In this paper we study the
very close connection between the k-th tensor product of the harmonic representation
ω of U(p,q) and the generalized unit disk 𝒟. We give a global version of ω realized on
the Fock space as an integral operator. Each irreducible component of ω is shown to
be equivalent in a natural way to a multiplier representation of U(p,q) acting on a
Hilbert space ℋ(𝒟,λ) of vector-valued holomorphic functions on 𝒟. The
intertwining operator between these realizations is then explicitly constructed.
We determine necessary and sufficient conditions for square integrability of
each component of ω and in this case derive the Hilbert space structure on
ℋ(𝒟,λ).
