On Generalized Fourier Transforms for Standard L-Functions

Shahidi, Freydoon

doi:10.1007/978-3-319-94833-1_12

Cited by 10 publications

(12 citation statements)

References 20 publications

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“…The reason for limiting to the symplectic case is threefold: (1) it is simpler; (2) in that case the formalisms of [12] and [39,17] coincide; (3) the doubling method for unitary or orthogonal groups will involve nonsplit or disconnected groups which deviates from our general framework, albeit harmlessly. We sketch the general case of doubling following [39,17]; see also [51]. Consider a field extension E|F with [E : F ] ≤ 2.…”

Section: Remarks On the General Casementioning

confidence: 99%

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Zeta Integrals, Schwartz Spaces and Local Functional Equations

2018

Lecture Notes in Mathematics

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According to Sakellaridis, many zeta integrals in the theory of automorphic forms can be produced or explained by appropriate choices of a Schwartz space of test functions on a spherical homogeneous space, which are in turn dictated by the geometry of affine spherical embeddings. We pursue this perspective by developing a local counterpart and try to explicate the functional equations. These constructions are also related to the L 2 -spectral decomposition of spherical homogeneous spaces in view of the Gelfand-Kostyuchenko method. To justify this viewpoint, we prove the convergence of p-adic local zeta integrals under certain premises, work out the case of prehomogeneous vector spaces and re-derive a large portion of Godement-Jacquet theory. Furthermore, we explain the doubling method and show that it fits into the paradigm of L-monoids developed by L. Lafforgue, B. C. Ngô et al., by reviewing the constructions of Braverman and Kazhdan (2002). In the global setting, we give certain speculations about global zeta integrals, Poisson formulas and their relation to period integrals.

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Section: Remarks On the General Casementioning

confidence: 99%

“…• The zeta integrals also have an interpretation as integration of matrix coefficients of G against the good sections in doubling method. There are subtleties related to normalization of intertwining operators; we refer to [51] for details.…”

Section: Remarks On the General Casementioning

confidence: 99%

Zeta Integrals, Schwartz Spaces and Local Functional Equations

2018

Lecture Notes in Mathematics

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“…The same operator is denoted by λ ! (η s ′ ψ ) in [BK02,Sha18]. To extend the domain of definition of λ !…”

Section: The Schwartz Space Of Y Pp ′ (F )mentioning

confidence: 99%

Schubert Eisenstein series and Poisson summation for Schubert varieties

Choie¹,

Getz²

2021

Preprint

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Schubert Eisenstein series are defined by restricting the summation in a degenerate Eisenstein series to a particular Schubert variety. In the case of GL 3 over Q they proved that these Schubert Eisenstein series have meromorphic continuations in all parameters and conjectured the same is true in general. We revisit their conjecture and relate it to the program of Braverman, Kazhdan, Lafforgue, Ngô, and Sakellaridis aimed at establishing generalizations of the Poisson summation formula. We prove the Poisson summation formula for certain schemes closely related to Schubert varieties and use it to refine and establish the conjecture of Bump and the first author in many cases. Contents 2.1. Braverman-Kazhdan spaces 2.2. The Y P 2.3. Examples 2.4. Function spaces 2.5. Measures 2.6. Quasi-characters 2.7. The unramified setting 3. The Schwartz space of Y P,P ′ (F ) 3.1. Induced representations

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“…We will not check this because it would make the current paper unnecessarily long. One ought to be able to obtain the precise relationship using the recent preprint of Shahidi [Sha17b] and its appendix by Li [Li17a].…”

Section: The Schwartz Space Of Xmentioning

confidence: 99%

A refined Poisson summation formula for certain Braverman-Kazhdan spaces

Getz

Liu

2020

Sci. China Math.

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Braverman and Kazhdan [BK00] introduced influential conjectures generalizing the Fourier transform and the Poisson summation formula. Their conjectures should imply that quite general Langlands L-functions have meromorphic continuations and functional equations as predicted by Langlands' functoriality conjecture. As evidence for their conjectures, Braverman and Kazhdan considered a setting related to the so-called doubling method in a later paper [BK02] and proved the corresponding Poisson summation formula under restrictive assumptions on the functions involved. The connection between the two papers is made explicit in [Li15]. In this paper we consider a special case of the setting of [BK02], and prove a refined Poisson summation formula that eliminates the restrictive assumptions of loc. cit. Along the way we provide analytic control on the Schwartz space we construct; this analytic control was conjectured to hold (in a slightly different setting) in [BK02]. Contents

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On Generalized Fourier Transforms for Standard L-Functions

Cited by 10 publications

References 20 publications

Zeta Integrals, Schwartz Spaces and Local Functional Equations

Zeta Integrals, Schwartz Spaces and Local Functional Equations

Schubert Eisenstein series and Poisson summation for Schubert varieties

A refined Poisson summation formula for certain Braverman-Kazhdan spaces

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