ be the image under ξ ′′ of the restriction of τ1 to R2 . Then τ3 ∈ Γ3 , τ3′ ∈ Γ′3 and ′

xτ3 = projR3 (xτ1 )

= projR3 (projR1 (xτ )) = projR3 (xτ )

by 21.6(iii)

τ3

=x for all chambers x ∈ R3 . Hence Γ3 = Γ′3 as claimed. We know that Γ3 is strictly semi-linear. Thus if we now suppose that either TJ3 = Bn,1 for some n ≥ 2 or TJ3 = Dn,1 or 2Dn,1 for some n ≥ 3, then by 33.27, we can apply 35.13 and 35.14 to the pair (R3 , Γ3 ) to obtain information about the Moufang set ˜ 3 , (MΓ3 )). (R R3 Here are four examples: (a) Suppose that T is the second affine Tits index in Table 36.1 (which occurs in numerous other places in Tables 36.1–36.4). If s1 and s2 are the first two circled elements of S (in either order), then ΠJ = F4 , J = J1 , J2′ = J2 = J3 and TJ3 = TJ2 = B3,1 . Thus by 35.13, the Moufang sets in (36.8) are in D∗ (4) (as defined in 36.4) for all s˜1 -panels P˜ . (b) Suppose that T is the last affine Tits index in Table 36.1 (which also occurs in numerous other places in Tables 36.1–36.4). If s1 and s2 are the second and first circled elements of S, respectively, then ΠJ = B3 × A1 , ΠJ1 = B3 , and TJ3 = B2,1 and the Moufang sets in (36.8) are in D∗ (2) for all s˜1 -panels P˜ . (c) Suppose that T is the first affine Tits index in Table 36.2 and that s1 and s2 are the elements of S contained in the second and third circles, respectively (reading from left to right). In this case, ΠJ = D6 × A1 , ΠJ1 = D6 and TJ3 = D5,1 (as defined in 35.12). Thus by 35.13, the Moufang sets in (36.8) are in M(8) (as defined in 36.4) for all s˜1 -panels P˜ . (d) Suppose that T is the second affine Tits index in Table 36.6 and s1 and s2 are any two of the circled elements of S. Then ΠJ = A5 × A1 , ΠJ1 = A5 , ΠJ2 = A3 × A1 and TJ3 = D3,1 . Thus by 35.13, the Moufang sets in (36.8) are in M(4) for all s˜1 -panels P˜ .

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∗

∗

∗

We start now with the problem of determining the Tits index of Γ in each case of 34.13. We can assume that Ξ is not itself residually pseudo-split; see 36.13 below. ˜ = C˜2 and organize our investigations We begin with the case that Π according to the type of ΞL (i.e. the Coxeter diagram Π). In each case of 14.3 and 17.3, these two theorems give us a description of the irreducible ˜ = C˜2 , every panel of Ξ is a gemrank 2 residues of Ξ. Note that since Π ˜ of Ξ, 14.3 and 17.3 panel. Thus for each vertex i of the Coxeter diagram Π provide (in each case) information about the structure of the Moufang sets MΞ,P defined in 32.9, where P is an arbitrary panel of Ξ of type i (see 32.10). Independently, we can apply 35.13 and 35.14 as explained in 36.9 and 36.10 to obtain information about the Moufang sets MΞΓL ,P , where P is an arbitrary panel of ΞΓL of type i for certain circles i of the various Tits indices which occur in Tables 36.1–36.4. By 36.2, the middle circle of the Tits index of Γ corresponds to the middle vertex of the relative Coxeter diagram Π. This information (together with 36.1) will suffice to carry out the determination of the Tits index of Γ in each case of 14.3 and 17.3, as we now show. By 36.1, the only affine Tits indices with relative diagram C˜2 are those appearing in Tables 36.1–36.3. It will be useful to give an ad hoc name to each of these affine Tits indices: The affine Tits index in the same row as the case (x.y.z) of 14.3 will be referred to as [x.y.z]. Thus the first affine Tits index in Table 36.1, for example, is herewith named [1.i.a], the two affine indices in Table 36.4 are [1.i.b] and [1.ii.b] and [1.ii.b]=[1.iii.a]. Proposition 36.11. If |Γ| = 2, then the Tits index of Γ cannot be [2.iv.a] or [3.iii.a]. Proof. Suppose that |Γ| = 2 and that its Tits index T is [2.iv.a] or [3.iii.a], let R be a Γ-chamber, let θ be the permutation of the type A of R induced by the non-trivial element τ in Γ and let ΠR be the Coxeter diagram of the building R. Then θ 6= 1 and ΠR = A1 × A1 × A1 × A1

