Unitary representations of the SL (2, C) group in horospheric basis

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Abstract
The matrix elements of unitary representations of the S L (2, C) group are derived in a basis defined by two-dimensional momenta corresponding to the horospheric subgroup (1β01). A parametrization fitting the above basis well is chosen on the analogy of the rotation group by sandwiching a complex rotation about the y axis with two horospheric translations. In this way the two outer subgroups can be factored out immediately in terms of plane waves which are counterparts to the exponentials formed by the Euler angles φ and ψ occuring in the rotation group. Finally, matrix elements of complex rotations about y axis are found by solving the simultaneous eigenvalue problem of the two Casimir operators. Unitary representations obtained in this way are expressed in a rather simple form in terms of Bessel functions.