# The Structure of Affine Buildings. (AM-168)

Richard M. Weiss
Pages: 384
https://www.jstor.org/stable/j.ctt7s9s3

1. Front Matter
(pp. i-iv)
(pp. v-vi)
3. Preface
(pp. vii-xii)
4. Chapter One Affine Coxeter Diagrams
(pp. 1-12)

By the results summarized in Chapter VI, Section 4.3, of [3], affine Coxeter groups can be characterized as groups generated by reflections of an affine space (by which is meant a Euclidean space without a fixed coordinate system or even a fixed origin). In an effort to get to affine buildings as quickly as possible, however, we will define them more succinctly, but with less motivation, simply as the Coxeter groups associated with the following Coxeter diagrams (and direct products of such groups):

Each Coxeter diagram¹ in Figure 1.1 has a name of the form${{{\text{\tilde X}}}_\ell }$, where

X=A,B,C,D,E,F or G...

5. Chapter Two Root Systems
(pp. 13-24)

In this chapter, we show that the Coxeter chamber systems corresponding to the connected affine Coxeter diagrams (Figure 1.1) have the three fundamental properties formulated in 1.3, 1.9 and 1.12. Our proofs rely on standard facts about root systems for which we will cite [3] and [17] as our references. With the representation of spherical and affine Coxeter chamber systems in terms of Weyl chambers and alcoves that we describe in this chapter, we are also setting the stage for Chapter 3.

Notation 2.1. Let$V$be a Euclidean space. For each non-zero$\alpha \; \in \;V$,we set

${s_\alpha }(\upsilon ) = \;\upsilon \; - \;2\frac{{\upsilon \; \cdot \;\alpha }}{{\alpha \; \cdot \;\alpha }}\alpha$

The elements${s_\alpha }$for...

6. Chapter Three Root Data with Valuation
(pp. 25-38)

In this chapter, we introduce the notion of a root datum with valuation. We will not need this notion until Chapter 13, but we prefer to introduce it now while the representation of spherical and affine Coxeter chamber systems in terms of root systems, Weyl chambers and alcoves as described in Chapter 2 is still fresh in the reader’s mind.

Throughout this chapter, we let$\Pi \; = \;{{\text{X}}_\ell }$be one of the spherical diagrams in Figure 1.3 with$\ell \; \geqslant \;2$(i.e. not A1), we let$I$denote the vertex set of$\Pi$, we let$\Delta$be an irreducible spherical building of type$\Pi$...

7. Chapter Four Sectors
(pp. 39-44)

We return now to the study of affine Coxeter chamber systems where we left off at the end of Chapter 1.¹ Our goal in this and the next chapter is to introduce sectors and faces. These are fundamental structures in the theory of affine buildings.

As in Chapter 1, we assume that$\Pi$is one of the affine Coxeter diagrams${{{\text{\tilde X}}}_\ell }$in Figure 1.1 and that$\Sigma \; = \;{\Sigma _\Pi }$is the corresponding Coxeter chamber system as defined in 29.4. Let$W\; = \;{W_\Pi }$be the Coxeter group associated with$\Pi$(as defined in 29.5) which we identify with${\text{Aut}}^\circ \,{\text{(}}\Sigma {\text{)}}$as described in 1.14. Let$T\; = \;{T_\Pi }$...

8. Chapter Five Faces
(pp. 45-52)

Roughly speaking, a face is the border between two “adjacent” sectors. Faces play a central role in the construction of the building at infinity, as we will see in Chapter 8.

We continue to denote by$\Pi$one of the affine Coxeter diagrams${{{\text{\tilde X}}}_\ell}$in Figure 1.1 and by$\Sigma$the corresponding Coxeter chamber system.

Definition 5.1. Let$R$be a gem, let$P\; = \;\{ d,\;e\}$be a panel contained in$R$, let$S\, = \;\sigma (R,\;d)$and let$\alpha$be the unique root of$\Sigma$containing$d$but not$e$. Let$\mu (\alpha )$be the wall of$\alpha$(as defined in 29.22), so$Q \cap \;\alpha {\kern 1pt} |\; = \;1$and hence

$Q \cap \;S{\kern 1pt} |\; \leqslant \;1$for...

9. Chapter Six Gems
(pp. 53-58)

In this chapter we give a characterization of affine Coxeter chamber systems in terms of their gems (6.6 below). We will need this result in Chapter 12.

