${\alpha}\in{\Delta}$, denote by $\omega_{\alpha}$ the corresponding fundamental weight.

If ${\lambda}\in {\Lambda}$, recall the definition of its *support*: $$\operatorname{Supp}({\lambda}) = \{{\alpha}\in {\Delta}{\, : \,}\langle {\lambda}, {\alpha}^\vee \rangle \neq 0 \}.$$

If $I \subset {\Delta}$, define its *border* $\partial{I}$, its *interior* $I^\circ$ and its *closure* ${\overline}{I}$ as follows: $$\partial{I} = 
\{ {\alpha}\in {\Delta}{\smallsetminus}I {\, : \,}{\exists\,}{\beta}\in I \mathrm{ \; such \,  that \; } 
\langle {\beta},{\alpha}^\vee \rangle \neq 0\};$$ $$I^\circ = I {\smallsetminus}\partial{({\Delta}{\smallsetminus}I)};$$ $$\overline{I} = I \cup \partial{I}.$$

For ${\lambda}\in \Lambda$, denote by ${\mathcal L}_{\lambda}$ the line bundle on $G/B$ whose $T$-weight in the point fixed by $B$ is $-{\lambda}$. For ${\lambda}$ dominant, ${V({{\lambda}})} = {\Gamma}(G/B,{\mathcal L}_{\lambda})^*$ is an irreducible $G$-module of highest weight ${\lambda}$; when we deal with different groups we will use the notation ${V_G({{\lambda}})}$.

Denote by ${\Pi({{\lambda}})}$ the set of weights occurring in ${V({{\lambda}})}$ and set ${\Pi^+({{\lambda}})}={\Pi({{\lambda}})}\cap {\Lambda}^+$. Let ${\lambda}\mapsto {\lambda}^*$ be the linear involution of ${\Lambda}$ defined by $({V({{\lambda}})})^*{\simeq}{V({{\lambda}^*})}$, for any dominant weight ${\lambda}$.

The weight lattice ${\Lambda}$ is endowed with the dominance order ${\leqslant}$ defined as follows: $\mu {\leqslant}{\lambda}$ if and only if ${\lambda}- \mu \in
{\mathbb N}{\Delta}$. If ${\beta}= \sum_{{\alpha}\in {\Delta}} n_{\alpha}{\alpha}\in {\mathbb Z}{\Delta}$, define its *support over* ${\Delta}$ (not to be confused with the previous 