by D. Timashev in [@Ti] in the case of a linear projective compactification of a reductive group.

As a corollary of Theorem B, we get that $X_{\lambda}$ is smooth if and only if its normalization is smooth.

The paper is organized as follows. In the first section we introduce the wonderful compactification of $G_\mathrm{ad}$ and the normalization of the variety $X_{\lambda}$. In the second section we prove Theorem A, and in the third section Theorem B. In the last section we discuss some possible generalizations of our results.

Preliminaries {#sez:preliminari}
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Notation {#ssez:notazioni}
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Recall that $G$ is semisimple and simply connected. Fix a Borel subgroup $B\subset G$, a maximal torus $T\subset B$ and let $U$ denote the unipotent radical of $B$. Lie algebras of groups denoted by upper-case latin letters ($G,U,L,\ldots$) will be denoted by the corresponding lower-case german letter (${\mathfrak g}, \mathfrak u, \mathfrak l,\ldots$). Let $\Phi$ denote the set of roots of $G$ relatively to $T$ and ${\Delta}\subset \Phi$ the basis associated to the choice of $B$. For all ${\alpha}\in {\Delta}$ let $e_{\alpha}, {\alpha}^{{\vee}},f_{\alpha}$ be an $\mathfrak{sl}(2)$-triple of $T$-weights ${\alpha},0,-{\alpha}$. Let ${\Lambda}$ denote the weight lattice of $T$ and ${\Lambda}^+$ the subset of dominant weights. For all 