Exercise 12, § 2

Let (G, B, N, S) be a Tits system, W = N/(B ∩ N) its Weyl group, I the (W, S)-building associated to (G, B) (Exerc. 10) and C = C_e the canonical chamber of I. Let A_0 be the apartment associated to the Coxeter system (W, S) (§ 1, Exerc. 16). For all g ∈ G, let φ_g be the map from A_0 to I that takes a point wW^{(s)} of A_0 (w ∈ W, s ∈ S) to the point gwG^{(s)} of I.

Show that, for all g ∈ G, the map φ_g is an isomorphism of numbered buildings from A_0 to a subset of I that is the union of chambers gn(C_e) for n ∈ N. Show that I, equipped with the set Λ of φ_g(A_0) for g ∈ G, is a structured building (§ 1, Exerc. 24) (to prove (SB 2), remark that, if g', g'' ∈ G, there exist b', b'' ∈ B and n ∈ N such that g'^{-1}g'' = b'nb''; then put g = g'b'n and show that g'(C) and g''(C) are contained in φ_g(A_0). To prove (SB 3), reduce to the case of two apartments A' = φ_e(A_0) and

$A'' = \varphi_b(A_0)$, with $b \in B$, and show that the map $a \mapsto b(a)$ leaves the points of $A' \cap A''$ fixed by using Prop. 2 of no. 5). Show that the Coxeter system $(W, S)$ and the numbering of I are adapted to $(I, \mathscr{U})$ (§ 1, Exerc. 24 e)) and that the set $\mathscr{J}$ of allowed isomorphisms from $A_0$ to the various elements of $\mathscr{U}$ is the set of $\varphi_g$ for $g \in G$.