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  • Bulletin of the Australian Mathematical Society, Volume 25, Issue 1
  • February 1982, pp. 1-28

On orbits of algebraic groups and Lie groups

  • R.W. Richardson (a1)
  • DOI: http://dx.doi.org/10.1017/S0004972700005013
  • Published online: 01 April 2009
Abstract

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.

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[2]A. Borel et J.-P. Serre , “Théorèmes de finitude en cohomologie galoisienne”, Conment. Math. Helv. 39 (1964/1965), 111164.

[3]A. Borel and T.A. Springer , “Rationality properties of linear algebraic groups II”, Tôhoku Math. J. 20 (1968), 443497.

[4]Armand Borel et Jacques Tits , “Groupes rédictifs”, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55150.

[7]Jean A. Dieudonné and James B. Carrell , “Invariant theory, old and new, Adv. in Math. 4 (1970), 180.

[8]George R. Kempf , “Instability in invariant theory”, Ann. of Math. (2) 108 (1978), 299316.

[9]D. Luna , “Adhérences d'orbite et invariants”, Invent. Math. 29 (1975), 231238.

[10]David Mumford , Geometric invariant theory (Ergebnisse der Mathematik und ihrer Grenzgebiete, 34. Springer-Verlag, Berlin, Heidelberg, New York, 1965).

[12]M.S. Raghunathan , Discrete subgroups of Lie groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, 68. Springer-Verlag, Berlin, Heidelberg, New York, 1972).

[13]R.W. Richardson , “Conjugacy classes in Lie algebras and algebraic groups”, Ann. of Math. (2) 86 (1967), 115.

[14]R.W. Richardson , “Affine coset spaces of reductive algebraic groups”, Bull. London Math. Soc. 9 (1977), 3841.

[17]Robert Steinberg , “Regular elements of semisimple algebraic groups”, Inst. Routes Études Sci. Publ. Math. 25 (1965), 4980.

[20]Hassler Whitney , “Elementary structure of real algebraic varieties”, Ann. of Math. (2) 66 (1957), 545556.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
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