Proposition 7

Assume that the group W is essential and finite. Then: (i) There exists a unique point a of E invariant under W. (ii) The family (c_H)_{H \in \mathscr{M}} is a basis of T. (iii) The chamber C is the open simplicial cone with vertex a defined by the basis (e'_H)_{H \in \mathscr{M}} of T such that (e_H|e'_H') = \delta_{HH'}.

(i) By Prop. 4 of no. 6, there exists a point $a \in E$ invariant under $W$. Let $t \in T$ be such that $t + a$ is invariant under $W$. For all $w \in W$,

so $U(w).t = t$; since $W$ is essential, this implies that $t = 0$, showing the uniqueness of $a$.

(ii) Since $W$ is essential, $T = T_1$ in the notation of no. 7, and Prop. 5, (iv) shows that the family $(e_H)_{H \in \mathscr{M}}$ generates the vector space $T$. The existence of a point of $E$ invariant under $W$ shows that the family $(e_H)_{H \in \mathscr{M}}$ is free (no. 6, Prop. 4).

(iii) Let $a$ be the unique point of $E$ invariant under $W$. Since $(e_H)_{H \in \mathscr{M}}$ is a basis of $T$, and since the scalar product is a non-degenerate bilinear form on $T$, there exists a unique basis $(e'_H)_{H \in \mathscr{M}}$ of $T$ such that $(e_H|e'_{H'}) = \delta_{HH'}$ for $H, H'$ in $\mathscr{M}$. Every point $x$ of $E$ can be written uniquely in the form $x = t + a$ with $t = \sum_{H \in \mathscr{M}} \xi_H e'_H$ and the $\xi_H$ real. Then $x$ belongs to $C$ if and only if, for every hyperplane $H \in \mathscr{M}$, $x$ is on the same side of $H$ as $e_H$, or in other words $(t|e_H) = \xi_H$ is strictly positive. Hence (iii).