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In this note, we compute the action of $\\mathcal{H}$ on $\\textrm{H}^1(I_1,C)$ and $\\textrm{H}^{\\textrm{top}}(I_1,C)$ when the root system of $G$ is irreducible. We also give some partial results in the general case."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1708.03013","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-08-09T21:04:50Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"2e6dc6a33eeecb78c82eaf70f383de1be840b6a9d69729bbf7052c3cbb903e1f","abstract_canon_sha256":"2b821e6246019d8544a4decd9bd21b594d2aa569266335a22e49585676926e98"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:16.896101Z","signature_b64":"1zJby2bfmeUycqqPnwejF9e5m4oRMPjzpHwOHlVlmAQxxVvxSaUcIz1x06AmjnDap6DSE+V3FEmPwojpfMVlBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b3081894c1a4c3aec5ac0a4215a719385fb80c2868eee5137ec005e81d144bf","last_reissued_at":"2026-05-18T00:20:16.895459Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:16.895459Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hecke module structure on first and top pro-$p$-Iwahori cohomology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.RT","authors_text":"Karol Koziol","submitted_at":"2017-08-09T21:04:50Z","abstract_excerpt":"Let $p\\geq 5$ be a prime number, $G$ a split connected reductive group defined over a $p$-adic field, and $I_1$ a choice of pro-$p$-Iwahori subgroup. 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