Irreducible unitary representations of SUp,q I: The discrete series

A class of irreducible unitary representations in the discrete series of SU p,q is explicitly determined in a a space of holomorphic functions of three complex matrices. The discrete series, which is the set of all square inte grable representations, corresponds to a compact subgroup of SUp,q. The rel evant algebraic properties of the group SUp,q are discussed in detail. For a degenerate irreducible unitary representation an explicit construction of the infinitesimal generators of the Lie algebra su(p,q) in terms of differ ential operators is given.