Let be an involution of the finite dimensional semisimple Lie algebra and be the associated Cartan decomposition. The nilpotent commuting variety of consists in pairs of nilpotent elements of such that . It is conjectured that this variety is equidimensional and that its irreducible components are indexed by the orbits of distinguished elements. This conjecture was established by A. Premet in the case where . In this work we prove the conjecture in a significant number of other cases.
Soit une involution de l’algèbre de Lie semi-simple de dimension finie et la décomposition de Cartan associée. La variété commutante nilpotente de l’algèbre de Lie symétrique est formée des paires d’éléments nilpotents de tels que . Il est conjecturé que cette variété est équidimensionnelle et que ses composantes irréductibles sont indexées par les orbites d’éléments -distingués. Cette conjecture a été démontrée par A. Premet dans le cas avec . Dans ce travail, nous la prouvons dans un grand nombre d’autres cas.
Mot clés : algèbre de Lie semi-simple, paire symétrique, variété commutante, orbite nilpotente
Keywords: Semisimple Lie algebra, symmetric pair, commuting variety, nilpotent orbit
@article{AIF_2009__59_1_37_0, author = {Bulois, Micha\"el}, title = {Composantes irr\'eductibles de la vari\'et\'e commutante nilpotente d{\textquoteright}une alg\`ebre {de~Lie} sym\'etrique semi-simple}, journal = {Annales de l'Institut Fourier}, pages = {37--80}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {59}, number = {1}, year = {2009}, doi = {10.5802/aif.2426}, zbl = {1189.17008}, mrnumber = {2514861}, language = {fr}, url = {http://archive.numdam.org/articles/10.5802/aif.2426/} }
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 SP - 37 EP - 80 VL - 59 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2426/ DO - 10.5802/aif.2426 LA - fr ID - AIF_2009__59_1_37_0 ER -
%0 Journal Article %A Bulois, Michaël %T Composantes irréductibles de la variété commutante nilpotente d’une algèbre de Lie symétrique semi-simple %J Annales de l'Institut Fourier %D 2009 %P 37-80 %V 59 %N 1 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2426/ %R 10.5802/aif.2426 %G fr %F AIF_2009__59_1_37_0
Bulois, Michaël. Composantes irréductibles de la variété commutante nilpotente d’une algèbre de Lie symétrique semi-simple. Annales de l'Institut Fourier, Volume 59 (2009) no. 1, pp. 37-80. doi : 10.5802/aif.2426. http://archive.numdam.org/articles/10.5802/aif.2426/
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