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Let $N$ (respectively, $C$) be the normalizer (respectively, centralizer) of $T$ in $G$, and let $W$ be the Weyl group $N/C$ for $T$. We prove that there is a natural bijective correspondence between the following two:\n  Torus subbundles $\\mathbb T$ of ${\\rm Ad}(E_G)$ such that for some (hence every) $x\\, \\in\\, M$, the fiber ${\\mathbb T}_x$ li"},"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":"1906.05364","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-06-12T20:04:34Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"a8afd2437b71d6adf5d7e609647307a0281f495a45828eedbe03f317222c0675","abstract_canon_sha256":"6c6230ac49a52d89b35e16f6abe1e2ce074df00165b2cfab3f39ed8ced30cad9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:25.479586Z","signature_b64":"w7WXNFEEXSFWpV0R3v9q5VmXa/bgq2f3BPmL+m6bKynfa2RZcJnvA3Uf4ys0fx7SzG/Ih/KsXInAz3M7dw7GBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d33891b79741cde00bd5f99d7d2d55a00c6eb2e34acc94629615a213fc8ea27e","last_reissued_at":"2026-05-17T23:43:25.479125Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:25.479125Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Automorphism group of principal bundles, Levi reduction and invariant connections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AG","authors_text":"Francois-Xavier Machu, Indranil Biswas","submitted_at":"2019-06-12T20:04:34Z","abstract_excerpt":"Let $M$ be a compact connected complex manifold and $G$ a connected reductive complex affine algebraic group. Let $E_G$ be a holomorphic principal $G$--bundle over $M$ and $T\\, \\subset\\, G$ a torus containing the connected component of the center of $G$. Let $N$ (respectively, $C$) be the normalizer (respectively, centralizer) of $T$ in $G$, and let $W$ be the Weyl group $N/C$ for $T$. We prove that there is a natural bijective correspondence between the following two:\n  Torus subbundles $\\mathbb T$ of ${\\rm Ad}(E_G)$ such that for some (hence every) $x\\, \\in\\, M$, the fiber ${\\mathbb T}_x$ li"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.05364","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1906.05364","created_at":"2026-05-17T23:43:25.479180+00:00"},{"alias_kind":"arxiv_version","alias_value":"1906.05364v1","created_at":"2026-05-17T23:43:25.479180+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.05364","created_at":"2026-05-17T23:43:25.479180+00:00"},{"alias_kind":"pith_short_12","alias_value":"2M4JDN4XIHG6","created_at":"2026-05-18T12:33:07.085635+00:00"},{"alias_kind":"pith_short_16","alias_value":"2M4JDN4XIHG6AC6V","created_at":"2026-05-18T12:33:07.085635+00:00"},{"alias_kind":"pith_short_8","alias_value":"2M4JDN4X","created_at":"2026-05-18T12:33:07.085635+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/2M4JDN4XIHG6AC6V7GOX2LKVUA","json":"https://pith.science/pith/2M4JDN4XIHG6AC6V7GOX2LKVUA.json","graph_json":"https://pith.science/api/pith-number/2M4JDN4XIHG6AC6V7GOX2LKVUA/graph.json","events_json":"https://pith.science/api/pith-number/2M4JDN4XIHG6AC6V7GOX2LKVUA/events.json","paper":"https://pith.science/paper/2M4JDN4X"},"agent_actions":{"view_html":"https://pith.science/pith/2M4JDN4XIHG6AC6V7GOX2LKVUA","download_json":"https://pith.science/pith/2M4JDN4XIHG6AC6V7GOX2LKVUA.json","view_paper":"https://pith.science/paper/2M4JDN4X","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1906.05364&json=true","fetch_graph":"https://pith.science/api/pith-number/2M4JDN4XIHG6AC6V7GOX2LKVUA/graph.json","fetch_events":"https://pith.science/api/pith-number/2M4JDN4XIHG6AC6V7GOX2LKVUA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2M4JDN4XIHG6AC6V7GOX2LKVUA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2M4JDN4XIHG6AC6V7GOX2LKVUA/action/storage_attestation","attest_author":"https://pith.science/pith/2M4JDN4XIHG6AC6V7GOX2LKVUA/action/author_attestation","sign_citation":"https://pith.science/pith/2M4JDN4XIHG6AC6V7GOX2LKVUA/action/citation_signature","submit_replication":"https://pith.science/pith/2M4JDN4XIHG6AC6V7GOX2LKVUA/action/replication_record"}},"created_at":"2026-05-17T23:43:25.479180+00:00","updated_at":"2026-05-17T23:43:25.479180+00:00"}