Orbits in Degenerate Compactifications of Symmetric Varieties

Let G be a simply connected semisimple algebraic group and let H^0 be the subgroup of points fixed under an involution of G. If V is an irreducible representation with a line L of vectors fixed by H^0 we consider the closure of the G-orbit of L in P(V). We describe the G-orbits of this closure and we prove that the normalization of this variety is homeomorphic to the variety itself