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On the relative Langlands duality for $\operatorname{Sp}_{2n} \backslash \operatorname{GL}_{2n+1}$ (with an appendix by Zeyu Wang)

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A period identity verifies the predicted Langlands duality for Sp(2n)\GL(2n+1).

desk verdict A concrete, largely sound numerical and geometric check of a BZSV dual pair, held back by a genuine characteristic-2 gap in the orbit classification and a slightly overstated abstract. read the letter →

arxiv 2504.18774 v1 pith:DMZLWN75 submitted 2025-04-26 math.RT math.NT

Weixiao Lu , Zeyu Wang , Guodong Xi This is my paper · ORCID
classification math.RTmath.NT MSC 11F7022E5511F6614D24
keywords relativeLanglandsdualityBZSVconjecturesymplecticperiodsEisensteinserieshypersphericalvarietiesGrassmannianorbitclassificationRankin-SelbergL-functionsgeometric
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes a period identity for symplectic periods of Eisenstein series on $\mathrm{GL}_{2n+1}$ and reads it as numerical and geometric evidence for the relative Langlands duality conjecture (BZSV duality) in the case of the hyperspherical variety $T^*(\mathrm{Sp}_{2n}\backslash \mathrm{GL}_{2n+1})$. The main theorem says that for the Eisenstein representation attached to cuspidal data $(\Pi_n,\Pi_{n+1})$ on $\mathrm{GL}_n\times \mathrm{GL}_{n+1}$, the normalized symplectic period squared equals a ratio of completed zeta values, a ratio of $L$-functions, and a product of local normalized periods, with the global $L$-factor exactly the one attached to the conjectured dual variety $T^*(\mathrm{GL}_n\times \mathrm{GL}_{n+1}\backslash \mathrm{GL}_{2n+1})$. The paper also proves vanishing of symplectic periods for cuspidal representations and for Eisenstein series attached to other maximal parabolics, and a geometric \'etale analogue in the appendix: the cuspidal part of the corresponding period functor vanishes. If correct, this supplies one of the few explicit, global checks of a duality conjecture that organizes many period--$L$-function relations.

What carries the argument

The machinery is the orbit decomposition of the Grassmannian $\mathrm{Gr}(n,2n+1)(F)$ under $\mathrm{Sp}_{2n}(F)$, which indexes the double cosets $P_{n,n+1}(F)\backslash \mathrm{GL}_{2n+1}(F)/\mathrm{Sp}_{2n}(F)$. The orbits split into four types (I--IV) according to the Witt index of the subspace and the position of the line spanned by $e_{2n+1}$; for every orbit except the isotropic type-I orbit, whose stabilizer is the Siegel parabolic, the stabilizer contains a normal unipotent subgroup $S_\gamma\cdot U_\gamma$ whose adelic quotient integral vanishes by cuspidality or by a Klingen-mirabolic vanishing lemma. The remaining main-orbit contribution is the intertwining period $J(\varphi,\lambda)$, which unfolds to a Whittaker-model integral; at unramified places that integral evaluates to the Rankin--Selberg factor $L(s+1,\Pi_n^\vee\times \Pi_{n+1})$, which supplies the $L$-function in the main identity.

What would settle it

For $n=2$ over a global field of characteristic 2, explicitly list the $\mathrm{Sp}_4(F)$-orbits on $\mathrm{Gr}(2,5)(F)$; if any orbit beyond the four types I--IV exists, or any listed stabilizer lacks a normal unipotent subgroup whose period integral vanishes, then Proposition 6.2.1 fails. Alternatively, compute both sides of (1.1.2) for one explicit unramified datum in that setting and check whether the equality holds.

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Extended reading notes

Core claim

The central discovery is the exact formula $$ \frac{|P(E(\cdot,\varphi))|^2}{\langle f,f\rangle_{\mathrm{Pet}}} = \frac{\zeta^*(1)\zeta(3)\cdots\zeta(2n+1)}{\zeta(2)\cdots\zeta(2n)} \cdot \frac{L^*(1,\Pi,\hat\rho_0)}{L^*(1,\Pi,\mathrm{Ad})^2} \cdot \prod_v \frac{|P^\natural_v(f_v)|^2}{\langle f_v,f_v\rangle^\natural} $$ for the $\mathrm{Sp}_{2n}$-period of the Eisenstein series attached to a cuspidal representation $\Pi = \Pi_n \boxtimes \Pi_{n+1}$. The paper interprets the $L$-ratio as the BZSV-predicted spectral factor for the dual pair $(\mathrm{GL}_{2n+1},\mathrm{GL}_n\times \mathrm{GL}_{n+1})$, and it proves that all non-main orbits in the unfolding contribute zero, leaving only the term attached to the Siegel parabolic. It further shows that symplectic periods of genuine cusp forms on $\mathrm{GL}_{2n+1}$ vanish, and that the same vanishing holds in the \'etale geometric setting for the cuspidal part of the functor $\Gamma_c(\pi_*F)$.

