{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:ZLKVQQ5GYPPPZDW3KX3LPXINAE","short_pith_number":"pith:ZLKVQQ5G","canonical_record":{"source":{"id":"0912.4190","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2009-12-21T15:26:41Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"fe518006a4bb449ab506790036d2fa2de48f5032be0244071c8053eafed97497","abstract_canon_sha256":"b6290e260efa7efd1520b2d03e8751cd3b01bcf23571f470b8d4423f444d7394"},"schema_version":"1.0"},"canonical_sha256":"cad55843a6c3defc8edb55f6b7dd0d013ca946245ccf06ae2d957b21fd7675a2","source":{"kind":"arxiv","id":"0912.4190","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0912.4190","created_at":"2026-05-18T04:05:57Z"},{"alias_kind":"arxiv_version","alias_value":"0912.4190v2","created_at":"2026-05-18T04:05:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0912.4190","created_at":"2026-05-18T04:05:57Z"},{"alias_kind":"pith_short_12","alias_value":"ZLKVQQ5GYPPP","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"ZLKVQQ5GYPPPZDW3","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"ZLKVQQ5G","created_at":"2026-05-18T12:26:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:ZLKVQQ5GYPPPZDW3KX3LPXINAE","target":"record","payload":{"canonical_record":{"source":{"id":"0912.4190","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2009-12-21T15:26:41Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"fe518006a4bb449ab506790036d2fa2de48f5032be0244071c8053eafed97497","abstract_canon_sha256":"b6290e260efa7efd1520b2d03e8751cd3b01bcf23571f470b8d4423f444d7394"},"schema_version":"1.0"},"canonical_sha256":"cad55843a6c3defc8edb55f6b7dd0d013ca946245ccf06ae2d957b21fd7675a2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:57.919846Z","signature_b64":"Ryv7Uw40D1l01wKtd8aJQB003SjhLuh9TJKdL9Bee2GrdPCrWAFVLVJJoX0zCw6BRMGzaWno8VsF7U9v5tY6Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cad55843a6c3defc8edb55f6b7dd0d013ca946245ccf06ae2d957b21fd7675a2","last_reissued_at":"2026-05-18T04:05:57.919261Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:57.919261Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0912.4190","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:05:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oTcvgU2qcZkvN8rS7PAE8fFxfxx94wbSm1BLiDHzI1Y2+ON/q2XMTlb5XQgYFEgHY2d40UHuXKRf8IerPWnDCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T11:17:20.458260Z"},"content_sha256":"4f843fb01602902286f20bccac3e2126b22478cba67a758600abe36238d62215","schema_version":"1.0","event_id":"sha256:4f843fb01602902286f20bccac3e2126b22478cba67a758600abe36238d62215"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:ZLKVQQ5GYPPPZDW3KX3LPXINAE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hilbert modules associated to parabolically induced representations of semisimple Lie groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.OA","authors_text":"Pierre Clare","submitted_at":"2009-12-21T15:26:41Z","abstract_excerpt":"Given a measured space X with commuting actions of two groups G and H satisfying certain conditions, we construct a Hilbert C*(H)-module E(X) equipped with a left action of C*(G), which generalises Rieffel's construction of inducing modules. Considering G to be a semisimple Lie group and H to be the Levi component L of a parabolic subgroup P=LN, the Hilbert module associated to X=G/N encodes the P-series representations of G coming from parabolic subgroups associated to P. We provide several descriptions of this Hilbert module, corresponding to the classical pictures of P-series. We then chara"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.4190","kind":"arxiv","version":2},"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":"2026-05-18T04:05:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vBTiFeq4+IrmvPxNWKSqmiOW3Xtpe4iUJ7fVm+za8A8GSgEQ2pPedE+xClDW7Z5YRDixxoGZHuCUZmlferp0Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T11:17:20.458631Z"},"content_sha256":"e7077896c0083399f6b58a91c63346c5d68986a555058dcf1d2eed836b248924","schema_version":"1.0","event_id":"sha256:e7077896c0083399f6b58a91c63346c5d68986a555058dcf1d2eed836b248924"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZLKVQQ5GYPPPZDW3KX3LPXINAE/bundle.json","state_url":"https://pith.science