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In this paper, we consider the case when $K=O(n)$. In this case, the Gelfand space $(O(n),F(n)$ is equipped with the Godement-Plancherel measure, and the spherical transform $\\land:L_{O(n)}^{2}(F(n))\\rightarrow L^{2}(\\Delta (O(n),F(n)))$ is an isometry. I will prove the Gelfand space $\\Delta (O(n),F(n))$ is equipped with the Go"},"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":"1610.00826","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-10-04T02:42:07Z","cross_cats_sorted":[],"title_canon_sha256":"5dc696ad95575a81212b30cd87a9c5b2a8e5be89338a52c834cb872ae1e25f84","abstract_canon_sha256":"dec967e3d4384bf3d5f8708cc9995ae2048f088934fa1604e8689c032de12a5c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:13.107905Z","signature_b64":"/JLA+Oq9HkUnQ8pCmSryWpE9pxTs0uzR3ltBNIj1wAvs5Za3y/nUqhOrMJsGIBsb/Rr3N3cBNfBQxV8lsqLQBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c0b96090fbaf57c7932d3eae59dfd13fe7f7b6f5671406f4c6e3a21b65140770","last_reissued_at":"2026-05-18T01:03:13.107526Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:13.107526Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The spherical transform of a Schwartz function on the free two step nilpotent lie group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Jingzhe Xu","submitted_at":"2016-10-04T02:42:07Z","abstract_excerpt":"Let $F(n)$ be a connected and simply connected free 2-step nilpotent lie group and $K$ be a compact subgroup of Aut($F(n)$). We say that $(K,F(n))$ is a Gelfand pair when the set of integrable $K$-invariant functions on $F(n)$ forms an abelian algebra under convolution. In this paper, we consider the case when $K=O(n)$. In this case, the Gelfand space $(O(n),F(n)$ is equipped with the Godement-Plancherel measure, and the spherical transform $\\land:L_{O(n)}^{2}(F(n))\\rightarrow L^{2}(\\Delta (O(n),F(n)))$ is an isometry. I will prove the Gelfand space $\\Delta (O(n),F(n))$ is equipped with the Go"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.00826","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":"1610.00826","created_at":"2026-05-18T01:03:13.107582+00:00"},{"alias_kind":"arxiv_version","alias_value":"1610.00826v1","created_at":"2026-05-18T01:03:13.107582+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.00826","created_at":"2026-05-18T01:03:13.107582+00:00"},{"alias_kind":"pith_short_12","alias_value":"YC4WBEH3V5L4","created_at":"2026-05-18T12:30:53.716459+00:00"},{"alias_kind":"pith_short_16","alias_value":"YC4WBEH3V5L4PEZN","created_at":"2026-05-18T12:30:53.716459+00:00"},{"alias_kind":"pith_short_8","alias_value":"YC4WBEH3","created_at":"2026-05-18T12:30:53.716459+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/YC4WBEH3V5L4PEZNH2XFTX6RH7","json":"https://pith.science/pith/YC4WBEH3V5L4PEZNH2XFTX6RH7.json","graph_json":"https://pith.science/api/pith-number/YC4WBEH3V5L4PEZNH2XFTX6RH7/graph.json","events_json":"https://pith.science/api/pith-number/YC4WBEH3V5L4PEZNH2XFTX6RH7/events.json","paper":"https://pith.science/paper/YC4WBEH3"},"agent_actions":{"view_html":"https://pith.science/pith/YC4WBEH3V5L4PEZNH2XFTX6RH7","download_json":"https://pith.science/pith/YC4WBEH3V5L4PEZNH2XFTX6RH7.json","view_paper":"https://pith.science/paper/YC4WBEH3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1610.00826&json=true","fetch_graph":"https://pith.science/api/pith-number/YC4WBEH3V5L4PEZNH2XFTX6RH7/graph.json","fetch_events":"https://pith.science/api/pith-number/YC4WBEH3V5L4PEZNH2XFTX6RH7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YC4WBEH3V5L4PEZNH2XFTX6RH7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YC4WBEH3V5L4PEZNH2XFTX6RH7/action/storage_attestation","attest_author":"https://pith.science/pith/YC4WBEH3V5L4PEZNH2XFTX6RH7/action/author_attestation","sign_citation":"https://pith.science/pith/YC4WBEH3V5L4PEZNH2XFTX6RH7/action/citation_signature","submit_replication":"https://pith.science/pith/YC4WBEH3V5L4PEZNH2XFTX6RH7/action/replication_record"}},"created_at":"2026-05-18T01:03:13.107582+00:00","updated_at":"2026-05-18T01:03:13.107582+00:00"}