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Then one can ask if there is an algebraic map $\\Phi\\colon V\\to V$ which lifts $\\phi$, i.e., $\\pi(\\Phi(v))=\\phi(\\pi(v))$ for all $v\\in V$. In \\cite{Kuttler} the case is treated where $V=r\\lieg$ is a multiple of the adjoint representation of $G$. It is shown that, for $r$ sufficiently large (often $r\\geq 2$ will do), any $\\phi$ has a lift.\n  We"},"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":"1201.6369","kind":"arxiv","version":9},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-01-30T21:00:47Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"ed34e884c87116e6558765ec89272eba2393d242db788110608fb440557146eb","abstract_canon_sha256":"cc365563967456b15c0884e8f8f78d6c9b82586551b792f0f1d3197a5208bb49"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:53.853435Z","signature_b64":"OgfW/0Rfhd3bT/OMVoesW6wffNFQYlM4jJn0c4O6qU7fHqw+jOMitd4plskU+SFVW/RIZMn2otB56c0CnFeJBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ebd251dddfbc52dd9b824c91ceb840cbe630c5736826a732317bb32ac754fd7","last_reissued_at":"2026-05-18T02:57:53.852881Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:53.852881Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Quotients, automorphisms and differential operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.GR","authors_text":"Gerald W. Schwarz","submitted_at":"2012-01-30T21:00:47Z","abstract_excerpt":"Let $V$ be a $G$-module where $G$ is a complex reductive group. Let $Z:=\\quot VG$ denote the categorical quotient and let $\\pi\\colon V\\to Z$ be the morphism dual to the inclusion $\\O(V)^G\\subset\\O(V)$. Let $\\phi\\colon Z\\to Z$ be an algebraic automorphism. Then one can ask if there is an algebraic map $\\Phi\\colon V\\to V$ which lifts $\\phi$, i.e., $\\pi(\\Phi(v))=\\phi(\\pi(v))$ for all $v\\in V$. In \\cite{Kuttler} the case is treated where $V=r\\lieg$ is a multiple of the adjoint representation of $G$. It is shown that, for $r$ sufficiently large (often $r\\geq 2$ will do), any $\\phi$ has a lift.\n  We"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.6369","kind":"arxiv","version":9},"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":"1201.6369","created_at":"2026-05-18T02:57:53.852968+00:00"},{"alias_kind":"arxiv_version","alias_value":"1201.6369v9","created_at":"2026-05-18T02:57:53.852968+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.6369","created_at":"2026-05-18T02:57:53.852968+00:00"},{"alias_kind":"pith_short_12","alias_value":"P26SKHO57PCS","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_16","alias_value":"P26SKHO57PCS3WNY","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_8","alias_value":"P26SKHO5","created_at":"2026-05-18T12:27:18.751474+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/P26SKHO57PCS3WNYETERZ24EBS","json":"https://pith.science/pith/P26SKHO57PCS3WNYETERZ24EBS.json","graph_json":"https://pith.science/api/pith-number/P26SKHO57PCS3WNYETERZ24EBS/graph.json","events_json":"https://pith.science/api/pith-number/P26SKHO57PCS3WNYETERZ24EBS/events.json","paper":"https://pith.science/paper/P26SKHO5"},"agent_actions":{"view_html":"https://pith.science/pith/P26SKHO57PCS3WNYETERZ24EBS","download_json":"https://pith.science/pith/P26SKHO57PCS3WNYETERZ24EBS.json","view_paper":"https://pith.science/paper/P26SKHO5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1201.6369&json=true","fetch_graph":"https://pith.science/api/pith-number/P26SKHO57PCS3WNYETERZ24EBS/graph.json","fetch_events":"https://pith.science/api/pith-number/P26SKHO57PCS3WNYETERZ24EBS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P26SKHO57PCS3WNYETERZ24EBS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P26SKHO57PCS3WNYETERZ24EBS/action/storage_attestation","attest_author":"https://pith.science/pith/P26SKHO57PCS3WNYETERZ24EBS/action/author_attestation","sign_citation":"https://pith.science/pith/P26SKHO57PCS3WNYETERZ24EBS/action/citation_signature","submit_replication":"https://pith.science/pith/P26SKHO57PCS3WNYETERZ24EBS/action/replication_record"}},"created_at":"2026-05-18T02:57:53.852968+00:00","updated_at":"2026-05-18T02:57:53.852968+00:00"}