Application of restriction of Fourier transforms to an example from representation theory

This paper uses restriction of Fourier transforms to construct explicit realizations of certain irreducible unitary representations of SU (n, n). The realizations begin with generalizations of the classical Szego map. Boundary values of these Szego maps naturally lead to certain restrictions of Fourier transforms. The image of these restrictions provide concrete constructions of unitary representations as L-2 spaces on certain orbits. The SU (n, n) invariance of the L-2 spaces and inner products follows immediately from the restriction maps and the natural pairing between certain degenerate principal series.