{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:3DN5AEOZTHOLK7AGACUT4U6D2V","short_pith_number":"pith:3DN5AEOZ","canonical_record":{"source":{"id":"1306.6392","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-06-27T01:57:40Z","cross_cats_sorted":[],"title_canon_sha256":"413c828c21e958bdc643a2fcc5c1598079c3e04d6cc7271410ca96a0bdee48cc","abstract_canon_sha256":"113093fa40adf2c221a256f9545a70ec541ccd49a1b716a4679e8d16c4c07e17"},"schema_version":"1.0"},"canonical_sha256":"d8dbd011d999dcb57c0600a93e53c3d543ffdc73e175a8ffa057bcac2a4d0e12","source":{"kind":"arxiv","id":"1306.6392","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.6392","created_at":"2026-05-18T03:04:22Z"},{"alias_kind":"arxiv_version","alias_value":"1306.6392v3","created_at":"2026-05-18T03:04:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.6392","created_at":"2026-05-18T03:04:22Z"},{"alias_kind":"pith_short_12","alias_value":"3DN5AEOZTHOL","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"3DN5AEOZTHOLK7AG","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"3DN5AEOZ","created_at":"2026-05-18T12:27:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:3DN5AEOZTHOLK7AGACUT4U6D2V","target":"record","payload":{"canonical_record":{"source":{"id":"1306.6392","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-06-27T01:57:40Z","cross_cats_sorted":[],"title_canon_sha256":"413c828c21e958bdc643a2fcc5c1598079c3e04d6cc7271410ca96a0bdee48cc","abstract_canon_sha256":"113093fa40adf2c221a256f9545a70ec541ccd49a1b716a4679e8d16c4c07e17"},"schema_version":"1.0"},"canonical_sha256":"d8dbd011d999dcb57c0600a93e53c3d543ffdc73e175a8ffa057bcac2a4d0e12","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:04:22.451889Z","signature_b64":"iTj6prDt1HYXTvQfPC4xaDal9LsTfcNbFjYdohRsC1TypW26cOCiWL7HP72nX5IPvmydV2JHmRBPL5acVkUoDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d8dbd011d999dcb57c0600a93e53c3d543ffdc73e175a8ffa057bcac2a4d0e12","last_reissued_at":"2026-05-18T03:04:22.451353Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:04:22.451353Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1306.6392","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:04:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2zPZ/CeRXWhG4BzrsY+UQAZolO399vuXEsU+N3XBVljMl2SsUsti0CbrKWCT6Zv8yG8Uj0IfT2fiWGgeA0omDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T14:25:32.538159Z"},"content_sha256":"a4d9cf8a0fcec7226e8d06823233459ecc8a881964a0efb204f7d87fc4f127d1","schema_version":"1.0","event_id":"sha256:a4d9cf8a0fcec7226e8d06823233459ecc8a881964a0efb204f7d87fc4f127d1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:3DN5AEOZTHOLK7AGACUT4U6D2V","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Plancherel Formula for Minimal Parabolic Subgroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Joseph A. Wolf","submitted_at":"2013-06-27T01:57:40Z","abstract_excerpt":"In a recent paper we found conditions for a nilpotent Lie group to be foliated into subgroups that have square integrable unitary representations that fit together to form a filtration by normal subgroups. That resulted in explicit character formulae, Plancherel formulae and multiplicity formulae. We also showed that nilradicals $N$ of minimal parabolic subgroups $P = MAN$ enjoy that \"stepwise square integrable\" property. Here we extend those results from $N$ to $P$. The Pfaffian polynomials, which give orthogonality relations and Plancherel density for $N$, also give a semiinvariant different"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.6392","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:04:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xXnE7cDjrmMEzFrjQb8skVmOrIVNa4QxInyaRpM1klUEwUdcVrRzD+r+dWcbvxdaqeA5Qg28q3a4/qDlWG1VDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T14:25:32.538678Z"},"content_sha256":"e9c0f41d9219e69f2c493fdc8024810cdc1d09d7e173171f3a599988e84ad220","schema_version":"1.0","event_id":"sha256:e9c0f41d9219e69f2c493fdc8024810cdc1d09d7e173171f3a599988e84ad220"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3DN5AEOZTHOLK7AGACUT4U6D2V/bundle.json","state_url":"https://pith.science/