Irreducible Components of the Nilpotent Commuting Variety of a Symmetric Semisimple Lie Algebra

Michaël Bulois[1]

  • [1] Université de Brest Département de mathématiques 29238 Brest cedex 3 (France)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 1, page 37-80
  • ISSN: 0373-0956

Abstract

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Let θ be an involution of the finite dimensional semisimple Lie algebra 𝔤 and 𝔤 = 𝔨 ⊕ 𝔭 be the associated Cartan decomposition. The nilpotent commuting variety of ( 𝔤 , θ ) consists in pairs of nilpotent elements ( x , y ) of 𝔭 such that [ x , y ] = 0 . It is conjectured that this variety is equidimensional and that its irreducible components are indexed by the orbits of 𝔭 distinguished elements. This conjecture was established by A. Premet in the case ( 𝔤 × 𝔤 , θ ) where θ ( x , y ) = ( y , x ) . In this work we prove the conjecture in a significant number of other cases.

How to cite

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Bulois, Michaël. "Composantes irréductibles de la variété commutante nilpotente d’une algèbre de Lie symétrique semi-simple." Annales de l’institut Fourier 59.1 (2009): 37-80. <http://eudml.org/doc/10396>.

@article{Bulois2009,
abstract = {Soit $\theta $ une involution de l’algèbre de Lie semi-simple de dimension finie $\mathfrak\{g\}$ et $\mathfrak\{g\}=\mathfrak\{k\}\oplus \mathfrak\{p\}$ la décomposition de Cartan associée. La variété commutante nilpotente de l’algèbre de Lie symétrique $(\mathfrak\{g\},\theta )$ est formée des paires d’éléments nilpotents $(x,y)$ de $\mathfrak\{p\}$ tels que $[x,y]=0$. Il est conjecturé que cette variété est équidimensionnelle et que ses composantes irréductibles sont indexées par les orbites d’éléments $\mathfrak\{p\}$-distingués. Cette conjecture a été démontrée par A. Premet dans le cas $(\mathfrak\{g\}\times \mathfrak\{g\},\theta )$ avec $\theta (x,y)=(y,x)$. Dans ce travail, nous la prouvons dans un grand nombre d’autres cas.},
affiliation = {Université de Brest Département de mathématiques 29238 Brest cedex 3 (France)},
author = {Bulois, Michaël},
journal = {Annales de l’institut Fourier},
keywords = {Semisimple Lie algebra; symmetric pair; commuting variety; nilpotent orbit},
language = {fre},
number = {1},
pages = {37-80},
publisher = {Association des Annales de l’institut Fourier},
title = {Composantes irréductibles de la variété commutante nilpotente d’une algèbre de Lie symétrique semi-simple},
url = {http://eudml.org/doc/10396},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Bulois, Michaël
TI - Composantes irréductibles de la variété commutante nilpotente d’une algèbre de Lie symétrique semi-simple
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 1
SP - 37
EP - 80
AB - Soit $\theta $ une involution de l’algèbre de Lie semi-simple de dimension finie $\mathfrak{g}$ et $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$ la décomposition de Cartan associée. La variété commutante nilpotente de l’algèbre de Lie symétrique $(\mathfrak{g},\theta )$ est formée des paires d’éléments nilpotents $(x,y)$ de $\mathfrak{p}$ tels que $[x,y]=0$. Il est conjecturé que cette variété est équidimensionnelle et que ses composantes irréductibles sont indexées par les orbites d’éléments $\mathfrak{p}$-distingués. Cette conjecture a été démontrée par A. Premet dans le cas $(\mathfrak{g}\times \mathfrak{g},\theta )$ avec $\theta (x,y)=(y,x)$. Dans ce travail, nous la prouvons dans un grand nombre d’autres cas.
LA - fre
KW - Semisimple Lie algebra; symmetric pair; commuting variety; nilpotent orbit
UR - http://eudml.org/doc/10396
ER -

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