Proposition 9

Let V be a real vector space of finite dimension l, and let G be a finite subgroup of $\mathbf{GL}(V)$, generated by reflections and essential. Give V a scalar product invariant under G. The following conditions are equivalent:

(i) There exists a discrete subgroup of V of rank l that is stable under G. (ii) There exists a $\mathbf{Q}$-structure on V (Algebra, Chap. II, § 8, no. 1, Def. 1) invariant under G. (iii) There exists a root system in V whose Weyl group is G. (iv) There exists a discrete group $G'$ of displacements of V, acting properly on V, and generated by reflections, such that $G'$ is the semi-direct product of G and a group of translations of rank l.

(ii) $\Longrightarrow$ (i): let $V' \subseteq V$ be a $\mathbf{Q}$-structure on V invariant under G. Let A be a finite subset of $V'$ generating the $\mathbf{Q}$-vector space $V'$. Replacing A by $\bigcup_{s \in G} s(A)$, we can assume that A is stable under G. Let B be the subgroup of V generated by A. Then B is stable under G, of finite type and torsion-free, so has a basis over $\mathbf{Z}$ which is both a basis of $V'$ over $\mathbf{Q}$ and a basis of V over $\mathbf{R}$.

(iii) $\Longrightarrow$ (ii): this follows from Prop. 1 of § 1, no. 1, for example. (iv) $\Longrightarrow$ (iii): let $G'$ be a group satisfying condition (iv). The group of translations of $G'$ is of rank l, and 0 is a special point for $G'$ by Prop. 9 of Chap. V, § 3, no. 10. Prop. 6 shows that there exists a reduced root system $R_0$ in $V^*$ such that $G'$ is identified with $W_\alpha(R_0)$; the group G is then the Weyl group of the inverse root system of $R_0$.

(i) $\Longrightarrow$ (iv): assume that G leaves stable a discrete subgroup M of V, of rank l. For any reflection $s \in G$, $s(x) - x \in M$ for all $x \in M$, so the line $D_s$ orthogonal to $H_s$ meets M; let $\alpha_s, -\alpha_s$ be the generators of the cyclic group $D_s \cap M$; the set A of the $\alpha_s$ and $-\alpha_s$ is stable under G, hence generates a subgroup $M'$ of M stable under G; the discrete group $M'$ is of rank l because G is essential. Let $G'$ be the group of affine transformations of V that is the semi-direct product of G and the group of translations whose vectors belong to $M'$. Let $G'_1$ be the subgroup of $G'$ generated by the reflections of $G'$. We shall show that $G'_1 = G'$, which will complete the proof. First, $G'_1 \supseteq G$ since G is generated by reflections. On the other hand, for any reflection s of G, let $t_s$ be the translation with vector $\alpha_s$. The transformation $s \circ t_s$ is a reflection, and $s \circ t_s \in G'$; thus $t_s$ is a product of two reflections of $G'$; this being true for every reflection s of G, the translations whose vector belongs to $M'$ are all in $G'_1$.