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 $\End(V(\lambda))$ 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.