Proposition 3

The special points (Chap. V, § 3, no. 10, Def. 1) of W_a are the weights of R'.

Let x_0 \in E and let \alpha \in R. The hyperplane L parallel to Ker \alpha and passing through x_0 has equation \langle \alpha, x \rangle = \langle \alpha, x_0 \rangle. To be equal to some L_{\beta,k}, it is necessary on the one hand that \alpha and \beta are proportional, and so, since R is reduced, that \beta = \pm \alpha, and on the other hand that \langle \alpha, x_0 \rangle is an integer. It follows immediately that x_0 is a special point of W_a if and only if \langle \alpha, x_0 \rangle \in \mathbf{Z} for all \alpha \in R, in other words, if and only if x_0 \in P(R') (§ 1, no. 9).