If $G$ is a simply-connected, non-compact complex Lie group, and if $(\mathcal{H}, \pi)$ is a unitary representation of $G$ on Hilbert space $\mathcal{H}$, does it follow that $\pi$ is reducible, i.e it can be written as a direct sum of infinite-dimensional irreducible representations?
In the post Infinite-dimensional Unitary representions that are not completely reducible, my question seems to have been already addressed as follows:
Non-compact semisimple Lie groups in general have many interesting irreducible infinite-dimensional representations, some but not all of which are unitarizable. It's still true that unitary representations are completely reducible (and the proof is the same), but often there are no nontrivial finite-dimensional ones: for example, if $G$ is a noncompact simple Lie group such as $PSL_2(\mathbb{R})$, it can't embed into any unitary group $U(n)$ (all of which are compact).