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On the Howe correspondence for symplectic-orthogonal dual pairs. (English) Zbl 1084.22008

Recall that a pair \(( G,G^{\prime}) \) of subgroups of \(Sp=Sp_{2N}( \mathbb{R}) \) is called a reductive dual pair if \(G\) and \(G^{\prime}\) are centralizers of each other in the group \(Sp\), and both \(G\) and \(G^{\prime}\) act absolutely reductively on \(\mathbb{R}^{2N}\), where \(Sp_{2N}( \mathbb{R}) \) is the real symplectic group of rank \(N\). Let \(\mathcal{R}( \tilde{H}) \) be the set of infinitesimal equivalence classes of continuous irreducible admissible representations of \(\tilde{H}\) on locally convex spaces for a reductive subgroup \(H\) of \(Sp\), where \(\tilde{H}\) is the two-fold cover of \(H\) in the unique nontrivial two-fold covering group \(\widetilde{Sp}\) of \(Sp\). Let \(\mathcal{R}( \tilde{H},\omega) \) consist of (classes of) representations \(\pi \in\mathcal{R}( \tilde{H}) \) which are quotients of the oscillator representation \(\omega\) of \(\widetilde{Sp}\) constructed by Shale and Weil.
Howe established that the condition \(\rho\otimes\rho^{\prime}\in\mathcal{R}( \tilde{G}\tilde{G}^{\prime},\omega) \) on the tensor product representation \(\rho\otimes\rho^{\prime}\in\mathcal{R}( \tilde{G} \times\tilde{G}^{\prime}) \cong\mathcal{R}( \tilde{G}\tilde {G}^{\prime}) \) for \(\rho\in\mathcal{R}( \tilde{G}) \) and \(\rho^{\prime}\in\mathcal{R}( \tilde{G}^{\prime}) \) characterizes a bijective correspondence \(\rho\to\rho^{\prime}\) between \(\mathcal{R}( \tilde{G},\omega) \) and \(\mathcal{R}( \tilde{G}^{\prime},\omega) \). The Howe correspondence has been computed for the reductive dual pair \(( Sp_{2n}( \mathbb{R}) ,O( p,q) ) \) of \(Sp_{2( p+q) n}( \mathbb{R}) \) with \(p+q\in\{ 2n,2n+2\} \) and both \(p\) and \(q\) even by C. Moeglin.
In this paper, the author fills in the missing cases by giving a complete detailed description of the Howe correspondence, in terms of Langlands parameters, for the dual pair \(( Sp_{2n}( \mathbb{R} ) ,O( p,q) ) \) with \(p+q\in\{ 2n,2n+2\} \) for both even \(p,q\) and odd \(p,q\).

MSC:

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
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