The harmonic representation of U(p,q) and its connection with the generalized unit disk
We study the very close connection between the kth tensor product of the harmonic representation \(\omega\) of U(p,q) and the generalized unit disk D. We give a global version of \(\omega\) realized on the Fock space as an integral operator. Each irreducible component of \(\omega\) is shown to be equivalent in a natural way to a multiplier representation of U(p,q) acting on a Hilbert space H(D,\(\lambda)\) 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 \(\omega\) and in this case derive the Hilbert space structure on H(D,\(\lambda)\).
scientific article; zbMATH DE number 3923368