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LECTURES ON BUILDINGSUpdated and Revised

Mark Ronan

THE UNIVERSITY OF CHICAGO PRESS CHICAGO AND LONDON

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CONTENTSIntroduction to the 2009 EditionIntroductionLeitfadenChapter 1 - CHAMBER SYSTEMS AND EXAMPLESO. Buildings and the Origin of Chamber Systems1. Chamber Systems2. Two Examples of Buildings

ExercisesChapter 2 - COXETER COMPLEXES1. Coxeter Groups and Complexes2. Words and Galleries3. Reduced Words and Homotopy4. Finite Coxeter Complexes5. Self-Homotopy

ExercisesChapter 3 - BUILDINGS1. A Defini tion of Buildings2. Generalized m-gons - the rank 2 case3. Residues and Apartments

ExercisesChapter4 - LOCAL PROPERTIES AND COVERINGS1. Chamber Systems of Type M2. Coverings and the Fundamental Group3. The Universal Cover4. Examples

Exercises

xxi

XlIl

1237

91217192224

27283035

3941444751

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V Ul CONTENTSChapter 5 - BN - PAIRS

1. Tits Systems and Buildings2. Parabolic Subgroups

Exercises555961

Chapter 6 - BUILDINGS OF SPHERICAL TYPE AND ROOT GROUPS1. Some Basic Lemmas2. Root Groups and the Moufang Property3. Commutator Relations4. Moufang Buildings - the general case

ExercisesChapter 7 - A CONSTRUCTION OF BUILDINGS

1. Blueprints2. Natural Labellings of Moufang Buildings3. Foundations

Exercises

6366687380

83868991

Chapter 8 - THE CLASSIFICATION OF SPHERICAL, BUILDINGS1. A3 Blueprints and Foundations 922. Diagrams with Single Bonds 943. C3 Foundations 974. Cn Buildings for n 2: 4 985. Tits Diagrams and F4 Buildings 1006. Finite Buildings 106

ExercisesChapter 9 - AFFINE BUILDINGS I

1. Affine Coxeter Complexes and Sectors2. The Affine Building.An _ 1(K, v)3. The Spherical Building at Infinity4. The Proof of (9.5)

ExercisesChapter 10 - AFFINE BUILDINGS II

1. Apartment Systems, Trees and Projective Valuations2. Trees associated to Walls and Panels at Infinity3. Root Groups with a Valuation4. Construction of an Affine BN-Pair5. The Classification6. AnApplication

Exercises

108

110114120124127

129133138144152157160

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CONTENTS ~Chapter 11 - TWIN BUILDINGS1. Twin Buildings and Kac-Moody Groups 1622. Twin Trees 1653. Twin Apartments 1674. An Example: Mfine Twin Buildings 1705. Residues, Rigidity, and Proj. 1726. 2-Spherical Twin Buildings 1747. The Moufang Property and Root Group Data 1788. Twin Trees Again 182

Appendix 1 - MOUFANG POLYGONS1. The m-function 1842. The Natural Labelling for a Moufang Plane 1873. The Non-existence Theorem 190

Appendix 2 - DIAGRAMS FOR MOUFANG POLYGONS 197Appendix 3 - NON -DISCRETE BUILDINGS 202Appendix 4 - TOPOLOGY AND THE STEINBERG

REPRESENTATION 206Appendix 5 - FINITE COXETER GROUPS 212Appendix6 - FINITE BUILDINGS AND GROUPS OF

LIE TYPE 213BIBLIOGRAPHY 216INDEX OF NOTATION 223INDEX 225

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2-spherical 174An apartment 4Affine

apartment system 129, 202BN-Pair 144building 110classification 152Coxeter complex 110diagrams 110root 129, 146twin building 170

anisotropic 101apartment 4, 31, 34, 129, 203asymptote class 133asymptotic 133automorphism 11BN-pair 55bipartite graph 28Blueprint 83

