An irreducible unitary affine module has irreducible ground space

Among the nonzero $\mathfrak{k}$-invariant subspaces of finite-dimensional $H_0$, choose one, $V$, of least positive dimension. Its orthogonal complement is invariant: if $w\perp V$ and $v\in V$, then $⟨X(0)w,v⟩=-⟨w,X(0)v⟩=0$. Thus $V$ is an irreducible $\mathfrak{k}$-module.

Every positive current mode annihilates $V$, since it lowers the nonnegative energy below zero; zero modes preserve $V$. Any finite word in current modes can be reordered into negative modes, then zero modes, then positive modes by repeatedly using $AB=BA+[A,B]$. Each commutator term has shorter word length by [F2], so induction on word length and the finite number of inversions gives a finite reordering. Positive modes then kill $V$, and negative modes raise energy. Thus the current submodule generated by $V$ has ground space exactly $V$. It is nonzero, so irreducibility of $H$ makes it all of $H$; hence $H_0=V$.