Proposition 8

Assume that $\mathfrak{g}$ is simple. Denote by $\alpha_1, . . . , \alpha_l$ the elements of B. Let $\varpi_1, . . . , \varpi_l$ be the corresponding fundamental weights. Let $H$ = $n_1H_{\alpha_1}+\cdots +n_lH_{\alpha_l}$ be the highest root of $R^{\vee}$, and J the set of $i\in \{1, . . . , l\}$ such that $n_i= 1$. Let $\lambda \in P_{++}$ -- $\{0\}$. Then conditions (i), (ii) and (iii) of Prop. 6 are equivalent to each of the following conditions:

(v) $\lambda (H) = 1$;

(vi) there exists $i\in J$ such that $\lambda =\varpi_i$.

The $\varpi_i$, for $i\in J$, form a system of representatives in P(R) of the non-zero elements of $P(R)/Q(R)$.

Let $\lambda =u_1\varpi_1+\cdots +u_l\varpi_l$, where $u_1, . . . , u_l$ are integers $\geq 0$ and not all zero. Then $\lambda (H) =u_1n_1+\cdots +u_ln_l$ and $n_1\geq 1, . . . , n_l\geq 1$, which gives the equivalence of (v) and (vi) immediately. On the other hand, $\lambda (H) =$ sup$_{\alpha\in R_+}\lambda (H_{\alpha})$, and $\lambda (H)>0$ since $\lambda$ is a non-zero element of $P_{++}$. Hence

condition (v) is equivalent to the condition $\lambda (H_{\alpha})\in \{0,1\}$ for all $\alpha \in R$, in other words to condition (iii) of Prop. 6.

The last assertion of the proposition follows from Chap. VI, §2, no. 3, Cor. of Prop. 6.