one) as follows: $$\operatorname{Supp}_{\Delta}({\beta}) = \{ {\alpha}\in {\Delta}{\, : \,}n_{\alpha}\neq 0 \}.$$

We introduce also some notations about the multiplication of sections. Notice that, for all ${\lambda},\mu \in\Lambda$, ${\mathcal L}_{\lambda}\otimes {\mathcal L}_\mu ={\mathcal L}_{{\lambda}+\mu}$. Therefore, if ${\lambda},\mu$ are dominant weights and $n\in {\mathbb N}$, the multiplication of sections defines maps as follows: $$m_{{\lambda},\mu}:{V({{\lambda}})}\times {V({\mu})} {\rightarrow}{V({{\lambda}+\mu})} \; \text{ and }
m_{\lambda}^n : {V({{\lambda}})} {\rightarrow}{V({n{\lambda}})}.$$ We will also write $uv$ for $m_{{\lambda},\mu}(u,v)$ and $u^n$ for $m^n_{\lambda}(u)$. Since $G/B$ is irreducible, $m_{{\lambda},\mu}$ and $m^n_{\lambda}$ induce the following maps at the level of projective spaces: $$\psi_{{\lambda},\mu}: {\mathbb P}({V({{\lambda}})}) \times {\mathbb P}({V({\mu})}) {\rightarrow}{\mathbb P}({V({{\lambda}+ \mu})})
\;{\text{ and }}\; \psi^n_{\lambda}: {\mathbb P}({V({{\lambda}})}) {\rightarrow}{\mathbb P}({V({n{\lambda}})}).$$

The following lemma is certainly well known; however we do not know any reference.

\[lem:immersioni\] Let ${\lambda},\mu$ be dominant weights.

1.  If $\operatorname{Supp}({\lambda}) \cap \operatorname{Supp}(\mu) = {\varnothing}$, then the map $\psi_{{\lambda},\mu}\colon {\mathbb P}({V({{\lambda}})}) \times {\mathbb P}({V({\mu})}) \to {\mathbb P}({V({{\lambda}+
      \mu})})$ is a closed embedding.

2.  For any $n>0$, the map $\psi_{\lambda}^n: {\mathbb P}({V({{\lambda}})})  {\rightarrow}{\mathbb P}({V({n{\lambda}})})$ is a closed embedding.

$i)$. Fix highest weight vectors $v_{\lambda}\in {V({{\lambda}})}$, $v_\mu \in {V({\mu})}$ and $v_{{\lambda}+\mu}= v_{\lambda}v_\mu \in {V({{\lambda}+\mu})}$.

If $V$ is irreducible, then ${\mathbb P}(V)$ has a unique closed orbit, namely the orbit of the highest 