Proposition 10

There exists a special point for $W$.

By Prop. 6 of no. 8, we need only consider the case when $W$ is essential.

The group $U(W)$ of automorphisms of $T$ is finite (cf. no. 6, Th. 3) and $U(s_H)$ is an orthogonal reflection for every hyperplane $H$; moreover, $U(W)$ is generated by the family $(U(s_H))_{H \in \mathfrak{H}}$. By Prop. 7 of no. 9, there exists a basis $(e_i)_{i \in I}$ of $T$ such that the group $U(W)$ is generated by the set of reflections $(s_i)_{i \in I}$ defined by

The Cor. of Th. 1 of no. 2 shows that every reflection $s \in U(W)$ is of the form $s = U(s_H)$ with $H \in \mathfrak{H}$. We can thus find in $\mathfrak{H}$ a family of hyperplanes $(H_i)_{i \in I}$ such that $s_i = U(s_{H_i})$ for all $i$. Since the vectors $e_i$ are linearly independent, there exists $a \in E$ such that $a \in H_i$ for all $i \in I$. We have $s_{H_i} \in W_a$, so $U(W) = U(W_a)$, which means that $W = W_a . L$ since $L$ is the kernel of $U$. Thus, $a$ is special.

When $W$ is finite and essential, there is only one special point for $W$, namely the unique point invariant under $W$. Thus, the consideration of special points is interesting mainly when $W$ is infinite.