Exercise 15, § 2

We use the hypotheses and notation of Exerc. 10. Assume moreover that the Weyl group W of $(G, B)$ is finite and denote by $w_0$ the longest element of W (§ 1, Exerc. 22). Two chambers C and $C'$ are said to be opposed if $t(C, C') = w_0$.

a) Show that if C and C' are opposed, so are C' and C. Show that there exists a chamber opposed to any given chamber C. Show that the stabilizer of a chamber C in G acts transitively on the set of chambers opposed to C.

b) Let C and C' be two opposed chambers. Show that for all $w \in W$, there exists a unique chamber $C_w$ having the following property: if $(s_1, \ldots, s_k)$ (resp. $(s'_1, \ldots, s'_h)$) is a reduced decomposition of w (resp. of $w' = w_0 w^{-1}$), there exists a minimal gallery $(C_0 = C, C_1, \ldots, C_n = C')$ of type $(s_1, \ldots, s_k, s'_1, \ldots, s'_h)$ (with $n = h + k$) such that $C_{w'} = C_k$ (use Exerc. 22 of §1 and Exerc. 10, d) and e)). Show that $C_w$ and $C_{w w_0}$ are opposed.

c) Let $\mathscr{M}$ be the set of pairs of opposed chambers. For $m = (C, C') \in \mathscr{M}$, let $A_m$ be the union of the chambers $C_w$ constructed above. Show that I equipped with the set $\mathscr{U}$ of $A_m$ for $m \in \mathscr{M}$ is a structured building ($§ 1$, Exerc. 24) and that the Coxeter system $(W, S)$ and the numbering of I are adapted to $(I, \mathscr{U})$; define a canonical bijection from $\mathscr{M}$ to the set denoted by $J$ in Exerc. 24 of $§ 1$. We identify these two sets.

d) Let $m = (C, C') \in \mathscr{M}$ with $C = C_e$, and let N be the stabilizer of $A_m$ in G. Show that $N/(B \cap N)$ can be identified with W and that $(G, B, N, S)$ is a saturated Tits system (use Exerc. 13).