The Descent Map from Automorphic Representations of GL(n) to Classical Groups

Ginzburg, David; Rallis, Stephen; Soudry, David

doi:10.1142/7742

Cited by 91 publications

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“…For π ∈ M(G) and a nilpotent orbit O ⊂ g denote The study of Whittaker and generalized Whittaker models for representations of reductive groups over local fields evolved in connection with the theory of automorphic forms (via their Fourier coefficients), and has found important applications in both areas. See for example [Sh74,NPS73,Kos78,Kaw85,Ya86,Wall88a,Gin06,Jia07,GRS11]. From the point of view of representation theory, the space of generalized Whittaker models may be viewed as one kind of nilpotent invariant associated to smooth representations.…”

Section: General Resultsmentioning

confidence: 99%

Generalized and degenerate Whittaker models

Gomez

Gourevitch

Sahi³

2017

Compositio Math.

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Abstract. We study generalized and degenerate Whittaker models for reductive groups over local fields of characteristic zero (archimedean or non-archimedean). Our main result is the construction of epimorphisms from the generalized Whittaker model corresponding to a nilpotent orbit to any degenerate Whittaker model corresponding to the same orbit, and to certain degenerate Whittaker models corresponding to bigger orbits. We also give choice-free definitions of generalized and degenerate Whittaker models. Finally, we explain how our methods imply analogous results for Whittaker-Fourier coefficients of automorphic representations.For GLn(F) this implies that a smooth admissible representation π has a generalized Whittaker model WO(π) corresponding to a nilpotent coadjoint orbit O if and only if O lies in the (closure of) the wavefront set WF(π). Previously this was only known to hold for F nonarchimedean and O maximal in WF(π), see [MW87]. We also express WO(π) as an iteration of a version of the Bernstein-Zelevinsky derivatives [BZ77,AGS15a]. This enables us to extend to GLn(R) and GLn(C) several further results from [MW87] on the dimension of WO(π) and on the exactness of the generalized Whittaker functor.

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Section: General Resultsmentioning

confidence: 99%

Generalized and degenerate Whittaker models

Gomez

Gourevitch

Sahi³

2017

Compositio Math.

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“…The twisted automorphic descent. The twisted automorphic descents was introduced in [17], which extends the automorphic descent of Ginzburg-Rallis-Soudry ( [10]) to much more general situation. Following [17] and [10], we introduce a family of Bessel-Fourier coefficients that defines the descent.…”

Section: 3mentioning

confidence: 99%

“…It has been seen in many previous works (see [10,14,17]) that L ℓ,β (A)-modules π ℓ,β satisfy the so-called tower property when the depth ℓ varies. That is, there exists an ℓ * such that π ℓ * ,β = 0 for some choice of data, and π ℓ,β = 0 for all ℓ * < ℓ < 2n.…”

Section: 3mentioning

confidence: 99%

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A reciprocal branching problem for automorphic representations and global Vogan packets

Jiang

Liu

Xu³

2019

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Let G be a group and H be a subgroup of G. The classical branching rule (or symmetry breaking) asks: For an irreducible representation π of G, determine the occurrence of an irreducible representation σ of H in the restriction of π to H. The reciprocal branching problem of this classical branching problem is to ask: For an irreducible representation σ of H, find an irreducible representation π of G such that σ occurs in the restriction of π to H. For automorphic representations of classical groups, the branching problem has been addressed by the well-known global Gan-Gross-Prasad conjecture. In this paper, we investigate the reciprocal branching problem for automorphic representations of special orthogonal groups using the twisted automorphic descent method as developed in [17]. The method may be applied to other classical groups as well.

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“…For example, Whittaker-Fourier coefficients play an essential role in the theory of constructing automorphic L-functions, either by Rankin-Selberg method or by Langlands-Shahidi method. In general, there is a framework of attaching Fourier coefficients to nilpotent orbits (see [GRS03,G06,J14,GGS17a], and also §2 for details), which has also been used in theory of automorphic descent (see [GRS11]). Let F be a number field and A be its ring of adeles.…”

Section: Introductionmentioning

confidence: 99%

On top Fourier coefficients of certain automorphic representations of $${\mathrm {GL}}_n$$

Liu

2020

manuscripta math.

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In this paper, we study top Fourier coefficients of certain automorphic representations of GL n (A). In particular, we prove a conjecture of Jiang on top Fourier coefficients of isobaric automorphic representations of GL n (A) of formwhere ∆(τ i , b i )'s are Speh representations in the discrete spectrum of GL aibi (A) with τ i 's being unitary cuspidal representations of GL ai (A), and n = r i=1 a i b i . Endoscopic lifting images of discrete spectrum of classical groups form a special class of such representations. The result of this paper will facilitate the study of automorphic forms of classical groups occurring in the discrete spectrum.

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The Descent Map from Automorphic Representations of GL(n) to Classical Groups

Cited by 91 publications

References 0 publications

Generalized and degenerate Whittaker models

Generalized and degenerate Whittaker models

A reciprocal branching problem for automorphic representations and global Vogan packets

On top Fourier coefficients of certain automorphic representations of $${\mathrm {GL}}_n$$

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