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We say that a simple $(g; k)$-module $M$ is of finite type if all $k$-isotypic components of $M$ are finite-dimensional. To a simple $g$-module $M$ one assigns interesting invariants V$(M)$, $\\EuScript V(M)$ and L$(M)$ reflecting the 'directions of growth of M'. In this work we prove that, for a given pair $(g; k)$, the set of possible such invariants is finite.\n  Let $K$"},"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":"1105.5020","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2011-05-25T13:30:48Z","cross_cats_sorted":["math.SG"],"title_canon_sha256":"3ca7380b3a7ed624a22469d4f656b53ebe336d586d31cc928261d2abb4ff5623","abstract_canon_sha256":"ff0cc50da8a504cb6902569a476ebf4e36b9ba20805f2c74c38a146b385e0a86"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:56:57.901432Z","signature_b64":"3gGoDt8ObmXwJUqm5YsCKt/jI3FPCNd7w+0wzKX5BjcANo0XaoLGWHP7acGChDMAnenbqkllykGwVopjy3d8BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4116e598c273d781385bb59e36f5ed0c61d4a24235ecbeb809c2d027ea233eba","last_reissued_at":"2026-05-18T00:56:57.900923Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:56:57.900923Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A geometric approach to (g, k)-modules of finite type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SG"],"primary_cat":"math.RT","authors_text":"Alexey Petukhov","submitted_at":"2011-05-25T13:30:48Z","abstract_excerpt":"Let $g$ be a semisimple Lie algebra over $\\mathbb C$ and $k$ be a reductive in $g$ subalgebra. 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