{"paper":{"title":"Signs, involutions and Jacquet modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Alan Roche, Steven Spallone","submitted_at":"2012-04-20T20:57:14Z","abstract_excerpt":"Let $G$ be a connected reductive $p$-adic group and let $\\theta$ be an automorphism of $G$ of order at most two. Suppose $\\pi$ is an irreducible smooth representation of $G$ that is taken to its dual by $\\theta$. The space $V$ of $\\pi$ then carries a non-zero bilinear form $(\\mspace{7mu},\\mspace{6mu})$, unique up to scaling, with the invariance property $(\\pi(g)v, \\pi({}^{\\theta}g)w) = (v,w)$, for $g \\in G$ and $v, w \\in V$. The form is easily seen to be symmetric or skew-symmetric and we set $\\varepsilon_\\theta(\\pi) = \\pm1$ accordingly. We use Cassleman's pairing (in commonly observed circums"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.4746","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"}