Harmonic analysis on the infinite-dimensional unitary group

The goal of harmonic analysis on the infinite-dimensional unitary group is to decompose a certain family of unitary representations of this group substituting the nonexisting regular representation and depending on two complex parameters (Olshanski, 2003). In the case of nonintegral parameters, the decomposing measure is described in terms of determinantal point processes (Borodin and Olshanski, 2005). The aim of the present paper is to describe the decomposition for integer parameters; in this case, a spectrum of decomposition leaps. A similar result was earlier obtained for the infinite symmetric group (Kerov, Olshanski, Vershik, 2004), but the case of the unitary group turned out to be much more complicated. In the proof we use Gustafson's multilateral summation formula for hypergeometric series.