Normality and smoothness of simple linear group compactifications

Given a semisimple algebraic group $G$ , we characterize the normality and the smoothness of its simple linear compactifications, namely those equivariant $G\times G$ -compactifications possessing a unique closed orbit which arise in a projective space of the shape $\mathbb{P }(\mathrm{End}(V))$ , where $V$ is a finite dimensional rational $G$ -module. Both the characterizations are purely combinatorial and are expressed in terms of the highest weights of $V$ . In particular, we show that ${\mathrm{Sp}}(2r)$ (with $r \geqslant 1$ ) is the unique non-adjoint simple group which admits a simple smooth compactification.