if T = [2.iv.a] and θ = 1 and ΠR = A3 × A3 if T = [3.iii.a]. Since Γ stabilizes no proper residues of R, it follows from 25.10 and 25.15 that there exists an apartment Σ of R stabilized by Γ such that τ sends each chamber of Σ to its opposite. By [62, 5.12], therefore, θ(s) = wA swA for all s ∈ A, where wA is as in 20.4(i). By 19.6, however, this implies that θ = 1 if ΠR = A1 ×A1 ×A1 ×A1 and θ 6= 1 if ΠR = A3 × A3 . With these contradictions, we conclude that the Tits index of Γ cannot be [2.iv.a] or [3.iii.a].

INDICES FOR THE EXCEPTIONAL BRUHAT-TITS BUILDINGS

315

˜6 . This We are now ready to begin. We assume first that ΞL is of type E happens when Ξ is in case (1.i.a) and (1.ii.a) of 14.3. Our goal is to show that if Ξ is in case (1.i.a), the Tits index of Γ is [1.i.a] and if Ξ is in case (1.ii.a), the Tits index of Γ is [1.ii.a]. First we observe that in case (1.i.a) of 14.3, ˜ have nonthe Moufang sets of Ξ corresponding to the one end vertex of Π abelian root groups and the Moufang sets corresponding to the other end vertex have abelian root groups and abelian tori (by 4.5 and 4.8). By 35.14, on the other hand, the Moufang sets of ΞΓL corresponding to the leftmost circle of [1.ii.a] have abelian root groups but non-abelian tori. Therefore the Tits index corresponding to case (1.i.a) cannot be the Tits index [1.ii.a]. The root groups of the Moufang sets of Ξ corresponding to the middle vertex in case (1.ii.a) are non-abelian, whereas the root groups of the Moufang sets of ΞΓL corresponding to the middle circle of [1.i.a] are abelian by 35.14. Hence the Tits index corresponding to (1.ii.a) cannot be [1.i.a]. By 36.1, [1.i.a] and ˜6 and relative type C˜2 . We [1.ii.a] are the only two indices of absolute type E can thus conclude that the Tits index of Γ in case (1.i.a) is [1.i.a] and the Tits index of Γ in the case (1.ii.a) is [1.ii.a]. This concludes the case that ΞL ˜6 . is of type E ˜7 . In case (2.i.a), the Moufang sets of Suppose next that ΞL is of type E Ξ corresponding to one end vertex have non-abelian root groups and the Moufang sets of Ξ corresponding to the other end vertex have abelian root groups and abelian tori (again by 4.5 and 4.8). By 35.14, on the other hand, the Moufang sets of ΞΓL corresponding to the leftmost circle of both [2.ii.a] and [2.iii.a] has abelian root groups but non-abelian tori. In case (2.i.a), |Γ| = 2 (see 34.10), so the Tits index of Γ cannot be [2.iv.a] by 36.11. Hence (by 36.1 again) the Tits index of Γ in case (2.i.a) is [2.i.a]. In case (2.ii.a), the Moufang sets of Ξ corresponding to the middle vertex have non-abelian root groups and both end vertices correspond to Moufang sets in M(4). By 35.14, on the other hand, the Moufang sets of ΞΓL corresponding to the middle circles of [2.i.a] have abelian root groups and by 35.13(i), the leftmost circle of [2.iii.a] correspond to Moufang sets in M(6). Thus the Tits index of Γ cannot be [2.i.a] or [2.iii.a]. By 36.11, [2.iv.a] is also ruled out. Hence (by 36.1 and 36.5) the Tits index of Γ in case (2.ii.a) is [2.ii.a]. In case (2.iii.a), the Moufang sets of Ξ corresponding to the middle vertex have non-abelian root groups and those corresponding to one of the end vertices are in M(6). By 35.14(i), the Moufang sets of ΞΓL corresponding to the middle circles of [2.i.a] have abelian root groups and the circles at both ends in [2.ii.a] correspond to Moufang sets in M(4). Applying 36.11 (as well as 36.1 and 36.5) once again, we can also rule out [2.iv.a]. Hence the Tits index corresponding to (2.iii.a) is [2.iii.a]. In case (2.iv.a), both the middle vertex and, by 36.3, one end vertex correspond to Moufang sets in M(4) and the other end vertex corresponds to Moufang sets with abelian tori. By 35.13(i), on the other hand, the middle circle of [2.i.a] and the leftmost circle of [2.iii.a] correspond to Moufang sets in M(8) and M(6), respectively, and by 35.14(ii), the leftmost and the

