Proposition 11

Let $W$ be a group of displacements of a euclidean affine space $A$, satisfying the assumptions of § 3. Assume that $W$ is infinite, essential and irreducible. Then the form $B_M$ attached to the Coxeter matrix $M$ of $W$ is positive degenerate and there exists a unique isomorphism from the affine space $A_M$ associated to $M$ to $A$, commuting with the action of $W$. This isomorphism transforms the scalar product of $A_M$ into a multiple of the scalar product of $A$.

Let $C_0$ be a chamber of $A$ and let $S$ be the set of orthogonal reflections with respect to the walls of $C_0$. If $\eta_s$ denotes the unit vector orthogonal to the hyperplane $N_s$ associated to the reflection $s$ and on the same side of $N_s$ as $C_0$ (§ 3, Prop. 3), the form $B_M$ is such that $B_M(e_s, e_t) = (\eta_s | \eta_t)$ for $s, t \in S$. It is thus positive. Since the $\eta_s$ are linearly dependent (§ 3, no. 9, Prop. 8), it is degenerate.

We can thus apply the preceding constructions to $M$. With the same notation as above, $(\varepsilon_s | \varepsilon_t) = (\eta_s | \eta_t)$ and there exists a unique isomorphism $\varphi$ of Hilbert spaces from $T$ to the space of translations of $A$ such that $\varphi(\varepsilon_s) = \eta_s$. Let $a$ and $b$ be two distinct vertices of $C_0$ and $s_0$ the reflection in $S$ such that $a \notin N_{s_0}$. Put $\lambda = (\eta_s | a - b)$ and let $\psi$ be the affine bijection from $A_M$ to $A$ defined by

It is then immediate that $\psi(L_s) = N_s$ for all $s \in S$ and that $\psi$ transforms the scalar product of $A_M$ into a multiple of that on $A$. It follows at once that $\psi$ commutes with the action of $W$. Finally, the uniqueness of $\psi$ is evident, for $a_s$ for example is the unique point of $A_M$ invariant under the reflections $t \in S, t \neq s$.