Proposition 4

Let C' be a chamber of R'.

(i) There exists a unique alcove C contained in C' such that 0 \in \overline{C}. (ii) The union of the w(\overline{C}) for w \in W is a neighbourhood of 0 in E. (iii) Every wall of C' is a wall of C.

This follows from Prop. 11 of Chap. V, § 3, no. 10.

Assume now that R is irreducible. Let (\alpha_i)_{i \in I} be a basis of R (§ 1, no. 5, Def. 2), and let (\overline{\omega}_i)_{i \in I} be the dual basis. The \overline{\omega}_i are the fundamental weights of R' for the chamber D' of R' corresponding to the basis (\alpha_i). Let

be the highest root of R (§ 1, no. 8), and let J be the set of i \in I such that n_i = 1.