Tensor products of holomorphic representations and bilinear differential operators
type: Item
instance of: scholarly article
title: Tensor products of holomorphic representations and bilinear differential operators
zbMATH Open document ID: 1050.22020
author: Genkai Zhang
author: Lizhong Peng
published in: Journal of Functional Analysis
publication date: 2004-08-16T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 32M15
Mathematics Subject Classification ID: 47B32
zbMATH DE Number: 2091743
zbMATH Keywords: weighted Bergman spaces
zbMATH Keywords: reproducing kernel spaces
zbMATH Keywords: tensor products
zbMATH Keywords: bounded symmetric domains
zbMATH Keywords: bilinear differential operators
zbMATH Keywords: representations of Lie groups
zbMATH Keywords: homogeneous tuples of operators
reviewed by: Gabriela Paola Ovando
MaRDI profile type: Publication
full work available at URL: j.jfa.2003.09.006
OpenAlex ID: W2008560990
cites work: Q4840745
cites work: Homogeneous multiplication operators on bounded symmetric domains.
cites work: Differential operators and highest weight representations
cites work: On quotient modules -- the case of arbitrary multiplicity
cites work: Q3992983
cites work: Function spaces and reproducing kernels on bounded symmetric domains
cites work: Q3331451
cites work: Remarks on Classical Invariant Theory
cites work: Q3849170
cites work: Tensor products, reproducing kernels, and power series
cites work: Restrictions and expansions of holomorphic representations
cites work: A New Generalization of Hankel Operators (the Case of Higher Weights)
cites work: Q4142012
cites work: Q4509277
cites work: Hankel forms of arbitrary weight over a symmetric domain via the transvectant
cites work: Q4840758
cites work: Q3719028
cites work: Tensor Products of Holomorphic Discrete Series Representations
cites work: Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen
cites work: Arithmetic of differential operators on symmetric domains
cites work: Differential operators, holomorphic projection, and singular forms
cites work: HOMOGENEOUS VECTOR BUNDLES AND COWEN-DOUGLAS OPERATORS
cites work: Nearly holomorphic functions and relative discrete series of weighted \(L^2\)-spaces on bounded symmetric domains
cites work: Tensor Products of Weighted Bergman Spaces and Invariant Ha-Plitz Operators.
cites work: Shimura invariant differential operators and their eigenvalues
cites work: Tensor products of Minimal Holomorphic Representations
DOI: 10.1016/J.JFA.2003.09.006
description: scientific article; zbMATH DE number 2091743