{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:XSTB654X5PMZZVEZGYH5PMGNSZ","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":"1b8869c5913e827344bd314f25ee720ffff93ed0e582a62d2bfc6b90eb121075","cross_cats_sorted":["math.AC","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-10-30T11:11:52Z","title_canon_sha256":"d1653a1a4a634eadf24c26ea163ef99d56e2a824876519d5a248f43b6c6d957e"},"schema_version":"1.0","source":{"id":"1210.7962","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.7962","created_at":"2026-05-18T03:41:57Z"},{"alias_kind":"arxiv_version","alias_value":"1210.7962v1","created_at":"2026-05-18T03:41:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.7962","created_at":"2026-05-18T03:41:57Z"},{"alias_kind":"pith_short_12","alias_value":"XSTB654X5PMZ","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XSTB654X5PMZZVEZ","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XSTB654X","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:98d49490b3930a84abcd826acd71620a2faeb0f06744ccdd7aa3d0dbcda5c11c","target":"graph","created_at":"2026-05-18T03:41:57Z","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":"Let (N,K) be a nilpotent Gelfand pair, i.e., N is a nilpotent Lie group, K a compact group of automorphisms of N, and the algebra D(N)^K of left-invariant and K-invariant differential operators on N is commutative. In these hypotheses, N is necessarily of step at most two. We say that (N,K) satisfies Vinberg's condition if K acts irreducibly on $n/[n,n]$, where n= Lie(N).\n  Fixing a system D of d formally self-adjoint generators of D(N)^K, the Gelfand spectrum of the commutative convolution algebra L^1(N)^K can be canonically identified with a closed subset S_D of R^d. We prove that, on a nilp","authors_text":"Fulvio Ricci, Oksana Yakimova, Veronique Fischer","cross_cats":["math.AC","math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-10-30T11:11:52Z","title":"Nilpotent Gelfand pairs and spherical transforms of Schwartz functions III. Isomorphisms between Schwartz spaces under Vinberg's condition"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.7962","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":"sha256:6317bf48a73e4a8a3d6cb9de613e6149e4a2d2093af11e487fc688b56d6c1381","target":"record","created_at":"2026-05-18T03:41:57Z","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":"1b8869c5913e827344bd314f25ee720ffff93ed0e582a62d2bfc6b90eb121075","cross_cats_sorted":["math.AC","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-10-30T11:11:52Z","title_canon_sha256":"d1653a1a4a634eadf24c26ea163ef99d56e2a824876519d5a248f43b6c6d957e"},"schema_version":"1.0","source":{"id":"1210.7962","kind":"arxiv","version":1}},"canonical_sha256":"bca61f7797ebd99cd499360fd7b0cd967d1a045f319fb4198be567ad1426d63e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bca61f7797ebd99cd499360fd7b0cd967d1a045f319fb4198be567ad1426d63e","first_computed_at":"2026-05-18T03:41:57.506204Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:41:57.506204Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gvce462BTz9hOYzzWeTBMSwgL24zHtUC484ZGm/ejBgsDyzy4BQl0SEvYkyzQtWljtFsMYLtMP8kz2hqJvh8Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:41:57.506820Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.7962","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6317bf48a73e4a8a3d6cb9de613e6149e4a2d2093af11e487fc688b56d6c1381","sha256:98d49490b3930a84abcd826acd71620a2faeb0f06744ccdd7aa3d0dbcda5c11c"],"state_sha256":"1c6b195390728250c813db551870ac34ff90084d9f99d5993829fbe455628af7"}