Load-bearing premise

The load-bearing premise is the completeness of the Section 5 orbit classification: the Darboux-basis lemma used there is proved only in characteristic not 2, while the main theorem is stated for all global fields, so in characteristic 2 a missed orbit or miscomputed stabilizer would invalidate the unfolding and change the period identity.

Editorial extensions

If this is right

  • The global identity (1.1.2) confirms, in the tested Eisenstein family, the BZSV prediction that $T^*(\mathrm{Sp}_{2n}\backslash \mathrm{GL}_{2n+1})$ and $T^*(\mathrm{GL}_n\times \mathrm{GL}_{n+1}\backslash \mathrm{GL}_{2n+1})$ are dual Hamiltonian varieties.
  • Symplectic periods of cuspidal automorphic forms on $\mathrm{GL}_{2n+1}$, and of Eisenstein series attached to maximal parabolics of type other than $(n,n+1)$, vanish identically.
  • Because the local normalized periods are 1 at unramified places, the unramified form of the identity is a pure equality of completed zeta values and $L$-functions.
  • In the geometric setting, the cuspidal part of the period functor $\Gamma_c(\pi_*F)$ vanishes for $X=\mathrm{GL}_{2n+1}/\mathrm{Sp}_{2n}$ and $X=\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}$, matching the geometric BZSV conjecture on the irreducible locus.
  • Together with the companion linear-period results for the dual pair, the duality is verified in both directions: the linear period computes an $L$-value, and the symplectic period computes the dual $L$-ratio.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the period identity extends from this single-Eisenstein family to a full spectral expansion, the same $L$-ratio should control the symplectic-period Plancherel formula for $\mathrm{GL}_{2n+1}$, giving a spectral interpretation of the duality beyond the cuspidal case.
  • The characteristic-2 gap is testable in small rank: a direct enumeration of $\mathrm{Sp}_4(F)$-orbits on $\mathrm{Gr}(2,5)(F)$ over $\mathbb{F}_2(t)$ would either locate a missing orbit or confirm that the Darboux-basis lemma has a characteristic-2 analogue, and either outcome sharpens the range of the theorem.
  • The same unfolding mechanism---one main orbit plus vanishing normal subgroups on all other orbits---should apply to other hyperspherical pairs $(G,H)$, yielding explicit period identities of the same shape for orthogonal or unitary periods.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies the BZSV relative Langlands duality conjecture for the hyperspherical Hamiltonian variety T*(Sp_{2n}\GL_{2n+1}). Its main numerical result, Theorem 1.1.3, asserts that Sp_{2n}-periods of cuspidal automorphic forms on GL_{2n+1} vanish, and that for Eisenstein series attached to cuspidal data (Π_n, Π_{n+1}) the normalized square period equals an explicit product of zeta values, a ratio of L-values, and a product of local normalized periods. The proof structure is: absolute convergence via Harder--Narasimhan truncation in the function-field case and Zydor truncation in the number-field case (§3); vanishing of certain Klingen-mirabolic and symplectic periods (§4); classification of Sp_{2n}-orbits on the relevant Grassmannian (§5); and unfolding of the period integral to the main orbit, followed by a local Rankin--Selberg computation (§6). An appendix by Zeyu Wang formulates and proves étale geometric analogues: cuspidal sheaves on Bun_{GL_{2n+1}} or Bun_{GL_{2n}} have zero integral over the corresponding Sp_{2n}-stack.

