the normalization of $X_{\lambda}$ {#ssez:meravigliosa}
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When ${\lambda}$ is a regular weight (i.e. $\operatorname{Supp}({\lambda})={\Delta}$) the variety $X_{\lambda}$ is called the wonderful compactification of $G_\mathrm{ad}$ and it has been studied by C. De Concini and C. Procesi in [@CP]. We will denote this variety by $M$: it is smooth and the complement of its open orbit is the union of smooth prime divisors with normal crossings whose intersection is the closed orbit. The closed orbit of $M$ is isomorphic to $G/B \times G/B$ and the restriction of line bundles determines an embedding of $\operatorname{Pic}(M)$ into $\operatorname{Pic}(G/B\times G/B)$, that we identify with ${\Lambda}\times {\Lambda}$ as before; the image of this map is the set of weights of the form $({\lambda},{\lambda}^*)$. Therefore $\operatorname{Pic}(M)$ is identified with ${\Lambda}$ and we denote by ${\mathcal M}_{{\lambda}}$ a line bundle on $M$ whose restriction to $G/B \times G/B$ is isomorphic to ${\mathcal L}_{\lambda}\boxtimes {\mathcal L}_{{\lambda}^*}$. If $D\subset M$ is a $G\times G$-stable prime divisor then the line bundle defined by $D$ is of the form ${\mathcal M}_{{\alpha}_D}$, where ${\alpha}_D$ is a simple root. The map $D\mapsto {\alpha}_D$ defines a bijection between the set of $G\times G$-stable prime divisors and ${\Delta}$, and we denote by $M_{\alpha}$ the prime 