dominant weight $\mu{\leqslant}{\lambda}$ there exists $n$ such that $$s^{{\lambda}-\mu}{E({\mu+(n-1){\lambda}})}^*\subset B_n(\Sigma);$$ in particular ${\varphi}={\varphi}_\mu {\varphi}_{\lambda}^{n-1}\in
B_n(\Sigma)$. Let’s prove that ${\varphi}/{\varphi}_{\lambda}\in \bar B({\Sigma})$ for every ${\varphi}\in s^{{\lambda}-\mu}{E({\mu})}^*$; this implies the thesis since $\bar A({\lambda})$ is generated in degree one. If ${\varphi}={\varphi}_\mu$ this is clear. Using the action of the Lie algebra ${\mathfrak g}\oplus{\mathfrak g}$ on $\bar
B(\Sigma)$, let’s show that if ${\varphi}/{\varphi}_{\lambda}\in \bar B(\Sigma)$ then $f_{\alpha}({\varphi})/{\varphi}_{\lambda}\in \bar B(\Sigma)$: indeed we have $$\frac{f_{\alpha}({\varphi})}{{\varphi}_{\lambda}} = f_{\alpha}(\frac{{\varphi}}{{\varphi}_{\lambda}}) +
\frac{{\varphi}}{{\varphi}_{\lambda}} \cdot \frac{f_{\alpha}({\varphi}_{\lambda})}{{\varphi}_{\lambda}}$$ and the claim follows since $f_{\alpha}({\varphi}_{\lambda})\in {E({{\lambda}})}^*\subset B_1(\Sigma)$.

We can describe the set $B_n(\Sigma)$ more explicitly. Indeed, as in [@DC] or in [@Ka], it is possible to identify sections of a line bundle on $M$ with functions on $G$ and use the description of the multiplication of matrix coefficients. Recall that as a $G\times G$-module we have ${\Bbbk}[G]=\bigoplus_{{\lambda}\in {\Lambda}^+}{E({{\lambda}})}^* {\simeq}\bigoplus_{{\lambda}\in
  {\Lambda}^+}{V({{\lambda}})}^*\otimes {V({{\lambda}})}$. More explicitly if $V$ is a representation of $G$, define $c_V:V^*\otimes V {\longrightarrow}{\Bbbk}[G]$ as usual by $c_V(\psi \otimes v)(g)= \langle \psi, gv\rangle$. If we multiply functions in ${\Bbbk}[G]$ of this type then we get $$c_{V}( \psi \otimes v) \cdot c_{W}(\chi \otimes w) =
c_{V\otimes W} \big((\psi\otimes\chi)\otimes(v\otimes w)\big):$$ in particular we get that the image of the multiplication ${E({{\lambda}})}^*
\otimes {E({\mu})}^* {\longrightarrow}{\Bbbk}[G]$ is 