  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.
