Normality and non-normality of group compactifications in simple projective spaces

If $G$ is a complex simply connected semisimple algebraic group and if $\lambda$ is a dominant weight, we consider the compactification $X_{\lambda }$ in the projectivisation of $\text{End}(V(\lambda ))$ obtained as the closure of the $G\times G$-orbit of the identity and we give necessary and sufficient conditions on the support of $\lambda$ so that $X_{\lambda }$ is normal; as well, we give necessary and sufficient conditions on the support of $\lambda$ so that $X_{\lambda }$ is smooth.