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We show that every irreducible Casselman-Wallach representation of $G'$ occurs with multiplicity at most one in every irreducible Casselman-Wallach representation of $G$. Similar results are proved for the Jacobi groups $\\GL_{n}(\\R)\\ltimes \\oH_{2n+1}(\\R)$, $\\GL_{n}(\\C)\\ltimes"},"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":"0903.1413","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-03-08T13:06:53Z","cross_cats_sorted":[],"title_canon_sha256":"8be9f0d5d6989cecfd863e9a9996efde1150c51c8c04a9e14e68fc90499569a6","abstract_canon_sha256":"8870e148038ce38479a509bb1801c63b4f1601738fd7ff9e5b73d66c03cc5257"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:42:24.232242Z","signature_b64":"wuV+VrEMAl+GRg6cymVbVMv6EfktucC+sGo5hvEb0r/t5FKcgyZvEeIwpWwupz4IYII63ykGLZsDt54+lwIeAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d304a139bde16e8af7c93154462995744c2a56a402fa105447238a1336bcfb4","last_reissued_at":"2026-05-18T03:42:24.231770Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:42:24.231770Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multiplicity one theorems: the Archimedean case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Binyong Sun, Chen-bo Zhu","submitted_at":"2009-03-08T13:06:53Z","abstract_excerpt":"Let $G$ be one of the classical Lie groups $\\GL_{n+1}(\\R)$, $\\GL_{n+1}(\\C)$, $\\oU(p,q+1)$, $\\oO(p,q+1)$, $\\oO_{n+1}(\\C)$, $\\SO(p,q+1)$, $\\SO_{n+1}(\\C)$, and let $G'$ be respectively the subgroup $\\GL_{n}(\\R)$, $\\GL_{n}(\\C)$, $\\oU(p,q)$, $\\oO(p,q)$, $\\oO_n(\\C)$, $\\SO(p,q)$, $\\SO_n(\\C)$, embedded in $G$ in the standard way. We show that every irreducible Casselman-Wallach representation of $G'$ occurs with multiplicity at most one in every irreducible Casselman-Wallach representation of $G$. Similar results are proved for the Jacobi groups $\\GL_{n}(\\R)\\ltimes \\oH_{2n+1}(\\R)$, $\\GL_{n}(\\C)\\ltimes"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.1413","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0903.1413","created_at":"2026-05-18T03:42:24.231836+00:00"},{"alias_kind":"arxiv_version","alias_value":"0903.1413v2","created_at":"2026-05-18T03:42:24.231836+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.1413","created_at":"2026-05-18T03:42:24.231836+00:00"},{"alias_kind":"pith_short_12","alias_value":"BUYEUE433YLO","created_at":"2026-05-18T12:25:58.837520+00:00"},{"alias_kind":"pith_short_16","alias_value":"BUYEUE433YLORL34","created_at":"2026-05-18T12:25:58.837520+00:00"},{"alias_kind":"pith_short_8","alias_value":"BUYEUE43","created_at":"2026-05-18T12:25:58.837520+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BUYEUE433YLORL34SMKUIYUZK5","json":"https://pith.science/pith/BUYEUE433YLORL34SMKUIYUZK5.json","graph_json":"https://pith.science/api/pith-number/BUYEUE433YLORL34SMKUIYUZK5/graph.json","events_json":"https://pith.science/api/pith-number/BUYEUE433YLORL34SMKUIYUZK5/events.json","paper":"https://pith.science/paper/BUYEUE43"},"agent_actions":{"view_html":"https://pith.science/pith/BUYEUE433YLORL34SMKUIYUZK5","download_json":"https://pith.science/pith/BUYEUE433YLORL34SMKUIYUZK5.json","view_paper":"https://pith.science/paper/BUYEUE43","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0903.1413&json=true","fetch_graph":"https://pith.science/api/pith-number/BUYEUE433YLORL34SMKUIYUZK5/graph.json","fetch_events":"https://pith.science/api/pith-number/BUYEUE433YLORL34SMKUIYUZK5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BUYEUE433YLORL34SMKUIYUZK5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BUYEUE433YLORL34SMKUIYUZK5/action/storage_attestation","attest_author":"https://pith.science/pith/BUYEUE433YLORL34SMKUIYUZK5/action/author_attestation","sign_citation":"https://pith.science/pith/BUYEUE433YLORL34SMKUIYUZK5/action/citation_signature","submit_replication":"https://pith.science/pith/BUYEUE433YLORL34SMKUIYUZK5/action/replication_record"}},"created_at":"2026-05-18T03:42:24.231836+00:00","updated_at":"2026-05-18T03:42:24.231836+00:00"}