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rightmost circle of [2.ii.a] both correspond to Moufang sets with non-abelian tori. We conclude that the Tits index of Γ in case (2.iv.a) can only be [2.iv.a]. ˜7 . This completes the case that ΞL is of type E ˜8 . In case (3.i.a), the Moufang sets Suppose next that ΞL is of type E corresponding to one end vertex have abelian tori and |Γ| = 2. By 35.14(ii), the leftmost circle and the rightmost circle of [3.ii.a] correspond to Moufang sets with non-abelian tori and by 36.11, the corresponding Tits index cannot be [3.iii.a]. Hence the Tits index of Γ in case (3.i.a) is [3.i.a]. In case (3.ii.a), both end vertices correspond to Moufang sets with nonabelian tori and again |Γ| = 2. By 36.9(b), therefore, the corresponding Tits index cannot be [3.i.a]. By 36.11 again, it cannot be [3.iii.a]. Hence the Tits index of Γ in case (3.ii.a) is [3.ii.a]. In case (3.iii.a), the Moufang sets corresponding to one end vertex have abelian tori and the middle vertex corresponds to Moufang sets in M(6). By the last comment in 36.9, the middle circle of [3.i.a] corresponds to Moufang sets in M(12) and by 35.14(ii), the leftmost and rightmost circles of [3.ii.a] both correspond to Moufang sets with non-abelian tori. Hence the Tits index of Γ in case (3.iii.a) is [3.iii.a]. This completes the case that ΞL is of type ˜8 . E There is only one case in 14.3 and one Tits index in the tables where ΞL is of type C˜3 . Thus the Tits index of Γ in case (2.i.c) can only be [2.i.c]. ˜3 . In case (2.ii.c), the middle vertex corresponds Suppose that ΞL is of type B to Moufang sets with non-abelian tori and in case (2.iv.c), the middle vertex corresponds to Moufang sets with abelian tori. By 35.14(ii), the middle circle of [2.ii.c] corresponds to Moufang sets with non-abelian tori and by 36.9(a), the middle vertex of [2.iv.c] corresponds to Moufang sets with abelian tori. Hence the Tits index corresponding to (2.x.c) is [2.x.c] for both x = ii and ˜ 3 for X = B and x = iv. This completes the case that ΞL is of type X X = C. Suppose that ΞL is of type F˜4 . This occurs in all the remaining cases of 14.3 as well as in 33.21. There are only two Tits indices, [1.i.b] and [1.ii.b], of absolute type F˜4 and relative type C˜2 . (We will explain the numerical decoration on some of these Tits indices below.) In some of the relevant cases of 14.3 and 17.3, the middle vertex corresponds to Moufang sets in D(4) and in the rest, the middle vertex corresponds to Moufang sets which are in D∗ (2). By 35.13(ii), the analogous assertions hold for the circle in the middle of [1.i.b] and [1.ii.b]. This is enough to determine which Tits index corresponds to each case. This completes the pairings that occur in the Tables 36.1–36.4. In particular, we have considered all the cases (A), (B) and (C) of 33.16. We have thus finished with all the cases of 14.3 and 17.3. We turn now to all the remaining cases in 34.13. Suppose first that Ξ ∼ = I ˜ X3 (Λ) for some honorary involutory set Λ = (C, F, σ) (so ∆ = Ξ∞ is as in 34.6), where X = B if C is ramified and X = C if C is unramified (by [65, 23.27]). Since Ξ is not itself residually pseudo-split, either C is ramified and

