On the adjoint quotient of Chevalley groups over arbitrary base schemes

2008/05/14 by Pierre-Emmanuel Chaput, Chaput, Pierre-Emmanuel, Matthieu Romagny +1
Mathematics · #13A50 #20G15 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:13A50 #msc:20G15

paper · pdf · doi:10.48550/arxiv.0805.2140

27 pages

arxiv created 2008/05/14 · arxiv updated 2009/12/01

Abstract

For a split semisimple Chevalley group scheme G with Lie algebra g over an arbitrary base scheme S, we consider the quotient of g by the adjoint action of G. We study in detail the structure of g over S. Given a maximal torus T with Lie algebra t and associated Weyl group W, we show that the Chevalley morphism t/W -> g/G is an isomorphism except for the group Sp_2n over a base with 2-torsion. In this case this morphism is only dominant and we compute it explicitly. We compute the adjoint quotient in some other classical cases, yielding examples where the formation of the quotient g -> g/G commutes, or does not commute, with base change on S.

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