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 x G-compactifications possessing a unique closed orbit which arise in a projective space of the shape P(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 Sp(2r) (with r >= 1) is the unique non-adjoint simple group which admits a simple smooth compactification