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Towards the Relative Langlands Duality for Orthosymplectic Pairs

T0 review · 1 major / 1 minor · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read The S-dual of SO_{2n}×Sp_{2n} acting on C_+^{2n}⊗C_-^{2n} equals SO_{2n+1}×SO_{2n} acting on T^*SO_{2n+1}.

desk verdict Mezer proves the (SO(2n), Sp(2n)) case of the relative Langlands duality conjecture and links it to theta correspondence for the derived Satake isomorphism. read the letter →

arxiv 2606.03187 v1 pith:ILHLEJ6Q submitted 2026-06-02 math.RT math.AG

Dor Mezer This is my paper · ORCID
classification math.RTmath.AG
keywords relativeLanglandsdualityorthosymplecticpairsS-dualcategoryequivalencethetacorrespondenceSatakeisomorphismrepresentationtheory
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

The paper proves a conjectured equivalence of categories that identifies the S-dual of one orthosymplectic group pair with another. Specifically, the action of SO_{2n} and Sp_{2n} on the tensor product of two 2n-dimensional spaces is shown to be dual to the action of SO_{2n+1} and SO_{2n} on the cotangent bundle of SO_{2n+1}. This instance is presented as a case of the non-polarized local relative Langlands duality. The result also establishes that the theta correspondence realizes Langlands functoriality for the derived Satake isomorphism between Sp_{2n} and SO_{2n}. The approach extends with modifications to the general even orthosymplectic setting.

What carries the argument

The S-dual equivalence of categories in the relative Langlands duality framework applied to orthosymplectic pairs.

What would settle it

An explicit computation for n=1 showing that the two categories have different numbers of irreducible objects or different endomorphism rings would disprove the claimed equivalence.

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Extended reading notes

Core claim

We prove that the S-dual of SO_{2n}×Sp_{2n} acting on C_+^{2n}⊗C_-^{2n} is SO_{2n+1}×SO_{2n} acting on T^*SO_{2n+1}. This equivalence is a particular case of the non-polarized version of the local relative Langlands duality, building on earlier results for pairs such as (SO_{2n+1}, Sp_{2n}) and (GL_n, GL_m).

Load-bearing premise

The result assumes the relative Langlands duality framework holds in its non-polarized form and that prior results for related pairs extend to this orthosymplectic setting without further justification.

Editorial extensions

If this is right

  • Langlands functoriality of the Derived Satake isomorphism for Sp_{2n} and SO_{2n} is realized by the theta correspondence.
  • The method applies with modifications to the general even orthosymplectic case of osp(2m|2n).
  • The same type of category equivalence was previously established for the pairs (SO_{2n+1}, Sp_{2n}) and (GL_n, GL_m).

Reading between the lines

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

  • The equivalence may allow transfer of representation-theoretic questions from symplectic to orthogonal sides via the theta correspondence.
  • Similar dualities could be tested for other supergroup pairs beyond the even orthosymplectic case.
  • The categorical statement might imply matching of certain geometric invariants or characters between the two sides.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 1 minor

Summary. The manuscript proves a conjectured equivalence of categories showing that the S-dual of SO_{2n}×Sp_{2n} acting on C_+^{2n}⊗C_-^{2n} equals SO_{2n+1}×SO_{2n} acting on T^*SO_{2n+1}. This is presented as a particular case of a non-polarized version of the local relative Langlands duality of Ben Zvi, Sakellaridis and Venkatesh. Similar results for other pairs were proved earlier; the converse direction was also known. As a consequence, Langlands functoriality of the Derived Satake isomorphism for (Sp_{2n}, SO_{2n}) is realized by the theta correspondence. The approach extends (with modifications) to the general even orthosymplectic case osp(2m|2n).

Significance. If the central equivalence holds, the work supplies a new family of examples in the relative Langlands program, connecting orthosymplectic duality to the theta correspondence and to prior results on classical groups. It would give concrete support for the non-polarized BSV framework and yield a functoriality statement that is directly testable via known theta lifts.

major comments (1)
  1. [Abstract] Abstract: the claim that the stated equivalence 'follows as a particular case' of the non-polarized BSV duality after 'appropriate modifications' for osp(2m|2n) is load-bearing for the main theorem, yet the text supplies no explicit check that the requisite geometric or categorical hypotheses (non-polarized duality conditions, Satake isomorphism compatibility, theta correspondence identification) hold for these groups; the cited prior results address different pairs and the converse direction.
minor comments (1)
  1. The symbol '∘learrowright' for the action should be defined or replaced by standard notation on first use.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the stated equivalence 'follows as a particular case' of the non-polarized BSV duality after 'appropriate modifications' for osp(2m|2n) is load-bearing for the main theorem, yet the text supplies no explicit check that the requisite geometric or categorical hypotheses (non-polarized duality conditions, Satake isomorphism compatibility, theta correspondence identification) hold for these groups; the cited prior results address different pairs and the converse direction.

    Authors: We agree that the abstract claim is load-bearing and that an explicit verification of the hypotheses would strengthen the manuscript. The body of the paper proves the equivalence directly by adapting the BSV framework to the orthosymplectic setting via the modifications described (particularly in the sections treating the non-polarized case and the theta correspondence). However, a separate, consolidated check confirming that the non-polarized duality conditions, Satake compatibility, and theta identification hold for these specific groups is not provided. In the revised version we will add a dedicated subsection (in the introduction or a new section on the BSV connection) that supplies this verification, explicitly distinguishing the result from the cited works on different pairs and from the known converse direction. This revision will make the 'particular case' statement fully substantiated. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; result framed as case of external BSV framework by different authors

full rationale

The abstract explicitly positions the main theorem as 'a particular case of a non-polarized version of the (local) relative Langlands duality of Ben Zvi, Sakellaridis and Venkatesh' and cites independent prior results by Braverman-Finkelberg-Kazhdan-Travkin and Fu for related pairs. No self-citations appear, no parameters are fitted then renamed as predictions, and no equations reduce by construction to inputs. The derivation chain is presented as building on external conjectures rather than internal self-reference or ansatz smuggling.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The claim rests on the relative Langlands duality conjecture of Ben Zvi, Sakellaridis and Venkatesh and on prior category equivalences proved by Braverman, Finkelberg, Kazhdan, Travkin and Fu; no free parameters or invented entities are mentioned in the abstract.

assumptions (1)
  • standard math Standard results from representation theory and algebraic geometry underlying the relative Langlands duality framework
    Invoked as the ambient setting for the non-polarized version of the duality.

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Cite this review

Pith. "Pith review of Towards the Relative Langlands Duality for Orthosymplectic Pairs." pith.science (2026). https://pith.science/paper/ILHLEJ6Q

@misc{pith2026260603187,
  author       = {Pith},
  title        = {Pith review of: Towards the Relative Langlands Duality for Orthosymplectic Pairs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ILHLEJ6Q}},
  note         = {Machine review of arXiv:2606.03187}
}
abstract

In this paper we prove a conjectured equivalence of categories, showing that the S-dual of $\mathrm{SO}_{2n}\times \mathrm{Sp}_{2n}$ acting on $\mathbb{C}_+^{2n}\otimes \mathbb{C}_-^{2n}$ is equal to $\mathrm{SO}_{2n+1}\times \mathrm{SO}_{2n}\circlearrowright T^*\mathrm{SO}_{2n+1}$. This result is a particular case of a non-polarized version of the (local) relative Langlands duality of Ben Zvi, Sakellaridis and Venkatesh. Similar results for the pairs $(\mathrm{SO}_{2n+1}, \mathrm{Sp}_{2n})$ and $(\mathrm{GL}_n, \mathrm{GL}_m)$ were proved by Braverman, Finkelberg, Kazhdan and Travkin and by Fu respectively, whereas the converse result was proved by Braverman, Finkelberg, and Travkin. As a consequence of our main result, we prove that Langlands functoriality of the Derived Satake isomorphism for the pair $\mathrm{Sp}_{2n},\mathrm{SO}_{2n}$ is given by the theta correspondence. Our approach works (with appropriate modifications) in the general even orthosymplectic case of $\mathfrak{osp}(2m|2n)$.

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