Exercise 2, § 4

a) Let $\Gamma_w$ be the set of extremal generators of the cone $w(C)$ and let $A$ be the union of the $\Gamma_w$ for $w \in W$. Show that $A$, equipped with the set $\{\Gamma_w | w \in W\}$, is a building (Chap. IV, § 1, Exerc. 15). Show that the map $j$ from the apartment $A_0$ associated to the Coxeter system $(W, S)$ (Chap. IV, § 1, Exerc. 16) to $A$ that transforms the point $wW^{(s)}$ of $A_0$ (for $w \in W$ and $s \in S$) to the generator $w(\mathbf{Re}_s)$, is an isomorphism from $A_0$ to $A$, compatible with the action of $W$. Show that, if $t = wsw^{-1}$, with $w \in W$ and $s \in S$, the image under $j$ of the wall $L_t$ defined by $t$ (resp. of a half of $A_0$ defined by $L_t$) (\emph{loc. cit.}) is the set of elements of $A$ contained in the hyperplane (resp. the closed half-space) that is the transform by $\sigma^*(w)$ of the hyperplane $e_s = 0$ (resp. of one of the closed half-spaces $e_s \geq 0$ or $e_s \leq 0$).

b) Show that $W$ is finite if and only if there exists an element $w_0 \in W$ such that $w_0(C) = -C$. This element $w_0$ is then unique and is the longest element of $W$ (use Exerc. 22 of Chap. IV, § 1). Show that in that case $j(-a) = -j(a)$ for all $a \in A_0$ (cf. Exerc. 22 c) of Chap. IV, § 1).

c) Show that $W$ is finite if and only if the cone $U$ formed by taking the union of the cone closures $w(\overline{C})$ for $w \in W$ is equal to the whole of $E^*$ (if $W$ is finite, use b) and the convexity of $U$. If $U = E^*$, consider an element $w \in W$ such that $w(\overline{C}) \cap (-C) \neq \varnothing$ and show that $w(C) = -C$.

d) Let $H$ be a finite subgroup of $W$. Show that there exists a subset $X$ of $S$ such that $W_X$ is finite and contains a conjugate of $H$. (Argue by induction on $\mathrm{Card}(S)$. Let $x \in C$ and $\overline{x} = \sum_{h \in H} h(x)$. Show, by using c) above, that $W$ is finite if $\overline{x} = 0$. If $\overline{x} \neq 0$, there exist $w \in W$ and $Y \subset S, Y \neq S$, such that $w(\overline{x})$ belongs to $C_Y$ (in the notation of no. 6), and hence $H \subset w^{-1}W_Yw$; apply the induction hypothesis to $Y$.)