{"paper":{"title":"Adjoint quotients of reductive groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Ting-Yu Lee","submitted_at":"2012-11-15T10:20:16Z","abstract_excerpt":"Let $\\rG$ be a reductive group over a commutative ring $k$. In this article, we prove that the adjoint quotient $\\adqG$ is stable under base change. Moreover, if $\\rG$ has a maximal torus $\\rT$, then the adjoint quotient of the torus $\\rT$ by its Weyl group will be isomorphic to $\\adqG$. Then we focus on the semisimple simply connected group $\\rG$ of the constant type. In this case, $\\adqG$ is isomorphic to the Weil restriction $\\underset{\\rD/\\spec k}{\\prod}\\aff^{1}_\\rD$, where $\\rD$ is the Dynkin scheme of $\\rG$. Then we prove that for such $\\rG$, the Steinberg's cross-section can be defined "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3559","kind":"arxiv","version":1},"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"}