  An analog of the Paley-Wiener isomorphism for the Hardy space with an
invariant measure over infinite-dimensional unitary groups is described. This
allows us to investigate on such space the shift and multiplicative groups, as
well as, their generators and intertwining operators. We show applications to
the Gauss-Weierstrass semigroups and to the Weyl-Schr\"odinger irreducible
representations of complexified infinite-dimensional Heisenberg groups.
