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In this paper we study the very close connection between the kth tensor product of the harmonic representation ω of U(p,q) and the generalized unit disk S). 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 Jί?(@,λ) of vector-valued holomorphic functions on 3). 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 3^{β, λ).

The harmonic representation ofU(p,q) and its connection with the generalized unit disk