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

This paper uses restriction of Fourier transforms to con-struct explicit realizations of certain irreducible unitary rep-resentations of SU(n, n). The realizations begin with general-izations 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 L2 spaces on cer-tain orbits. The SU(n, n) invariance of the L2 spaces and inner products follows immediately from the restriction maps and the natural pairing between certain degenerate principal series. 1. Introduction. Calculating explicit and natural constructions of unitary representations– especially singular ones–has been a very fruitful field of study in represen-tation theory. Consider, for instance, the Metaplectic representation an