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By using \"many\" characters $\\{\\alpha\\}$ of $G$ and \"many\" flat line bundles $\\{E_{\\alpha}\\}$ over $G/\\Gamma$, we show that an isomorphism \\[\\bigoplus_{\\{\\alpha\\}} H^{\\ast}(\\g, V_{\\alpha}\\otimes V_{\\rho})\\cong \\bigoplus_{\\{E_{\\alpha}\\}} H^{\\ast}(G/\\Gamma, E_{\\alpha}\\otimes E_{\\rho})\\] holds. This isomorphism is a generalization of the well-known fact:\""},"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":"1207.3988","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-07-17T13:38:41Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"7bb5ab02c37ec17b275b17637b40a92c96c255797f54011f9188f1cf84c0a044","abstract_canon_sha256":"771bd888ba4393067de072117a3d1a8e5203e96d1a00134fb21655a6efbabb39"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:17.722184Z","signature_b64":"8MXLIfcvJ8/OnnnWUQe7tWD4w89Nj/rZS4tjp2oNyefdvxtIQb7vEcjO3LR3hSk/cnNx0wKtHGIszJVPhy2JAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7817ed65c6db9416f8cf569e0ec0c389b2812af1165ddc89c254c1bc4d3ecb7e","last_reissued_at":"2026-05-18T02:38:17.721553Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:17.721553Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"de Rham and Dolbeault Cohomology of solvmanifolds with local systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.DG","authors_text":"Hisashi Kasuya","submitted_at":"2012-07-17T13:38:41Z","abstract_excerpt":"Let $G$ be a simply connected solvable Lie group with a lattice $\\Gamma$ and the Lie algebra $\\g$ and a representation $\\rho:G\\to GL(V_{\\rho})$ whose restriction on the nilradical is unipotent. Consider the flat bundle $E_{\\rho}$ given by $\\rho$. By using \"many\" characters $\\{\\alpha\\}$ of $G$ and \"many\" flat line bundles $\\{E_{\\alpha}\\}$ over $G/\\Gamma$, we show that an isomorphism \\[\\bigoplus_{\\{\\alpha\\}} H^{\\ast}(\\g, V_{\\alpha}\\otimes V_{\\rho})\\cong \\bigoplus_{\\{E_{\\alpha}\\}} H^{\\ast}(G/\\Gamma, E_{\\alpha}\\otimes E_{\\rho})\\] holds. 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