{"paper":{"title":"Counting Parabolic Principal G-bundles with Nilpotent Sections over $\\mathbb{P}^{1}$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.AG","authors_text":"Rahul Singh","submitted_at":"2021-11-22T23:54:07Z","abstract_excerpt":"Let $G$ be a split connected reductive group over $\\mathbb{F}_q$ and let $\\mathbb{P}^1$ be the projective line over $\\mathbb{F}_q$. Firstly, we give an explicit formula for the number of $\\mathbb{F}_{q}$-rational points of generalized Steinberg varieties of $G$. Secondly, for each principal $G$-bundle over $\\mathbb{P}^1$, we give an explicit formula counting the number of triples consisting of parabolic structures at $0$ and $\\infty$ and a compatible nilpotent section of the associated adjoint bundle. In the case of $GL_{n}$ we calculate a generating function of such volumes re-deriving a resu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2111.11583","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2111.11583/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}