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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{","authors_text":"Alexander Braverman, Michael Finkelberg, Roman Travkin","cross_cats":["math.RT","math.SG"],"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","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2026-04-15T16:59:00Z","title":"Relative Langlands duality and Koszul duality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.14085","kind":"arxiv","version":3},"verdict":{"created_at":"2026-05-10T12:10:20.776934Z","id":"e3a272bb-a6f2-4062-95c5-48d69054b75d","model_set":{"reader":"grok-4.3"},"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","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","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}.","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)."}},"verdict_id":"e3a272bb-a6f2-4062-95c5-48d69054b75d"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ac006dcb4b382204483763c920a2df525b87a3270fc984c2191a274ce91b776d","target":"record","created_at":"2026-05-22T02:04:41Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ce1665d3791c92a2332e270333d65ebf7cb431e1cd5e8f1037ae30d674bc11fc","cross_cats_sorted":["math.RT","math.SG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2026-04-15T16:59:00Z","title_canon_sha256":"b44c0382377292d6acdc721d55a4e550c7527d8056e32f249c0d397832455d79"},"schema_version":"1.0","source":{"id":"2604.14085","kind":"arxiv","version":3}},"canonical_sha256":"cd568bea7fb81faeaf8358f7d22778ad8f858d65f493cf2049d5d9313eeff363","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cd568bea7fb81faeaf8358f7d22778ad8f858d65f493cf2049d5d9313eeff363","first_computed_at":"2026-05-22T02:04:41.055626Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-22T02:04:41.055626Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YWLz/VmYbpZV2/Fte35PPr7sSDCAJbdlTnshXZpMoW+N2L2WusyNLrFvLbuLk1Mu0BAjt+cs/3EgWrkQAgg/DA==","signature_status":"signed_v1","signed_at":"2026-05-22T02:04:41.056543Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.14085","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ac006dcb4b382204483763c920a2df525b87a3270fc984c2191a274ce91b776d","sha256:1372b971cb1a846c390137ed74a6fa4924c6ce661c1f2e41b39300b10441604e"],"state_sha256":"d034cd15b1b91048aa7bd44f40a3383da08cd04dc53422e77dc153a73f55dfc1"}