{"paper":{"title":"Spherical functions on the space of $p$-adic unitary hermitian matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Yasushi Komori, Yumiko Hironaka","submitted_at":"2012-07-26T07:47:23Z","abstract_excerpt":"We investigate the space $X$ of unitary hermitian matrices over $\\frp$-adic fields through spherical functions. First we consider Cartan decomposition of $X$, and give precise representatives for fields with odd residual characteristic, i.e., $2\\notin \\frp$. In the latter half we assume odd residual characteristic, and give explicit formulas of typical spherical functions on $X$, where Hall-Littlewood symmetric polynomials of type $C_n$ appear as a main term, parametrization of all the spherical functions. By spherical Fourier transform, we show the Schwartz space $\\SKX$ is a free Hecke algebr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.6189","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"}