$X_\Sigma$ is a compactification of $G_{\mathrm{ad}}$, while if $G_{\mathrm{ad}}$ is not simple we can only say that is a compactification of a group which is a quotient of $G_{\mathrm{ad}}$.

We say that $\Sigma$ is *simple* if there exists ${\lambda}\in\Sigma$ such that $\Sigma \subset {\Pi^+({{\lambda}})}$ or equivalently if ${\Sigma}$ contains a unique maximal element with respect to the dominance order ${\leqslant}$. Notice also that if ${\lambda}\in \Sigma$ is such that for all $\mu \in \Sigma$ different from ${\lambda}$ the vector $\mu - {\lambda}$ is not in ${\mathbb Q}_{{\geqslant}0}[{\Delta}]$ then is easy to construct a cocharacter $\chi
: {\Bbbk}^*{\longrightarrow}G\times G$ such that $\lim_{t\to 0}\chi(t)x_\Sigma $ is the highest weight line in ${\mathbb P}(E({\lambda}))$. In particular $X_\Sigma$ is a simple $G\times G$ semi-compactification of $G_{\mathrm{ad}}$ if and only if $\Sigma$ is simple.

By the description of the normalization of $X_{\lambda}$ is $\Sigma$ is simple and ${\lambda}\in {\Sigma}$ is the maximal element, then we get $$\xymatrix{\widetilde X_{\lambda}\ar[r]^{r} & X_\Sigma \ar[r] & X_{\lambda}}$$ In particular, it follows that $r=r_\Sigma:\widetilde X_{\lambda}{\longrightarrow}X_\Sigma$ is the normalization of $X_\Sigma$.

If ${\Sigma}$ is simple, denote $B(\Sigma)=\bigoplus_{n=0}^\infty
B_n(\Sigma)$ the projective coordinate ring of $X_\Sigma\subset
{\mathbb P}({E({\Sigma})})$: it is the subalgebra of $A({\lambda})$ generated by ${E({\Sigma})}^*\subset {\Gamma}(M,{\mathcal M}_{\lambda})$.

The discussion above and 