fix a maximal torus $H \subset B$, and take the opposite Borel subgroup $B^{-}$ of $G$ that contains $H$. In the following, for an (arbitrary) algebraic group $E$ over ${\mathbb{C}}$, we denote its Lie algebra ${\mathop{\rm Lie}\nolimits}(E)$ by the corresponding German letter ${\mathfrak{e}}$; in particular, we write ${\mathfrak{g}}={\mathop{\rm Lie}\nolimits}(G)$, ${\mathfrak{b}}={\mathop{\rm Lie}\nolimits}(B)$, ${\mathfrak{n}}={\mathop{\rm Lie}\nolimits}(N)$, and ${\mathfrak{h}}={\mathop{\rm Lie}\nolimits}(H)$. Thus, ${\mathfrak{g}}$ is a finite-dimensional simple Lie algebra over ${\mathbb{C}}$ with Cartan subalgebra ${\mathfrak{h}}$. Denote by $\{ \alpha_{i}^{\vee} \}_{i \in I}$ and $\{ \alpha_{i} \}_{i \in I}$ the set of simple coroots and simple roots of ${\mathfrak{g}}$, respectively, and set $Q := \bigoplus_{i \in I} {\mathbb{Z}}\alpha_i$, $Q^{+} := \sum_{i \in I} {\mathbb{Z}}_{\ge 0} \alpha_i$, and $Q^{\vee} := \bigoplus_{i \in I} {\mathbb{Z}}\alpha_i^{\vee}$, $Q^{\vee,+} := \sum_{i \in I} {\mathbb{Z}}_{\ge 0} \alpha_i^{\vee}$; for $\xi,\,\zeta \in Q^{\vee}$, we write $\xi \ge \zeta$ if $\xi-\zeta \in Q^{\vee,+}$. Let $\Delta$ and $\Delta^{+}$ be the set of roots and positive roots of ${\mathfrak{g}}$, respectively, with $\theta \in \Delta^{+}$ the highest root of ${\mathfrak{g}}$. For a root $\alpha \in \Delta$, we denote by $\alpha^{\vee}$ its dual root. We set $\rho:=(1/2) \sum_{\alpha \in \Delta^{+}} \alpha$ and $\rho^{\vee}:= (1/2) \sum_{\alpha \in \Delta^{+}} \alpha^{\vee}$. Also, let ${\varpi}_{i}$, $i \in I$, denote the fundamental weights for 