The explicit duality correspondence of (Sp(p,q),O∗(2n))

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We reformulate some of Moeglin's results on the correspondence for the dual pairs (Sp(2n, R), O(p, q)) with p and q even, and fill in the cases where p and q are both odd. We arrive at a complete and detailed description, in terms of Langlands parameters, of the dual pair correspondence for the cases p + q = 2n and p + q = 2n + 2. In addition, we point out and suggest a way to correct an error in Moeglin's paper.

“…The proof is very much like that of Corollary 3.21 of [2], Theorem 4.5.5 of [16], and Theorem 4.20 of [13], using ideas of [11] (see also Corollary III.8 of [14]). …”

Section: The Induction Principlementioning

confidence: 87%

On the Howe correspondence for symplectic–orthogonal dual pairs

Paul

2005

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“…This map depends on a choice of one of the two oscillator representations for Mp: Because of its role in the construction of automorphic forms, it is of considerable interest to compute the map y as explicitly as possible. This has been accomplished in some cases, most notably when G and G 0 are complex [AB1]; when, roughly speaking, G and G 0 have the same size [AB2,M,P1,P2]: whenever n5p þ q for the ðSpðp; qÞ; O n ð2nÞÞ dual pair [LPTZ]; and for type II dual pairs [AB1,LPTZ,M]. In complete generality, very little is known explicitly.…”

Section: Introductionmentioning

confidence: 99%

One-Dimensional Representations of U (p,q) and the Howe Correspondence

Paul

Trapa

2002

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We explicitly determine the theta lifts of all one-dimensional representations of Uðp; qÞ in terms of Langlands parameters, and determine exactly which lifts are unitary. Moreover, we show that such a lift is unitary if and only if it is a weakly fair derived functor module of the form A q ðlÞ: Finally, we show that the correspondence of unitary representations behaves well with respect to associated cycles. # 2002 Elsevier Science (USA)

“…For complex groups in dual pairs or real dual pairs of type II, the correspondence is completely and explicitly described in [Moeg89,AB95,LPTZ03]. For other real dual pairs of type I, explicit results up to now are mainly for compact cases [EHW83] and "(almost) equal rank" cases [AB98,Pau98,Pau05], with the exception of [LTZ01,LPTZ03] for a few non-equalrank cases. In general, however, a full and explicit description of the local theta correspondence remains elusive.…”

mentioning

confidence: 99%

Explicit induction principle and symplectic-orthogonal theta lifts

Fan

2017

Abstract. During the last two decades, great efforts have been devoted to the calculation of the local theta correspondence for reductive dual pairs. However, uniform formulas remain elusive for real dual pairs of type I. The purpose of this paper is twofold: to formulate an explicit version of induction principle for dual pairs (O(p, q), Sp(2n, R)) with p + q even, and to apply it to obtain a complete and explicit description of the local theta correspondence when p + q = 4. Our approach is very elementary by analysis on the infinitesimal characters and K-types under the correspondence.