Theorem 3

(i) The set of walls of a chamber is finite. (ii) The set of directions of hyperplanes belonging to $\mathfrak{H}$ is finite. (iii) The group $U(W)$ is finite.

Assertion (i) follows immediately from Prop. 3, (iii) and Lemma 7.

We prove (ii). Let $C$ be a chamber and $M$ the set of its walls. The facets of $\overline{C}$ (relative to $\mathfrak{H}$) are the same as those relative to $M$ (§ 1, no. 4, Prop. 9).

Since $\mathfrak{M}$ is finite, they are finite in number. Since a facet meets only finitely many hyperplanes belonging to $\mathfrak{H}$, the set of hyperplanes belonging to $\mathfrak{H}$ and meeting $\overline{C}$ is finite, hence so is the set $A(C)$ of unit vectors in $T$ orthogonal to some hyperplane belonging to $\mathfrak{H}$ and meeting $\overline{C}$. Consequently, there exists a real number $\lambda < 1$ such that $(a|a') \leq \lambda$ for $a, a' \in A(C)$ and $a \neq a'$.

Let $A$ be the set of unit vectors in $T$ orthogonal to a hyperplane belonging to $\mathfrak{H}$. Let $a, a' \in A$ with $a \neq a'$. If $a$ and $a'$ are parallel, then $a = -a'$ and $(a|a') = -1$. Otherwise, let $H \in \mathfrak{H}$ (resp. $H' \in \mathfrak{H}$) be such that $a$ (resp. $a'$) is orthogonal to $H$ (resp. $H'$). We have $H \cap H' \neq \varnothing$, and if $x \in H \cap H'$ there exists an element $w \in W$ such that $x \in w(\overline{C})$. The vectors $U(w).a$ and $U(w).a'$ then belong to $A(C)$, we have

and the set $A$ is finite by Lemma 7. Hence (ii).

Now let $w \in W$ be such that $U(w).a = a$ for all $a \in A$. Then $U(w).t = t$ for all $t$ belonging to the subspace of $T$ generated by $A$. On the other hand, if $t \in T$ is orthogonal to $A$, we have $U(s_H).t = t$ for all $H \in \mathfrak{H}$, hence $U(w).t = t$ and finally $U(w) = 1$. Since $U(w)(A) = A$ for all $w \in W$, we deduce that $U(W)$ is isomorphic to a group of permutations of the finite set $A$, hence (iii).