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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, (pierre-emmanuel.chaput@math.univ-nantes.fr)
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, (romagny@math.jussieu.fr)
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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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References

1.Anantharaman, S., Schémas en groupes, espaces homogènes et espaces algébriques sur une base de dimension 1, Bull. Soc. Math. France 33 (1973), 5–79.Google Scholar
2.Artin, M., Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165–189.CrossRefGoogle Scholar
3.Berthelot, P., Breen, L. and Messing, W., Théorie de Dieudonné cristalline, Volume II, Lecture Notes in Mathematics, Volume 930 (Springer, 1982).Google Scholar
4.Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 21 (Springer, 1990).Google Scholar
5.Bourbaki, N., Groupes et algèbres de Lie, Chapters 4–6 (Hermann, Paris, 1968).Google Scholar
6.Chevalley, C., Invariants of finite groups generated by reflections, Am. J. Math. 77 (1955), 778–782.CrossRefGoogle Scholar
7.Demazure, M., Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287–301.CrossRefGoogle Scholar
8.Demazure, M. and Grothendieck, A., Schémas en groupes, III: structure des schémas en groupes réductifs, Séminaire de Géométrie Algébrique du Bois Marie 1962/1964 (SGA 3), Lecture Notes in Mathematics, Volume 153 (Springer, 1962/1964).Google Scholar
9.Dieudonné, J. and Grothendieck, A., Éléments de géométrie algébrique, Publ. Math. IHÉS 4, 8, 11, 17, 20, 24, 28, 32 (1961–1967).Google Scholar
10.Farnsteiner, R., Varieties of tori and Cartan subalgebras of restricted Lie algebras, Trans. Am. Math. Soc. 356(10) (2004), 4181–4236.CrossRefGoogle Scholar
11.Humphreys, J., Linear algebraic groups, Graduate Texts in Mathematics, No. 21 (Springer, 1975).Google Scholar
12.Ikai, H., Spin groups over a commutative ring and the associated root data, Monatsh. Math. 139(1) (2003), 33–60.CrossRefGoogle Scholar
13.Ikai, H., Spin groups over a commutative ring and the associated root data (odd rank case), Beit. Alg. Geom. 46(2) (2005), 377–395.Google Scholar
14.Matsumura, H., Commutative ring theory, Cambridge Studies in Advanced Mathematics, Volume 8 (Cambridge University Press, 1989).Google Scholar
15.Serre, J. P., Algèbres de Lie semi-simples complexes (W. A. Benjamin, New York/Amsterdam, 1966).Google Scholar
16.Springer, T. A. and Steinberg, R., Conjugacy classes, in Seminar on Algebraic Groups and Related Finite Groups, IAS, Princeton, NJ, 1968/1969 (ed. Borel, E. et al. ), pp. 167–266, Lecture Notes in Mathematics, Volume 131 (Springer, 1970).Google Scholar
17.Steinberg, R., Regular elements of semisimple algebraic groups, Publ. Math. IHES 25 (1965), 49–80.CrossRefGoogle Scholar
18.Strade, H., Simple Lie algebras over fields of positive characteristic, I, Structure theory, de Gruyter Expositions in Mathematics, Volume 38 (de Gruyter, Berlin, 2004).Google Scholar
19.Taylor, D., The geometry of the classical groups, Sigma Series in Pure Mathematics, No. 9 (Heldermann, 1992).Google Scholar
20.Veldkamp, F. D., The center of the universal enveloping algebra of a Lie algebra in characteristic p, Annales Scient. Éc. Norm. Sup. 5 (1972), 217–240.CrossRefGoogle Scholar