On orbit closures of spherical subgroups in flag varieties

Let F be the flag variety of a complex semi-simple group G,let H be an alge braic subgroup of G acting on F with finitely many orbits, and let V be an H -orbit closure in F. Expanding the cohomology class of V in the basis of Schubert classes defines a union V-0 of Schubert varieties in F with positi ve multiplicities. If G is simply-laced, we show that these multiplicities are equal to the same power of 2. For arbitrary G,we show that V-0 is conne cted in codimension 1. Ii moreover all multiplicities are 1, we show that t he singularities of V are rational and we construct a Aat degeneration of V to V-0 in F. Thus, for any effective line bundle L on F, the restriction m ap H-0(F, L) --> H-0(V,L) is surjective, and H-n(V, L) = 0 for all n greate r than or equal to 1.