/pith/ZLKVQQ5GYPPPZDW3KX3LPXINAE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZLKVQQ5GYPPPZDW3KX3LPXINAE/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":"2026-06-23T11:17:20Z","links":{"resolver":"https://pith.science/pith/ZLKVQQ5GYPPPZDW3KX3LPXINAE","bundle":"https://pith.science/pith/ZLKVQQ5GYPPPZDW3KX3LPXINAE/bundle.json","state":"https://pith.science/pith/ZLKVQQ5GYPPPZDW3KX3LPXINAE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZLKVQQ5GYPPPZDW3KX3LPXINAE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:ZLKVQQ5GYPPPZDW3KX3LPXINAE","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":"b6290e260efa7efd1520b2d03e8751cd3b01bcf23571f470b8d4423f444d7394","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2009-12-21T15:26:41Z","title_canon_sha256":"fe518006a4bb449ab506790036d2fa2de48f5032be0244071c8053eafed97497"},"schema_version":"1.0","source":{"id":"0912.4190","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0912.4190","created_at":"2026-05-18T04:05:57Z"},{"alias_kind":"arxiv_version","alias_value":"0912.4190v2","created_at":"2026-05-18T04:05:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0912.4190","created_at":"2026-05-18T04:05:57Z"},{"alias_kind":"pith_short_12","alias_value":"ZLKVQQ5GYPPP","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"ZLKVQQ5GYPPPZDW3","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"ZLKVQQ5G","created_at":"2026-05-18T12:26:02Z"}],"graph_snapshots":[{"event_id":"sha256:e7077896c0083399f6b58a91c63346c5d68986a555058dcf1d2eed836b248924","target":"graph","created_at":"2026-05-18T04:05:57Z","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":"Given a measured space X with commuting actions of two groups G and H satisfying certain conditions, we construct a Hilbert C*(H)-module E(X) equipped with a left action of C*(G), which generalises Rieffel's construction of inducing modules. Considering G to be a semisimple Lie group and H to be the Levi component L of a parabolic subgroup P=LN, the Hilbert module associated to X=G/N encodes the P-series representations of G coming from parabolic subgroups associated to P. We provide several descriptions of this Hilbert module, corresponding to the classical pictures of P-series. We then chara","authors_text":"Pierre Clare","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2009-12-21T15:26:41Z","title":"Hilbert modules associated to parabolically induced representations of semisimple Lie groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.4190","kind":"arxiv","version":2},"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":"sha256:4f843fb01602902286f20bccac3e2126b22478cba67a758600abe36238d62215","target":"record","created_at":"2026-05-18T04:05:57Z","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":"b6290e260efa7efd1520b2d03e8751cd3b01bcf23571f470b8d4423f444d7394","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2009-12-21T15:26:41Z","title_canon_sha256":"fe518006a4bb449ab506790036d2fa2de48f5032be0244071c8053eafed97497"},"schema_version":"1.0","source":{"id":"0912.4190","kind":"arxiv","version":2}},"canonical_sha256":"cad55843a6c3defc8edb55f6b7dd0d013ca946245ccf06ae2d957b21fd7675a2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cad55843a6c3defc8edb55f6b7dd0d013ca946245ccf06ae2d957b21fd7675a2","first_computed_at":"2026-05-18T04:05:57.919261Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:05:57.919261Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ryv7Uw40D1l01wKtd8aJQB003SjhLuh9TJKdL9Bee2GrdPCrWAFVLVJJoX0zCw6BRMGzaWno8VsF7U9v5tY6Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T04:05:57.919846Z","signed_message":"canonical_sha256_bytes"},"source_id":"0912.4190","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4f843fb01602902286f20bccac3e2126b22478cba67a758600abe36238d62215","sha256:e7077896c0083399f6b58a91c63346c5d68986a555058dcf1d2eed836b248924"],"state_sha256":"9f6a4b0eacea74182bb90c8328a34db0053de26005ccb22022b896a797222715"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FUbkzSAm1Niekw2/qiE5qkTdSej0non5NVdwOPduzpXNSyuqJSb1xkvQdSXjoaJRSd2Mp2qJwISDPPrLbleTDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T11:17:20.460607Z","bundle_sha256":"4d866f5f21768d0aef959986d59440d79c38d51e3c15d839b40cff02be8ddbf7"}}