conform to 84realisable 84

Borel subgroup 59bounded subgroup 117Bruhat decomposition 55Building 27

at infinity 122, 129, 170, 204affine 110Moufang 66, 74spherical 63twin 162, 164

Cn apartment 6Cn building 5,98

INDEX

carrefour 130Cayley plane 92, 97, 197chamber 1, 2chamber system 2

oftypeM39cloison de quartier 113codistance 164, 165commutator relations 68, 181, 182condition (co) 176contractible 207convex 14cotype 3cover 41Covering

topological 41universal 44

Coxeterbuilding 168complex 9, 10group 3, 9

of spherical type 212of affine type 110

system 3cross-polytope 6D4 building 95Dn building 96Desarguesian plane 92, 197Diagram

Coxeter 9of spherical type 212of affine type 110

Tits 100, 197

' IIi

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226 LECTURES ON BUILDINGSdiameter 19, 28Dieudonne determinant 116direct product 3, 33direction 121, 130, 170distance 13, 27En building 96Elementary

contraction 12equivalence 85expansion 12homotopy 12homotopy of galleries 42homotopy, strict 39

end 130,166equipollent valuations 143equivalent valuations 144exchange property 24F4 building 103Finite

Coxeter complex 19Coxeter group 20, 212

building 106flag 4flag complex 1folding 15Form

alternating 5,98Hermitian 98pseudo-quadratic 99quadratic 99sesquilinear 98symmetric bilinear 99symplectic 5,98trace-valued 99

foundation 89free constructions 67, 182fundamental group 42Gallery 3, 12

closed 42J-3length of13minimal 13

Generalised m-gon 28, 36, 68, 196hexagons 200octagons 201quadrangles 36, 197

geometric realisation 6girth 28groups of Lie type 106, 213half-apartment 14, 129, 202homology 207Homotopy

of galleries 42of words 12, 17strict 32type 206

interior chamber, panel, residue 178irreducible components 25isometry 31, 164isomorphism 3isotropic 5Iwahori subgroup 126junction 130J-reduced 173Kac-Moody group 74, 80, 162label 83Labelling 83

natural 88of a foundation 89

lattice 115, 172length 13m-function 184morphism3Moufang

building 66,74hexagons 200plane 81, 97, 102, 187, 197polygons 184, 197octagons 201quadrangles 197

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twin building 178twin tree 182

non-fragile 177opposite 14, 19, 35, 63, 64, 165, 166opposition involution 64oriflamme 96p-adics 74,120,157,170panel 1, 3parabolic subgroup 59parahoric subgroup 126parallel 121, 204parameter set 83parameter system 83parameters 30polarity 36polyhedra 24polysimplicial complex 110, 206positive roots 76, 148prenilpotent 73, 181proj 16, 32, 173projective module 209projective plane 36 .projective valuation 131projectivity 137quartier 113R-tree 203RGD-system 180reduced word 12, 17reflection 13residue 13residue field 114retraction 32Root 14, 31data with valuation 139

group 66, 74, 119, 178group data 178, 181twin 178

s-extended 164s-reduced 164

INDEXsap 130second interior 179Sector 112, 120, 202

base chamber of 113vertex of 113direction 124face of 121, 129,202panel 113, 202

self-homotopy 22simple-transitive 11simplex 1simply-connected 42spherical type 20, 63special vertex 112Steinberg representation 207strict homotopy 39strongly transitive 56supports 89thick 29thin 29Tits diagram 100, 197Tits system 55topology 161, 206totally isotropic 5,99totally singular 99trace at infinity 121tree 36, 130,203

with sap 130Twinapartment 167building 162, 164Coxeter building 168ends 167root 178tree 165, 182

type 1,3typeM9, 10,27,39uniformizer 114universal cover 44valency 29Valuation

discrete 114, 131, 140, 170

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228 LECTURES ON BUILDINGS

Valuation (continued) wall 13, 31, 129, 146, 202, 203of root groups 138 Weyl chambers 20ring 114 Witt index 99special 161 word 3, 12

W-codistance 163, 164W-distance 27