Simple linear compactifications of spherical homogeneous spaces

Let G be a simply connected semisimple algebraic group over an algebraically closed field k of characteristic 0 and let V be a rational simple G-module. If G/H is a spherical orbit in P(V) and if X is its closure, then we describe the orbits of X and those of its normalization X. If moreover the wonderful completion of G/H is strict, then we give necessary and sufficient combinatorial conditions so that the normalization morphism X → X is a homeomorphism. Such conditions are trivially fulfilled if G is simply laced or if H is a symmetric subgroup. In the special case of an odd orthogonal group G regarded as a GxG variety, we give an explicit classification of its simple linear compactifications, namely those equivariant compactifications with a unique closed orbit which are obtained by taking the closure of the GxG-orbit of the identity in a projective space P(End(V)), where V is a finite dimensional rational G-module.