= n v_{\lambda}^{n-1} v.$$ Thus $d (\psi^n_{\lambda})_{x}$ is injective and $\psi_{\lambda}^n$ is smooth.

The variety $X_{\lambda}$ {#ssez:Xlambda}
-------------------------

If ${\lambda}$ is a dominant weight, denote by ${E({{\lambda}})}$ the $G\times
G$-module $\operatorname{End}({V({{\lambda}})})$ and set $X_{\lambda}$ the closure of the $G\times G$-orbit of $[I_{\lambda}] \in {\mathbb P}({E({{\lambda}})})$. More generally if ${\lambda}_1,\dots,{\lambda}_m$ are dominant weights we define $$X_{{\lambda}_1, \ldots, {\lambda}_m}
= {\overline}{G\times G([I_{{\lambda}_1}], \ldots, [I_{{\lambda}_m}])} \subset
{\mathbb P}({E({{\lambda}_1})}) \times \cdots \times {\mathbb P}({E({{\lambda}_m})}).$$ Since ${E({{\lambda}})}$ is an irreducible $G\times G$-module of highest weight $({\lambda},{\lambda}^*)$, as a consequence of Lemma \[lem:immersioni\] we get that if ${\lambda}$ and $\mu$ have non-intersecting supports and if $n\in {\mathbb N}$ then $$X_{{\lambda}+\mu}\simeq X_{{\lambda},\mu} \qquad \mathrm{ and } \qquad X_{n{\lambda}}{\simeq}X_{\lambda}.$$ As a consequence we get the following proposition:

\[prp:supporto\]Let ${\lambda},\mu$ be dominant weights. Then $X_{\lambda}\simeq  X_\mu$ as $G\times G$-varieties if and only if ${\lambda}$ and $\mu$ have the same support. Moreover, if $\operatorname{Supp}({\lambda})=\{{\alpha}_1,\dots,{\alpha}_m\}$ then $$X_{\lambda}{\simeq}X_{\omega_{{\alpha}_1},\dots,\omega_{{\alpha}_m}}.$$

By the discussion above we have to prove only that the condition is necessary. This follows by noticing that if $X_{\lambda}$ and $X_\mu$ are $G\times G$-isomorphic then also their closed $G\times G$-orbits are isomorphic, which is equivalent to the fact that ${\lambda}$ and $\mu$ have the same support.

The wonderful compactification of $G_{\mathrm{ad}}$ and 