Symplectic projective orbits of unimodular exponential Lie groups

For an exponential Lie group $G$ and an irreducible unitary representation $(\pi ,{\mathcal{H}}_{\pi })$ of $G$, we consider the natural action defined by $\pi$ on the projective space of ${\mathcal{H}}_{\pi }$, and show that the stabilisers of this action coincide with the projective kernel of $\pi$. Using this, we prove that, if $G\mathrm{/}\mathrm{p}\mathrm{k}\mathrm{e}\mathrm{r}(\pi )$ is unimodular, then $\pi$ admits a symplectic projective orbit if and only if $\pi$ is square-integrable modulo its projective kernel $\mathrm{p}\mathrm{k}\mathrm{e}\mathrm{r}(\pi )$.