The explicit duality correspondence of (\(\text{Sp}(p,q),\text{O}^{*}(2n)\))
type: Item
instance of: scholarly article
title: The explicit duality correspondence of (\(\text{Sp}(p,q),\text{O}^{*}(2n)\))
zbMATH Open document ID: 1028.22013
DOI: 10.1016/S0022-1236(02)00079-4
author: Jian-Shu Li
author: Annegret Paul
author: Eng-Chye Tan
author: Chen-Bo Zhu
published in: Journal of Functional Analysis
publication date: 2003-05-25T00:00:00Z
review text: The main result of this paper is the fact that whenever \(p+q\leq n\), the theta correspondence for the dual pair \((\text{Sp}(p,q),\text{O}^*(2n))\) gives rise to an injection of the unitary dual of \(\text{Sp}(p,q)\) into the unitary dual of \(\text{O}^*(2n)\). This injection is described explicitly in terms of Langlands parameters. As an application, the authors obtain many previously unknown small unitary representations of \(\text{O}^*(2n)\), corresponding to known unitary representations of \(\text{Sp}(p,q)\) for \(p+q\leq n/2\) (the stable range). The latter include the trivial representation, the tempered representations and the unitary representations with nonzero cohomology. For \(n\geq 8\), this gives a complete classification of irreducible unitary representations of \(\text{O}^*(2n)\) with rank (in the sense of Howe) at most 3. The method employed is to use some known results about correspondences in the equal rank case (i.e., \(p+q=n\) or \(n-1\)), the induction principle of Kudla and Moeglin, and a careful analysis of lowest \(K\)-types. For the sake of completeness, the authors also work out the correspondence for the type I dual pairs over quaternions, \((\text{GL}(m,\mathbb{H}),\text{GL}(n,\mathbb{H}))\).
reviewed by: Pavle Pandžić
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 1915621
zbMATH Keywords: dual pairs
zbMATH Keywords: theta correspondences
zbMATH Keywords: unitary representations
MaRDI profile type: Publication
cites work: Reductive dual pair correspondence for complex groups
cites work: Q4209242
cites work: The unitary dual of \(Sp(n,1)\), \(n\geq 2\)
cites work: Q5734920
cites work: Q3774088
cites work: Transcending Classical Invariant Theory
cites work: Q3741813
cites work: Q4840147
cites work: On the local theta-correspondence
cites work: Singular unitary representations of classical groups
cites work: Theta lifting for unitary representations with nonzero cohomology
cites work: Correspondance de Howe pour les paires reductives duales: Quelques calculs dans le cas archimédien. (Howe correspondence for dual reductive pairs: Some calculations in the Archimedean case)
cites work: Howe correspondence for real unitary groups
cites work: On Howe's duality theorem
cites work: On some explicit formulas in the theory of Weil representation
cites work: Reducibility of generalized principal series representations
cites work: Representations of real reductive Lie groups
cites work: Unitarizability of certain series of representations
full work available at URL: s0022-1236(02)00079-4
OpenAlex ID: W2039667105
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description: scientific article; zbMATH DE number 1915621