Homogeneous spaces admitting transitive semigroups

Let G be a semi-simple Lie group with finite center and S ⊂ G a semigroup with intS 6= Ø . A closed subgroup L ⊂ G is said to be S -admissible if S is transitive in G/L . In [10] it was proved that a necessary condition for L to be S -admissible is that its action in B (S) is minimal and contractive where B (S) is the flag manifold associated with S , as in [9]. It is proved here, under an additional assumption, that this condition is also sufficient provided S is a compression semigroup. A subgroup with a finite number of connected components is admissible if and only if its component of the identity is admissible, and if L is a connected admissible group then L is reductive and its semi-simple component E is also admissible. Moreover, E is transitive in B (S) which turns out to be a flag manifold of E .