{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:RMXGU4JK7FBPI35YZPEZ3WXFYE","short_pith_number":"pith:RMXGU4JK","canonical_record":{"source":{"id":"1511.09064","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-11-29T19:05:24Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"d126748ef4725ca8a31761358ca17e57ab7bb91c45c881a53c58ba32dc3e4d49","abstract_canon_sha256":"15e8aaad55966418b18ab0eeb211488f995111eca3232d32b047d9cc359714a7"},"schema_version":"1.0"},"canonical_sha256":"8b2e6a712af942f46fb8cbc99ddae5c1022a0fa8a6592e239557a3c2d22afd44","source":{"kind":"arxiv","id":"1511.09064","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.09064","created_at":"2026-05-18T01:25:41Z"},{"alias_kind":"arxiv_version","alias_value":"1511.09064v1","created_at":"2026-05-18T01:25:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.09064","created_at":"2026-05-18T01:25:41Z"},{"alias_kind":"pith_short_12","alias_value":"RMXGU4JK7FBP","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"RMXGU4JK7FBPI35Y","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"RMXGU4JK","created_at":"2026-05-18T12:29:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:RMXGU4JK7FBPI35YZPEZ3WXFYE","target":"record","payload":{"canonical_record":{"source":{"id":"1511.09064","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-11-29T19:05:24Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"d126748ef4725ca8a31761358ca17e57ab7bb91c45c881a53c58ba32dc3e4d49","abstract_canon_sha256":"15e8aaad55966418b18ab0eeb211488f995111eca3232d32b047d9cc359714a7"},"schema_version":"1.0"},"canonical_sha256":"8b2e6a712af942f46fb8cbc99ddae5c1022a0fa8a6592e239557a3c2d22afd44","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:41.311831Z","signature_b64":"yPlQ1KdG2WCG06S1tb4W8DwXbKSMLUcPnO8ct7eHBUetkGg9ycjFnCWQQpUSSziBOPMLy1Ru6d1tvKRMyNdpAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8b2e6a712af942f46fb8cbc99ddae5c1022a0fa8a6592e239557a3c2d22afd44","last_reissued_at":"2026-05-18T01:25:41.311182Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:41.311182Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1511.09064","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:25:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f32S9/pUAKwLDoKcwx736M00taJBhs0sZ7zIGOPkncOSq5+9zkyk6YsjclfGaF4cJkcAjyEKD7mRG3Nv2Q2fBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T19:27:15.488665Z"},"content_sha256":"4cd5a9c1736a4ca06e2b35aa7e7a66b0acc24009ab158cd92721f6ca908798a1","schema_version":"1.0","event_id":"sha256:4cd5a9c1736a4ca06e2b35aa7e7a66b0acc24009ab158cd92721f6ca908798a1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:RMXGU4JK7FBPI35YZPEZ3WXFYE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stepwise Square Integrable Representations: the Concept and Some Consequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.RT","authors_text":"Joseph A. Wolf","submitted_at":"2015-11-29T19:05:24Z","abstract_excerpt":"There are some new developments on Plancherel formula and growth of matrix coefficients for unitary representations of nilpotent Lie groups. These have several consequences for the geometry of weakly symmetric spaces and analysis on parabolic subgroups of real semisimple Lie groups, and to (infinite dimensional) locally nilpotent Lie groups. Many of these consequences are still under development. In this note I'll survey a few of these new aspects of representation theory for nilpotent Lie groups and parabolic subgroups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.09064","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:25:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BKHW7xdD4hrnIytabZZa+TqxXiP/hwoT3ff0Y6yMofmFKhvfV4Ccq6BR9PDzxJIr0iTow5g+wIQqZJu4xeAnAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T19:27:15.489005Z"},"content_sha256":"37235fe205433df0759d1f0cd453214f2ea835d3da843f7ad4710243feab852f","schema_version":"1.0","event_id":"sha256:37235fe205433df0759d1f0cd453214f2ea835d3da843f7ad4710243feab852f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RMXGU4JK7FBPI35YZPEZ3WXFYE/bundle.json","state_url":"https://pith.science/pith/RMXGU4JK7FBPI35YZPEZ3WXFYE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RMXGU4JK7FBPI35YZPEZ3WXFYE/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":"2026-06-26T19:27:15Z","links":{"resolver":"https://pith.science/pith/RMXGU4JK7FBPI35YZPEZ3WXFYE","bundle":"https://pith.science/pith/RMXGU4JK7FBPI35YZPEZ3WXFYE/bundle.json","state":"https://pith.science/pith/RMXGU4JK7FBPI35YZPEZ3WXFYE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RMXGU4JK7FBPI35YZPEZ3WXFYE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:RMXGU4JK7FBPI35YZPEZ3WXFYE","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":"15e8aaad55966418b18ab0eeb211488f995111eca3232d32b047d9cc359714a7","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-11-29T19:05:24Z","title_canon_sha256":"d126748ef4725ca8a31761358ca17e57ab7bb91c45c881a53c58ba32dc3e4d49"},"schema_version":"1.0","source":{"id":"1511.09064","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.09064","created_at":"2026-05-18T01:25:41Z"},{"alias_kind":"arxiv_version","alias_value":"1511.09064v1","created_at":"2026-05-18T01:25:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.09064","created_at":"2026-05-18T01:25:41Z"},{"alias_kind":"pith_short_12","alias_value":"RMXGU4JK7FBP","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"RMXGU4JK7FBPI35Y","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"RMXGU4JK","created_at":"2026-05-18T12:29:39Z"}],"graph_snapshots":[{"event_id":"sha256:37235fe205433df0759d1f0cd453214f2ea835d3da843f7ad4710243feab852f","target":"graph","created_at":"2026-05-18T01:25:41Z","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":"There are some new developments on Plancherel formula and growth of matrix coefficients for unitary representations of nilpotent Lie groups. These have several consequences for the geometry of weakly symmetric spaces and analysis on parabolic subgroups of real semisimple Lie groups, and to (infinite dimensional) locally nilpotent Lie groups. Many of these consequences are still under development. In this note I'll survey a few of these new aspects of representation theory for nilpotent Lie groups and parabolic subgroups.","authors_text":"Joseph A. Wolf","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-11-29T19:05:24Z","title":"Stepwise Square Integrable Representations: the Concept and Some Consequences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.09064","kind":"arxiv","version":1},"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":"sha256:4cd5a9c1736a4ca06e2b35aa7e7a66b0acc24009ab158cd92721f6ca908798a1","target":"record","created_at":"2026-05-18T01:25:41Z","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":"15e8aaad55966418b18ab0eeb211488f995111eca3232d32b047d9cc359714a7","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-11-29T19:05:24Z","title_canon_sha256":"d126748ef4725ca8a31761358ca17e57ab7bb91c45c881a53c58ba32dc3e4d49"},"schema_version":"1.0","source":{"id":"1511.09064","kind":"arxiv","version":1}},"canonical_sha256":"8b2e6a712af942f46fb8cbc99ddae5c1022a0fa8a6592e239557a3c2d22afd44","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8b2e6a712af942f46fb8cbc99ddae5c1022a0fa8a6592e239557a3c2d22afd44","first_computed_at":"2026-05-18T01:25:41.311182Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:25:41.311182Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yPlQ1KdG2WCG06S1tb4W8DwXbKSMLUcPnO8ct7eHBUetkGg9ycjFnCWQQpUSSziBOPMLy1Ru6d1tvKRMyNdpAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:25:41.311831Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.09064","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4cd5a9c1736a4ca06e2b35aa7e7a66b0acc24009ab158cd92721f6ca908798a1","sha256:37235fe205433df0759d1f0cd453214f2ea835d3da843f7ad4710243feab852f"],"state_sha256":"5ba257abc84b9b27a343f58df02809031f071524ab8f16523782c7c8ac20c21e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XvcFHCZg+/eZJ8rJD9EAzFVGVlljmuTnv6HqBRsWOL/0NbvDUa4LNde7/5ZtEzARItgaDoNmtbEdp9d5rnJVCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T19:27:15.490818Z","bundle_sha256":"89106e121c01bc0fafe125914e724d92b953dbb2baeeb26fd1bbedb7e5d04f4f"}}