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Descent in Buildings
By Bernhard Mühlherr, Holger P. Petersson, Richard M. Weiss PRINCETON UNIVERSITY PRESS
Copyright © 2015 Princeton University Press
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ISBN: 978-1-4008-7401-9
CHAPTER 1
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 Δ, 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 Δ. More generally, the rank of a J-residue of Δ 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 Δ, the corresponding root group Uα is the subgroup of Aut(Δ) 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 Δ 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 a, 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.)
Definition 1.2. Let Δ be a building, let R be a residue of Δ and let c be an arbitrary chamber of Δ. 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 [member of] R. This unique nearest chamber z is called the projection of c to R and is denoted by projR(c). The projection map projR: Δ ->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 [member of] X}), where X is a set with |X | ≥ 3 and for each x [member of] X , Ux is a subgroup of Sym(X) (where we compose from right to left) such that the following hold:
(i) For each x [member of] X , Ux fixes x and acts sharply transitively on X \{x}.
(ii) For all x, y [member of] X and each g [member of] Ux, gUy g–1 = Ug(y).
The groups Ux for x [member of] X are called the root groups of the Moufang set.
Definition 1.6. Let M = (X, {Ux | x [member of] X}) be a Moufang set and let G = .
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 [member of] U*x := Ux\{1}, there exist a unique element µxy(g) in the double coset
UygUy
that interchanges x and y. Thus µ := µxy is a map from U*x 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 [member of] 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.
Definition 1.7. Two Moufang sets
(X, {Ux | x [member of] X}) and (X', {Ux | x [member of] 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 [member of] X}) and M' = (X', {Ux | x [member of] 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 U'ψ(x) such that
(1.9) φ(uµ(a)µ(b)) = φ(u)µ'(φ(a))µ'(φ(b))
for all u [member of] Ux and all a, b [member of] U*x, 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 x1y1-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 U'x, such that (1.9) holds for all u [member of] Ux and all a, b [member of] U*x with µ = µxy and µ' = µx'y'. Then the map ψ from X to X' which sends x to x' and yu to (y')φ(u) for all u [member of] Ux is an xy-isomorphism from M to M'.
Notation 1.11. Let M = (X, {Ux | x [member of] X}) be a Moufang set, choose distinct points x, y in X, let µ = µxy be the map from U*x to Aut(M) defined in 1.6, choose a [member of] U*x and let m = µ(a). There exists a unique permutation ρ of U*x such that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
for all u [member of] U*x. Therefore
(1.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
for all u [member of] U*x since m interchanges x and y. We identify Ux with the set X\{x} via the map u -> yu, then we identify ρ with the permutation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 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 from (1.12) that the permutations m and ρ of X are the same. In particular <Ux, ρ> = <Ux, Uy>. Since this group acts transitively on X, it acts transitively on the set of root groups {Uz | z [member of] 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.
(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 Δ and let E2(c) denote the subgraph spanned by all the irreducible rank 2 residues of Δ. Another fundamental result of Tits ([62, 10.16]) says that Δ 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 Δ, 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) or CI2(Λ), BQ2(Λ) or CQ2(Λ), BD2(Λ) or CD2(Λ), BP2(Λ) or CP2(Λ), BE2(Λ) or CE2(Λ), BF2(Λ) or CF2(Λ), and G2(Λ).
(Continues...)
Excerpted from Descent in Buildings by Bernhard Mühlherr, Holger P. Petersson, Richard M. Weiss. Copyright © 2015 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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