# Do cyclic product vectors generatating irreducible representation of a Lie group come from a unique orbit?

Consider a Hilbert space $\mathcal{H}$ which is a carrier space of a unitary, irreducible and strongly continuous representation $\Pi$ of a Lie group $G$. Let $\Pi\otimes \Pi$ denote the corresponding representation of $G$ in $\mathcal{H}\otimes \mathcal{H}$.

Assume that $\phi\in\mathcal{H}$ is such that $\phi\otimes \phi$ is a cyclic vector of some irreducible representation $\tilde{\Pi}$ (with a carrier space $\tilde{\mathcal{H}}\subset \mathcal{H}\otimes\mathcal{H}$) of the group $G$. Let $\psi\in\mathcal{H}$ be another vector with this property.

Question: Are vectors $\phi$ and $\psi$ related by the action of $G$? More precisely, do we have

$\Pi(g)\mathbb{P}_\phi\Pi(g)^\dagger=\mathbb{P}_\psi$,

where $g\in G$ and $\mathbb{P}_\phi$, $\mathbb{P}_\psi$ are projectors onto one dimensional subspaces spanned by $\phi$ and $\psi$ respectively. (In order to disregard the trivial problem of the "global phase" I passed to the projective space).

Motivation I am interested in this problem because it gives the possibility of characterising (in some cases) particular orbits of $G$ in the projective space via single homogenous polynomial condition. In the compact case the situation is easy. I want to generalise what I have to the non-compact cases. In particular I am interested in groups like the Heisenberg Group, SU(1,1) or the Metaplectic group.

Remark: In the case of compact connected group $G$ the answer is yes by the structural theory of representations of semisimple Lie groups.