Spherical orbit closures in simple projective spaces and their normalizations

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 = G/H is its closure, then we describe the orbits of X and those of its normalization Y. If, moreover, the wonderful completion of G/H is strict, then we give necessary and sufficient combinatorial conditions so that the normalization morphism Y --> X is a homeomorphism. Such conditions are trivially fulfilled if G is simply laced or if H is a symmetric subgroup.