Significance. If the main theorem is correct, this is a substantial, parameter-free verification of the BZSV duality in a new hyperspherical family, with exact global constants and local normalizations. The derivation is structural rather than fitted: the local L-factor appears from an unramified Rankin--Selberg computation, the zeta constants arise from Tamagawa measures, and the BZSV prediction is used only at the end to name the L-factor L(1,Π,ρ0). The geometric appendix is a valuable independent piece of evidence. However, the proof as written does not cover characteristic 2 function fields, despite the theorem being stated without such a restriction, and one displayed zeta constant is at best ambiguous. These issues are local and repairable, which is why I recommend major revision rather than rejection.

major comments (3)
  1. [§5.1, Lemma 5.1.1; Theorem 1.1.3; Appendix A] Theorem 1.1.3 is stated for a global field F with no characteristic restriction, and the abstract advertises evidence over function fields generally. However, the orbit classification used in Proposition 6.2.1 relies on Lemma 5.1.1, which is explicitly stated only for char(F)≠2, and the proof of Proposition 6.2.1 depends on the complete list of representatives and stabilizers in §5.2. A missed orbit or a different stabilizer in characteristic 2 would change the period identity (1.1.2). The geometric appendix is also affected when the base field has characteristic 2. Please either give a characteristic-2 proof of Lemma 5.1.1 and of the orbit classification, or restrict the main theorem and the appendix to char(F)≠2 and state this restriction in the abstract and introduction. The sentence in §6 saying that the function-field case follows by a similar, easier argument should also be expanded, since the theorem claims all global fields.
  2. [Theorem 1.1.3(2) and §6.4] The zeta constant displayed in (1.1.2) is not the one obtained from the Tamagawa computation in §6.4. From Δ*_G=ζ*(1)ζ(2)⋯ζ(2n+1) and Δ*_H=ζ(2)ζ(4)⋯ζ(2n), the ratio Δ*_G/(Δ*_H)^2 simplifies to ζ*(1)ζ(3)ζ(5)⋯ζ(2n+1)/(ζ(2)ζ(4)⋯ζ(2n)), with the denominator running over even integers. As printed, "ζ(2)⋯ζ(2n)" is naturally read as the product over all integers 2,...,2n, which is different for n≥2. Please correct the denominator, or explicitly define the product as running over even indices, and check the identity for n=2.
  3. [§5.1] The parameter r is called the "Witt index" of W, but for an n-dimensional subspace W of a 2n-dimensional symplectic space the usual Witt index lies in {ceil(n/2),...,n}, never in {0,...,[n/2]}. The quantity actually used in the representatives is the half-rank of the restricted symplectic form, i.e. the number of hyperbolic pairs. The classification statements should be phrased in terms of half-rank, and the resulting ranges for types I–IV should be restated accordingly. This is more than a terminological point, because the orbit parametrization is load-bearing for Proposition 6.2.1, and the distinction is especially important in characteristic 2.
minor comments (5)
  1. [§2.2.1] The product for L_{Sp_{2n}}(s) should be written as ζ(s+2)ζ(s+4)⋯ζ(s+2n); in the current text "ζ(s+2)⋯ζ(s+2n)" is ambiguous, and the same ambiguity propagates to the zeta denominator in (1.1.2).
  2. [§5.2] The displayed matrices for γhγ^{-1} in the Type I–IV stabilizer computations are difficult to parse and several blocks are misaligned in the submitted text; please typeset them with explicit block indices and sizes.
  3. [Proposition 6.2.1] After listing the subgroups Sγ·Uγ, the proof says "Using cuspidality or Corollary 4.3.3, the integral on this normal subgroup already vanishes" without explicitly identifying, for each orbit type, the flag data that matches Sγ·Uγ to the subgroups appearing in Corollary 4.3.3. A table or a short case-by-case verification would make the unfolding argument checkable.
  4. [Theorem 1.1.3 and §6.4] The local normalized inner product ⟨φ_v,φ_v⟩♮ is introduced only in §6.4, after Theorem 1.1.3 is stated; adding a forward reference in the theorem would help the reader.
  5. [Remark 4.3.2] The remark makes a strong claim that a local vanishing statement in [AGR93] is incorrect. The paper's own computation supports the intended local nonvanishing, but the remark should either give a precise local Hom statement with a reference to the local BZSV conjecture or be trimmed.

