The harmonic representation of U(p,q) and its connection with the generalized unit disk
type: Item
instance of: scholarly article
title: The harmonic representation of U(p,q) and its connection with the generalized unit disk
zbMATH Open document ID: 0577.43003
author: Mark G. Davidson
published in: Pacific Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: We study the very close connection between the kth tensor product of the harmonic representation \(\omega\) of U(p,q) and the generalized unit disk D. We give a global version of \(\omega\) realized on the Fock space as an integral operator. Each irreducible component of \(\omega\) is shown to be equivalent in a natural way to a multiplier representation of U(p,q) acting on a Hilbert space H(D,\(\lambda)\) of vector-valued holomorphic functions on D. The intertwining operator between these realizations is then explicitly constructed. We determine necessary and sufficient conditions for square integrability of each component of \(\omega\) and in this case derive the Hilbert space structure on H(D,\(\lambda)\).
Mathematics Subject Classification ID: 43A22
Mathematics Subject Classification ID: 46E20
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 43A65
zbMATH DE Number: 3923368
zbMATH Keywords: tensor product
zbMATH Keywords: harmonic representation
zbMATH Keywords: U(p,q)
zbMATH Keywords: generalized unit disk
zbMATH Keywords: Fock space
zbMATH Keywords: multiplier representation
zbMATH Keywords: Hilbert space
zbMATH Keywords: vector-valued holomorphic functions
zbMATH Keywords: intertwining operator
zbMATH Keywords: Segal-Shale-Weil representation
zbMATH Keywords: oscillator representation
MaRDI profile type: Publication
full work available at URL: pjm.1987.129.33
OpenAlex ID: W2081882566
DOI: 10.2140/PJM.1987.129.33
description: scientific article; zbMATH DE number 3923368