INDICES FOR THE EXCEPTIONAL BRUHAT-TITS BUILDINGS

317

C¯ is a quaternion division algebra or C is unramified and C¯ is an octonion division algebra (by [65, 26.15]). In both cases, ramified and unramified, ˜7 with ΞL is of type E˜7 . By 36.1, the two Tits indices of absolute type E ˜3 or C˜3 are the two Tits indices in Table 36.5. relative Coxeter diagram B ˜3 . Thus if C The first has relative type C˜3 and the second has relative type B is unramified, the corresponding Tits index is the first one in Table 36.5 and if C is ramified, the corresponding Tits index is the second one in Table 36.5. ˜ 2 (C) for some octonion division algebra C (so Now suppose that Ξ ∼ = A ∞ ∆ = Ξ is as in 34.7). In this case, ΞL has absolute type E˜6 . By [65, 26.20], C¯ is either a quaternion or an octonion division algebra (as in the previous paragraph). By [65, 18.30], all the irreducible rank 2 residues of Ξ are ¯ By 30.11, therefore, the Moufang sets corresponding isomorphic to A2 (C). to the three vertices of the Coxeter diagram are contained in M(4) if C¯ is quaternion and in M(8) if it is octonion. By 36.1, the corresponding Tits index is one of the two indices in Table 36.6. By 35.13(i), we conclude that the first of these two indices corresponds to the case that C¯ is octonion and the second to the case that C¯ is quaternion. ˜4 (C, F ) for some composition algebra (C, F ) Suppose, finally, that Ξ ∼ =F such that the composition algebra C¯ is not of type (i) as defined in [65, 30.17] (so either ∆ = Ξ∞ is as in 34.8 or C¯ is of type (ii), i.e. C¯ = F¯ ). We think of the Coxeter diagram F˜4 of Ξ drawn with the special vertex on the left and we number the vertices from left to right. Let q denote its norm of the composition algebra (C, F ). Suppose first that C = F or C/F is a separable ¯ F¯ is a separable quadratic extension or quadratic extension. Then either C/ ¯ ¯ ˜6 . By 36.1, the second index in C = F . In the first case, ΞL is of type E Table 36.7 is the only Tits index of absolute type E˜6 and relative type F˜4 . If C¯ = F¯ , then Ξ is residually split. We have been assuming that Ξ is not residually split in these investigations, but we have decided to include this case along with the split affine Tits index in Table 36.7 anyway. In fact, this Tits index occurs twice in Table 36.7, once for the case that C = F and a second time for the case that C/F is a ramified quadratic extension; see 36.15. Suppose next that C is a quaternion division algebra, in which case F is the center of C. Applying now [65, 26.15–26.16] and our assumption that Ξ is not residually split, we conclude that C¯ is either a quaternion division ¯ F¯ is a separable quadratic extension. Let P be a panel algebra over F¯ or C/ of Ξ whose type is the vertex 4 of the Coxeter diagram F˜4 . By [65, 26.17], ¯ ¯ ¯0 ) MΞ,P ∼ = BQ 1 (F , C, q if C¯ is quaternion and MΞ,P ∼ = A1 (F¯ ) if C¯ is commutative. Hence MΞ,P is in M(4) and thus has non-abelian tori if C¯ is quaternion and has abelian tori if C¯ is commutative. By 36.1, the only ˜7 and relative type F˜4 are the fourth and fifth Tits indices of absolute type E Tits indices in Table 36.7. By 35.14, the Moufang set associated with the

318

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fourth circle in the first of these two Tits indices is in M(4) and by 36.9(c), the Moufang set associated with the fourth circle in the second of these two Tits indices has abelian tori. We conclude that if C¯ is quaternion, the Tits ¯ F¯ is a separable index of Γ is the fourth Tits index in Table 36.7 and if C/ quadratic extension, the Tits index of Γ is the fifth. Suppose that C is an octonion division algebra with center F . By [65, 26.15–26.16] (and our assumption that Ξ is not residually split), C¯ is either an octonion or a quaternion division algebra over F¯ . This time we let P be a panel of Ξ whose type is the vertex 3 of the Coxeter diagram F˜4 . By [65, 26.17], MΞ,P ∼ = A1 (F¯ ) if C¯ is octonion and ¯ MΞ,P ∼ = A1 (C) if C¯ is quaternion. Hence MΞ,P has abelian tori if C¯ is octonion and nonabelian tori if C¯ is quaternion. By 36.1, the only Tits indices of absolute ˜8 and relative type F˜4 are the last two Tits indices in Table 36.7. By type E 36.9(d), the Moufang set associated with the third circle in the first of these two Tits indices has abelian tori and by 35.13, the Moufang set associated with the third circle in the second of these two Tits indices has non-abelian tori. We conclude that if C¯ is octonion, the Tits index of Γ is the sixth Tits index in Table 36.7 and if C¯ is a quaternion, the Tits index of Γ is the seventh. This completes the determination of the Tits index for all exceptional ˜ 2. Bruhat-Tits buildings of type other than G ˜ 2 . Suppose that Ξ is of type G ˜ 2 . Given 36.12. The Table for Type G 34.15, we can also deduce that the Tits indices for the various cases in 34.9 must be as in Table 36.8, but we cannot justify every step with the methods developed in this monograph and will not say anything more about this table. 36.13. Exceptional Affine Indices. If Ξ itself is residually pseudosplit, then the corresponding Tits index is simply the split Tits index of type Π, where Π is the Coxeter diagram of Ξ, which can be either one of ˜ 4 or one of the diagrams C˜2 , C˜3 , the exceptional affine diagrams including D ˜ ˜ B3 or A2 . In light of this, we modify the standard notion of an exceptional affine diagram by adding these four diagrams to the usual list consisting of ˜6 , E˜7 , E ˜8 , F˜4 and G ˜ 2 ; see 20.44. E ˜3 , C˜3 and 36.14. Numerical Decorations. Suppose that ΞL is of type B ˜ F4 and the Tits index is one of the Tits indices in Tables 36.1–36.3 other ¯ must be two. and than [1.iii.a]. In all of these cases, the characteristic of L the building ΞL is residually split (but the building at infinity ∆L = Ξ∞ L is not split). Let S denote the vertex set of the Coxeter diagram Π of ΞL , let J be a subset of S spanning a subdiagram of type C3 (there are exactly two

