Exercise 17, § 2

We retain the hypotheses and notation of Exerc. 15 and 16. Choose an element $\varphi \in \mathscr{J}$ taking the canonical chamber $C$ of $A_0$ ($§ 1$, Exerc. 16) to the canonical chamber $C_e$ of $I$ and, as in Exerc. 15 $d$), denote by $N$ the stabilizer of the apartment $\varphi(A_0) \in \mathscr{U}$ in the group $G$. For any subset $D$ of $A_0$, denote by $B_D$ the subgroup of $G$ leaving fixed all the points of $\varphi(D)$: we have $B = B_C$ and $B \cap N = B_{A_0}$.

a) Let $\alpha$ be a half of $A_0$ containing $C$. Show that $B_\alpha$ acts transitively on the semi-apartments of $I$ distinct from $\varphi(\alpha)$ whose wall is the wall $L$ of $\varphi(\alpha)$ (let $X$ be such a semi-apartment, $F$ a panel contained in $L$ and $C'$ a chamber of $\varphi(\alpha)$ containing $F$; show, by using Exerc. 16 $c$) and Exerc. 13 $a$), that there exists $g \in G$ such that $g(X \cup \varphi(\alpha)) = \varphi(A_0)$ and $g(C') = C'$; show that $g(F) = F$ and deduce from Exerc. 22 $c$ of $§ 1$ that $g \in B_\alpha$). Deduce that $B_L$ acts doubly transitively on the semi-apartments with wall $L$ and that $(B_L, B_\alpha, B_L \cap N)$ is a Tits system whose Weyl group is of order 2 (cf. Exerc. 7).

b) Let $D_1$ and $D_2$ be two convex subsets of $A_0$ such that $C \subset D_1 \subset D_2$ and that there exists a unique half $\alpha$ of $A_0$ with $D_1 \subset \alpha$ and $D_2 \not\subset \alpha$: then $D_1 = D_2 \cap \alpha$ ($§ 1$, Exerc. 21 $c$). Show that $B_{D_1} = B_\alpha B_{D_2}$ and that $B_\alpha \cap B_{D_2} = B \cap N$ (by considering a gallery of smallest possible length having one extremity equal to $C$ and the other containing a point $a \in D_2 - D_1$); show that there exist two neighbouring chambers $C'_1$ and $C'_2$ of $A_0$ such that $C'_1 \subset D_1, C'_2 \subset D_2, C'_2 \not\subset D_1$ and $C'_1 \cap C'_2$ contained in the wall $L'$ of $\alpha$. Put $C_i = \varphi(C'_i), F = \varphi(C'_1 \cap C'_2)$ and $L = \varphi(L')$. Show that the convex hull of $D_1 \cup C'_2$ is equal to $D_2$. Let $b \in B_{D_1}$: then $b(C_1) = C_1$, so $b(F) = F$ and $b(C_2) \neq C_1$. Deduce that the convex hull of $b(C_2) \cup L$ is a semi-apartment $X$ distinct from $\varphi(\alpha)$ and with wall $L$, and that there exists $b' \in B_\alpha$ such that

Show that $b'b(a) = a$ for all $a \in \varphi(D_1 \cup C'_2)$ and that $b'b \in B_{D_2}$.

c) Let $(\alpha_i)_{1 \leq i \leq q}$ be the halves of $A_0$ containing $C$, numbered so that the map

is strictly decreasing (cf. $§ 1$, Exerc. 17). Show that $B = B_{\alpha_1} \ldots B_{\alpha_q}$ and that if $b_i, b'_i \in B_{\alpha_i}$ with $b_1 \ldots b_q = b'_1 \ldots b'_q$, then $b'_i \in b_i(B \cap N)$ for $1 \leq i \leq q$.