$p\,(v_{\lambda}\otimes v_\mu) \in {V({{\lambda}})} \otimes {V({\mu})}$ is a maximal vector of weight $\nu$. Since $\operatorname{Supp}_{\Delta}({\lambda}+ \mu - \nu) \subset {\Delta}'$, we may assume that the only $f_{\alpha}$’s appearing in $p$ are those with ${\alpha}\in {\Delta}'$; therefore $p\,(v_{\lambda}\otimes v_\mu) \in {V_L({{\lambda}})}
\otimes {V_L({\mu})}$ and it generates a simple $L$-module of highest weight $\nu$.

\[lem:traslazione\] Fix ${\lambda}, \mu, \nu \in {\Lambda}^+$ such that ${V({\nu})} \subset {V({{\lambda}})}
\otimes {V({\mu})}$. Then, for any $\nu' \in {\Lambda}^+$, it also holds $${V({\nu + \nu'})} \subset {V({{\lambda}+ \nu'})} \otimes {V({\mu})}.$$

Fix a maximal vector $v_{\nu'} \in {V({\nu'})}$ and consider the $U$-equivariant map $$\begin{array}{cccc}
\phi: & {V({{\lambda}})} \otimes {V({\mu})} & {\longrightarrow}& {V({{\lambda}+ \nu'})}\otimes {V({\mu})}\\
& w_1 \otimes w_2 & \longmapsto & m_{{\lambda},\nu'}(w_1,v_{\nu'}) \otimes w_2
\end{array}$$ The claim follows since, if $v_\nu \in {V({{\lambda}})}\otimes {V({\mu})}$ is a $U$-invariant vector of weight $\nu$, then $\phi(v_\nu) \in {V({{\lambda}+
\nu'})} \otimes {V({\mu})}$ is a $U$-invariant vector of weight $\nu +
\nu'$.

We now describe some more explicit results. When we deal with explicit irreducible root systems, unless otherwise stated, we always use the numbering of simple roots and fundamental weights of Bourbaki [@Bo].

In order to describe the simple subsets ${\Sigma}\subset {\Lambda}^+$ which give rise to a non-normal variety $X_{\Sigma}$, we will 