INDICES FOR THE EXCEPTIONAL BRUHAT-TITS BUILDINGS

319

of them) and let R be a J-residue of ΞL . We define the short vertex of J to be the unique element x of J that is contained in the edge of ΠJ with label 4 but not in the other edge of Π. Then R∼ = BQ (Λ) 3

¯ For one of the for some totally wild anisotropic quadratic space Λ over L. two choices of J, the dimension Λ has a well determined value (we denote this choice of J by J1 and this dimension by m) and for the other choice of J it does not. We decorate Π by putting a small m over the short vertex of J1 . Here is an example. Suppose that Ξ is in case (1.i.b) of 14.3, so Π = F˜4 , and let J consist of the second, third and fourth vertex of Π as drawn in Table 36.1 (so the second vertex is the short vertex in J). By 33.5(i), L/K is an a separable quadratic extension such that qL is the norm of a wild separable ˜4 (E, L). By [65, quadratic extension E/L and by 33.16 and 33.20, ΞL = F 26.15], the composition algebra (E, L) is unramified and hence by [65, 26.17], ¯ ¯ ¯0 ), where q is the norm of the extension E/L. Since E/L is R∼ = BQ 3 (L, E, q ¯ = 2 and we have thus labeled the second vertex in wild, we have dimL¯ (E) this diagram with a 2. If J ′ is the other choice of a subset of S that spans a subdiagram of type C3 and R′ is a J ′ -residue of ΞL , then by 28.15, ∼ BQ (E ¯ 2 , L, ¯ x 7→ x2 ). R′ = 3

¯ over E ¯ 2 , we do not Since we cannot say anything about the dimension of L put any decoration over the third vertex of [1.i.b]. We observe that in every case in the Tables 36.1–36.3 where Π = F˜4 , the label is over a vertex contained in a subdiagram of type A3 if and only if the composition algebra defining ΞL is ramified. In cases (2.i.b) and (2.ii.b), for example, this composition algebra is an unramified quaternion algebra and in cases (2.iii.b) and (2.iv.b), it is a ramified quaternion algebra. ˆ are as in 33.18, we cannot say anything definite about the If L and L degree of either extension ˆ or L/ ˆ L ¯ 2. ¯ L L/

For this reason we have not endowed the indices in Table 36.4 with any decorations. 36.15. Arrows. Suppose that we are in case (1.iii.a) of 14.3. In this case, ∆L = Ξ∞ L is a split building. Let J1 be the subset of S consisting of the first three vertices of Π and let J2 denote the other subset of S spanning a C3 2 ¯ ¯ subdiagram of Π. The J1 -residues of ΠL are isomorphic to BQ 3 (L, L, x 7→ x ) I ¯ ¯ and the J2 -residues of ΠL are isomorphic to B3 (L, L, id). In [56], Tits draws this index with an arrow •

•

•

•

•

¯ = 2, then to distinguish J1 from J2 . If char(L) ¯ L, ¯ id) ∼ ¯ 2 , L, ¯ x 7→ x2 ) BI (L, = BQ (L 3

