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We can view this isomorphism as a version of the geometric Langlands duality in the formal neighborhood of the trivial local system, for the group GL_r. We extend this to all reductive groups and we compute the image, under our correspondence, of the skyscraper sheaf supported on the trivial local system."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1009.0678","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2010-09-03T14:09:12Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"5224ad1f8def922aa41a7ecb975e6fa96f5f7b97908dbd187c331f78f4c756c8","abstract_canon_sha256":"3d53af1ad828434caf280c005418f1d715fa30b6ed25678ee79af746c4e6bf8d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:25.698728Z","signature_b64":"zp78Co2rDPvr9CFrDl+5fW+r+b/yolx7q53+yWla1wdFYuIkecwlejBk6wwgFqlDe1Lu8QgFHWb6bZS4jrdIAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a987f65c2b238ffdb5df123a2eb0e82a25ddf76d783c06b399592faedc5c05b0","last_reissued_at":"2026-05-18T04:41:25.698102Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:25.698102Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hall algebras of curves, commuting varieties and Langlands duality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.QA","authors_text":"E. 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