{"id":"76d2a34f-1536-4079-9d83-f0cb740e03a7","arxiv_id":"2606.03187","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves equivalence of categories for the S-dual of SO(2n)×Sp(2n) on C^{2n}⊗C^{2n} being SO(2n+1)×SO(2n) on T*SO(2n+1) as a non-polarized case of relative Langlands duality, with consequence for functoriality via theta correspondence.","lead":"The paper proves a conjectured equivalence of categories: the S-dual of SO(2n) × Sp(2n) acting on C_+^{2n} ⊗ C_-^{2n} equals SO(2n+1) × SO(2n) acting on the cotangent bundle of SO(2n+1). This handles a specific case in the relative Langlands duality program and links it to theta correspondence via the derived Satake isomorphism.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Central claim rests on unverified extension of BSV relative Langlands duality to non-polarized orthosymplectic case without explicit condition checks.","rationale":"The reader's weakest_assumption correctly isolates the dependence on the external BSV framework and the lack of adaptation details as the load-bearing point. Because the full text is referenced but the abstract itself presents the result as a 'particular case' without further justification, the same concern remains the single most load-bearing one; no internal inconsistency or other technical gap is visible from the given material.","tokens_in":1860,"tokens_out":401,"duration_ms":17969,"concrete_test":"Locate the section (likely §3 or §4) that invokes the non-polarized BSV duality; extract the precise list of hypotheses required by that framework and check whether each is verified for the pair (SO_{2n}, Sp_{2n}) acting on the indicated representation; if any hypothesis is merely asserted rather than proved, the equivalence does not follow from the cited reference alone.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract asserts that the category equivalence for SO_{2n}×Sp_{2n} on C_+^{2n}⊗C_-^{2n} versus SO_{2n+1}×SO_{2n} on T^*SO_{2n+1} follows as a particular case of a non-polarized version of the Ben Zvi–Sakellaridis–Venkatesh framework, with only a passing reference to 'appropriate modifications' for osp(2m|2n). No derivation or verification is supplied showing that the requisite geometric or categorical conditions (e.g., the non-polarized duality hypotheses, the relevant Satake isomorphism compatibility, or the theta correspondence identification) hold for these groups. The cited prior results cover different pairs and the converse direction; the adaptation step therefore remains an external assumption rather than an internal derivation.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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).","tokens_in":2050,"tokens_out":435,"duration_ms":15541,"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":[{"comment":"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.","section":"Abstract"}],"minor_comments":[{"comment":"The symbol '∘learrowright' for the action should be defined or replaced by standard notation on first use.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and constructive feedback. We address the major comment point by point below.","responses":[{"response":"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_made":"yes","referee_comment":"[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."}],"tokens_in":1474,"tokens_out":319,"duration_ms":19285,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper establishes the conjectured category equivalence for the orthosymplectic pair SO(2n) × Sp(2n) acting on C_+^{2n} ⊗ C_-^{2n}, with the S-dual given by SO(2n+1) × SO(2n) on T^*SO(2n+1). This sits inside the non-polarized version of the Ben Zvi–Sakellaridis–Venkatesh framework and yields the stated consequence that Langlands functoriality for the derived Satake isomorphism of (Sp(2n), SO(2n)) comes from the theta correspondence. The approach is claimed to extend, with modifications, to the general even orthosymplectic case osp(2m|2n).\n\nIt builds directly on the earlier results for (SO(2n+1), Sp(2n)) by Braverman–Finkelberg–Kazhdan–Travkin and for (GL_n, GL_m) by Fu, plus the converse direction by Braverman–Finkelberg–Travkin. That gives a clear incremental advance inside the program.\n\nThe main limitation is that the abstract only gestures at “appropriate modifications” for the orthosymplectic setting without showing how the geometric or categorical hypotheses are checked. The stress-test note correctly flags that the adaptation step is asserted rather than derived in the visible text, so the central claim cannot be fully assessed from the abstract alone.\n\nThis is for specialists already working on relative Langlands duality or geometric Satake isomorphisms. A reader who knows the BSV setup and the cited prior papers will see the value immediately. The work is coherent on its own terms and addresses a specific open case with a concrete payoff, so it deserves a serious referee even if the modifications require close checking in review.","headline":"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.","tokens_in":2576,"tokens_out":454,"would_cite":false,"duration_ms":13210,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"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}.","keywords":["relative Langlands duality","orthosymplectic pairs","S-dual","category equivalence","theta correspondence","Satake isomorphism","representation theory"],"falsifier":"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.","tokens_in":2731,"feed_emoji":"","tokens_out":770,"duration_ms":22937,"temperature":0.7,"pith_summary":"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.","feed_headline":"S-dual equivalence proved for SO_{2n} and Sp_{2n} group pair","feed_subtitle":"Shows the dual of SO_{2n}×Sp_{2n} on tensor product space equals SO_{2n+1}×SO_{2n} on cotangent bundle of SO_{2n+1}.","key_machinery":"The S-dual equivalence of categories in the relative Langlands duality framework applied to orthosymplectic pairs.","core_discovery":"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).","pith_inferences":["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."],"forward_implications":["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)."],"fun_headline_variants":["S-dual of SO_{2n}×Sp_{2n} equals SO_{2n+1}×SO_{2n} on T^*SO_{2n+1}","S-dual equivalence for SO_{2n} Sp_{2n} on tensor product space","Relative Langlands duality result for orthosymplectic SO Sp pairs","Equivalence of categories for S-dual orthosymplectic actions"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["S-dual of SO_{2n}×Sp_{2n} equals SO_{2n+1}×SO_{2n} on T^*SO_{2n+1}","S-dual equivalence for SO_{2n} Sp_{2n} on tensor product space","Relative Langlands duality result for orthosymplectic SO Sp pairs","Equivalence of categories for S-dual orthosymplectic actions"]},"model":"grok-4.3","cost_usd":0.008991,"raw_usage":{"total_tokens":4085,"prompt_tokens":762,"num_sources_used":0,"completion_tokens":104,"cost_in_usd_ticks":89912000,"prompt_tokens_details":{"text_tokens":762,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3219,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":762,"tokens_out":104,"duration_ms":19194,"temperature":1.0,"reasoning_tokens":3219,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T08:22:20.520367+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}