Transverse groups preserving proper domains in flag manifolds

Given a semisimple Lie group G and a self-opposite flag manifold F of G, we establish a necessary condition for an infinite subgroup H of G to preserve a proper domain in F. In the case where G is a Hermitian Lie group of tube type, we introduce and study a notion of causal convexity in the Shilov boundary Sb (G) of the symmetric space of G, inspired by the one already existing in conformal Lorentzian geometry. We show that subgroups H of G that are transverse with respect to a parabolic subgroup of G defining Sb (G) and that preserve a proper domain in Sb (G) satisfy a geometric property with respect to this causal convexity, close to the strong projective convex cocompactness defined by Danciger--Guéritaud--Kassel. This result highlights the spatial nature of the dynamics of H. We construct Zariski-dense examples of such transverse subgroups.