Tensor products of holomorphic representations and bilinear differential operators

Let Hν be the weighted Bergman space on a bounded symmetric domain D=G/K. It has analytic continuation in the weight ν and for ν in the so-called Wallach set Hν still forms unitary irreducible (projective) representations of G. We give the irreducible decomposition of the tensor product Hν1⊗Hν2 of the representations for any two unitary weights ν 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 SL(2,C). As applications, we find a generalization of the Bolʹs lemma and we characterize the multiplication operators by the coordinate functions on the quotient space of the tensor product Hν1⊗Hν2 modulo the subspace of functions vanishing of certain degree on the diagonal.