 Finite-dimensional unitary representations of the massless little group have trivial translations

In a finite-dimensional unitary representation of $ISO(2)$, its two commuting Hermitian translation generators have a finite joint spectrum. Conjugating by every planar rotation rotates that spectrum. A finite set invariant under all rotations contains only the origin, so both generators vanish. An irreducible representation is then a one-dimensional rotation character, giving a helicity state. Nontrivial translations instead require the infinite-dimensional continuous-spin representation. The same argument applies to the physical double cover of this little group.