We continue to let$\Pi$be one of the affine Coxeter diagrams in Figure 1.1 and to denote by$\Sigma\;=\;{\Sigma_\Pi}$the corresponding Coxeter chamber system.

Definition 6.1. Let$R$and${R_1}$be two gems of$\Sigma$. We will say that$R$and${R_1}$areadjacentif$R\: \cap \:{R_1}\: = \:\not 0$and there exist chambers in$R$adjacent to chambers in${R_1}$.

Note that if$R$and${R_1}$are adjacent gems of$\Sigma$and$d\;\in\;R$and$e\: \in \:{R_1}$are adjacent chambers,...

10. Chapter Seven Affine Buildings
(pp. 59-66)

We now turn to affine buildings. Our goal in this chapter is to prove 7.24. This result is crucial for the construction (described in Chapter 8) of the building at infinity of an affine building. It was first proved in 2.9.6 of [6].

Let$\Delta$be a thick irreducible affine building. More precisely, we suppose that$\Delta$is a thick building of type$\Pi$(as defined in 29.10), where$\Pi$is one of the affine Coxeter diagrams in Figure 1.1. Let$W\: = \:{W_\Pi }$be the Coxeter group associated with$\Pi$; we identify$W$with${\text{Aut}}^\circ{\text{(}}{\Sigma_\Pi})$as described in 29.4. Let$\delta$be...

11. Chapter Eight The Building at Infinity
(pp. 67-76)

In this chapter we show that the “boundary” of an affine building is a building of spherical type. This “building at infinity” plays a central role in the study of affine buildings. The notion of the building at infinity generalizes, as we will see, the notion of the celestial sphere of a Euclidean space. The main results of this chapter are 8.24 and 8.36.

We continue to assume that$\Pi$is one of the affine Coxeter diagrams in Figure 1.1 and that$\Delta$is a thick building of type$\Pi$.

Definition 8.1. Let$A$be an apartment of$\Delta$, let$R$...

12. Chapter Nine Trees with Valuation
(pp. 77-88)

Affine buildings of rank 1 are the same thing as thick trees, as was explained in 7.32. In this chapter we pause in our investigation of affine buildings of arbitrary rank to introduce an important connection between trees and valuations of fields. The principal results in this chapter are 9.14, 9.24 and 9.30.

Notation 9.1. The apartments of a thick tree are its thin subtrees. We will refer to the sectors of thick trees asrays. (This will be helpful later on when we talk about rays in trees and sectors in affine buildings of higher rank at the same...

13. Chapter Ten Wall Trees
(pp. 89-100)

We return now to our assumptions in Chapter 8:$\Pi$is one of the affine Coxeter diagrams in Figure 1.1 with vertex set$I$,$o\: \in \:I$is a special vertex of$\Pi$,$\Delta$is a thick building of type$\Pi$,$\mathcal{A}$is a system of apartments of$\Delta$and$\Delta_\mathcal{A}^\infty$is the building at infinity of the pair$(\Delta,\;\mathcal{A})$as constructed in 8.9 and 8.25. Our main goal in this chapter is to introduce a family of trees with sap, one for each wall of$\Delta_\mathcal{A}^\infty$. We also prove (in 10.24) for each gem$R$of$\Delta$the existence of a...

14. Chapter Eleven Panel Trees
(pp. 101-106)

In this chapter, which is a companion to the previous chapter on wall trees, we introduce a second, related family of trees with sap, one for each panel of the building at infinity$\Delta_\mathcal{A}^\infty$.

We continue to assume that$\Pi$is one of the affine Coxeter diagrams in Figure 1.1 with vertex set$I$, that$\Delta$is a thick building of type$\Pi$and that$\mathcal{A}$is a system of apartments as defined in 8.4.

Conventions 11.1. From now on, we identify the set of panels of$\Delta_\mathcal{A}^\infty$with the set of parallel classes of$\mathcal{A}$-faces via the map$[f]\mapsto{f^\infty}$and we...

15. Chapter Twelve Tree-Preserving Isomorphisms
(pp. 107-118)

The goal of this chapter is to prove 12.3. This fundamental result gives a necessary and sufficient condition expressed in terms of wall and panel trees for an isomorphism from one building at infinity to another to extend to an isomorphism of affine buildings.¹ In 12.31, we deduce as a corollary that all the elements of all the root groups of the building at infinity extend to automorphisms of the affine building itself.

