Proposition 5

Let V be a finite dimensional simple $\mathfrak{g}$-module, $\omega$ its highest weight, $\mathscr{X}$ its set of weights. Choose a W-invariant non-degenerate positive symmetric bilinear form $(\cdot |\cdot )$ on $\mathfrak{h}^*_{\mathbf{R}}$, and let $\lambda \rightarrow \|\lambda \|= (\lambda |\lambda )^{1/2}$be the corresponding norm.

(i) $\mathscr{X}$ is the smallest R-saturated subset of P containing $\omega$.

(ii) The R-extremal elements of $\mathscr{X}$ are the W-transforms of $\omega$.

(iii) If $\mu\in \mathscr{X}$, we have $\|\mu\| \leq \|\omega \|$. If, in addition, $\mu\not=\omega$, we have $\|\mu+\rho \|<\|\omega +\rho \|$. If $\mu$ is not R-extremal in $\mathscr{X}$, then $\|\mu\|<\|\omega \|$.

(iv) We have $\mathscr{X}= W.(\mathscr{X}\cap P_{++})$. An element $\lambda$ of $P_{++}$ belongs to $\mathscr{X}\cap P_{++}$ if and only if $\omega -\lambda \in Q_+$.

(i) Let $\mathscr{X}'$ be the smallest R-saturated subset of P containing $\omega$. We have $\mathscr{X}'\subset \mathscr{X}$ (Prop. 4). Assume that $\mathscr{X}\not=\mathscr{X}'$. Let $\lambda$ be a maximal element of $\mathscr{X}$ -- $\mathscr{X}'$. Since $\lambda \not=\omega$, there exists $\alpha \in B$ such that $\lambda +\alpha \in \mathscr{X}$. Introduce $p$ and $q$ as in Prop. 3. Since $\lambda$ is maximal in $\mathscr{X}$ -- $\mathscr{X}',\lambda +p\alpha \in \mathscr{X}'$. By Prop. 3 (ii), $\lambda -q\alpha \in \mathscr{X}'$ since $\mathscr{X}'$ is stable under W. Hence $\lambda +u\alpha \in \mathscr{X}'$ for every integer $u$ in the interval $-q, p$. This contradicts $\lambda \notin \mathscr{X}'$ and proves (i).

(ii) It is clear that $\omega$ is an R-extremal element of $\mathscr{X}$; its W-transforms are therefore also R-extremal in $\mathscr{X}$. Let $\lambda$ be an R-extremal element of $\mathscr{X}$; we shall prove that $\lambda \in W.\omega$. Since there exists $w\in W$ such that $w\lambda \in P_{++}$ (Chap. VI, §1, no. 10), we can assume that $\lambda \in P_{++}$. Let $\alpha \in B$; introduce $p$ and $q$ as in Prop. 3. Since $\lambda$ is R-extremal, either $p= 0$ or $q= 0$. Since

we cannot have $p >0$. Hence $p= 0$ and $\lambda =\omega$.

(iii) Let $\mu\in \mathscr{X}\cap P_{++}$. Then $\omega +\mu\in P_{++}$ and $\omega -\mu\in Q_+($§6, no. 1, Prop. 1), so $0\leq (\omega -\mu|\omega +\mu) = (\omega |\omega )-(\mu|\mu)$; hence, $(\mu|\mu)\leq (\omega |\omega )$, and this extends to all $\mu\in \mathscr{X}$ by using the Weyl group. If $\mu\in \mathscr{X}$ -- $\{\omega \}$,

Now $\omega -\mu=\sum_{\alpha\in B}n_{\alpha}\alpha$ with integers $n_{\alpha}\geq 0$ not all zero, so $(\omega -\mu|\rho )>0$ since $(\rho |\alpha )>0$ for all $\alpha \in B$ (Chap. VI, §1, no. 10, Prop. 29 (iii)). If $\mu$ is not R-extremal in $\mathscr{X}$, there exists $\alpha \in R$ such that $\mu+\alpha \in \mathscr{X}$ and $\mu-\alpha \in \mathscr{X}$; then

$\|\mu\|<$ sup($\|\mu+\alpha \|,\|\mu-\alpha \|$)$\leq$ sup$_{\lambda\in\mathscr{X}}\|\lambda \|$

and this last upper bound is $\|\omega \|$ by the preceding.

(iv) We have $\mathscr{X}= W.(\mathscr{X}\cap P_{++})$ by Chap. VI, §1, no. 10. If $\lambda \in \mathscr{X}$, then $\omega -\lambda \in Q_+($§6, no. 1, Prop. 1). If $\lambda \in P_{++}$ and $\omega -\lambda \in Q_+$, then $\lambda \in \mathscr{X}$ (Cor. 2 of Prop. 3).