Symplectic projective orbits of unimodular exponential Lie groups

For an exponential Lie group G and an irreducible unitary representation (π,Hπ) of G, we consider the natural action defined by π on the projective space of Hπ, and show that the stabilisers of this action coincide with the projective kernel of π. Using this, we prove that, if G/pker(π) is unimodular, then π admits a symplectic projective orbit if and only if π is square-integrable modulo its projective kernel pker(π).