Proposition 3

Let V be a finite dimensional $\mathfrak{g}$-module and $\mathscr{X}$ its set of weights. Let $\lambda \in \mathscr{X},\alpha \in R$, I the set of $t\in \mathbf{Z}$ such that $\lambda +t\alpha \in \mathscr{X},p$ (resp. $-q$) the largest (resp. smallest) element of I. Let $m_t$ be the multiplicity of $\lambda +t\alpha$.

(i) I = $-q, p$ and $q-p=\lambda (H_{\alpha})$.

(ii) For any integer $u\in 0, p+q,\lambda + (p-u)\alpha$ and $\lambda + (-q+u)\alpha$ are conjugate under $s_{\alpha}$, and $m_{-q+u}=m_{p-u}$.

(iii) If $t\in \mathbf{Z}$ and $t <(p-q)/2$, $(X_{\alpha})_V$ maps $V^{\lambda+t\alpha}$ injectively into $V^{\lambda+(t+1)\alpha}$.

(iv) The function $t \rightarrow m_t$ is increasing on $-q,(p-q)/2$ and decreasing on $(p-q)/2, p$.

Let $\alpha \in B$. Give V the $\mathfrak{s}\mathfrak{l}(2, k$)-module structure defined by the elements $X_{\alpha}, X_{-\alpha}, H_{\alpha}$ of $\mathfrak{g}$. Every non-zero element of $V^{\lambda+p\alpha}$ is then primitive. Consequently, $(\lambda +p\alpha )(H_{\alpha})\geq 0$ and $(X_{-\alpha})^rV^{\lambda+p\alpha}\not= 0$ for

(§1, no. 2, Prop. 2). It follows that $V^{\lambda+t\alpha}\not= 0$ for $p\geq t\geq p-(\lambda (H_{\alpha}) + 2p)$, so $p+q\geq \lambda (H_{\alpha}) + 2p$. Applying this result to $-\alpha$ gives

Hence $q-p=\lambda (H_{\alpha})$ and $\lambda +t\alpha \in \mathscr{X}$ for $p\geq t\geq -q$, which proves (i).

We have $s_{\alpha}(\alpha ) =-\alpha$, and $s_{\alpha}(\mu)\in \mu+k\alpha$ for all $\mu\in \mathfrak{h}^*$. Since W leaves $\mathscr{X}$ stable (Cor. 2 of Prop. $2$)$,s_{\alpha}$ leaves $\{\lambda -q\alpha , \lambda -q\alpha +\alpha , . . . , \lambda +p\alpha \}$ stable and takes $\lambda -q\alpha +u\alpha$ to $\lambda +p\alpha -u\alpha$ for all $u\in k$. Using Cor. 2 of Prop. 2 again, we see that $m_{-q+u}=m_{p-u}$ for every integer $u\in 0, p+q$. This proves (ii).

By §1, Cor. of Prop. $2, (X_{\alpha})_V|V^{\lambda+t\alpha}$ is injective for $t <(p-q)/2$. Now $(X_{\alpha})_V$ maps $V^{\lambda+t\alpha}$ to $V^{\lambda+(t+1)\alpha}$. Hence $m_{t+1}\geq m_t$ for $t <(p-q)/2$. Changing $\alpha$ to $-\alpha$, we see that $m_{t+1}\leq m_t$ for $t >(p-q)/2$. This proves (iii) and (iv).