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On the adjoint quotient of Chevalley groups over arbitrary base schemes

Part of: General commutative ring theory

Published online by Cambridge University Press:  16 April 2010

Pierre-Emmanuel Chaput  and
Matthieu Romagny
Pierre-Emmanuel Chaput
Affiliation:
Laboratoire de Mathématiques Jean Leray, UMR 6629 du CNRS, UFR Sciences et Techniques, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 03, France, ([email protected])
Matthieu Romagny
Affiliation:
Institut de Mathématiques, Théorie des Nombres, Université Pierre et Marie Curie, Case 82, 4, place Jussieu, F-75252 Paris Cedex 05, France, ([email protected])
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Abstract

For a split semisimple Chevalley group scheme G with Lie algebra over an arbitrary base scheme S, we consider the quotient of by the adjoint action of G. We study in detail the structure of over S. Given a maximal torus T with Lie algebra and associated Weyl group W, we show that the Chevalley morphism π : /W → /G is an isomorphism except for the group Sp2n 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 commutes, or does not commute, with base change on S.

Keywords

Chevalley groupsadjoint actioninvariant theorybase change

MSC classification

Secondary: 13A50: Actions of groups on commutative rings; invariant theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 9 , Issue 4 , October 2010 , pp. 673 - 704
Copyright
Copyright © Cambridge University Press 2010

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