Zeta Integrals, Schwartz Spaces and Local Functional Equations

Li, Wen-Wei

doi:10.1007/978-3-030-01288-5

Cited by 19 publications

(46 citation statements)

References 49 publications

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“…Some similar zeta integrals have been studied in [6], which is based on case-by-case discussion with explicit computations. By the way, the results here also complement for some missing details in [28] in the Archimedean setting.…”

Section: Introductionsupporting

confidence: 66%

Towards Generalized Prehomogeneous Zeta Integrals

2018

Lecture Notes in Mathematics

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Let X be a prehomogeneous vector space under a connected reductive group G over R.Assume that the open G-orbit X + admits a finite covering by a symmetric space. We study certain zeta integrals involving (i) Schwartz functions on X, and (ii) generalized matrix coefficients on X + (R) of Casselman-Wallach representations of G(R), upon a twist by complex powers of relative invariants. This merges representation theory with prehomogeneous zeta integrals of Igusa et al. We show their convergence in some shifted cone, and prove their meromorphic continuation via the machinery of bfunction together with V. Ginzburg's results on admissible D-modules. This provides some evidence for a broader theory of zeta integrals associated to affine spherical embeddings.

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Section: Introductionsupporting

confidence: 66%

Towards Generalized Prehomogeneous Zeta Integrals

2018

Lecture Notes in Mathematics

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“…We begin by reviewing the basic set-up about prehomogeneous vector spaces from [21]; see also [22,Chapter 6] or [27,14]. The following assumptions will remain in force throughout this article.…”

Section: Prehomogeneous Vector Spaces 21 Relative Invariants and Regu...mentioning

confidence: 99%

“…In [22], the author proposed a general framework to define zeta integrals whenever one has a spherical homogeneous G-space X + , an equivariant embedding X + ֒→ X together with a reasonable notion of Schwartz space and Fourier transform. That project is largely speculative, the only accessible case being the setting of prehomogeneous vector spaces mentioned above.…”

Section: Introductionmentioning

confidence: 99%

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Generalized zeta integrals on certain real prehomogeneous vector spaces

Li¹

2019

Preprint

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Let X be a real prehomogeneous vector space under a reductive group G, such that X is an absolutely spherical G-variety with affine open orbit. We define local zeta integrals that involve the integration of Schwartz-Bruhat functions on X against generalized matrix coefficients of admissible representations of G(R), twisted by complex powers of relative invariants. We establish the convergence of these integrals in some range, the meromorphic continuation as well as a functional equation in terms of abstract γ-factors. This subsumes the Archimedean zeta integrals of Godement-Jacquet, those of Sato-Shintani (in the spherical case), and the previous works of Bopp-Rubenthaler. The proof of functional equations is based on Knop's results on Capelli operators. MSC (2010) 11S40; 11S90 43A85 KeywordsZeta integrals, prehomogeneous vector spaces, Capelli operators where η ∈ N π (X + ), v ∈ V π , and ξ ∈ S(X). The goal of this article is to prove three basic properties of these integrals, in increasing level of difficulty:Convergence (Theorem 3.10) The integral Z λ (η, v, ξ) converges for Re(λ) ≥ X κ for some κ ∈ Λ R depending only on π and (G, ρ, X), and it is jointly continuous in (v, ξ) in that range.Meromorphic continuation (Theorem 3.12) Z λ (η, v, ξ) admits a meromorphic continuation to all λ ∈ Λ C . To be precise, there exists a holomorphic function L(η, λ) on Λ C for any given η, not identically zero, such that LZ λ (η, v, ξ) := L(η, λ)Z λ (η, v, ξ) extends holomorphically to all λ ∈ Λ C .Functional equation (Theorem 3.13) Fix an additive character ψ and denote the integral for (G, ρ, X) as Žλ . There is then a unique meromorphic family of C-linear maps γ(π, λ) :for all η ∈ N π ( X+ ), v ∈ V π and ξ ∈ S(X), where both sides are viewed as meromorphic families in λ ∈ Λ C .Moreover, one can obtain slightly more information on the "denominator" L(η, λ), and describe the dependence of γ(λ, π) on ψ; it turns out that the γ-factor, which is actually a linear transform, is generically invertible (Proposition 3.14). We refer to the cited Theorems for the precise statements.Note that our formalism is non-trivial only when N π (X + ) = {0}; in other words, π must be distinguished by X + (R). Distinguished representations and their generalized matrix coefficients are the main concerns of harmonic analysis on spherical varieties.The same result hold for prehomogeneous vector spaces over C; see §3.4.

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“…As a result, the Schwartz space S(X P (F )) admits a natural inductive limit topology under which it is nuclear, barreled, and separable (see [Li18, §4.1].) These topological properties of S(X P (F )) are stated as axioms in [Li18] to validate Weil's interpretation of zeta integrals as a unique family of tempered distributions. We expect an analogous result in the archimedean setting can be derived similarly.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics of Schwartz functions: nonarchimedean

Hsu¹

2021

Preprint

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Let G be a split, simply connected, simple group, and let P ≤ G be a maximal parabolic. Braverman and Kazhdan in [BK02] defined a Schwartz space on the affine closure XP of X • P := P der \G. An alternate, more analytically tractable definition was given in [GHL21], following several earlier works. In the nonarchimedean setting when G is a classical group or G2, we clarify the relationship between the two definitions and prove several previously conjectured properties of the Schwartz space that will be useful in applications. In addition, we prove that the quotient of the Schwartz space by the space of compactly supported smooth functions on the open orbit is of finite length and we describe its subquotients. Finally, we use our work to study the set of possible poles of degenerate Eisenstein series under certain assumption at archimedean places.

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Zeta Integrals, Schwartz Spaces and Local Functional Equations

Cited by 19 publications

References 49 publications

Towards Generalized Prehomogeneous Zeta Integrals

Towards Generalized Prehomogeneous Zeta Integrals

Generalized zeta integrals on certain real prehomogeneous vector spaces

Asymptotics of Schwartz functions: nonarchimedean

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