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, which is a substitute for the nonexisting regular representations and depends on two complex parameters (Olshanski, 2003). In the case of noninteger parameters, the decomposing measure is described in terms of determinantal point processes (Borondin and Olshanski, 2005). The aim of the present paper is to describe the decomposition for integer parameters; in this case, the spectrum of the decompositions changes drastically. A similar result was earlier obtained for the infinite symmetric group (Kerov, Olshanski, and 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. Bibliography: 6 titles.