Semigroups in 3-graded Lie groups and endomorphisms of standard subspaces

Let V be a standard subspace in the complex Hilbert space H and U : G \to U(H) unitary representation of finite dimensional Lie group. We assume existence an element h algebra such that U(exp th) is modular group involution J_V normalizes U(G). want to determine semigroup $S_V = \{ g\in U(g)V \subseteq V\}.$ In previous work we have seen its infinitesimal generators span on which ad defines 3-grading, here completely S_V under assumption 3-grading. Concretely, show h-eigenspaces for eigenvalue $\pm 1$ contain closed convex cones $C_\pm$, exp(C_+) G_V exp(C_-)$, where $G_V$ stabilizer G. To obtain this result compare several subsemigroups specified by grading positive cone $C_U$ U. particular, orbit U(G)V, endowed with inclusion order, ordered symmetric covering adjoint $Ad(G)h$, partial order defined by~$C_U$.