On symplectic projective orbits of unimodular exponential Lie groups

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