  The infinite-dimensional unitary group U(infinity) is the inductive limit of
growing compact unitary groups U(N). In this paper we solve a problem of
harmonic analysis on U(infinity) stated in the previous paper math/0109193. The
problem consists in computing spectral decomposition for a remarkable
4-parameter family of characters of U(infinity). These characters generate
representations which should be viewed as analogs of nonexisting regular
representation of U(infinity).
  The spectral decomposition of a character of U(infinity) is described by the
spectral measure which lives on an infinite-dimensional space Omega of
indecomposable characters. The key idea which allows us to solve the problem is
to embed Omega into the space of point configurations on the real line without
2 points. This turns the spectral measure into a stochastic point process on
the real line. The main result of the paper is a complete description of the
processes corresponding to our concrete family of characters. We prove that
each of the processes is a determinantal point process. That is, its
correlation functions have determinantal form with a certain kernel. Our
kernels have a special `integrable' form and are expressed through the Gauss
hypergeometric function.
  In simpler situations of harmonic analysis on infinite symmetric group and
harmonic analysis of unitarily invariant measures on infinite hermitian
matrices similar results were obtained in our papers math/9810015,
math/9904010, math-ph/0010015.
