Adjoint quotients of reductive groups

2012/11/15 by Ting-Yu Lee, Lee, Ting-Yu
Mathematics · #13A50 #14L15 #14L24 #14L30 #20G05 #20G35 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.1211.3559

openalex publication_date 2012/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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∏\aff¹_\rD, where \rD is the Dynkin scheme of \rG. Then we prove that for such \rG, the Steinberg's cross-section can be defined over k if \rG is quasi-split and without \rA_2m-type components

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