Tensor products of Minimal Holomorphic Representations

Abstract

Let D = G / K D=G/K be an irreducible bounded symmetric domain with genus p p and H ν ( D ) H^{u }(D) the weighted Bergman spaces of holomorphic functions for ν > p − 1 u >p-1 . The spaces H ν ( D ) H^u (D) form unitary (projective) representations of the group G G and have analytic continuation in ν u ; they give also unitary representations when ν u in the Wallach set, which consists of a continuous part and a discrete part of r r points. The first non-trivial discrete point ν = a 2 u =\frac a2 gives the minimal highest weight representation of G G . We give the irreducible decomposition of tensor product H a 2 ⊗ H a 2 ¯ H^{\frac a2}\otimes \overline {H^{\frac a2}} . As a consequence we discover some new spherical unitary representations of G G and find the expansion of the corresponding spherical functions in terms of the K K -invariant (Jack symmetric) polynomials, the coefficients being continuous dual Hahn polynomials.