Proposition 5

Let C be the alcove contained in C' and containing 0 in its closure (Prop. 4).

(i) $C$ is the set of $x \in E$ such that $\langle \alpha_i, x \rangle > 0$ for all $i \in I$ and $\langle \tilde{\alpha}, x \rangle < 1$.

(ii) The set $\overline{C} \cap P(R^-)$ consists of 0 and the $\overline{\omega}_i$ for $i \in J$.

Let $D$ be the set of $x \in E$ such that $\langle \tilde{\alpha}, x \rangle < 1$ and set $C_1 = C' \cap D$. Since $0 \in \overline{C}$, we have $C \subseteq D$ and hence $C \subseteq C_1$. We are going to show that, for all $\alpha \in R$ and all $k \in \mathbf{Z}$, the sets $C$ and $C_1$ are on the same side of the hyperplane $L_{\alpha,k}$. This will prove that $C_1 \subseteq C$ and so will establish assertion (i). If $k = 0$, the whole of the chamber $C'$ is on one side of $L_{\alpha,0}$, which establishes our assertion in this case. If $k \neq 0$, we may, by replacing $\alpha$ by $-\alpha$, assume that $k > 0$. Then $\langle \alpha, x \rangle < k$ on $C$, since $0 \in \overline{C}$. On the other hand, $\tilde{\alpha} - \alpha$ is positive on $C'$ (§ 1, no. 8, Prop. 25). Thus, for $y \in C_1$, we have $\langle \alpha, y \rangle \leq \langle \tilde{\alpha}, y \rangle < 1 \leq k$. Consequently, $C$ and $C_1$ are on the same side of $L_{\alpha,k}$.

Now let $\overline{\omega} \in P(R^-)$. Then $\overline{\omega} = \sum_i p_i \overline{\omega}_i$, with $p_i \in \mathbf{Z}$ (§ 1, no. 10), and $\overline{\omega} \in \overline{C'}$ if and only if the integers $p_i$ are positive. If $\overline{\omega} \in \overline{C'}$, then $\overline{\omega} \in \overline{C}$ if and only if $\langle \tilde{\alpha}, \overline{\omega} \rangle = \sum_i n_i p_i$ is $\leq 1$, hence (ii).