Lemma 1

Let $\alpha \in R$ and $t\in k^*$. Let $\varphi$ be the homomorphism $\lambda \rightarrow t^{\lambda(H_{\alpha})}$ from Q(R) to $k^*$. Then $f(\varphi ) =\theta_{\alpha}(t)\theta_{\alpha}(-1)$.

Let $\rho$ be the representation of $\mathfrak{s}\mathfrak{l}(2, k)$ on $\mathfrak{g}$ associated to $X_{\alpha}$. Let $\pi$ be the representation of $\mathbf{S}\mathbf{L}(2, k)$ compatible with $\rho$. Introduce the notations $\theta (t), h(t)$ of §1, no. 5. Since $\rho (H) =$ ad $H_{\alpha}$, the elements of $\mathfrak{g}^{\lambda}$ are of weight $\lambda (H_{\alpha})$ for $\rho$. By §2, no. $2,\theta_{\alpha}(t)\theta_{\alpha}(-1) =\pi (\theta (t)\theta (-1)) =\pi (h(t))$. Hence the restriction of $\theta_{\alpha}(t)\theta_{\alpha}(-1)$ to $\mathfrak{g}^{\lambda}$ is the homothety of ratio $t^{\lambda(H_{\alpha})}($§1, no. 5, Prop. 6), hence the lemma.