The main result of this chapter was first proved by Tits in [35]. In fact, Theorem 2 in Section 8 of [35] is more general: It applies...

16. Chapter Thirteen The Moufang Property at Infinity
(pp. 119-130)

In the last chapter we saw (in 12.3) that a pair$(\Delta,\;\mathcal{A})$is uniquely determined by its building at infinity$\Delta_\mathcal{A}^\infty$and its “tree structure.” In this chapter, we introduce the assumption that the building at infinity$\Delta_\mathcal{A}^\infty$has the Moufang property (defined in 29.15) and show how the tree structure of$(\Delta,\;\mathcal{A})$can be translated into a valuation of the root datum of$\Delta_\mathcal{A}^\infty$as defined in 3.21. Our main results are 13.30 and 13.31.

Definition 13.1. ABruhat-Tits pairis a pair$(\Delta,\;\mathcal{A})$such that$\Delta$is a building whose Coxeter diagram (or type) is one of the affine...

17. Chapter Fourteen Existence
(pp. 131-146)

In this chapter we are concerned with the existence part of the classification of Bruhat-Tits buildings. Our main result is 14.47. Combining this existence result with the uniqueness result 13.31, we obtain as a corollary (in 14.54) one of our major conclusions: that Bruhat-Tits buildings are classified by root data (of Moufang spherical buildings of rank$\ell \; \geqslant \;2$) with valuation (up to equipollence).

This chapter is essentially a continuation of Chapter 3. We have included it at this point so that we can make the link to 13.8 in 14.47.iii and to 13.31 in 14.54.

The results in this chapter were...

18. Chapter Fifteen Partial Valuations
(pp. 147-158)

In 14.54, we concluded that Bruhat-Tits buildings are classified by root data with valuation. Our principal goal for the rest of this book is to determine when the root datum of a spherical building has a valuation. In this chapter, we show (in 15.21) that it suffices to consider this question “locally” in the sense spelled out in 15.1 and 15.4.

We adopt all the notation in 3.4, 3.5 and 3.12. In particular,$\Delta$is a building satisfying the Moufang property whose Coxeter diagram$\Pi$is one of the spherical diagrams${{\text{X}}_\ell}$in Figure 1.3 with$\ell \; \geqslant \;2$,$\Sigma$is an apartment...

19. Chapter Sixteen Bruhat-Tits Theory
(pp. 159-166)

We continue to adopt all the notation in 3.4 and 3.12. In particular,$\Delta$is a building whose Coxeter diagram$\Pi$is one of the spherical diagrams${{\text{X}}_\ell }$in Figure 1.3 with$\ell \; \geqslant \;2$,$\Delta$satisfies the Moufang property,$\Sigma$is an apartment of$\Delta$,$({G^\dag },\:\xi )$is the root datum of$\Delta$based at$\Sigma$and$\iota$is a root map of$\Sigma$with target$\Phi$.

From now on, we rely on the classification of Moufang spherical buildings ([32] and [36]). More precisely, we rely on the description of the root data of Moufang spherical buildings as summarized in 30.14.¹

In each case...

20. Chapter Seventeen Completions
(pp. 167-174)

Before we begin to study the problem described at the conclusion of Chapter 16, we would like to consider an important issue. In 14.54, we saw that Bruhat-Tits pairs$(\Delta,\;\mathcal{A})$are classified by root data with valuation (up to equipollence). Suppose, however, that we are interested in classifying Bruhat-Titsbuildings(as defined in 13.1) rather than Bruhat-Titspairs. To investigate the question when two root data with valuation correspond to the same Bruhat-Tits building$\Delta$, but not necessarily the same Bruhat-Tits pair$(\Delta,\;\mathcal{A})$, we need to introduce completions.

Definition 17.1. Let$(\Delta,\;\mathcal{A})$be a Bruhat-Tits pair as defined in 13.1...

21. Chapter Eighteen Automorphisms and Residues
(pp. 175-188)

With this chapter we again postpone attacking the problem described at the conclusion of Chapter 16, this time in order to examine more closely the structure of the residues of a Bruhat-Tits building. In particular, we prove various results about the automorphism group of a Bruhat-Tits pair (see 18.4–18.8), use them to show (in 18.18 and 18.19) that the proper irreducible residues of rank at least 2 of a Bruhat-Tits building satisfy the Moufang condition and produce a general method for determining the structure of the gems of a Bruhat-Tits building in 18.25.

