On Singular Holomorphic Representations *

In this article the full set of irreducible unitary holomorphic representations of U (p, q) is determined. For a general Hermitian symmetric space G/K of the noncompact type the set of irreducible unitary representations of G on scalar valued holomorphic functions has been found by Wallach [14] and by Rossi and Vergne [11]. For the groups Mp(n, ~) and SU(n, n) unitary irreducible representations on vector valued holomorphic functions have been obtained by Gross and Kunze [1] from the decomposition of tensor products of the harmonic (Segal-Shale-Weil) representation L. Later the complete description of these tensor products for the groups Mp(n, lR) and U(p,q) was given by Kashiwara and Vergne [8] (see also [4]), and it was conjectured that any irreducible unitary representation with highest weight appears in the k-th fold tensor product of L for some k. As we shall see, for the groups U(p, q) this is indeed so. The case of G=SU(2,2) has been treated in [9] and [2]. In both cases some rather technical computations of F-functions were successfully completed by means of the detailed knowledge of Clebsch-Gordan and Racah coefficients for U(2), available through the physics literature. Based on results in [10] a different proof for SU(2, 2) was recently given [15]. For G=SU(n,n) the representations are of the form (U(g)f)(z) =J(g-l ,z)-l f((az+b)/(cz+d)) where the automorphic factor J(g,z) is a product of an automorphic factor J0(g, z), which does not contain det(c z + d) to any power, and det(cz+d) to an integer power. Let U = Uso, k if the power is k. The key observation we make is: If Ujo,k and Uso, k_ 1 are unitary and the Hilbert space Hjo,k of Ujo, k is annihilated by constant coefficient differential operators, then USo,k_ ~ is annihilated by constant coefficient differential operators of one order less. To establish this fact and to make full use of it we need the theory of tensor products of holomorphic representations as developed in [-5] and [6].