The U n,1 and IUn Representation Matrix Elements

We construct a realization of the Un,1 and IUn groups as multiplier representations of the space of functions on the Un group manifold. Making use of the orthogonality and completeness of the Un unitary irreducible representation matrix elements (UIRME's), we are able to express the Un,1 boost and IUn translation matrix elements (the generalized Wigner d-functions) of the principal series of UIR's as an integral over a compact domain (unit disc) of two Un d-functions, phases, and the multiplier. This is an extension to the unitary groups of a method previously used [J. Math. Phys. 12, 197 (1971)] to find the SOn, SOn,1, and ISOn UIRME's in a recursive fashion. We establish a number of symmetry properties, the asymtotic (Regge-like) and contraction (Un,1 → IUn) behavior of these functions.