Tensor products of holomorphic representations and bilinear differential operators
Let \(\mathcal H_{\nu}\) be the weighted Bergman space of holomorphic functions on a bounded symmetric domain \(D\subset V\). In this paper the authors study the irreducible decomposition of the tensor product \(\mathcal H_{\nu_1}\otimes \mathcal H_{\nu_2}\) for any \(\nu_1\) and \(\nu_2\) in the Wallach set. The idea is roughly the following: the highest weight vectors are of the form of the heighest weight vectors in the space of polynomials of \((z -w)\) on \(V\); by using the explicit K-type structure they can determine which of them actually appear. Next they find the explicit intertwining operators that are certain bilinear differential operators generalizing the classical transvectants in the case of the unit disk in the complex plane. As applications they give generalizations of the Bol's lemma and they characterize the multiplication operators by the coordinate functions on the quotient space of the tensor product \(\mathcal H_{\nu_1} \otimes \mathcal H_{\nu_2}\) modulo the subspace of functions of a certain degree vanishing on the diagonal. Roughly speaking the restriction to the diagonal gives a realization of the quotient module as a reproducing kernel Hilbert space with an explicitly determined kernel.
scientific article; zbMATH DE number 2091743
scientific article