Massless induced representation of the Poincare double cover

For a future null standard momentum $k$, choose Lorentz transformations $L(p)k=p$ and a unitary irreducible representation $d$ of its little group. The Wigner rotation $R(\Lambda ,p)=L(\Lambda p{)}^{-1}\Lambda L(p)$ acts on the internal state. The resulting induced representation acts on wavefunctions with invariant measure $d^{3}p/(2p^0)$. Trivial little-group translations give fixed helicity; nontrivial translations give a continuous-spin representation with infinitely many polarization states.