On the dual pairs \((O(p,q), SL(2,\mathbb{R}))\), \((U(p,q), U(1,1))\) and \((Sp(p,q), O^*(4))\)
type: Item
instance of: scholarly article
title: On the dual pairs \((O(p,q), SL(2,\mathbb{R}))\), \((U(p,q), U(1,1))\) and \((Sp(p,q), O^*(4))\)
zbMATH Open document ID: 0930.22015
author: Annegret Paul
author: Eng-Chye Tan
published in: Pacific Journal of Mathematics
publication date: 2000-02-07T00:00:00Z
review text: Let \(Sp(2k,\mathbb R)\) denote the symplectic group acting on \(\mathbb R^{2k}\) and consider a dual pair \((G,G')\), i.e., two reductive subgroups of \(Sp(2k,\mathbb R)\) which are their mutual centralizers. In addition, we assume that \(G\) and \(G'\) are invariant under the standard Cartan involution of \(G\), so that \(K := G \cap U(k)\) and \(K' := G' \cap U(k)\) are their maximal compact subgroups. If \(H\) is a reductive subgroup of \(Sp(2k,\mathbb R)\) with maximal compact subgroup \(K_H \subseteq U(k)\) and \(\widetilde K_H\) the corresponding subgroup of the twofold covering group \(\widetilde Sp(2k,\mathbb R)\) of \(Sp(2k,\mathbb R)\), then we write \({\mathcal R}(\mathbf h,\widetilde K_H, \omega)\) for the set of equivalence classes of simple \((\mathbf h,\widetilde K_H)\) modules which can be obtained as simple quotients of submodules of the oscillator representation \(\omega\) on the space \({\mathcal P}\) of holomorphic polynomials on \(\mathbb C^k\) (here we use the realization in the Fock model). The theta correspondence between \({\mathcal R}(\mathbf g,\widetilde K, \omega)\) and \({\mathcal R}(\mathbf g',\widetilde K', \omega)\) is now established as follows. For \(\rho \in {\mathcal R}(\mathbf g',\widetilde K', \omega)\) we consider the space \(\Omega(\rho) := {\mathcal P}/{\mathcal N}_\rho\), where \({\mathcal N}_\rho\) is the intersection of all \((\mathbf g', \widetilde K')\)-submodules \({\mathcal N}\) of \({\mathcal P}\) such that \({\mathcal P}/{\mathcal N} \cong \rho\). According to a result of R.\ Howe, we then have \(\Omega(\rho) \cong \rho \otimes \rho'\), where \(\rho'\) is a \((\mathbf g,\widetilde K)\)-module of finite length with a unique simple quotient \(\theta(\rho)\) (the so-called theta lift of \(\rho\)). A central problem in the theory of dual pairs is to make the theta correspondence, i.e., the map \(\theta\), more explicit. In the paper under review, the authors use quite elementary methods and explicit computations with po…
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 13A50
zbMATH DE Number: 1344301
zbMATH Keywords: dual pair
zbMATH Keywords: oscillator representation
zbMATH Keywords: theta lift
zbMATH Keywords: theta correspondence
zbMATH Keywords: reductive group
reviewed by: Karl-Hermann Neeb
MaRDI profile type: Publication
full work available at URL: pjm.1999.187.349
OpenAlex ID: W2058362338
DOI: 10.2140/PJM.1999.187.349
description: scientific article; zbMATH DE number 1344301