On the local factors of representations of classical groups

Lapid, Erez; Rallis, Stephen

doi:10.1515/9783110892703.309

Cited by 97 publications

(99 citation statements)

References 24 publications

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“…The condition (b) above ensures that Π ′ is cuspidal. To show that Π ′ is nonzero, we consider the standard L-function L(s, Π) of Π defined using the doubling zeta integral of Piatetski-Shapiro-Rallis [46], [40]. Observe that the partial L-function L S∪{v 0 } (s, Π) agrees with the partial standard L-function L S∪{v 0 } (s, Σ) of Σ, so that …”

Section: 7mentioning

confidence: 99%

The Gross–Prasad conjecture and local theta correspondence

Gan

Ichino

2016

Invent. math.

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Abstract. We establish the Fourier-Jacobi case of the local Gross-Prasad conjecture for unitary groups, by using local theta correspondence to relate the Fourier-Jacobi case with the Bessel case established by Beuzart-Plessis. To achieve this, we prove two conjectures of D. Prasad on the precise description of the local theta correspondence for (almost) equal rank unitary dual pairs in terms of the local Langlands correspondence. The proof uses Arthur's multiplicity formula and thus is one of the first examples of a concrete application of this "global reciprocity law".

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Section: 7mentioning

confidence: 99%

The Gross–Prasad conjecture and local theta correspondence

Gan

Ichino

2016

Invent. math.

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“…However, if one assumes the existence of a theory of γ-factors satisfying the usual properties (such as those listed as the "Ten Commandments" in [LR05]), then we can show that (v) holds for all representations, in which case the map L will be uniquely characterized by (iii) and (v) (with r ≤ 2 in (v)).…”

Section: Introductionmentioning

confidence: 99%

The local Langlands conjecture for GSp(4)

2011

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“…We note here a typo in [LR,(25)], where on the right-hand side of the above equation they had τ (−1) instead of (τ ⊗ χ)(−1). Moreover, it was shown in [LR] that…”

Section: Relation To Linear Casementioning

confidence: 99%

Doubling zeta integrals and local factors for metaplectic groups

Gan¹

2012

Nagoya Mathematical Journal

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Abstract. We develop the theory of the doubling zeta integral of PiatetskiShapiro and Rallis for metaplectic groups Mp 2n , and we use it to give precise definitions of the local γ-factors, L-factors, and -factors for irreducible representations of Mp 2n × GL1, following the footsteps of Lapid and Rallis. §1. Introduction Some twenty-five years ago, Piatetski-Shapiro and Rallis [GPSR, part A] discovered a Rankin-Selberg integral representation for the standard L-function for G × GL 1 , where G is a classical group. This Rankin-Selberg integral is called the doubling zeta integral. The associated local zeta integral can be used,à la Tate, to define the local γ-factor γ(s, π × χ, ψ) for an irreducible representation π χ of G × GL 1 . This was sketched in [PSR], but the definitive treatment only appeared in the recent work of Lapid and Rallis [LR], which treated the case of symplectic, orthogonal, and unitary groups. There are, however, two classes of classical groups to which the theory of doubling zeta integrals applies but which were not treated in [LR]. The first of these is the class of metaplectic groups Mp 2n , which are the unique 2-fold covers of the symplectic groups Sp 2n . The other is the class of quaternion unitary groups, which are inner forms of symplectic and orthogonal groups (this case has in fact been treated in a recent preprint of Yamana [Y]).The purpose of this paper is to treat the theory of the doubling zeta integral for the metaplectic group Mp 2n , following the footsteps of Lapid and Rallis [LR]. In particular, given an irreducible genuine representation

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On the local factors of representations of classical groups

Cited by 97 publications

References 24 publications

The Gross–Prasad conjecture and local theta correspondence

The Gross–Prasad conjecture and local theta correspondence

The local Langlands conjecture for GSp(4)

Doubling zeta integrals and local factors for metaplectic groups

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