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On orbits of algebraic groups and Lie groups

Published online by Cambridge University Press:  17 April 2009

R.W. Richardson
R.W. Richardson
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, PO Box 4, Canberra, ACT 2600, Australia.
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Abstract

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In this paper we will be concerned with orbits of a closed subgroup Z of an algebraic group (respectively Lie group) G on a homogeneous space X for G. More precisely, let D be a closed subgroup of G and let X denote the coset space G/D. Let S be a subgroup of G and let Z denote (GS)0 the identity component of GS, the centralizer of S in G. We consider the orbits of Z on XS, the set of fixed points of S on X. We also treat the more general situation in which S is an algebraic group (respectively Lie group) which acts on G by automorphisms and acts on X compatibly with the action of G; again we consider the orbits of (GS)0 on XS.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 25 , Issue 1 , February 1982 , pp. 1 - 28
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Borel, Armand, Linear algebraic groups (Benjamin, New York, Amsterdam, 1969).Google Scholar
[2]Borel, A. et Serre, J.-P., “Théorèmes de finitude en cohomologie galoisienne”, Conment. Math. Helv. 39 (1964/1965), 111–164.CrossRefGoogle Scholar
[3]Borel, A. and Springer, T.A., “Rationality properties of linear algebraic groups II”, Tôhoku Math. J. 20 (1968), 443–497.CrossRefGoogle Scholar
[4]Borel, Armand et Tits, Jacques, “Groupes rédictifs”, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55–150.CrossRefGoogle Scholar
[5]Dieudonné, J., Cours de géométrie algebrique, 2 (Presses universitaires de France, Paris, 1974).Google Scholar
[6]Dieudonné, J., Éléments d'analyse. Tome III: Chapitres XVI et XVII (Cahiers Scientifiques, Fasc. 33. Gauthier-Villars Éditeur, Paris, 1970).Google Scholar
[7]Dieudonné, Jean A. and Carrell, James B., “Invariant theory, old and new, Adv. in Math. 4 (1970), 1–80.CrossRefGoogle Scholar
[8]Kempf, George R., “Instability in invariant theory”, Ann. of Math. (2) 108 (1978), 299–316.CrossRefGoogle Scholar
[9]Luna, D., “Adhérences d'orbite et invariants”, Invent. Math. 29 (1975), 231–238.CrossRefGoogle Scholar
[10]Mumford, David, Geometric invariant theory (Ergebnisse der Mathematik und ihrer Grenzgebiete, 34. Springer-Verlag, Berlin, Heidelberg, New York, 1965).CrossRefGoogle Scholar
[11]Nagata, Masayoshi, “Complete reducibility of rational representations of a matric group”, J. Math. Kyoto Univ. 1 (1961–1962), 87–99.Google Scholar
[12]Raghunathan, M.S., Discrete subgroups of Lie groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, 68. Springer-Verlag, Berlin, Heidelberg, New York, 1972).CrossRefGoogle Scholar
[13]Richardson, R.W., “Conjugacy classes in Lie algebras and algebraic groups”, Ann. of Math. (2) 86 (1967), 1–15.CrossRefGoogle Scholar
[14]Richardson, R.W., “Affine coset spaces of reductive algebraic groups”, Bull. London Math. Soc. 9 (1977), 38–41.CrossRefGoogle Scholar
[15]Serre, J-P., “Applications algébriques de la cohomologie des groupes. I”, Cohomologie des groupes, suite spectrale, faisceaux, Exposé 5 (Séminaire Henri Cartan de L'Ecole Normale Supérieure, 3e année, 1950/1951. Secrétariat Mathématique, Paris, 1955).Google Scholar
[16]Springer, T.A., Linear algebraic groups (Birkhauser, Boston, Basel, Stuttgard, 1981).Google Scholar
[17]Steinberg, Robert, “Regular elements of semisimple algebraic groups”, Inst. Routes Études Sci. Publ. Math. 25 (1965), 49–80.CrossRefGoogle Scholar
[18]Винберг, З.Б. [É.B. Vinberg] “Группа Вейля градуированной алгебры ли”, [The Weyl group of a graded Lie algebra], Izv. Akad. Nauk SSSR 40 (1976), 488–526. English translation: Math. USSR-Izv. 10 (1976), 463–495 (1977).Google ScholarPubMed
[19]Vust, Thierry, “Opération de groupes réductifs dans un type de cônes presque homogènes”, Bull. Soc. Math. France 102 (1974), 317–333.CrossRefGoogle Scholar
[20]Whitney, Hassler, “Elementary structure of real algebraic varieties”, Ann. of Math. (2) 66 (1957), 545–556.CrossRefGoogle Scholar