In 18.30 and 18.31, we apply...

22. Chapter Nineteen Quadrangles of Quadratic Form Type
(pp. 189-204)

In this chapter, we consider the case$X\;={\mathcal{Q}_\mathcal{Q}}$of the problem framed at the end of Chapter 16. Our main result is 19.18. We then use this result to complete the classification of Bruhat-Tits pairs whose building at infinity is${\rm{B}}_\ell ^{\cal Q}(\Lambda )$for some anisotropic quadratic space$\Lambda$in 19.23. See 12.2 and 12.4 in [36] for the definition of an anisotropic quadratic space and 30.15 for the definition of the building${\rm{B}}_\ell ^{\cal Q}(\Lambda )$.

In this and the next seven chapters, we also investigate the completions and residues of each family of Bruhat-Tits pairs under consideration using 17.4 and 17.9 (for completions)...

23. Chapter Twenty Quadrangles of Indifferent Type
(pp. 205-208)

In this chapter, we consider the case$X\; = \;{{\cal Q}_{\cal D}}$of the problem framed at the end of Chapter 16. Our main results are 20.2 and 20.6 in which we give the classification of Bruhat-Tits pairs whose building at infinity is${\rm{B}}_2^{\cal D}(\Lambda)$for some indifferent set$\Lambda$. See 10.1 in [36] for the definition of an indifferent set and 30.15 for the definition of the building${\rm{B}}_2^{\cal D}(\Lambda)$.

Definition 20.1. Let$(K,\;{K_0},\;{L_0})$be an indifferent set and let$\nu$be a discrete valuation of$K$. By 10.1 and 10.2 in [36],$1\; \in \;{L_0}\; \subset \;{K_0},\;{K^2}{L_0} \subset \;{L_0}$and${K^2}{K_0} \subset \;{K_0}.$By 9.17.i, it follows that$\nu (K_0^*)\; = \;$or$2\mathbb{Z},\;\nu (L_0^*)\; = \;\mathbb{Z}$or$2\mathbb{Z}$...

24. Chapter Twenty One Quadrangles of Type ${E_6},{E_7}$ and $E_8$
(pp. 209-220)

In this chapter, we consider the case$X\; = \;{{\cal Q}_\varepsilon }$of the problem framed at the end of Chapter 16. Our main results are 21.27 and 21.35 in which we give the classification of Bruhat-Tits pairs whose building at infinity is${\rm{B}}_2^\varepsilon (\Lambda )$(as defined in 30.15) for some quadratic space Λ of type${E_6},{E_7}$or$E_8$(as defined in 21.1).

We begin by repeating the definition of a quadratic space of type${E_6},{E_7}$or$E_8$(taken from 12.31 of [36]):

Definition 21.1. Let$\Lambda \; = \;(K,\;L,\;q)$be a quadratic space. Then$\Lambda$is aquadratic space of type${E_k}$for$k\; = \;6,7$or 8 if$q$...

25. Chapter Twenty Two Quadrangles of Type ${F_{\mathbf{4}}}$
(pp. 221-228)

In this chapter, we consider the case$X\; = \;{\mathcal{Q}_\mathcal{F}}$of the problem framed at the end of Chapter 16. Our main results are 22.16 and 22.29 in which we give the classification of Bruhat-Tits pairs whose building at infinity is${\text{B}}_2^\mathcal{F}(\Lambda )$(as defined in 30.15) for some quadratic space Λ of type${F_{\mathbf{4}}}$. See 14.1 of [36] for the definition of a quadratic space of type${F_{\mathbf{4}}}$and 30.15 for the definition of the building${\text{B}}_2^\mathcal{F}(\Lambda )$.

Throughout this chapter, we make the following assumptions:

Notation 22.1. Let

$\Lambda \; = \;(K,\;L,\;q)$

be a quadratic space of type${F_{\mathbf{4}}}$and let...

26. Chapter Twenty Three Quadrangles of Involutory Type
(pp. 229-238)

In this chapter, we consider the case$X\; = \;{\mathcal{Q}_\mathcal{I}}$of the problem framed at the end of Chapter 16. Our main result is 23.3. We use this result to give the classification of Bruhat-Tits pairs whose building at infinity is${\text{B}}_\ell ^\mathcal{I}(\Lambda )$for some involutory set$\Lambda$in 23.10. See 11.1 in [36] for the definition of an involutory set and 30.15 for the definition of the building${\text{B}}_\ell ^\mathcal{I}(\Lambda )$.

