{"paper":{"title":"Relative Langlands duality and Koszul duality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Assuming the local conjecture for S-dual hyperspherical varieties holds and a polarization condition is met, S^1-equivariant localization produces an equivalence between the Z/2-graded B-equivariant D-modules on Y and the Z/2-graded unipotе","cross_cats":["math.RT","math.SG"],"primary_cat":"math.AG","authors_text":"Alexander Braverman, Michael Finkelberg, Roman Travkin","submitted_at":"2026-04-15T16:59:00Z","abstract_excerpt":"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{"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Using a variant of the S^1-equivariant localization of arxiv:0706.0322, we deduce an equivalence between the Z/2-graded B-equivariant category (D_ψ(Y)-mod)^{Z/2})^B and the Z/2-graded unipotent B^vee-monodromic category (Q(X^vee)-mod^{Z/2})^{B^vee,mon}.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The local conjecture of Ben-Zvi, Sakellaridis and Venkatesh holds for this pair of S-dual hyperspherical varieties, and X ≃ T^*_ψ(Y) is polarized so that Q(X)=D_ψ(Y).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Assuming the Ben-Zvi-Sakellaridis-Venkatesh local conjecture and polarization of X, a variant of S^1-equivariant localization yields an equivalence between the Z/2-graded B-equivariant (D_ψ(Y)-mod)^{Z/2} and the Z/2-graded unipotent B^vee-monodromic (Q(X^vee)-mod^{Z/2})^{B^vee,mon}.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Assuming the local conjecture for S-dual hyperspherical varieties holds and a polarization condition is met, S^1-equivariant localization produces an equivalence between the Z/2-graded B-equivariant D-modules on Y and the Z/2-graded unipotе","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"ab411cfbfb91fd0b5d161e4e636c5bf44af291417c7392d39a932f548bd6b77a"},"source":{"id":"2604.14085","kind":"arxiv","version":3},"verdict":{"id":"e3a272bb-a6f2-4062-95c5-48d69054b75d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T12:10:20.776934Z","strongest_claim":"Using a variant of the S^1-equivariant localization of arxiv:0706.0322, we deduce an equivalence between the Z/2-graded B-equivariant category (D_ψ(Y)-mod)^{Z/2})^B and the Z/2-graded unipotent B^vee-monodromic category (Q(X^vee)-mod^{Z/2})^{B^vee,mon}.","one_line_summary":"Assuming the Ben-Zvi-Sakellaridis-Venkatesh local conjecture and polarization of X, a variant of S^1-equivariant localization yields an equivalence between the Z/2-graded B-equivariant (D_ψ(Y)-mod)^{Z/2} and the Z/2-graded unipotent B^vee-monodromic (Q(X^vee)-mod^{Z/2})^{B^vee,mon}.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The local conjecture of Ben-Zvi, Sakellaridis and Venkatesh holds for this pair of S-dual hyperspherical varieties, and X ≃ T^*_ψ(Y) is polarized so that Q(X)=D_ψ(Y).","pith_extraction_headline":"Assuming the local conjecture for S-dual hyperspherical varieties holds and a polarization condition is met, S^1-equivariant localization produces an equivalence between the Z/2-graded B-equivariant D-modules on Y and the Z/2-graded unipotе"},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.14085/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}