pith/3DN5AEOZTHOLK7AGACUT4U6D2V/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3DN5AEOZTHOLK7AGACUT4U6D2V/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-26T14:25:32Z","links":{"resolver":"https://pith.science/pith/3DN5AEOZTHOLK7AGACUT4U6D2V","bundle":"https://pith.science/pith/3DN5AEOZTHOLK7AGACUT4U6D2V/bundle.json","state":"https://pith.science/pith/3DN5AEOZTHOLK7AGACUT4U6D2V/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3DN5AEOZTHOLK7AGACUT4U6D2V/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:3DN5AEOZTHOLK7AGACUT4U6D2V","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"113093fa40adf2c221a256f9545a70ec541ccd49a1b716a4679e8d16c4c07e17","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-06-27T01:57:40Z","title_canon_sha256":"413c828c21e958bdc643a2fcc5c1598079c3e04d6cc7271410ca96a0bdee48cc"},"schema_version":"1.0","source":{"id":"1306.6392","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.6392","created_at":"2026-05-18T03:04:22Z"},{"alias_kind":"arxiv_version","alias_value":"1306.6392v3","created_at":"2026-05-18T03:04:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.6392","created_at":"2026-05-18T03:04:22Z"},{"alias_kind":"pith_short_12","alias_value":"3DN5AEOZTHOL","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"3DN5AEOZTHOLK7AG","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"3DN5AEOZ","created_at":"2026-05-18T12:27:32Z"}],"graph_snapshots":[{"event_id":"sha256:e9c0f41d9219e69f2c493fdc8024810cdc1d09d7e173171f3a599988e84ad220","target":"graph","created_at":"2026-05-18T03:04:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In a recent paper we found conditions for a nilpotent Lie group to be foliated into subgroups that have square integrable unitary representations that fit together to form a filtration by normal subgroups. That resulted in explicit character formulae, Plancherel formulae and multiplicity formulae. We also showed that nilradicals $N$ of minimal parabolic subgroups $P = MAN$ enjoy that \"stepwise square integrable\" property. Here we extend those results from $N$ to $P$. The Pfaffian polynomials, which give orthogonality relations and Plancherel density for $N$, also give a semiinvariant different","authors_text":"Joseph A. Wolf","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-06-27T01:57:40Z","title":"The Plancherel Formula for Minimal Parabolic Subgroups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.6392","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a4d9cf8a0fcec7226e8d06823233459ecc8a881964a0efb204f7d87fc4f127d1","target":"record","created_at":"2026-05-18T03:04:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"113093fa40adf2c221a256f9545a70ec541ccd49a1b716a4679e8d16c4c07e17","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-06-27T01:57:40Z","title_canon_sha256":"413c828c21e958bdc643a2fcc5c1598079c3e04d6cc7271410ca96a0bdee48cc"},"schema_version":"1.0","source":{"id":"1306.6392","kind":"arxiv","version":3}},"canonical_sha256":"d8dbd011d999dcb57c0600a93e53c3d543ffdc73e175a8ffa057bcac2a4d0e12","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d8dbd011d999dcb57c0600a93e53c3d543ffdc73e175a8ffa057bcac2a4d0e12","first_computed_at":"2026-05-18T03:04:22.451353Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:04:22.451353Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iTj6prDt1HYXTvQfPC4xaDal9LsTfcNbFjYdohRsC1TypW26cOCiWL7HP72nX5IPvmydV2JHmRBPL5acVkUoDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:04:22.451889Z","signed_message":"canonical_sha256_bytes"},"source_id":"1306.6392","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a4d9cf8a0fcec7226e8d06823233459ecc8a881964a0efb204f7d87fc4f127d1","sha256:e9c0f41d9219e69f2c493fdc8024810cdc1d09d7e173171f3a599988e84ad220"],"state_sha256":"8cc5cb3e02bb25224b98768ccc1b45b090a93c77adb1352bf8062cfc1a407efc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aYEWxZFXl8Tw+2Jlizubtsye6w66oLVhpI00D8o+mp8ktxd29T19UlOMMp+BWMFfk6LSe+QpwXv0CkYLZpwEBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T14:25:32.541469Z","bundle_sha256":"e9903505091ad56926b2fd3d2cf8acd988fff7f922a0b928ee8424dd88b9afda"}}