Proposition 7

Assume that $R$ is irreducible. Let $B = \{ \alpha_1, \ldots, \alpha_l \}$ be a basis of $R$, $f$ the connection index of $R$ (§ 1, no. 9) and $\tilde{\alpha} = n_1 \alpha_1 + \cdots + n_l \alpha_l$ the highest root of $R$ (for the order defined by $B$). Then the order of $W$ is equal to

Let $(\overline{\omega}_1, \ldots, \overline{\omega}_l)$ be the basis of $P(R^-)$ dual to $B$. By the Cor. of Prop. 5, the open simplex $C$ with vertices $0, n_1^{-1} \overline{\omega}_1, \ldots, n_l^{-1} \overline{\omega}_l$ is an alcove of $E$. Choose a Haar measure $\mu$ on the additive group $V^*$. Let $A$ be the set of elements of $V^*$ of the form $\xi_1 \overline{\omega}_1 + \cdots + \xi_l \overline{\omega}_l$, with $0 < \xi_i < 1$ for $i = 1, \ldots, l$. By Cor. 2 of Prop. 15 of Integration, Chap. VII, § 1, no. 10,

On the other hand, let $A'$ be the set of elements of $V^*$ of the form with $0 < \xi_i < 1$ for $i = 1, \ldots, l$. Since $(\alpha_{\tilde{1}}, \ldots, \alpha_{\tilde{l}})$ is a basis of the $\mathbf{Z}$-module $Q(R^-)$, we can apply Lemma 1 with $X = V^*$, $G = W_a$, $G' = Q$, $U = C$ and $U' = A'$. This gives

Finally, Lemma 1 can be applied again, taking X = V*, G = P, G' = Q, U = A and U' = A'. This gives

The proposition now follows by comparing formulas (5), (6) and (7).