  If G is a complex semisimple algebraic group, we characterize the normality
and the smoothness of its simple linear compactifications, namely those
equivariant GxG-compactifications which possess a unique closed orbit and which
arise in a projective space of the shape P(End(V)), where V is 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 Sp(2r) (with r > 0) is the unique non-adjoint simple
group which admits a simple smooth compactification.
