Tensor Products of Holomorphic Representations and Bilinear Differential Operators

Let be weighted Bergman space on a bounded symmetric domain . It has analytic continuation in the weight and for in the so-called Wallach set still forms unitary irreducible (projective) representations of . We give the irreducible decomposition of the tensor product of the representation for any two unitary weights and we find the highest weight vectors of the irreducible components. We find also certain bilinear differential intertwining operators realizing the decomposition, and they generalize the classical transvectants in invariant theory of . As an application we also find an generalization of the Bol’s lemma.