3

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¯ = 1, but dimL¯ 2 L ¯ depends on the field L. To by 28.15. Of course dimL¯ (L) be consistent with the convention described in 36.14, we have preferred to distinguish J1 from J2 by decorating the third vertex in [1.iii.a] with a 1 rather than inserting an arrow. In Table 36.7, however, we have decorated two of the Tits indices with arrows used in the sense of [56]. Remark 36.16. We have determined the Tits indices and their decorations in our various tables in reference to the pairs (ΞL , Γ) which we have defined in 33.20–33.25. A deeper understanding of the theory of descent in residually pseudo-split Bruhat-Tits buildings ought to yield the result that these Tits indices and their decorations are, in fact, invariants of Ξ in every case. Remark 36.17. If we assume that Ξ is locally finite (i.e. all residues are finite), then the Tits index of Γ must be quasi-split (as defined in 20.10). The quasi-split Tits indices that occur in Tables 36.7 and 36.8 (together with the split indices of type E6 , E7 and E8 ) are precisely the indices in our tables which occur also in the tables at the end of Tits’ Corvallis Notes [56]. See also [65, Chapter 28]. (Note that all the Tits indices in the tables in [56] are split or quasi-split, so there was no need for Tits to include in his tables the circles which appear in our tables.) Remark 36.18. Many of the Tits indices in our tables appear in Tableau II at the end of [3] (with a very different interpretation). Tableau II contains, in fact, precisely those Tits indices in our tables which arise when the field of definition of Ξ is R((x)). (As observed in [60, 12.38], quadratic forms of type E7 and E8 do not exist over R.)

321

INDICES FOR THE EXCEPTIONAL BRUHAT-TITS BUILDINGS

Spherical Index

• •

• •

•

•

Case

Affine Index

(1.i.a)

• •

• •

(1.i.b)

•

(1.ii.a)

•

•

•

•

•

•

•

•

•

•

• •

• •

(1.ii.b)

•

•

•

•

•

(1.iii.a)

•

•

•

•

•

2

1

2

Table 36.1 Theorem 14.3: The Case E6

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CHAPTER 36

Spherical Index

Case (2.i.a)

Affine Index • •

(2.i.b)

•

•

•

•

(2.i.c)

4

•

•

•

•

•

•

•

•

8

•

•

•

•

•

4

•

•

•

• •

(2.ii.c)

•

•

4

(2.iii.a)

•

•

• •

(2.iii.b)

•

•

(2.iv.a)

•

•

(2.iv.b)

•

• •

•

(2.iv.c)

•

•

•

• •

(2.ii.b)

• 8

•

(2.ii.a) •

•

•

•

•

•

• • •

• •

•

•

•

• •

• •

• •

•

•

•

4

• •

2

2

Table 36.2 Theorem 14.3: The Case E7

•

323

INDICES FOR THE EXCEPTIONAL BRUHAT-TITS BUILDINGS

Spherical Index

Case

Affine Index •

(3.i.a)

•

•

•

(3.i.b)

•

•

•

•

•

•

•

•

(3.ii.b)

•

•

• •

•

•

•

•

•

•

•

•

• •

•

8

•

(3.ii.a) •

•

•

•

•

•

•

(3.ii.c)

• 4

•

•

•

•

•

•

•

8

•

(3.iii.a)

•

•

•

(3.iii.b)

•

•

• 4

•

•

• •

•

Table 36.3 Theorem 14.3: The Case E8

Spherical Index

•

•

•

Case

Affine Index

(i)

•

•

•

•

•

(ii)

•

•

•

•

•

•

Table 36.4 Theorem 17.3

Spherical Index

Affine Index •

• •

•

•

•

• •

•

•

•

•

(X = C) •

•

•

•

•

•

• •

•

•

•

(X = B)

˜ I3 (Λ), Λ honorary Table 36.5 Diagrams for X

324

CHAPTER 36

Spherical Index

Affine Index • • •

•

• •

.....

•

. ... ... ...... ....... .. ... .... ................................................................... ... ....... ......











