Proposition 10

The group W provided with the discrete topology acts properly on A; it is generated by orthogonal reflections; it is infinite, irreducible and essential (§ 3, no. 7). The intersection C ∩ A is a chamber of A for W. If Ls denotes the hyperplane of A formed by the intersection of A with the hyperplane of E* orthogonal to e_s, the L_s for s ∈ S are the walls of C ∩ A. If $\varepsilon_s$ is the unit vector of T orthogonal to L_s on the same side of L_s as C ∩ A, then $(\varepsilon_s|\varepsilon_t) = -\cos(\pi/m(s,t))$ (for s, t ∈ S) and the Coxeter matrix of W (§ 3, no. 4) is identified with M.*

By Cor. 3 of Th. 1, W is discrete in $\mathbf{GL}(E)$, and hence in G, and acts properly on A. Let $s \in S$. Since Card S ≥ 2, the hyperplane of E* orthogonal to $e_s$ is not orthogonal to v and its intersection L_s with A is indeed a hyperplane. The displacement corresponding to s is thus a displacement of order 2 leaving fixed all the points of L_s: it is necessarily the orthogonal reflection associated to L_s. It follows that W is generated by orthogonal reflections. Th. 2 now shows that it is infinite and Prop. 7 that it is essential and irreducible.

Since C is an open simplicial cone, whose walls are the hyperplanes with equations $\langle x^*, e_s \rangle = 0$ (for $s \in S$), the intersection C ∩ A is a convex, hence connected, open and closed subset of the complement of the union of the L_s in A. Moreover, C ∩ A is non-empty, for if $x^* \in C$ we have $\langle x^*, v \rangle = \sum_s v_s \langle x^*, e_s \rangle > 0$ and $\langle x^*, v \rangle^{-1} x^* \in C \cap A$. It follows that C ∩ A is a chamber of A relative to the system of the L_s. Moreover, $w(C \cap A) \cap L_s = \varnothing$ for all $w \in W$ (cf. no. 4, property (P_n)) and it follows that C ∩ A is a chamber of A relative to the system consisting of the transforms of the L_s by the elements of W; by the Cor. of Th. 1 of § 3, no. 2, it follows that C ∩ A is a chamber of A relative to W.

Let $a_s^*$ be the vertex of the simplex C ∩ A not in L_s. We have

for $s, t \in S$ and $s \neq t$, and

Let $\varepsilon_s$ be the vector in T defined by the relations:

The vector $\varepsilon_s$ is orthogonal to $L_s$ and is on the same side of the hyperplane $L_s$ as $C \cap A$. Moreover

which shows that $\varepsilon_s$ is the image of the class of $e_s$ under the isomorphism from $E/E^0$ to $T$ given by the quadratic form $B_M$. It follows that

Consequently $\varepsilon_s$ is a unit vector and the last assertion of Prop. 10 is proved.

The euclidean affine space $A$ provided with the group $W$ is called the space associated to the Coxeter matrix $M$ and we denote it by $A_M$.

Proposition 10 admits a converse: