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arxiv: 2604.14085 · v3 · pith:ZVLIX2T7new · submitted 2026-04-15 · 🧮 math.AG · math.RT· math.SG

Relative Langlands duality and Koszul duality

Alexander Braverman , Michael Finkelberg , Roman Travkin This is my paper

Pith reviewed 2026-05-22 10:38 UTC · model grok-4.3

classification 🧮 math.AG math.RTmath.SG
keywords relative Langlands dualityKoszul dualityhyperspherical varietiesS-dual varietiesequivariant localizationderived categoriesmonodromic categoriesBorel subgroups
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The pith

Assuming the local Ben-Zvi-Sakellaridis-Venkatesh conjecture, a variant of S^1-equivariant localization yields an equivalence between the Z/2-graded B-equivariant category of D_ψ(Y)-modules and the Z/2-graded unipotent B^vee-monodromic Q(X^

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

The paper establishes a categorical equivalence that links relative Langlands duality to Koszul duality for pairs of S-dual hyperspherical varieties equipped with quantizations. It assumes the local conjecture of Ben-Zvi, Sakellaridis and Venkatesh holds for the pair and that one variety X is polarized as a cotangent bundle so its quantization Q(X) equals D_ψ(Y). The authors then apply a modified form of S^1-equivariant localization to deduce that the Z/2-graded B-equivariant category of modules over D_ψ(Y) is equivalent to the Z/2-graded unipotent B^vee-monodromic category of modules over Q(X^vee). A reader would care because this gives an explicit bridge between equivariant and monodromic perspectives in geometric representation theory.

Core claim

For S-dual hyperspherical varieties G acting on X and G^vee acting on X^vee with equivariant quantizations, assuming the local Ben-Zvi-Sakellaridis-Venkatesh conjecture and polarization so that Q(X) = D_ψ(Y), a variant of S^1-equivariant localization implies an equivalence between the Z/2-graded B-equivariant category (D_ψ(Y)-mod^B)^{Z/2} and the Z/2-graded unipotent B^vee-monodromic category (Q(X^vee)-mod^{B^vee,mon})^{Z/2}.

What carries the argument

Variant of S^1-equivariant localization, which identifies the equivariant derived category on one side with the monodromic category on the dual side under the polarization and conjecture assumptions.

If this is right

  • Properties of B-equivariant D-modules on Y transfer to properties of unipotent B^vee-monodromic modules on X^vee.
  • The equivalence realizes a form of relative Langlands duality at the level of Z/2-graded categories.
  • Koszul duality phenomena appear explicitly through the localization relating the two sides.
  • The result applies whenever the input assumptions on the varieties and quantizations are met.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar localization arguments might produce equivalences for other graded versions or for non-unipotent monodromy conditions.
  • The equivalence could be used to compute Ext groups or characters in one category by transferring to the dual side.
  • Explicit examples with low-dimensional hyperspherical varieties would give concrete instances of the duality.
  • This approach may extend to settings without the Z/2-grading by refining the localization technique.

Load-bearing premise

The local conjecture of Ben-Zvi, Sakellaridis and Venkatesh holds for the pair of S-dual hyperspherical varieties, together with the polarization assumption that X is isomorphic to T^*_ψ(Y) so Q(X) equals D_ψ(Y).

What would settle it

A specific pair of hyperspherical varieties satisfying the polarization and local conjecture where the two Z/2-graded categories are not equivalent would falsify the claimed deduction from the localization variant.

read the original abstract

Consider a pair of $S$-dual hyperspherical varieties $G\circlearrowright X$ and $G^\vee\circlearrowright X^\vee$ equipped with equivariant quantizations $Q(X)$, $Q(X^\vee)$. Assume that the local conjecture of Ben-Zvi, Sakellaridis and Venkatesh holds for this pair, and also that $X\simeq T^*_\psi(Y)$ is polarized, so that $Q(X)=D_\psi(Y)$. Let $B\subset G$ (resp. $B^\vee\subset G^\vee$) be Borel subgroups. Then using a variant of the $S^1$-equivariant localization of arxiv:0706.0322, we deduce an equivalence between the ${\mathbb Z}/2$-graded $B$-equivariant category $(D_\psi(Y)\operatorname{-mod}^B)^{{\mathbb Z}/2}$ and the ${\mathbb Z}/2$-graded unipotent $B^\vee$-monodromic category $(Q(X^\vee)\operatorname{-mod}^{B^\vee,\operatorname{mon}})^{{\mathbb Z}/2}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript considers a pair of S-dual hyperspherical varieties G acting on X and G^vee acting on X^vee, equipped with equivariant quantizations Q(X) and Q(X^vee). Assuming the local conjecture of Ben-Zvi, Sakellaridis and Venkatesh holds for this pair and that X is polarized as T^*_ψ(Y) so that Q(X) = D_ψ(Y), the authors apply a variant of the S^1-equivariant localization theorem from arXiv:0706.0322 to deduce a Z/2-graded equivalence between the B-equivariant category (D_ψ(Y)-mod^B)^{Z/2} and the unipotent B^vee-monodromic category (Q(X^vee)-mod^{B^vee,mon})^{Z/2}.

