Let $G$ be a reductive group, $\Lambda$ a weight lattice, $\Lambda^{+}$ the monoid of dominating weights, $\omega_1,\dots,\omega_r\in \Lambda^{+}$ the fundamental weights and $\{\alpha_1,\dots, \alpha_r\}\subset\Lambda$ the simple roots, where $\alpha_i$ is orthogonal to $\omega_j$ is $i\neq j$. Just to keep the notational simple let us assume that $r = 3$.
Given a dominant weight $\lambda$ let $V_{\lambda}$ be the irreducible representation with highest weight $\lambda$. Consider the action of $G\times G$ on
$$\mathbb{P}(End(V_{\omega_1}))\times\mathbb{P}(End(V_{\omega_2}))\times\mathbb{P}(End(V_{\omega_3}))$$
given by $(g,g')\cdot (x,y,z) = (g^{-1}xg',g^{-1}yg',g^{-1}zg')$, and let $X$ be the closure of the orbit of $(Id_{V_{\omega_1}},Id_{V_{\omega_2}},Id_{V_{\omega_3}})\in \mathbb{P}(End(V_{\omega_1}))\times\mathbb{P}(End(V_{\omega_2}))\times\mathbb{P}(End(V_{\omega_3}))$.
As far as I understand the closure $X\subset\prod_{i=1}^{3}\mathbb{P}(End(V_{\omega_i}))$ is the wonderful compactification of $G$.
Then we may indentify $\Lambda$ with $Pic(X)$. Now, take $\overline{\lambda}\in\Lambda$ a non dominating weight corresponding to a big but not nef divisor on $X$, and consider the analogue of $f$ obtained by replacing for instance $\mathbb{P}(End(V_{\omega_2}))$ with $\mathbb{P}(End(V_{\overline{\lambda}}))$ and let $Y\subset\mathbb{P}(End(V_{\omega_1}))\times \mathbb{P}(End(V_{\overline{\lambda}}))\times \mathbb{P}(End(V_{\omega_3}))$ be the closure of the orbit of $(Id_{V_{\omega_1}},Id_{V_{\overline{\lambda}}},Id_{V_{\omega_3}})\in \mathbb{P}(End(V_{\omega_1}))\times\mathbb{P}(End(V_{\overline{\lambda}}))\times\mathbb{P}(End(V_{\omega_3}))$ with respect to the action of $G\times G$ on $$\mathbb{P}(End(V_{\omega_1}))\times\mathbb{P}(End(V_{\overline{\lambda}}))\times\mathbb{P}(End(V_{\omega_3}))$$ given by $(g,g')\cdot (x,\overline{y},z) = (g^{-1}xg',g^{-1}\overline{y}g',g^{-1}zg')$
What can we say about $Y$. For instance under which hypothesis $Y$ is a smooth variety? And what is its relation with the wonderful compactification $X$?
