divisor which corresponds to a simple root ${\alpha}$. We denote by $s_{\alpha}$ a section of ${\mathcal M}_{\alpha}$ whose associated divisor is $M_{\alpha}$; notice that such a section is $G\times G$-invariant. More generally if $\nu = \sum_{{\alpha}\in {\Delta}} n_{{\alpha}}{\alpha}\in {\mathbb N}{\Delta}$, set $s^\nu = \prod_{{\alpha}\in {\Delta}} s_{{\alpha}} ^{n_{{\alpha}}} \in
{\Gamma}(M,{\mathcal M}_\nu)$. Then, given any ${\lambda}\in {\Lambda}$, the multiplication by $s^\nu$ injects ${\Gamma}(M,{\mathcal M}_{{\lambda}- \nu})$ in ${\Gamma}(M,{\mathcal M}_{\lambda})$.

If ${\lambda}$ is a dominant weight, the map $G_\mathrm{ad}{\longrightarrow}{\mathbb P}({E({{\lambda}})})$ extends to a map $q_{\lambda}:M{\longrightarrow}{\mathbb P}({E({{\lambda}})})$ (see [@CP]) whose image is $X_{\lambda}$ and such that ${\mathcal M}_{\lambda}=q_{\lambda}^*({\mathcal O}_{{\mathbb P}({E({{\lambda}})})}(1))$. If we pull back the homogeneous coordinates of ${\mathbb P}({E({{\lambda}})})$ to $M$, we get then a submodule of ${\Gamma}(M,{\mathcal M}_{\lambda})$ which is isomorphic to ${E({{\lambda}})}^*$; by abuse of notation we will denote this submodule by ${E({{\lambda}})}^*$.

If ${\lambda}\in {\Lambda}$, in [@CP Theorem 8.3] the following decomposition of ${\Gamma}(M,{\mathcal M}_{\lambda})$ is given: $${\Gamma}(M,{\mathcal M}_{\lambda}) = \bigoplus_{\mu\in {\Lambda}^+ {\, : \,}\mu {\leqslant}{\lambda}} s^{{\lambda}-\mu}{E({\mu})}^*.$$

Consider the graded algebra $A({\lambda})=\bigoplus_{n=0}^\infty A_n({\lambda})$, where $A_n({\lambda})= {\Gamma}(M,{\mathcal M}_{n{\lambda}})$, and set $\widetilde{X}_{\lambda}=
\operatorname{Proj}A({\lambda})$. We have then a commutative diagram as follows: $$\xymatrix{M \ar@{->>}^{p_{\lambda}}[r] \ar@{->>}_{q_{\lambda}}[dr]
& \widetilde{X}_{\lambda}\ar@{->>}^{r_{\lambda}}[d]\\  & X_{\lambda}}$$

In [@Ka], it has been shown that $A({\lambda})$ is generated in degree $1$ and in 