Representations of the Lorentz group and generalization of helicity states

The principal series of unitary representations of the Lorentz group is obtained by complexification of the three-dimensional group of rotations and by the solution of the eigenvalue equation for the Casimir operators. The representation obtained can be expressed simply in terms of $D$ functions (of the first and second kind) of the group of rotations. The harmonic analysis of the functions on the group is discussed. Spherical functions on a two-dimensional complex sphere are constructed.