Mid-reversibility properties of semigroup actions on homogeneous spaces

In this paper we study mid-reversibility of subsemigroups acting on homogeneous spaces. The mid-reversor set of a subsemigroup is defined and it is described in terms of the invariant control sets for semigroups acting on certain homogeneous spaces. Let G be a connected noncompact semi-simple Lie group and L a subgroup of G. Assume that S is a subsemigroup of G with nonempty interior. We characterize the mid-reversibility of the S-action on G/L in terms of the actions of S and L on the flag manifolds of G. We show that the mid-reversibility of S in G/L is related to the reversibility of S in G/L. We also present sufficient conditions for S to generate G if S is mid-reversible in G/L.