{"paper":{"title":"A spectral decomposition of orbital integrals for $PGL(2,F)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"David Kazhdan, Stephen DeBacker","submitted_at":"2017-01-18T10:30:27Z","abstract_excerpt":"Let $F$ be a local non-archimedian field, $G$ a semisimple $F$-group, $dg$ a Haar measure on $G$ and $\\mathcal S(G)$ be the space of locally constant complex valued functions $f$ on $G$ with compact support. For any regular elliptic congugacy class $\\Omega =h^G\\subset G$ we denote by $I_\\Omega$ the $G$-invariant functional on $\\mathcal S (G)$ given by $$I_\\Omega (f)=\\int_G f(g^{-1}hg)dg$$ This paper provides the spectral decomposition of functionals $I_\\Omega$ in the case $G=PGL(2,F)$ and in the last section first steps of such an analysis for the general case."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.04999","kind":"arxiv","version":2},"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"}