{"paper":{"title":"The category of Whittaker modules over the Cartan Type Lie algebra $\\bar{S}_2$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Each block of Whittaker modules over the Cartan-type Lie algebra bar S_2 with finite-dimensional Whittaker vector spaces is equivalent to the finite-dimensional modules over its parabolic subalgebra bar S_2 to the non-negative part.","cross_cats":[],"primary_cat":"math.RT","authors_text":"Genqiang Liu, Xiaoyao Zheng, Yufang Zhao","submitted_at":"2026-04-28T03:43:53Z","abstract_excerpt":"The Lie algebra $\\bar{S}_2$ of polynomial vector fields on $\\mathbb{C}^2$ with constant divergence is an important Cartan type Lie algebra. In this paper, we study Whittaker $\\bar{S}_2$-modules that are locally finite\n  over $\\text{span}\\{\\frac{\\partial}{\\partial t_1}, \\frac{\\partial}{\\partial t_2}\\}$. We first show that each block $\\Omega^{\\widetilde{S}_2}_{\\mathbf{a}}$ of the category of $(A_2, \\bar{S}_2)$-Whittaker modules with finite-dimensional Whittaker vector spaces is equivalent to the category of finite-dimensional modules over the parabolic subalgebra $\\bar{S}_2^{\\geq 0}$. Then we cl"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Each block Ω^{~S_2}_a of the category of (A_2, bar S_2)-Whittaker modules with finite-dimensional Whittaker vector spaces is equivalent to the finite-dimensional module category over the parabolic subalgebra bar S_2^{≥0}; all simple Whittaker bar S_2-modules with finite-dimensional Whittaker vector spaces are classified using gl_2-modules; and Ω^{bar S_2}_1 is equivalent to H_1-fmod.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The Whittaker vector spaces are finite-dimensional; this restriction is essential for the block decomposition, the equivalences to parabolic and H_1 modules, and the classification via gl_2 to hold as stated.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Blocks of Whittaker modules over bar S_2 with finite-dimensional Whittaker spaces are equivalent to finite-dimensional modules over a parabolic subalgebra, with simples classified via gl_2-modules and one block equivalent to H_1-fmod.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Each block of Whittaker modules over the Cartan-type Lie algebra bar S_2 with finite-dimensional Whittaker vector spaces is equivalent to the finite-dimensional modules over its parabolic subalgebra bar S_2 to the non-negative part.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"95da618fb97507ea1f99f648f2ad1d95d711f572e1d8956f303ebe3ff8111ef1"},"source":{"id":"2604.25185","kind":"arxiv","version":2},"verdict":{"id":"f3efa1b9-ea4b-4841-aac6-e9c2033472f3","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T14:40:02.406308Z","strongest_claim":"Each block Ω^{~S_2}_a of the category of (A_2, bar S_2)-Whittaker modules with finite-dimensional Whittaker vector spaces is equivalent to the finite-dimensional module category over the parabolic subalgebra bar S_2^{≥0}; all simple Whittaker bar S_2-modules with finite-dimensional Whittaker vector spaces are classified using gl_2-modules; and Ω^{bar S_2}_1 is equivalent to H_1-fmod.","one_line_summary":"Blocks of Whittaker modules over bar S_2 with finite-dimensional Whittaker spaces are equivalent to finite-dimensional modules over a parabolic subalgebra, with simples classified via gl_2-modules and one block equivalent to H_1-fmod.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The Whittaker vector spaces are finite-dimensional; this restriction is essential for the block decomposition, the equivalences to parabolic and H_1 modules, and the classification via gl_2 to hold as stated.","pith_extraction_headline":"Each block of Whittaker modules over the Cartan-type Lie algebra bar S_2 with finite-dimensional Whittaker vector spaces is equivalent to the finite-dimensional modules over its parabolic subalgebra bar S_2 to the non-negative part."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.25185/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T05:37:50.593149Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T21:23:52.894546Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"80491cd52432514d4662e430db53c76c5327d0f7425e29f8fcbff2101ac561bc"},"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"}