We include honorary involutory sets and root group sequences

${\mathcal{Q}_\mathcal{I}}(\Lambda )$

for honorary involutory sets$\Lambda$(as defined in 30.22) in this chapter. It will be convenient in some places, therefore, to refer to...

(pp. 239-260)

In this chapter, we consider the case$X\; = \;{\mathcal{Q}_\mathcal{P}}$of the problem framed at the end of Chapter 16. Our main result is 24.31. We use this result to give the classification of Bruhat-Tits pairs whose building at infinity is${\text{B}}_\ell ^\mathcal{P}(\Lambda )$for some anisotropic pseudo-quadratic space$\Lambda$in 24.37. See 11.17 in [36] for the definition of an anisotropic pseudo-quadratic space and 30.15 for the definition of the building${\text{B}}_\ell ^\mathcal{P}(\Lambda )$.

Proposition 24.1.Let

$\Lambda \; = \;(K,\;{K_0},\;\sigma ,\;L,\;q)$

be a pseudo-quadratic space as defined in 11.17 of [36] (so$\Lambda$is not necessarily anisotropic), let f be the associated skew-hermitian form and suppose...

28. Chapter Twenty Five Hexagons
(pp. 261-274)

In this chapter, we consider the case$X\; = \;\mathcal{H}$of the problem framed at the end of Chapter 16. Our main results are 25.21 and 25.25 in which we give the classification of Bruhat-Tits pairs of type${{{\text{\tilde G}}}_2}$.

We begin with the analogue to 19.1, 21.8 and 24.1:

Proposition 25.1.Let$\Lambda \; = \;(J,\;K,\;N,\;\# ,\;T,\; \times ,\;1)$be an hexagonal system as defined in 15.15 in [36],¹ let E be a field containing K and let${J_E}\; = \;J\;{ \otimes _K}E$. Let${T_E}$be the unique bilinear form on${J_E}$and let${ \times _E}$be the unique bilinear map from${J_E}\; \times \;{J_E}$to${J_E}$such that

(25.2)${T_E}(u\; \otimes \;s,\;v\; \otimes \;t)\; = \;T(u,\;v)st$...

29. Chapter Twenty Six Assorted Conclusions
(pp. 275-288)

We have now finished answering the question posed at the end of Chapter 16. To complete the classification of Bruhat-Tits pairs, it remains only to say a few more words about Bruhat-Tits pairs of type${{{\text{\tilde F}}}_4}$. We do this in the first part of this chapter. In the second part of this chapter, we describe the classification of algebraic Bruhat-Tits buildings (as defined in 26.2) and in the third part we prove a few more facts about the automorphism group of a Bruhat-Tits pair.

The principal results in this chapter are 26.12, 26.25–26.27, 26.37 and 26.39.

Before going on,...

30. Chapter Twenty Seven Summary of the Classification
(pp. 289-296)

Looking back, we can observe that the classification of Bruhat-Tits pairs falls naturally into three parts. Part I is covered in Chapters 1–2 and 4–12. It includes the construction (starting with an arbitrary irreducible affine building) of the building at infinity and, in particular, the key result 7.24¹ that makes this construction possible, as well as the construction of the wall and panel trees$({T_m},\;{\mathcal{A}_m})$and$({T_F},\;{\mathcal{A}_F})$. Part I culminates in the result 12.3 that an affine building is uniquely determined by its building at infinity together with its tree structure. This part of the classification is valid...

31. Chapter Twenty Eight Locally Finite Bruhat-Tits Buildings
(pp. 297-320)

A building is calledlocally finiteif each chamber is adjacent to only finitely many other chambers.¹ In this chapter we apply the results summarized in the previous chapter to produce a list (in 28.33–28.38) of all locally finite Bruhat-Tits buildings and their principal features.

As we will see, almost all locally finite Bruhat-Tits buildings are algebraic as defined in 26.2.² That is to say, they are the affine buildings associated with pairs$(G,F)$, where$F$is a local field (as defined in 28.1) and$G$is an absolutely simple algebraic group of$F$-rank$\ell \; \geqslant \;2$. These are precisely the pairs whose classification...

32. Chapter Twenty Nine Appendix A
(pp. 321-342)
33. Chapter Thirty Appendix B
(pp. 343-360)
34. Bibliography
(pp. 361-364)
35. Index
(pp. 365-368)