An extension of Rais' theorem and seaweed subalgebras of simple Lie algebras

Dmitri I. Panyushev[1]

  • [1] Independent university of Moscow, Bol'shoi Vlaservskii per.11, 119002 Moscow (Russia)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 3, page 693-715
  • ISSN: 0373-0956

Abstract

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We prove an extension of Rais' theorem on the coadjoint representation of certain graded Lie algebras. As an application, we prove that, for the coadjoint representation of any seaweed subalgebra in a general linear or symplectic Lie algebra, there is a generic stabiliser and the field of invariants is rational. It is also shown that if the highest root of a simple Lie algerba is not fundamental, then there is a parabolic subalgebra whose coadjoint representation do not have a generic stabiliser.

How to cite

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I. Panyushev, Dmitri. "An extension of Rais' theorem and seaweed subalgebras of simple Lie algebras." Annales de l’institut Fourier 55.3 (2005): 693-715. <http://eudml.org/doc/116204>.

@article{I2005,
abstract = {We prove an extension of Rais' theorem on the coadjoint representation of certain graded Lie algebras. As an application, we prove that, for the coadjoint representation of any seaweed subalgebra in a general linear or symplectic Lie algebra, there is a generic stabiliser and the field of invariants is rational. It is also shown that if the highest root of a simple Lie algerba is not fundamental, then there is a parabolic subalgebra whose coadjoint representation do not have a generic stabiliser.},
affiliation = {Independent university of Moscow, Bol'shoi Vlaservskii per.11, 119002 Moscow (Russia)},
author = {I. Panyushev, Dmitri},
journal = {Annales de l’institut Fourier},
keywords = {field of invariants; generic stabiliser; simple Lie algebra; seaweed subalgebra},
language = {eng},
number = {3},
pages = {693-715},
publisher = {Association des Annales de l'Institut Fourier},
title = {An extension of Rais' theorem and seaweed subalgebras of simple Lie algebras},
url = {http://eudml.org/doc/116204},
volume = {55},
year = {2005},
}

TY - JOUR
AU - I. Panyushev, Dmitri
TI - An extension of Rais' theorem and seaweed subalgebras of simple Lie algebras
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 3
SP - 693
EP - 715
AB - We prove an extension of Rais' theorem on the coadjoint representation of certain graded Lie algebras. As an application, we prove that, for the coadjoint representation of any seaweed subalgebra in a general linear or symplectic Lie algebra, there is a generic stabiliser and the field of invariants is rational. It is also shown that if the highest root of a simple Lie algerba is not fundamental, then there is a parabolic subalgebra whose coadjoint representation do not have a generic stabiliser.
LA - eng
KW - field of invariants; generic stabiliser; simple Lie algebra; seaweed subalgebra
UR - http://eudml.org/doc/116204
ER -

References

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  10. M. Raïs, L’indice des produits semi-directs E × ρ 𝔤 , C.R. Acad. Sc. Paris, Ser. A 287 (1978), 195-197 Zbl0387.17002MR506502
  11. R.W. Richardson, Principal orbit types for algebraic transformation spaces in characteristic zero, Invent. Math. 16 (1972), 6-14 Zbl0242.14010MR294336
  12. P. Tauvel, R. Yu, Indice et formes linéaires stables dans les algèbres de Lie, J. Alg. 273 (2004), 507-516 Zbl1088.17006MR2037708
  13. P. Tauvel, R. Yu, Sur l'indice de certaines algèbres de Lie, Ann. Inst. Fourier 54 (2004), 1793-1810 Zbl1137.17300MR2134224
  14. E.B. Vinberg, V.L. Popov, Invariant theory, Algebraic Geometry IV 55 (1994), 123-284, Springer, Berlin-Heidelberg-New York Zbl0789.14008

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