Exercise 22, § 1

We retain the notation of Exerc. 16 and 21.

a) Let $t \in T$ and $w \in W$. Show that the chambers $C$ and $w(C)$ are separated by the wall $L_t$ if and only if $l(tw) < l(w)$ (use the folding determined by the half $A^+(t)$).

b) Let $w_0 \in W$. Show that the following conditions are equivalent:

whenever $t \in T$, the chambers $C$ and $w_0(C)$ are separated by the wall $L_t$ (iii)

(Use Exerc. 16 h) to show that (iii) $\Longrightarrow$ (i).) Show that such an element $w_0$ is unique and exists if and only if $W$ is finite. It is then the element of greatest length of $W$ and is characterized by

Moreover, $w_0^2 = 1, w_0Sw_0 = S$ and $l(w_0) = \mathrm{Card}(T)$.

(c) Assume that $W$ is finite. Show that, for any chamber $C_0$, there exists a unique chamber $-C_0$ such that $C_0 \cup (-C_0)$ is not contained in any half of $A$. Show that there exists a unique involutive automorphism $\varphi$ of $A$ (not necessarily allowed such that $\varphi(C_0) = -C_0$ for any chamber $C_0$ and that $\varphi(L) = L$ for any wall $L$ of $A$. We put $\varphi(a) = -a$ for $a \in A$. If $F$ is a facet, the facet $-F = \varphi(F)$ will be said to be opposed to $F$.

d) Let $C_0$ be a chamber of $A$ and $F$ a partition of $C_0$. Show that the convex hull of $C_0 \cup (-F)$ is the half of $A$ determined by the wall $L$ that is the support of $F$ and that contains $C_0$.