On a smooth compactification of PSL(n, C)/T

Let $T$ be a maximal torus of $\mathrm{P}\mathrm{S}\mathrm{L}(n,\mathbb{C})$. For $n\ge 4$, we construct a smooth compactification of $\mathrm{P}\mathrm{S}\mathrm{L}(n,\mathbb{C})\mathrm{/}T$ as a geometric invariant theoretic quotient of the wonderful compactification $\overline{\mathrm{P}\mathrm{S}\mathrm{L}(n,\mathbb{C})}$ for a suitable choice of $T$–linearized ample line bundle on $\overline{\mathrm{P}\mathrm{S}\mathrm{L}(n,\mathbb{C})}$. We also prove that the connected component, containing the identity element, of the automorphism group of this compactification of $\mathrm{P}\mathrm{S}\mathrm{L}(n,\mathbb{C})\mathrm{/}T$ is ${\rm PSL}(n, \mathbb C)$ itself.