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It is known that (G, K) is a Gelfand pair as soon as G acts strongly transitively on $\\Delta$; this is in particular the case when G is a semi-simple algebraic group over a local field. We show a converse to this statement, namely: if (G, K) is a Gelfand pair and G acts cocompactly on $\\Delta$, then the action is strongly transitive. The proof uses the existence of strongly regular hyperbolic elements in"},"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":"1304.6210","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-04-23T09:09:47Z","cross_cats_sorted":["math.DS","math.GR"],"title_canon_sha256":"e32879f326d8efdda564db166d758dfa95076dc2fe31b45c3039571e0105a525","abstract_canon_sha256":"b8dc4f91fb7a4d7ac3c90d7d28ad87272e395eda0286ca06d2e6965a72380598"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:04:11.665461Z","signature_b64":"89wUQa2LHvnSs+KbQxi1iE4HKkfQypaFepwgYJcMJ1Si9D9oAx0Kyi+kLom7C1FD0K0oYOwsqcchimuiNgcDAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b35fe00b96da074920a41b98ee44aed4810792a4758bf199ce0890516df1a63","last_reissued_at":"2026-05-18T02:04:11.664504Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:04:11.664504Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gelfand pairs and strong transitivity for Euclidean buildings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.GR"],"primary_cat":"math.RT","authors_text":"Corina Ciobotaru, Pierre-Emmanuel Caprace","submitted_at":"2013-04-23T09:09:47Z","abstract_excerpt":"Let G be a locally compact group acting properly by type-preserving automorphisms on a locally finite thick Euclidean building $\\Delta$ and K be the stabilizer of a special vertex in $\\Delta$. It is known that (G, K) is a Gelfand pair as soon as G acts strongly transitively on $\\Delta$; this is in particular the case when G is a semi-simple algebraic group over a local field. We show a converse to this statement, namely: if (G, K) is a Gelfand pair and G acts cocompactly on $\\Delta$, then the action is strongly transitive. 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