{"paper":{"title":"On the Stability and Gelfand Property of Symmetric Pairs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Shachar Carmeli","submitted_at":"2015-11-04T16:19:11Z","abstract_excerpt":"A symmetric pair of reductive groups $(G,H,\\theta)$ is called stable, if every closed double coset of $H$ in $G$ is preserved by the anti-involution $g\\mapsto \\theta(g^{-1})$.\n  In this paper, we develop a method to verify the stability of symmetric pairs over local fields of characteristic 0 (Archimedean and $p$-adic), using non-abelian group cohomology. Combining our method with results of Aizenbud and Gourevitch, we classify the Gelfand pairs among the pairs \\begin{align*} &(SL_n(F), (GL_k(F) \\times GL_{n - k}(F)) \\cap SL_n(F)), (U(B_1 \\oplus B_2),U(B_1) \\times U(B_2)),\\\\ &(GL_n(F),O(B)), ("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.01381","kind":"arxiv","version":3},"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"}