Circularity Check

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No circular derivation: the period formula is unfolded from first principles; BZSV is invoked only to label L-factors after the computation.

full rationale

The central identity (Theorem 1.1.3(2)) is obtained through a genuine chain of independent computations: absolute convergence of the Eisenstein period (§3), vanishing of non-main double-coset contributions via cuspidality and the orbit classification (§§4–5), unfolding to an intertwining period and then to a global Whittaker integral (§6.3, eqs. (6.4.1)–(6.4.3)), and the unramified local identity P_{H,v}(φ°, λ) = L(sλ+1, Π∨_{n,v}×Π_{n+1,v}) taken from standard Rankin–Selberg theory [JS81], together with the Petersson norm formula from [JS81, §4]. No parameter is fitted: the local normalized periods P♮_{H,v} are defined by dividing the local integral by the Rankin–Selberg L-factor, and the same Whittaker-integral identity is asserted globally for all φ. The BZSV prediction enters only at the very end, where L*(1, Π, ρ_0) is introduced as a new name for the already-computed L*(1, Π_n⊞Π_{n+1}, Ad); this relabeling supplies no ingredient of the derivation. The cited [BSV24] conjecture is an external target statement, not a proof input, and no load-bearing argument reduces to a self-citation: the only self-authored citation ([LW25], by one of the appendix authors) appears in a remark about potential applications and plays no role in the proof. The characteristic-2 caveat around Lemma 5.1.1 is a genuine generality gap for the theorem as stated, but it is a completeness/correctness concern, not a circularity: no orbit or stabilizer is being taken as an assumption equivalent to the final formula.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No numerical fitting appears in the paper. The identity is fully determined by the cuspidal data and the fixed additive character and measures; the local normalized periods are defined so as to be 1 at unramified places. No new physical or mathematical entities are postulated; the period sheaf and L-sheaf used in the appendix already belong to the BZSV framework.

assumptions (6)
  • standard math Witt's theorem and the Darboux-basis lemma for flags in symplectic spaces (Lemma 5.1.1, stated for char(F) not equal to 2).
    Used to classify Sp(2n)(F)-orbits on the Grassmannian in Section 5; this classification is needed for the unfolding in Proposition 6.2.1.
  • standard math Meromorphic continuation, constant term formula, and intertwining operators for Eisenstein series (Langlands; Bernstein-Lapid; Moeglin-Waldspurger).
    Used throughout Sections 3, 4, and 6 to define and manipulate Eisenstein series and pseudo-Eisenstein series.
  • standard math Zydor's truncation theorem (Theorem 3.2.2, from Zydor 2019) that certain truncated functions are Schwartz.
    Establishes absolute convergence of the periods in the number field case in Section 3.2.
  • standard math The unramified local Rankin-Selberg computation of the symplectic and Whittaker period (Jacquet-Shalika [JS81]).
    Gives the local L-factor in Section 6.4 that turns the unfolded period into the global L-function ratio.
  • domain assumption The geometric Langlands equivalence ShvNilp(Bun_G) equivalent to IndCohNilp(Loc_Ghat), miraculous duality, and the Fourier transform functor of [FYZ23].
    The appendix uses these to define geometric symplectic periods and to compare with [BSV24, Conjecture 12.1.1].
  • ad hoc to paper The field characteristic is not 2 for the orbit classification in Lemma 5.1.1.
    The lemma is stated with char(F) not equal to 2, but the main theorem is stated for all global fields. If function fields of characteristic 2 are included, this assumption needs a separate argument.

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Pith. "Pith review of On the relative Langlands duality for $\operatorname{Sp}_{2n} \backslash \operatorname{GL}_{2n+1}$ (with an appendix by Zeyu Wang)." pith.science (2026). https://pith.science/paper/DMZLWN75

@misc{pith2026250418774,
  author       = {Pith},
  title        = {Pith review of: On the relative Langlands duality for $\operatornameSp_2n \backslash \operatornameGL_2n+1$ (with an appendix by Zeyu Wang)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DMZLWN75}},
  note         = {Machine review of arXiv:2504.18774}
}
abstract

We verify the relative Langlands duality conjecture proposed by Ben-Zvi, Sakellaridis, Venkatesh for the hyperspherical Hamiltonian variety $T^*(\operatorname{Sp}_{2n}\backslash \operatorname{GL}_{2n+1})$. We provide numerical (over number fields and function fields) and geometric (in the \'{e}tale setting) evidence that its dual Hamiltonian variety should be $T^*(\operatorname{GL}_n \times \operatorname{GL}_{n+1} \backslash \operatorname{GL}_{2n+1})$ as is predicted by Ben-Zvi, Sakellaridis, Venkatesh.

