Proposition 33

Assume that R is irreducible and reduced. Put $s_{\alpha_i} = s_i$, and let $\Gamma$ be the subgroup of W generated by $c = s_1 \ldots s_l$.

(i) Let $\theta_i = s_l s_{l-1} \ldots s_{i+1} (\alpha_i)$ ($i = 1, \ldots, l$). Then, $\theta_i > 0, c(\theta_i) < 0$. (ii) If $\alpha$ is a root $> 0$ such that $c(\alpha) < 0$, then $\alpha$ is equal to one of the $\theta_i$. (iii) The family $(\theta_i)_{1 \leq i \leq l}$ is a basis of Q(R). (iv) Let $\Omega_i$ be the orbit of $\theta_i$ under $\Gamma$. The sets $\Omega_i$ are pairwise disjoint, they are all the orbits of $\Gamma$ on R, and each has h elements.

Observe first that $(s_1, \ldots, s_l)$ is a reduced decomposition of $c$ (Chap. IV, § 1, no. 1) with respect to the set S of the $s_i$. Indeed, otherwise there would exist a subset $X = S - \{ j \}$ of $l - 1$ elements of S such that $c \in W_X$, which would contradict Cor. 2 of Prop. 7 of Chap. IV, § 1, no. 8.

Applying Cor. 2 of Prop. 17 of no. 6 to $c$ gives assertions (i) and (ii).

Let $Q_i$ be the subgroup of Q(R) generated by the $\alpha_j, j > i$. It is immediate that $Q_i$ is stable under the $s_j, j > i$, and that $s_j (\alpha_i) = \alpha_i$ mod. $Q_i$ for $j > i$. Hence:

In other words, there exist integers $c_{ij}$ such that

Part (iii) follows immediately.

Finally, let $\alpha$ be a root. The element $\sum_{k=0}^{h-1} c^k(\alpha)$ is invariant under $c$, and hence zero (Chap. V, § 6, no. 2). The $c^k(\alpha)$ cannot therefore all have the same sign, and there exists a $k$ such that $c^k(\alpha) > 0$ and $c^{k+1}(\alpha) < 0$. By (ii), $c^k(\alpha)$ is one of the $\theta_i$. Thus every orbit of $\Gamma$ on $R$ is one of the $\Omega_i$. Extend $(x|y)$ to a hermitian form on $V \otimes \mathbf{C}$. Each orbit of $\Gamma$ in $R$ has at most $h$ elements, and there are at most $l$ distinct orbits, by the above remarks. Now (Chap. V, § 6, no. 2, Theorem 2, ii)), the cardinal of $R$ is equal to $hl$, which immediately implies (iv).