Significance. If the stated assumptions hold, the result provides a conditional bridge between relative Langlands duality and Koszul duality for hyperspherical varieties by relating equivariant D-module categories to monodromic quantization categories via adapted localization. This could serve as a useful technical step in the geometric Langlands program, particularly when the BSV conjecture is verified for concrete pairs. The explicit conditioning on external results is a strength, as it avoids overclaiming.

major comments (1)
  1. [Main deduction (following the statement of the localization variant)] The central deduction relies on a variant of the S^1-equivariant localization from arXiv:0706.0322, but the manuscript does not provide a self-contained statement of the precise modifications needed to incorporate the Z/2-grading and the unipotent monodromy condition; this step is load-bearing for the equivalence and requires explicit verification that the variant preserves the relevant structures.
minor comments (2)
  1. [Introduction / Notation] The notation (D_ψ(Y)-mod^B)^{Z/2} and (Q(X^vee)-mod^{B^vee,mon})^{Z/2} is introduced without a preliminary definition of the Z/2-grading functor or the precise meaning of 'unipotent monodromic' in this context; adding a short paragraph on these conventions would improve readability.
  2. [Assumptions paragraph] The polarization assumption X ≃ T^*_ψ(Y) is invoked to identify Q(X) with D_ψ(Y), but a brief remark on how this interacts with the S-duality of the hyperspherical pair would clarify applicability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recommending minor revision. The result is conditional on the Ben-Zvi–Sakellaridis–Venkatesh local conjecture and on the polarization hypothesis, and we appreciate the referee’s recognition that the explicit conditioning is a strength. We address the single major comment below.

read point-by-point responses

  1. Referee: The central deduction relies on a variant of the S^1-equivariant localization from arXiv:0706.0322, but the manuscript does not provide a self-contained statement of the precise modifications needed to incorporate the Z/2-grading and the unipotent monodromy condition; this step is load-bearing for the equivalence and requires explicit verification that the variant preserves the relevant structures.

    Authors: We agree that the manuscript would benefit from a more explicit account of the variant. In the revised version we will add a short subsection immediately following the statement of the localization result. This subsection will record the precise modifications to the S^1-equivariant localization theorem of arXiv:0706.0322 that are needed to incorporate the Z/2-grading and the unipotent monodromy condition. We will verify that the adapted localization functor preserves the B-equivariant and B^vee-monodromic structures and induces an equivalence of the indicated Z/2-graded categories, by adapting the original arguments to the additional grading and monodromy data while keeping the same formal properties of the localization functor. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is conditional deduction from external inputs

full rationale

The paper states its central result as a direct deduction: assuming the Ben-Zvi-Sakellaridis-Venkatesh local conjecture holds for the given S-dual hyperspherical pair and that X ≃ T^*_ψ(Y) is polarized (so Q(X) = D_ψ(Y)), a variant of the S^1-equivariant localization from arXiv:0706.0322 yields the stated Z/2-graded equivalence between (D_ψ(Y)-mod^B)^{Z/2} and (Q(X^vee)-mod^{B^vee,mon})^{Z/2}. This structure relies on independent external hypotheses and a cited prior technique rather than any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. No equation or step reduces to its own input by construction, and the argument remains self-contained against the stated benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no free parameters, axioms, or invented entities are explicitly introduced or fitted in the provided text.

pith-pipeline@v0.9.0 · 5733 in / 1206 out tokens · 31385 ms · 2026-05-22T10:38:25.154956+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Assume that the local conjecture of Ben-Zvi, Sakellaridis and Venkatesh holds for this pair, and also that X ≃ T^*_ψ(Y) is polarized, so that Q(X) = D_ψ(Y). ... using a variant of the S^1-equivariant localization of [BZN], we deduce an equivalence between the Z/2-graded B-equivariant category (D_ψ(Y)-mod^B)^{Z/2} and the Z/2-graded unipotent B^vee-monodromic category (Q(X^vee)-mod^{B^vee,mon})^{Z/2}.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Theorem 1.3.1. (1) Assume the validity of Conjecture 1.1.2 ... Then the category (D(Y)^{Z/2})^B is equivalent to (Q_{ℏ=1}(X^vee)-mod^{Z/2})^{B^vee,mon}.

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unclear
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Works this paper leans on

22 extracted references · 22 canonical work pages · 1 internal anchor

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