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We prove that there is a discrete component $\\pi_{\\alpha+\\beta}$ for small parameters $\\alpha, \\beta$ (in our parametrization). We prove further that for $G=SO_0(n, 1)$ there are finitely many complementary series of the form $\\pi_{\\alpha+\\beta + 2j}$, $j=0, 1, \\cdots, k$, appearing in the tensor product $\\pi_{\\alpha} \\otimes \\pi_{\\beta} $ of two complementary series $\\pi_{\\alpha}$ and $\\p"},"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":"1402.2950","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2014-02-12T19:48:42Z","cross_cats_sorted":[],"title_canon_sha256":"a81e39e7ae47f096943ae2207125f809b8326976ebb89bb26691f186cd88fae3","abstract_canon_sha256":"c58b340ae1e6db55da4ff3821390083160d77111a6113580f5f03fda50968a83"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:34.042737Z","signature_b64":"w+5fqnD0/JydBJ53XgJJw4RJdG1CZLBIesVGoYHY+zj+rrJ8UsRfijNCtHyQ3pR6M/Th1AY7Hht/3ak1Y0b4Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ad80ed062ebe2748b1b50db6c4fa60f3d62311381e53293577bfc1e9c69399f0","last_reissued_at":"2026-05-18T00:44:34.042389Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:34.042389Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tensor products of complementary series of rank one Lie groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Genkai Zhang","submitted_at":"2014-02-12T19:48:42Z","abstract_excerpt":"We consider the tensor product $\\pi_{\\alpha}\\otimes \\pi_{\\beta}$ of complementary series representations $\\pi_{\\alpha}$ and $\\pi_{\\beta}$ of classical rank one groups $SO_0(n, 1)$, $SU(n, 1)$ and $Sp(n, 1)$. We prove that there is a discrete component $\\pi_{\\alpha+\\beta}$ for small parameters $\\alpha, \\beta$ (in our parametrization). We prove further that for $G=SO_0(n, 1)$ there are finitely many complementary series of the form $\\pi_{\\alpha+\\beta + 2j}$, $j=0, 1, \\cdots, k$, appearing in the tensor product $\\pi_{\\alpha} \\otimes \\pi_{\\beta} $ of two complementary series $\\pi_{\\alpha}$ and $\\p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.2950","kind":"arxiv","version":4},"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":"1402.2950","created_at":"2026-05-18T00:44:34.042446+00:00"},{"alias_kind":"arxiv_version","alias_value":"1402.2950v4","created_at":"2026-05-18T00:44:34.042446+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.2950","created_at":"2026-05-18T00:44:34.042446+00:00"},{"alias_kind":"pith_short_12","alias_value":"VWAO2BROXYTU","created_at":"2026-05-18T12:28:54.890064+00:00"},{"alias_kind":"pith_short_16","alias_value":"VWAO2BROXYTURMNV","created_at":"2026-05-18T12:28:54.890064+00:00"},{"alias_kind":"pith_short_8","alias_value":"VWAO2BRO","created_at":"2026-05-18T12:28:54.890064+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/VWAO2BROXYTURMNVBW3MJ6TA6P","json":"https://pith.science/pith/VWAO2BROXYTURMNVBW3MJ6TA6P.json","graph_json":"https://pith.science/api/pith-number/VWAO2BROXYTURMNVBW3MJ6TA6P/graph.json","events_json":"https://pith.science/api/pith-number/VWAO2BROXYTURMNVBW3MJ6TA6P/events.json","paper":"https://pith.science/paper/VWAO2BRO"},"agent_actions":{"view_html":"https://pith.science/pith/VWAO2BROXYTURMNVBW3MJ6TA6P","download_json":"https://pith.science/pith/VWAO2BROXYTURMNVBW3MJ6TA6P.json","view_paper":"https://pith.science/paper/VWAO2BRO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1402.2950&json=true","fetch_graph":"https://pith.science/api/pith-number/VWAO2BROXYTURMNVBW3MJ6TA6P/graph.json","fetch_events":"https://pith.science/api/pith-number/VWAO2BROXYTURMNVBW3MJ6TA6P/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VWAO2BROXYTURMNVBW3MJ6TA6P/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VWAO2BROXYTURMNVBW3MJ6TA6P/action/storage_attestation","attest_author":"https://pith.science/pith/VWAO2BROXYTURMNVBW3MJ6TA6P/action/author_attestation","sign_citation":"https://pith.science/pith/VWAO2BROXYTURMNVBW3MJ6TA6P/action/citation_signature","submit_replication":"https://pith.science/pith/VWAO2BROXYTURMNVBW3MJ6TA6P/action/replication_record"}},"created_at":"2026-05-18T00:44:34.042446+00:00","updated_at":"2026-05-18T00:44:34.042446+00:00"}