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Works this paper leans on

9 extracted references · 5 canonical work pages

  1. [1]

    Vanishing periods of cusp forms over modular symbols

    [AGR93] Avner Ash, David Ginzburg, and Steven Rallis. “Vanishing periods of cusp forms over modular symbols”. In: Mathematische Annalen 296.1 (Dec. 1993), pp. 709–723. [Ari+22a] D. Arinkin, D. Gaitsgory, D. Kazhdan, S. Raskin, N. Rozenblyum, and Y. Varshavsky. Duality for automorphic sheaves with nilpotent singular support

    work page 1993

  2. [3]

    A trace formula for reductive groups I terms associated to classes in G (Q)

    arXiv: 2010.01906 [math.AG]. [Art78] James G Arthur. “A trace formula for reductive groups I terms associated to classes in G (Q)”. In: (1978). [BCZ22] Rapha¨ el Beuzart-Plessis, Pierre-Henri Chaudouard, and Micha l Zydor. “The global Gan-Gross-Prasad conjecture for unitary groups: the endoscopic case”. In: Publ. Math. Inst. Hautes ´Etudes Sci. 135 (2022)...

    arXiv 1978

  3. [8]

    Residual spectrum of GL 2 n distinguished by the symplectic group

    arXiv: 2310.17837 [math.NT]. [Off06] Omer Offen. “Residual spectrum of GL 2 n distinguished by the symplectic group”. In: (2006). [Shm94] A. S. Shmel¨ ev. “Orbits of the symplectic group on the manifold of complete flags of a symplectic space”. In: Izv. Ross. Akad. Nauk Ser. Mat. 58.1 (1994), pp. 144–166. [Zyd19] Micha l Zydor. Periods of automorphic form...

    arXiv 2006

  4. [2017]

    Enhanced adic formalism and perverse t-structures for higher Artin stacks

    arXiv: 1404.1128 [math.AG]. [MO21] Arnab Mitra and Omer Offen. “On distinguished representations of the quasi-split unitary groups”. In: Journal of the Institute of Mathematics of Jussieu 20.1 (2021), pp. 225–276. [MW95] C Moeglin and J. L. Waldspurger. Spectral Decomposition and Eisenstein Series: A Paraphrase of the Scriptures. Cambridge Tracts in Mathe...

    work page Pith review arXiv 2021

  5. [2019]

    arXiv: 1903.01697 [math.NT]. Massachusetts Institute of Technology, Department of Mathematics, 77 Massachusetts A venue, Cambridge, MA 02139, USA Email address : weixiaol@mit.edu University of Minnesota, Department of Mathematics, 206 Church St. S.E., Minneapolis, MN 55455, USA Email address : xi000023@umn.edu 43

    work page Pith review arXiv 1903

  6. [2022]

    07665 [math.AG]

    arXiv: 2012 . 07665 [math.AG]. 41 [Ari+22b] D. Arinkin, D. Gaitsgory, D. Kazhdan, S. Raskin, N. Rozenblyum, and Y. Varshavsky. The stack of local systems with restricted variation and geometric Langlands theory with nilpotent singular support

    work page 2012

  7. [2023]

    On the motive of a reductive group

    arXiv: 2308.10979 [math.NT]. [Gro97] Benedict H Gross. “On the motive of a reductive group”. In: Inventiones mathematicae 130.2 (1997), pp. 287–314. [JR92a] Herv´ e Jacquet and Stephen Rallis. “Symplectic periods”. In: J. reine angew. Math 423.1992 (1992), pp. 175–197. [JR92b] Herv´ e Jacquet and Stephen Rallis. “Symplectic periods.” In: Journal f¨ ur die...

    arXiv 1997

  8. [2024]

    Linear periods

    arXiv: 2409.04677 [math.RT]. [FJ93] Solomon Friedberg and Herv´ e Jacquet. “Linear periods”. In: J. Reine Angew. Math. 443 (1993), pp. 91–139. [FYZ23] Tony Feng, Zhiwei Yun, and Wei Zhang. Modularity of higher theta series I: cohomology of the generic fiber

    arXiv 1993

Show all 9 references
  1. [2025]

    Local geometrised Rankin-Selberg method for GL(n)

    arXiv: 2504.00275 [math.NT]. [Lys02] Sergey Lysenko. “Local geometrised Rankin-Selberg method for GL(n)”. In: DUKE MATHEMATICAL JOURNAL 111.3 (2002). [Lys08] Sergey Lysenko. “Geometric Waldspurger periods”. In: Compositio Mathematica 144.2 (2008), pp. 377–438. [Lys21] S Lysenk...

    2002 arXiv

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