[@Ti], D. Timashev studied the more general situation of a sum of irreducible representations, giving necessary and sufficient conditions for the normality and smoothness of these compactifications; however the conditions for normality are not completely explicit. In this paper we give an explicit characterization of the normality of $X_{\lambda}$, which allows to simplify the conditions for the smoothness as well.

To explain our results we need some notation. Let ${\Delta}$ be the set of simple roots (w.r.t. $T\subset B$) and identify ${\Delta}$ with the vertices of the Dynkin diagram. Define the support of ${\lambda}$ as the set $\operatorname{Supp}({\lambda})=\{{\alpha}\in {\Delta}{\, : \,}\langle {\lambda}, {\alpha}^\vee \rangle \neq 0
\}$.

\[see Theorem \[teo:normalita\]\] The variety $X_{\lambda}$ is normal if and only if ${\lambda}$ satisfies the following property:

-   For every non-simply laced connected component $\Delta'$ of $\Delta$, if $\operatorname{Supp}({\lambda})\cap {\Delta}'$ contains a long root, then it contains also the short root which is adjacent to a long simple root.

In particular, if the Dynkin diagram of $G$ is simply laced then $X_{\lambda}$ is normal, for all ${\lambda}$. In the paper we will prove the theorem in a more general form, for simple (i.e. with a unique closed orbit) linear projective compactifications of an 