On special unipotent orbits and Fourier coefficients for automorphic forms on symplectic groups

Jiang, Dihua; Liu, Baiying

doi:10.1016/j.jnt.2014.03.002

Cited by 19 publications

(54 citation statements)

References 13 publications

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“…1 · · · p e l l ) with 2 ≤ k ≤ r also fail to support Θ 2r . Combining this vanishing with Propositions 3.2 and 3.3 of Jiang-Liu [33], we now see that nilpotent orbits of type O ′ = ((2k + 1) 2 p e 1 1 · · · p e l l ) for 2 ≤ k ≤ r also fail to support Θ 2r . This exhausts those orbits O ′ such that either O ′ > O Θ or O ′ and O Θ are not comparable.…”

Section: Vanishing Statementssupporting

confidence: 53%

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A generalized Theta lifting, CAP representations, and Arthur parameters

Leslie

2019

Trans. Amer. Math. Soc.

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We study a new lifting of automorphic representations using the theta representation Θ on the 4-fold cover of the symplectic group, Sp 2r (A). This lifting produces the first examples of CAP representations on higher-degree metaplectic covering groups. Central to our analysis is the identification of the maximal nilpotent orbit associated to Θ.We conjecture a natural extension of Arthur's parameterization of the discrete spectrum to Sp 2r (A). Assuming this, we compute the effect of our lift on Arthur parameters and show that the parameter of a representation in the image of the lift is non-tempered. We conclude by relating the lifting to the dimension equation of Ginzburg to predict the first non-trivial lift of a generic cuspidal representation of Sp 2r (A).Proof. See [29, Section7.1]. 7.2. Local Root Exchange. Suppose now that F is a non-archimedean local field, and let G be the F -rational points of a split algebraic group over F or finite cover thereof. As in the global setting, we consider unipotent subgroups C, X, Y ⊂ G, and a nontrivial characterAs before, set B = Y C, D = XC, and A = BX = DY . We also assume that these subgroups satisfy the local analogue of the six properties preceding Lemma 7.1 (For a precise statement and proof, see [13, Section 6.1]).Lemma 7.2. Suppose that π is a smooth representation of A, and extend the character ψ C trivially to ψ B on B and ψ D on D. Then we have an isomorphism of C-modules J B,ψ B (π) ∼ = J D,ψ D (π).Moreover, J C,ψ C (π) = 0 ⇐⇒ J B,ψ B (π) = 0 ⇐⇒ J D,ψ D (π) = 0.

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Section: Vanishing Statementssupporting

confidence: 53%

“…The proof relies on global results of Ginzburg-Rallis-Soudry[30] and Jiang-Liu[33]. If we can show that nilpotent orbits of the form O ′ = ((2k)1 2r−2k ) for 2 ≤ k ≤ n do not support Θ 2r , then by[30, Lemma 2.6] , it follows that all orbits of the form…”

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confidence: 99%

A generalized Theta lifting, CAP representations, and Arthur parameters

Leslie

2019

Trans. Amer. Math. Soc.

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“…Given a symplectic partition p of 2n (that is, odd parts occur with even multiplicities), Denote by p Sp 2n the Sp 2n -expansion of p, which is the smallest special symplectic partition that is bigger than p. In [JL15a], we proved the following theorem which provides a crucial reduction in the proof of Theorem 2.1. Theorem 2.3 (Theorem 4.1 [JL15a]). Let π be an irreducible automorphic representation of Sp 2n (A).…”

Section: The Main Resultsmentioning

confidence: 99%

“…For definitions of the unipotent group V p,2 and its character ψ p , see [JL15a,Section 2]. Note that the one-dimensional torus H p defined in [JL15a, (2.1)] has elements of the following form H p (t) = diag(A(t), A(t), .…”

Section: Proof Ofmentioning

confidence: 99%

On Fourier coefficients of certain residual representations of symplectic groups

Jiang

Liu

2016

Pacific J. Math.

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In the theory of automorphic descents developed by Ginzburg, Rallis and Soudry in [GRS11], the structure of Fourier coefficients of the residual representations of certain special Eisenstein series plays an essential role. Started from [JLZ13], the authors are looking for more general residual representations, which may yield more general theory of automorphic descents. In this paper, we investigate the structure of Fourier coefficients of certain residual representations of symplectic groups, associated with certain interesting families of global Arthur parameters. On one hand, the results partially confirm a conjecture proposed by the first named author in [J14] on relations between the global Arthur parameters and the structure of Fourier coefficients of the automorphic representations in the associated global Arthur packets. On the other hand, the results of this paper can be regarded as a first step towards more general automorphic descents for symplectic groups, which will be considered in our future work.

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“…This proves Parts (3) and (4) of the theorem. The last part (Part (6) of the theorem) follows from Part (5) as consequence of the Fourier coefficients associated to a composite of two partitions as discussed in [9] and [17], which will be briefly discussed before the end of Section 3.…”

Section: 2mentioning

confidence: 99%

The Jacquet–Langlands Correspondence via Twisted Descent

Jiang

Liu

et al. 2015

Int Math Res Notices

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Abstract. The existence of the well-known Jacquet-Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970 ([13]). An explicit construction of such a correspondence was obtained by Shimizu via theta series in 1972 ([30]). In this paper, we extend the automorphic descent method of Ginzburg-Rallis-Soudry ([10]) to a new setting. As a consequence, we recover the classical Jacquet-Langlands correspondence for PGL(2) via a new explicit construction.

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On special unipotent orbits and Fourier coefficients for automorphic forms on symplectic groups

Cited by 19 publications

References 13 publications

A generalized Theta lifting, CAP representations, and Arthur parameters

A generalized Theta lifting, CAP representations, and Arthur parameters

On Fourier coefficients of certain residual representations of symplectic groups

The Jacquet–Langlands Correspondence via Twisted Descent

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