Howe correspondence for real unitary groups
type: Item
instance of: scholarly article
title: Howe correspondence for real unitary groups
zbMATH Open document ID: 0924.22012
author: Annegret Paul
published in: Journal of Functional Analysis
publication date: 1999-03-22T00:00:00Z
review text: Let \((G,G')\) be a reductive dual pair in \(H\), i.e., a pair of real reductive Lie groups embedded in a large group \(H\) in such a way that they are each other's centralizers in \(H\). The restriction of a representation \(\rho\) of \(H\) to \(G\cdot G'\) may be regarded as a representation of \(G\times G'\) via the natural mapping \(G\times G'\to G\cdot G'\). Let \({\mathcal R}\) be the set of all equivalence classes of irreducible representations \(\pi\otimes\pi'\) of \(G\times G'\), where \(\pi\) and \(\pi'\) are, respectively, irreducible representations of \(G\) and \(G'\), such that, if \(Y\) is the carrier space of \(\rho\), there exists a \(G\times G'\)-invariant subspace \(X\) of \(Y\) with the action of \(\rho| G\times G'\) on the quotient space \(Y/X\) equivalent to \(\pi\otimes\pi'\). If the equivalence class of \(\pi\otimes\pi'\) belongs to \({\mathcal R}\), then \(\pi\) is said to correspond to \(\pi'\) and this dual pair correspondence for \((G,G')\) defines a relationship between some of the irreducible representations of \(G\) and \(G'\). If now \(H\) is the symplectic group \(Sp(2n,\mathbb{R})\) \((n\in\mathbb{N}^*)\), \(\widetilde{Sp}(2n,\mathbb{R})\) its nontrivial twofold covering, and \(\widetilde G,\widetilde G'\) the inverse images of \(G,G'\), respectively, under the covering mapping \(\widetilde{Sp}(2n,\mathbb{R})\to Sp(2n,\mathbb{R})\), then the dual pair correspondence for \((\widetilde G,\widetilde G')\) is a bijection between certain sets of irreducible representations of \(\widetilde G\) and \(\widetilde G'\) [\textit{R. Howe}, J. Am. Math. Soc. 2, 535-552 (1989; Zbl 0716.22006)]. In this paper, the author determines explicitly this bijection for the reductive dual pairs \((\widetilde U(p,q),\widetilde U(r, s))\) in \(\widetilde{Sp}(2(p+q)(r+s),\mathbb{R})\) with \(r+s=p+q\) and remarks that it may be interpreted as a bijection between representations of \(U(p,q)\) and \(U(r, s)\). She describes it in terms of Langlands parameters a…
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 1268052
zbMATH Keywords: reductive dual pair
zbMATH Keywords: real reductive Lie groups
zbMATH Keywords: irreducible representations
zbMATH Keywords: symplectic group
zbMATH Keywords: Langlands parameters
zbMATH Keywords: discrete series representations
zbMATH Keywords: induced representations
MaRDI profile type: Publication
full work available at URL: jfan.1998.3330
OpenAlex ID: W1981549357
cites work: Reductive dual pair correspondence for complex groups
cites work: Q4209242
cites work: The Langlands classification and irreducible characters for real reductive groups
cites work: Q4075116
cites work: Q3856974
cites work: Transcending Classical Invariant Theory
cites work: Theta dichotomy for unitary groups
cites work: Characters, asymptotic and n-homology of Harish-Chandra modules
cites work: On the Segal-Shale-Weil representations and harmonic polynomials
cites work: Q3741813
cites work: Q4840147
cites work: On the local theta-correspondence
cites work: Theta lifting for unitary representations with nonzero cohomology
cites work: Howe correspondences on a \(p\)-adic field
cites work: On some explicit formulas in the theory of Weil representation
cites work: Q3033995
cites work: Reducibility of generalized principal series representations
cites work: The algebraic structure of the representation of semisimple Lie groups. I
cites work: Representations of real reductive Lie groups
cites work: Unitarizability of certain series of representations
cites work: The unitary dual of \(\mathrm{GL}(n)\) over an archimedean field
cites work: Q3033996
cites work: Invariant distributions of classical groups
DOI: 10.1006/JFAN.1998.3330
Recommended article: Howe correspondence for real unitary groups II
Recommended article: Unitary representations of real reductive groups
Recommended article: Howe correspondence for reductive groups over finite fields
Recommended article: Unitary Shimura correspondences for split real groups
Recommended article: Q4681591
Recommended article: The unitary spectrum for real rank one groups
Recommended article: On unitary representations of disconnected real reductive groups
Recommended article: Howe correspondence and Springer correspondence for real reductive dual pairs
Recommended article: Theta correspondence for unitary groups
Recommended article: Q4011182
description: scientific article; zbMATH DE number 1268052