COMPRESSION SEMIGROUPS OF OPEN ORBITS ON REAL FLAG MANIFOLDS

Let (G,H) be an irreducible semisimple symmetric pair, P subset of or equal to G a parabolic subgroup. Suppose that the L-orbit of the base point in the flag manifold G/P is open and write S(L, P)= {g is an ele ment of G: gL subset of or equal to LP} for the compression semigroup of this orbit. We show that if P is minimal and S(L, P)= G, then (G, H ) is Riemannian and we give a geometric characterization of those case s where S(L, P) has non-empty interior different from G. If G/H is a s ymmetric space of regular type, then we show under certain additional assumptions that S(L, Q) is an Ol'shanskii semigroup.