Exercise 16, § 2

We retain the hypotheses and notation of Exerc. 15.

a) Let C and C' be two chambers. Show that there exists a chamber $C''$ opposed both to C and to C' (take $C''$ opposed to C such that $t(C, C')$ is of greatest possible length; if $t(C', C'') \neq w_0$, there exists a neighbouring chamber $C_1$ of $C''$ such that $l(t(C', C_1)) > l(t(C', C''))$; show by using condition (G) of Exerc. 10 that we can assume that $C_1 \notin A_{(C, C'')}$ and that $C_1$ is then opposed to C).

b) Let $a \in \mathscr{U}$; show that there exists a unique involutive automorphism $j_A$ (not necessarily allowed) that takes every chamber of A to the opposed chamber (use Exerc. 22 c) of $§ 1$). Let F and F' be two facets of I; show that if $F' = j_A(F)$ for some $A \in \mathscr{U}$ containing F and F', then the same is true for all $A \in \mathscr{U}$ containing F and F' (if $F, F' \in A \cap A'$, with $A, A' \in \mathscr{U}$, consider a chamber C (resp. C') of A (resp. A') containing F (resp. F') and an $A'' \in \mathscr{U}$ containing C and C' and use (SB 3)). Then F and F' are said to be opposed. Show that two facets have a common opposed facet if and only if they are of the same type T and that a facet opposed to a facet of type T is of type $w_0 T w_0^{-1}$.

c) Let $A_0$ be the apartment associated to a Coxeter system $(W, S)$ ($§ 1$, Exerc. 16) and let $\mathscr{J}$ be the set of allowed isomorphisms from $A_0$ to the various elements of $\mathscr{U}$. If $\alpha$ is a half of $A_0$, with wall L ($§ 1$, Exerc. 16) and if $\varphi \in \mathscr{J}$, we say that $\varphi(\alpha)$ is a semi-apartment of I, with wall $\varphi(L)$. Show that $\varphi(L)$ then depends only on $\varphi(\alpha)$ and not on the pair $(\varphi, \alpha)$. Let $D_1$ and $D_2$ be two distinct semi-apartments, with the same wall L: show that there exist $\varphi \in \mathscr{J}$ and a wall $L_0$ of $A_0$ such that $L = \varphi(L_0)$ and that the $D_i$ are the images under $\varphi$ of the two halves of $A_0$ determined by $L_0$ (choose a panel $F$ contained in $L$ and two chambers $C_1$ and $C_2$, one contained in $D_1$ and the other in $D_2$ and such that $C_1$ contains $F$ and $C_2$ contains the panel opposed to $F$ in $D_1$ (which is also opposed to $F$ in $D_2$) and consider the apartment of $I$ containing $C_1$ and $C_2$).