• • •

•

• • •

˜ 2 (C), C octonion Table 36.6 Diagrams for A

dimF C

Spherical Index

1

•

2

•

•

•

•

•

• •

• •

Affine Index •

•

•

•

•

•

•

•

• •

• •

•

•

•

•

•

• •

4

•

•

•

•

•

•

•

•

•

•

•

•

• •

• •

• •

•

•

•

• •

8

•

•

•

•

•

• •

•

•

•

•

•

•

•

•

•

•

• •

•

•

˜ 4 (C, F ) Table 36.7 Diagrams for F

•

•

•

325

INDICES FOR THE EXCEPTIONAL BRUHAT-TITS BUILDINGS

dimK J

Spherical Index •

1

Affine Index

•

•

•

•

3

• • • •

•

• • •

•

•

•

• • •

•

•

9 •

•

•

•

• • • • •

• •

• •

•

•

• • • •

9

•

• •

•

• • •

˜6) (3 E

•

• •

•

• • •

• •

˜6) (6 E

•

• •

•

27

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

• •

•

•

˜ 2 (J, K, #) Table 36.8 Diagrams for G

•

•

•

•

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Index absolute Coxeter system/diagram, 153, 160 ramification index, 120 rank, 160 type, 153, 160 value, 58 affine Coxeter system, 162 dimension, 221 hyperplane, 221 map, 221 span, 221 subspace, 221 Tits index, 163, 309 transformation, 221 alcove, 225 algebraic building, 9 anisotropic part, 19, 288 quadratic space, 13 Tits index, 153 Ant, 262 anti-isomorphism, 23 arctic/antarctic region, 147 arrows, 319 barycenter, 223 bent edges, 297 Bruhat-Tits building, 9 building, 11 exceptional, 9 fixed point, 188 indifferent, 243 at infinity, 9 irreducible, 3 mixed, 244 orthogonal, 303 pseudo-split, 275 reducible, 175, 189 residually split, 275 spherical, 3 split, 245 thick, 3 chamber, 11, 146 circle, 297, 306, 310 classical building, 9

classification of Bruhat-Tits buildings, 9 of Moufang polygons, 6 of spherical buildings, 7 Clifford invariant, 69 coherent, 176, 199 color, 11, 146, 165 compatible representation, 156 complete system of apartments, 9, 85 component, 149 composition algebra, 68, 139 cone point, 238 conical cell, 238 closure, 238 conventions, 58, 109, 153, 195, 297, 314 convex closure, 221 Coxeter chamber system, 146 diagram/system, 143 absolute/relative, 153, 160 affine, 162 exceptional, 318 cuts, 169 Davis complex, 189 defect, 19 defining extensions, 244 field, 244 descent group, 187 dimension affine, 221 of a simplex, 11, 226 of a simplex, 12 direct summand, 149 discrete subgroup, 222 Euclidean representation, 227 exceptional Bruhat-Tits building, 9, 296 Coxeter diagram, 318 Moufang quadrangle, 7 spherical building, 9 Tits index, 163, 318 facet, 226, 238

334

INDEX

field of definition, 244 fixed point building, 188 flag complex, 303 flexible, 26 form, 188 full rank lattice, 59, 65

isotropic quadratic space, 13 subspace, 19 Tits index, 153

gallery, 146 Galois action, 257 embedding, 257 unramified, 275 involution, 271 unramified, 275 map, 257 Γcell, 238 chamber, 181 form, 188 panel, 184 residue, 181 simplex, 233 gem, 86, 137 gem-panel, 277 generalized polygon, 6 generic, 110 genuine involutory set, 138

lattice, 223 length function, 143 linear automorphism, 257 linked tori, 25 linking, 255 locally finite Bruhat-Tits building, 320 set of affine subspaces, 237 locally opposite residues, 193 long case, 86 root, 265, 272 longest element, 145

half-space, 223 reflection, 226 true, 238 Hasse invariant, 69 hexagonal system, 7, 139 homothety, 19 homotopic, 143 honorary involutory set, 138 hyperbolic pair, 16 plane, 16 quadratic space, 16 index set, 11 indifferent building, 243 set, 6 proper, 245 type, 7 initial point, 238 involutory set, 6 genuine/honorary, 138 type, 7 irreducible building, 3 Coxeter system, 143 Tits index, 163 isometry, 13, 222 isomorphism, 5, 11, 23, 168

joint torus, 90, 277

M-automorphism, 209 metric Euclidean, 222 realization, 230 standard, 230 middle, 109, 277, 314 mixed building, 9, 244 morphism of simplicial complexes, 11 Moufang condition, 3 polygon, 6 set, 4, 10 structure, 205 standard, 208 multiplier, 13 µ-maps, 4, 8, 24 non-singular, 15 strictly, 15 non-trivial quadratic space, 15 norm of a composition algebra, 68 exponent, 126 splitting, 17 on a vector space, 58 norm splitting field, 17 numbering, 11 octagonal system, 7 octonion division algebra, 6 opposite map, 150 residues, 150, 170 locally, 193 root group sequence, 23 orientation, 309 oriflamme complex, 305

