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We prove that there exist finitely many discrete components in its restriction under the subgroup $H_{n-1}=\\SO_0(n-1, 1; \\mathbb F)$. This is proved by imbedding the complementary series into analytic continuation of holomorphic discrete series of $G_n=SU(n, 1)$, $SU(n, 1)\\times SU(n, 1)$ and $SU(2n, 2)$ and by the branching of holomorphic representations under the corresponding subgroup\n  $G_{n-1}$."},"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":"1304.2868","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-04-10T07:41:22Z","cross_cats_sorted":[],"title_canon_sha256":"65595a7c8c13f3cdecacaa0787a348b4a09c73f179c989d441f9b32f56630ed6","abstract_canon_sha256":"9411dc19d5c1de475c147f764b4d4b7340e521b44170ce26d8328c6135f986ca"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:28:28.074300Z","signature_b64":"Idaruwc4WT5S1nUxaxS3Gn7LMxW34eH9DPXs3Dw7bPEPtbEHwL1hXFV3Q1crbAfk+VbnxbFUcL0i142avdLkAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dc57027bd37216a68c2803f80fbaf457360bd333eea691c32a0a6954e506c8e9","last_reissued_at":"2026-05-18T03:28:28.073776Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:28:28.073776Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Restriction to symmetric subgroups of unitary representations of rank one semisimple Lie groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Birgit Speh, Genkai Zhang","submitted_at":"2013-04-10T07:41:22Z","abstract_excerpt":"We consider the spherical complementary series of rank one Lie groups $H_n=\\SO_0(n, 1; \\mathbb F)$ for $\\mathbb F=\\mathbb R, \\mathbb C, \\mathbb H$. We prove that there exist finitely many discrete components in its restriction under the subgroup $H_{n-1}=\\SO_0(n-1, 1; \\mathbb F)$. This is proved by imbedding the complementary series into analytic continuation of holomorphic discrete series of $G_n=SU(n, 1)$, $SU(n, 1)\\times SU(n, 1)$ and $SU(2n, 2)$ and by the branching of holomorphic representations under the corresponding subgroup\n  $G_{n-1}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.2868","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":"1304.2868","created_at":"2026-05-18T03:28:28.073876+00:00"},{"alias_kind":"arxiv_version","alias_value":"1304.2868v1","created_at":"2026-05-18T03:28:28.073876+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.2868","created_at":"2026-05-18T03:28:28.073876+00:00"},{"alias_kind":"pith_short_12","alias_value":"3RLQE66TOILK","created_at":"2026-05-18T12:27:32.513160+00:00"},{"alias_kind":"pith_short_16","alias_value":"3RLQE66TOILKNDBI","created_at":"2026-05-18T12:27:32.513160+00:00"},{"alias_kind":"pith_short_8","alias_value":"3RLQE66T","created_at":"2026-05-18T12:27:32.513160+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/3RLQE66TOILKNDBIAP4A7OXUK4","json":"https://pith.science/pith/3RLQE66TOILKNDBIAP4A7OXUK4.json","graph_json":"https://pith.science/api/pith-number/3RLQE66TOILKNDBIAP4A7OXUK4/graph.json","events_json":"https://pith.science/api/pith-number/3RLQE66TOILKNDBIAP4A7OXUK4/events.json","paper":"https://pith.science/paper/3RLQE66T"},"agent_actions":{"view_html":"https://pith.science/pith/3RLQE66TOILKNDBIAP4A7OXUK4","download_json":"https://pith.science/pith/3RLQE66TOILKNDBIAP4A7OXUK4.json","view_paper":"https://pith.science/paper/3RLQE66T","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1304.2868&json=true","fetch_graph":"https://pith.science/api/pith-number/3RLQE66TOILKNDBIAP4A7OXUK4/graph.json","fetch_events":"https://pith.science/api/pith-number/3RLQE66TOILKNDBIAP4A7OXUK4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3RLQE66TOILKNDBIAP4A7OXUK4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3RLQE66TOILKNDBIAP4A7OXUK4/action/storage_attestation","attest_author":"https://pith.science/pith/3RLQE66TOILKNDBIAP4A7OXUK4/action/author_attestation","sign_citation":"https://pith.science/pith/3RLQE66TOILKNDBIAP4A7OXUK4/action/citation_signature","submit_replication":"https://pith.science/pith/3RLQE66TOILKNDBIAP4A7OXUK4/action/replication_record"}},"created_at":"2026-05-18T03:28:28.073876+00:00","updated_at":"2026-05-18T03:28:28.073876+00:00"}