Uniqueness of Whittaker model for the metaplectic group

Szpruch, Dani

doi:10.2140/pjm.2007.232.453

Cited by 32 publications

(32 citation statements)

References 11 publications

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“…2r of the symplectic group Sp 2r , see [Szp2]. This uniqueness plays pivotal role in the work of Szpruch (cf.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Distinguished theta representations for certain covering groups

Gao

2017

Pacific J. Math.

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Abstract. For Brylinski-Deligne covering groups of an arbitrary split reductive group, we consider theta representations attached to certain exceptional genuine characters. The goal of the paper is to study the dimension of the space of Whittaker functionals of a theta representation. In particular, we investigate when the dimension is exactly one, in which case the theta representation is called distinguished. For this purpose, we first give effective lower and upper bounds for the dimension of Whittaker functionals for general theta representations. As a consequence, the dimension in many cases can be reduced to simple combinatorial computations, e.g., the Kazhdan-Patterson covering groups of the general linear groups, or covering groups whose complex dual groups (à la Finkelberg-Lysenko-McNamara-Reich) are of adjoint type. In the second part of the paper, we consider coverings of certain semisimple simply-connected groups and give necessary and sufficient condition for the theta representation to be distinguished. There are subtleties arising from the relation between the rank and the degree of the covering group. However, in each case we will determine the exceptional character such that its associated theta representation is distinguished.

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“…2r of the symplectic group Sp 2r , see [Szp2]. This uniqueness plays pivotal role in the work of Szpruch (cf.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Distinguished theta representations for certain covering groups

Gao

2017

Pacific J. Math.

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“…Details will appear in a forthcoming paper of Szpruch. The notions of non-degenerate characters and generic representations are also defined and the uniqueness of Whittaker model is proved in [Szp07]. The results of this paper as well as the proofs immediately carry over toG.…”

Section: Where [· ·] Is a Non-zero Invariant (Positive-definite) Innmentioning

confidence: 85%

On the asymptotics of Whittaker functions

Lapid

Mao

2009

Represent. Theory

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“…where c(g 1 , g 2 ) is the Rao cocycle defined in [Rao], cf [Sz1] for a brief review of the cocycle formulas.…”

Section: Fmentioning

confidence: 99%

Stability of Rankin–Selberg gamma factors for Sp(2n),Sp̃(2n) and U(n,n)

Zhang

2017

Int. J. Number Theory

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Abstract. Let F be a p-adic field and E/F be a quadratic extension. In this paper, we prove the stability of Rankin-Selberg gamma factors for Sp 2n (F ), ‹ Sp 2n (F ) and U E/F (n, n) when the characteristic of the residue field of F is not 2. IntroductionLet G n be Sp 2n , Sp 2n and U E/F (n, n), where E/F is a quadratic extension of local or global field. The global Rankin-Selberg zeta integrals for the generic irreducible cuspidal automorphic representations of G n twisted by generic irreducible cuspidal representations of GL m has been developed by Gelbart, Piatetski-Shapiro, Ginzburg, Rallis and Soudry, [GePS2,GiRS1,GiRS2]. Recently, the standard properties of such local γ-factors were established by Kaplan [Ka]. As a complimentary result of their work, in this paper, we prove the stability of the local gamma factor for a generic representation of G n (F ) when twisted by a sufficiently highly ramified character of GL 1 for a p-adic field F , when the residue field of F is not 2. More precisely, the main result of this paper is the following Theorem 0.1. Let F be a p-adic field such that the characteristic of its residue field is odd, E/F be a quadratic extension. Let ψ U be a generic character of a maximal unipotent subgroup of G n (F ) defined by a given nontrivial additive character ψ of F . Let π 1 , π 2 be two ψ U -generic irreducible smooth representations of G n (F ) with the same central character. If η is a highly ramified quasicharacter of F × , then γ(s, π 1 , η, ψ) = γ(s, π 2 , η, ψ).Here the γ-factors are the Rankin-Selberg gamma factors, see §1 for more details. We also notice that the main theorem also holds for U E/F if the residue characteristic of F is 2 and E/F is unramified.Here we remark that in the Sp 2n case, this result can be deduced from previous work. Cogdell, Kim, Piatetski-Shapiro and Shahidi proved the stability of gamma factors for classical groups (which at least includes Sp 2n and SO n ) in [CKPSS], where the gamma factors are defined using Langlands-Shahidi method. In [Ka], Kaplan proved that Rankin-Selberg gamma factors agree with the Langlands-Shahidi gamma factor. Thus our result in the Sp 2n case follows from the stability result in [CKPSS] and Kaplan's result on the agreement of the two type gamma factors. In the U E/F (n, n)-case, the stability of the Langlands-Shahidi gamma factors is proved in [KK]. Thus in principle, our result in the U E/F (n, n) case should follow from an agreement result of the two type γ-factors, which is unfortunately not included in [Ka].In this paper, we prove the stability of gamma factors for G n in the Rankin-Selberg context. Although one can deduce this by pulling back the Langlands-Shahidi gamma factors via [Ka], it is still important to have a proof of stability that remains within the context of integral representations, since there are L-functions that we have integral representations for that are not covered by the Langlands-Shahidi method. So developing methods that work in the integral representation context have an intrinsic valu...

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Uniqueness of Whittaker model for the metaplectic group

Cited by 32 publications

References 11 publications

Distinguished theta representations for certain covering groups

Distinguished theta representations for certain covering groups

On the asymptotics of Whittaker functions

Stability of Rankin–Selberg gamma factors for Sp(2n),Sp̃(2n) and U(n,n)

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