{"paper":{"title":"Explicit Formulas for the Casimir Eigenvalues of $SL(n,\\mathbb{Z})$-Maass Forms","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Maass forms for SL(n,Z) have their Casimir eigenvalues given explicitly by formulas in the Langlands parameters.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Vishal Muthuvel","submitted_at":"2026-05-16T04:15:50Z","abstract_excerpt":"Maass forms for $SL(n,\\mathbb{Z})$ are defined to be eigenfunctions of the Casimir operators $\\mathcal{D}_{m,n}$ of orders $1 \\leq m \\leq n$ for $GL(n,\\mathbb{R})$. For any $1 \\leq m \\leq n$ and Maass form $\\phi$ for $SL(n,\\mathbb{Z})$, we provide a formula for the eigenvalue of $\\mathcal{D}_{m,n}$ associated with $\\phi$ in terms of the Langlands parameters of $\\phi$. In the case $m=2$, we recover the formula for the Laplace eigenvalue of a Maass form due to Terras, the Casimir differential operator of order $2$ being the Laplacian. Our proof takes a graph-theoretic approach, relating the acti"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For any 1 ≤ m ≤ n and Maass form φ for SL(n,Z), we provide a formula for the eigenvalue of D_{m,n} associated with φ in terms of the Langlands parameters of φ.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The Maass forms for SL(n,Z) are eigenfunctions of the full set of Casimir operators D_{m,n} of orders 1 ≤ m ≤ n for GL(n,R), and the Langlands parameters are the standard ones that classify these forms.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Explicit formulas are derived for the eigenvalues of Casimir operators D_{m,n} on SL(n,Z)-Maass forms in terms of Langlands parameters, with a graph-theoretic proof that recovers the known Laplacian case for m=2.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Maass forms for SL(n,Z) have their Casimir eigenvalues given explicitly by formulas in the Langlands parameters.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9e93f2cb75fac98c9aacb4eca1b374af3da20b45ee56e8c3778074b86eed7b04"},"source":{"id":"2605.16803","kind":"arxiv","version":1},"verdict":{"id":"22780a9a-a32f-4d27-b568-94996ce8f048","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T20:06:29.048917Z","strongest_claim":"For any 1 ≤ m ≤ n and Maass form φ for SL(n,Z), we provide a formula for the eigenvalue of D_{m,n} associated with φ in terms of the Langlands parameters of φ.","one_line_summary":"Explicit formulas are derived for the eigenvalues of Casimir operators D_{m,n} on SL(n,Z)-Maass forms in terms of Langlands parameters, with a graph-theoretic proof that recovers the known Laplacian case for m=2.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The Maass forms for SL(n,Z) are eigenfunctions of the full set of Casimir operators D_{m,n} of orders 1 ≤ m ≤ n for GL(n,R), and the Langlands parameters are the standard ones that classify these forms.","pith_extraction_headline":"Maass forms for SL(n,Z) have their Casimir eigenvalues given explicitly by formulas in the Langlands parameters."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16803/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T20:31:19.133757Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T20:22:04.485470Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T19:01:56.285403Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.422297Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"49f137d150365d6b881fc6ebe76b04892d17d0f0b61b3ccedf969d2130a3587e"},"references":{"count":3,"sample":[{"doi":"","year":1979,"title":"Eisenstein Series and the Trace Formula","work_id":"9ffc8dcc-bbac-4e64-bf21-dbd95e3b38f3","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1951,"title":"On some applications of the universal en- veloping algebra of a semisimple Lie algebra","work_id":"704b5d8b-2d1d-4562-bb18-15337e55884d","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1956,"title":"Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces With Applications to Dirichlet Series","work_id":"b2699b1d-b56c-499e-bc8c-51b87fd213ad","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":3,"snapshot_sha256":"50dd6410de4bc3990e9abc7cffc99b0bd0583d1a67bedc0042d6cfe34a37c358","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"d1accc0b76fe2ce88a281830003d2211215c3cc89ac7c3ac17c599cf78262501"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}