Consider a compact Lie group (for simplicity let it be $SO(3)$). All unitary irreps of the corresponding Lie algebra are finite-dimensional.
However, the same is not true for the group. As a counter example, consider a vector space spanned by points on the 2-sphere (taken as linearly independent basis elements). $SO(3)$ obviously acts on that vector space, but there is no well defined $\mathfrak{so}_3$ action, because the representation is discontinuous. It is straightforward to show that the representation is unitary and irreducible.
However, the vector space in this counter example is nonseparable.
Are there any infinite-dimensional (discontinuous or non-differentiable) unitary irreps of $SO(3)$ acting on the separable Hilbert space? If not, which theorem forbids their existence?