Proposition 2

The group $W_a$, with the discrete topology, acts properly on E and permutes the hyperplanes $L_{\alpha,k}$ (for $\alpha \in \mathbf{R}$ and $k \in \mathbf{Z}$).

Since $Q(\mathbf{R}^\sim)$ is a discrete subgroup of $V^*$, the group Q acts properly on E. Hence, so does $W_a = W.Q$, since W is finite. Moreover, for $\alpha, \beta \in \mathbf{R}$ and $k \in \mathbf{Z}$:

where $n(\alpha,\beta) = \langle \beta^\sim, \alpha \rangle$ is an integer, hence the second assertion.

We can thus apply the results of Chapter V, § 3 to $W_a$ acting on E. To avoid any confusion with the chambers of the Weyl group W in $V^*$, we shall call the chambers determined by the system of hyperplanes $L_{\alpha,k}$ (for $\alpha \in \mathbf{R}$ and $k \in \mathbf{Z}$) in E alcoves. The group $W_a$ thus acts simply transitively on the set of alcoves and the closure of an alcove is a fundamental domain for $W_a$ acting on E (Chap. V, § 3, no. 2, Th. 1 and no. 3, Th. 2). It is clear that the Weyl group W is identified with the canonical image $U(W_a)$ of $W_a$ in the orthogonal group of V* (cf. Chap. V, § 3, no. 6). It follows that W_a is essential (Chap. V, § 3, no. 7) and that W_a is irreducible if and only if the root system R is (§ 1, no. 2, Cor. of Prop. 5). If R is irreducible, every alcove is an open simplex (Chap. V, § 3, no. 9, Prop. 8). In the general case, the canonical product decomposition of the affine space E (Chap. V, § 3, no. 8) corresponds to the decomposition of R into irreducible components. In particular, the alcoves are products of open simplexes.

Note also that the Cor. of Th. 1 of Chap. V, § 3, no. 2 shows that the s_{\alpha,k} are the only reflections in W_a.