335

INDEX orthogonal building, 303 complement, 16 sum, 14 panel, 147 parabolic subgroup, 10, 300 parallel residues, 166 Pfister form, 119 polar region, 272 space, 20 p-reduced, 143, 144 projection map, 4 projective space, 20 proper, 31 pseudo-quadratic form type, 7 space, 6 pseudo-reductive group, 249 pseudo-split building, 246, 275 group, 249 pseudo-splitting field, 19 quadrangle exceptional, 7 of indifferent type, 7 of involutory type, 7 of pseudo-quadratic form type, 7 of quadratic form type, 7 ramified, 91 semi-ramified, 91 unramified, 91 quadratic form type, 7 quadratic module, 13 quadratic space, 13 anisotropic, 13 base point of, 17 isotropic, 13 non-singular, 15 non-trivial, 15 pointed, 17 ramified, 58 round, 18 singular, 15 split, 17 splitting extension of, 17 splitting field of, 17 tame, 58 totally singular, 15 wild, 58 trim, 58 of type E5 , 76 of type E6 , E7 and E8 , 69 of type F4 , 79 dual of, 79

unital, 17 unramified, 58 wild, 58 quadratic submodule, 13 quasi-split Tits index, 154 quaternion division algebra, 32 ramified quadrangle, 91 quadratic extension, 58 space, 58 rank absolute, 160 of a building, 3 of a Coxeter system, 143 relative, 160 of a residue, 3, 12 ray, 238 reduced representation, 143 word, 165 reflection, 8, 147, 222 half-space, 226 hyperplane, 226 regular, 19 relative Coxeter system/diagram, 160 rank/type, 160 representation compatible, 157 reduced, 143 residual quadratic spaces, 57 residually pseudo-split building, 275 split building, 275 residue, 146 root, 3, 8, 147 long, 265 root group labeling, 7, 254 of a building, 3 of a Moufang set, 4 sequence, 6, 23 based at (c, Σ), 23 length of, 23 opposite, 23 subsequence of, 7 terms of, 23 round, 18 scalar extension, 14 semi-linear automorphism, 19, 257 similitude, 20 strictly, 21 semi-ramified quadrangle, 91 short

336 case, 86 vertex, 318 similarity, 13, 222 similitude, 13 simplex, 11, 226 simplicial complex, 11 numbered, 11 singular, 15 special vertex, 137 spherical building, 3 exceptional, 9 pseudo-split, 246 split, 245 Coxeter system, 143 descent group, 187 subset, 145 Tits index, 163, 309 split building, 245, 275 quadratic space, 17 Tits index, 154 splitting extension, 17 field, 17 stabilize, 153 standard involution, 32, 259 metric, 230 Moufang structure, 208 strictly non-singular, 15 semi-linear, 21 subbuilding, 8, 195 of split type, 195 subcomplex, 11 subform, 13 support, 149 Tapartment, 197 building/group, 195 tame quadratic extension, 58 quadratic space, 58 terms of a root group sequence, 23 thick, 3 Tits index, 153 affine, 163, 309 anisotropic, 153 conventions, 297 decorations, 318 of a descent group, 188 display of, 297 exceptional, 163, 318 irreducible, 163 isotropic, 153

INDEX quasi-split, 154 spherical, 163, 309 split, 154 torus joint, 90 of a Moufang set, 4 at Ui , 24 totally isotropic subspace, 19 singular, 15 wild quadratic space, 58 trace, 41 of a composition algebra, 68 exponent, 126 translation, 162, 222 tree, 10 trim, 58 true, 238 trunk, 23 typ/Typ, 165 type absolute/relative, 153, 160 of a building, 3 of a gallery, 149 of a residue, 165 of a simplex, 226 of a vertex, 226 type-preserving, 165 uniformizer, 57 unipotent group, 205 radical, 10, 209, 272 unitary building, 243 unramified Galois embedding, 275 involution, 275 group of automorphisms, 275 quadrangle, 91 quadratic extension, 58 space, 58 vertex, 11, 226 special, 137 wall of a half-space, 223 of a root, 8, 147 true, 238 weak isomorphism, 5 Weyl-distance, 146 wild quadratic extension, 58 quadratic space, 58 quaternion/octonion algebra, 288 Witt index, 19