Consider a separable irreducible unitary representation of $U(\mathcal{H})$ in the Hilbert space $V$. Assume that $\mathcal{H}$ is separable. My question is the following:
Is it true that all orbits of $U(\mathcal{H})$ in $\mathbb{P}\mathcal{V}$ are closed? If not, are there any conditions for given state $[v]\in\mathbb{P}\mathcal{H}$ that guarantee that the orbit $[U(\mathcal{H}).v]$ is closed?
Motivation: I want to generalize a well known result from the theory of representations of compact semisimple Lie groups that states that the orbit through the highest weight vector in $\mathbb{P}V$ is characterized by a certain a quadric. Due to the structure of separable representations of $U(\mathcal{H})$ (see here) one can hope that the similar characterization is valid for orbits through "highest weight vectors" in these representations. In order to complete the proof I need to know that orbits through these highest weight vectors are closed. I was able to prove closedness for "highest weight orbits" in cases that are most interesting from the physical perspective: $V=\bigwedge^L\mathcal{H}$ and $V=Sym^L(\mathcal{H})$ but I am not able to cope with the general case.
