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We consider the Heisenberg group $N(p,q,\\FF)$ defined as $N(p,q,\\FF)=\\FF^{n}\\times \\mathfrak{Im}\\FF$, with group law given by $$(v,\\zeta)(v',\\zeta')=(v+v', \\zeta+\\zeta'-{1/2} \\mathfrak{Im} B(v,v')),$$ where $B(v,w)=\\sum_{j=1}^{p} v_{j}\\bar{w_{j}} - \\sum_{j=p+1}^{n} v_{j}\\bar{w_{j}}$. Let $U(p,q,\\FF)$ be the group of $n\\times n$ matrices with coefficients in $\\FF$ that leave invariant the form $B$. In this work we compute explicit fundamental solut"},"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":"1205.5489","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-05-24T15:46:05Z","cross_cats_sorted":[],"title_canon_sha256":"b2f90f4ca9cc5a162227c2feafa396cd1bc7ab35562ea37fbb89b9b02940eb2c","abstract_canon_sha256":"6674b81849556834aaf8a600f34304de39d96cba7668ef638632415481ed4d9c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:55:00.956894Z","signature_b64":"72QCVfgNb8Z9L15fINF9vvrsLvoQh8PWLY62Cbqg+Mv8kXaFSh/yqJO5OMJerVtcKgEUPEpyOGf6r8YtEXjiBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7cff13640cd2bb307a0793cef63bd7065794a6c909b36140a31fd70b6a5dfccc","last_reissued_at":"2026-05-18T03:55:00.956339Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:55:00.956339Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Explicit fundamental solutions of some second order differential operators on Heisenberg groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Isolda Cardoso, Linda Saal","submitted_at":"2012-05-24T15:46:05Z","abstract_excerpt":"Let $p,q,n$ be natural numbers such that $p+q=n$. Let $\\FF$ be either $\\CC$, the complex numbers field, or $\\HH$, the quaternionic division algebra. We consider the Heisenberg group $N(p,q,\\FF)$ defined as $N(p,q,\\FF)=\\FF^{n}\\times \\mathfrak{Im}\\FF$, with group law given by $$(v,\\zeta)(v',\\zeta')=(v+v', \\zeta+\\zeta'-{1/2} \\mathfrak{Im} B(v,v')),$$ where $B(v,w)=\\sum_{j=1}^{p} v_{j}\\bar{w_{j}} - \\sum_{j=p+1}^{n} v_{j}\\bar{w_{j}}$. Let $U(p,q,\\FF)$ be the group of $n\\times n$ matrices with coefficients in $\\FF$ that leave invariant the form $B$. In this work we compute explicit fundamental solut"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.5489","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":"1205.5489","created_at":"2026-05-18T03:55:00.956417+00:00"},{"alias_kind":"arxiv_version","alias_value":"1205.5489v1","created_at":"2026-05-18T03:55:00.956417+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.5489","created_at":"2026-05-18T03:55:00.956417+00:00"},{"alias_kind":"pith_short_12","alias_value":"PT7RGZAM2K5T","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_16","alias_value":"PT7RGZAM2K5TA6QH","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_8","alias_value":"PT7RGZAM","created_at":"2026-05-18T12:27:18.751474+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/PT7RGZAM2K5TA6QHSPHPMO6XAZ","json":"https://pith.science/pith/PT7RGZAM2K5TA6QHSPHPMO6XAZ.json","graph_json":"https://pith.science/api/pith-number/PT7RGZAM2K5TA6QHSPHPMO6XAZ/graph.json","events_json":"https://pith.science/api/pith-number/PT7RGZAM2K5TA6QHSPHPMO6XAZ/events.json","paper":"https://pith.science/paper/PT7RGZAM"},"agent_actions":{"view_html":"https://pith.science/pith/PT7RGZAM2K5TA6QHSPHPMO6XAZ","download_json":"https://pith.science/pith/PT7RGZAM2K5TA6QHSPHPMO6XAZ.json","view_paper":"https://pith.science/paper/PT7RGZAM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1205.5489&json=true","fetch_graph":"https://pith.science/api/pith-number/PT7RGZAM2K5TA6QHSPHPMO6XAZ/graph.json","fetch_events":"https://pith.science/api/pith-number/PT7RGZAM2K5TA6QHSPHPMO6XAZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PT7RGZAM2K5TA6QHSPHPMO6XAZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PT7RGZAM2K5TA6QHSPHPMO6XAZ/action/storage_attestation","attest_author":"https://pith.science/pith/PT7RGZAM2K5TA6QHSPHPMO6XAZ/action/author_attestation","sign_citation":"https://pith.science/pith/PT7RGZAM2K5TA6QHSPHPMO6XAZ/action/citation_signature","submit_replication":"https://pith.science/pith/PT7RGZAM2K5TA6QHSPHPMO6XAZ/action/replication_record"}},"created_at":"2026-05-18T03:55:00.956417+00:00","updated_at":"2026-05-18T03:55:00.956417+00:00"}