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For generic values of $\\boldsymbol \\lambda =(\\lambda_1,\\lambda_2,\\lambda_3)$ in $\\mathbb C^3$, there exits a (essentially unique) trilinear form on $\\mathcal C^\\infty(S)\\times \\mathcal C^\\infty(S)\\times \\mathcal C^\\infty(S)$ which is invariant under $\\pi_{\\lambda_1}\\otimes \\pi_{\\lambda_2}\\otimes \\pi_{\\lambda_3}$. Using differential operators on the sphere $S$ which are covariant under the conformal group $SO_0(1,n)$, we constru"},"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":"1102.1861","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2011-02-09T13:43:39Z","cross_cats_sorted":[],"title_canon_sha256":"8af6f7847e4d48d56fb20a2c14af579b83188edad522387cd9106df1a860de01","abstract_canon_sha256":"77192d30b4154dba5709395d7cee828387ab353710fd542538604a64e89bd870"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:24.262782Z","signature_b64":"vH1gttlhdVfGYaPh03U1kObCEgacJrwlfWc5a9YmNswnGzzwYAazOjfDBEY+x3/2th+cPbEgus9/AqPG2PwJBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7852e6b4e269c0ee0fae756d4e0b5f9d06ee126322237486c1f49ebe0af30e5e","last_reissued_at":"2026-05-18T00:32:24.262085Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:24.262085Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Singular conformally invariant trilinear forms and covariant differential operators on the sphere","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Jean-Louis Clerc","submitted_at":"2011-02-09T13:43:39Z","abstract_excerpt":"Let $G=SO_0(1,n)$ be the conformal group acting on the $(n-1)$ dimensional sphere $S$, and let $(\\pi_\\lambda)_{\\lambda\\in \\mathbb C}$ be the spherical principal series. For generic values of $\\boldsymbol \\lambda =(\\lambda_1,\\lambda_2,\\lambda_3)$ in $\\mathbb C^3$, there exits a (essentially unique) trilinear form on $\\mathcal C^\\infty(S)\\times \\mathcal C^\\infty(S)\\times \\mathcal C^\\infty(S)$ which is invariant under $\\pi_{\\lambda_1}\\otimes \\pi_{\\lambda_2}\\otimes \\pi_{\\lambda_3}$. Using differential operators on the sphere $S$ which are covariant under the conformal group $SO_0(1,n)$, we constru"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.1861","kind":"arxiv","version":1},"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":"1102.1861","created_at":"2026-05-18T00:32:24.262185+00:00"},{"alias_kind":"arxiv_version","alias_value":"1102.1861v1","created_at":"2026-05-18T00:32:24.262185+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.1861","created_at":"2026-05-18T00:32:24.262185+00:00"},{"alias_kind":"pith_short_12","alias_value":"PBJONNHCNHAO","created_at":"2026-05-18T12:26:39.201973+00:00"},{"alias_kind":"pith_short_16","alias_value":"PBJONNHCNHAO4D5O","created_at":"2026-05-18T12:26:39.201973+00:00"},{"alias_kind":"pith_short_8","alias_value":"PBJONNHC","created_at":"2026-05-18T12:26:39.201973+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/PBJONNHCNHAO4D5OOVWU4C27TU","json":"https://pith.science/pith/PBJONNHCNHAO4D5OOVWU4C27TU.json","graph_json":"https://pith.science/api/pith-number/PBJONNHCNHAO4D5OOVWU4C27TU/graph.json","events_json":"https://pith.science/api/pith-number/PBJONNHCNHAO4D5OOVWU4C27TU/events.json","paper":"https://pith.science/paper/PBJONNHC"},"agent_actions":{"view_html":"https://pith.science/pith/PBJONNHCNHAO4D5OOVWU4C27TU","download_json":"https://pith.science/pith/PBJONNHCNHAO4D5OOVWU4C27TU.json","view_paper":"https://pith.science/paper/PBJONNHC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1102.1861&json=true","fetch_graph":"https://pith.science/api/pith-number/PBJONNHCNHAO4D5OOVWU4C27TU/graph.json","fetch_events":"https://pith.science/api/pith-number/PBJONNHCNHAO4D5OOVWU4C27TU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PBJONNHCNHAO4D5OOVWU4C27TU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PBJONNHCNHAO4D5OOVWU4C27TU/action/storage_attestation","attest_author":"https://pith.science/pith/PBJONNHCNHAO4D5OOVWU4C27TU/action/author_attestation","sign_citation":"https://pith.science/pith/PBJONNHCNHAO4D5OOVWU4C27TU/action/citation_signature","submit_replication":"https://pith.science/pith/PBJONNHCNHAO4D5OOVWU4C27TU/action/replication_record"}},"created_at":"2026-05-18T00:32:24.262185+00:00","updated_at":"2026-05-18T00:32:24.262185+00:00"}