{"paper":{"title":"The first pro-$p$-Iwahori cohomology of mod-$p$ principal series for $p$-adic $\\textrm{GL}_n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.RT","authors_text":"Karol Koziol","submitted_at":"2017-08-09T21:10:48Z","abstract_excerpt":"Let $p\\geq 3$ be a prime number and $F$ a $p$-adic field. Let $I_1$ denote the pro-$p$-Iwahori subgroup of $\\textrm{GL}_n(F)$, and $\\mathcal{H}$ the pro-$p$-Iwahori--Hecke algebra of $\\textrm{GL}_n(F)$ with respect to $I_1$ (over a coefficient field of characteristic $p$). We compute the structure of $\\textrm{H}^1(I_1,\\pi)$ as an $\\mathcal{H}$-module, where $\\pi$ is a mod-$p$ principal series representation of $\\textrm{GL}_n(F)$. We also give some partial results about the structure of $\\textrm{H}^1(I_1,\\pi)$ for a general split reductive group with irreducible root system."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.03014","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"}