Symplectic projective orbits of unimodular exponential Lie groups

Abstract

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