Composantes irréductibles de la variété commutante nilpotente d’une algèbre de Lie symétrique semi-simple
Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 37-80.

Soit θ une involution de l’algèbre de Lie semi-simple de dimension finie 𝔤 et 𝔤=𝔨𝔭 la décomposition de Cartan associée. La variété commutante nilpotente de l’algèbre de Lie symétrique (𝔤,θ) est formée des paires d’éléments nilpotents (x,y) de 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 𝔭-distingués. Cette conjecture a été démontrée par A. Premet dans le cas (𝔤×𝔤,θ) avec θ(x,y)=(y,x). Dans ce travail, nous la prouvons dans un grand nombre d’autres cas.

Let θ be an involution of the finite dimensional semisimple Lie algebra 𝔤 and 𝔤=𝔨𝔭 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.

DOI : 10.5802/aif.2426
Classification : 17B20, 14L30, 17B20
Mot clés : algèbre de Lie semi-simple, paire symétrique, variété commutante, orbite nilpotente
Keywords: Semisimple Lie algebra, symmetric pair, commuting variety, nilpotent orbit
Bulois, Michaël 1

1 Université de Brest Département de mathématiques 29238 Brest cedex 3 (France)
@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, Tome 59 (2009) no. 1, pp. 37-80. doi : 10.5802/aif.2426. http://archive.numdam.org/articles/10.5802/aif.2426/

[1] Baranovski, V. The variety of pairs of commuting nilpotent matrices is irreducible, Transform. Groups, Volume 6 (2001), pp. 3-8 | DOI | MR | Zbl

[2] Bourbaki, N. Groupes et algèbres de Lie. Chapitres 4, 5 et 6, Hermann, Paris, 1968 | MR | Zbl

[3] Djokovic, D. Z. Classification of nilpotent elements in simple exceptional real algebras of inner type and description of their centralizers, J. Algebra, Volume 112 (1988), pp. 503-524 | DOI | MR | Zbl

[4] Djokovic, D. Z. Classification of nilpotent elements in simple real lie algebras E 6(6) and E 6(-26) and description of their centralizers, J. Algebra, Volume 116 (1988), pp. 196-207 | DOI | MR | Zbl

[5] Djokovic, D. Z. Explicit Cayley triples in real forms of F 2 , G 4 and E 6 , Pacific J. Math., Volume 184 (1998), pp. 231-255 | DOI | Zbl

[6] Djokovic, D. Z. Explicit Cayley triples in real forms of E 8 , Pacific J. of Math., Volume 194 (2000), pp. 57-82 | DOI | MR | Zbl

[7] Djokovic, D. Z. The closure diagram for nilpotent orbits of the split real form of E 7 , Represent. Theory, Volume 5 (2001), pp. 284-316 | DOI | MR | Zbl

[8] Djokovic, D. Z. The closure diagrams for nilpotent orbits of real forms of E 6 , J. Lie Theory, Volume 11 (2001), pp. 381-413 | MR | Zbl

[9] Djokovic, D. Z. The closure diagram for nilpotent orbits of the split real form of E 8 , Centr. Europ. J. Math., Volume 4 (2003), pp. 573-643 | DOI | MR | Zbl

[10] Elashvili, E. B. The centralisers of nilpotent elements in semisimple Lie algebras, Trudy Tbiliss. Inst. Mat. Nats. Nauk Gruzin., Volume 46 (1975), pp. 109-132 (In Russian)

[11] Goodman, R.; Wallach, N. R. An algebraic group approach to compact symetric spaces, 1997 (http ://www.math.rutgers.edu/pub/goodman/symspace.pdf)

[12] Helgason, S. Differential geometry, Lie groups, and symmetric spaces, Pure and applied mathematics, Academic press, 1978 | MR | Zbl

[13] Jackson, S. G.; Noel, A. G. Prehomogeneous spaces associated with nilpotent orbits, 2005 (http://www.math.umb.edu/~anoel/publications/tables)

[14] Jantzen, J. C. Nilpotent orbits in representation theory, Lie Theory (Progr. Math.), Volume 228, Birkhäuser, 2004, pp. 1-211 | MR

[15] Kawanaka, N. Orbits and stabilizers of nilpotent elements of a graded semisimple Lie algebra, J. Fac. Sci. Univ. Tokyo, Volume 34 (1987), pp. 573-597 | MR | Zbl

[16] King, D. R. The component groups of nilpotents in exceptionnal simple real Lie algebras, Comm. Algebra, Volume 20 (1992), pp. 219-284 | DOI | MR | Zbl

[17] Kostant, B.; Rallis, S. Orbits and representations associated with symmetric spaces, Amer. J. Math., Volume 93 (1971), pp. 753-809 | DOI | MR | Zbl

[18] Otha, T. The singularities of the closure of nilpotent orbits in certain symmetric pairs, Tôhoku Math. J., Volume 38 (1986), pp. 441-468 | DOI | MR | Zbl

[19] Otha, T. The closure of nilpotent orbits in the classical symmetric pairs and their singularities, Tôhoku Math. J., Volume 43 (1991), pp. 161-211 | DOI | MR | Zbl

[20] Panyushev, D. I. The Jacobian modules of a representation of a Lie algebra and geometry of commuting varieties, Compositio Math., Volume 94 (1994), pp. 181-199 | Numdam | MR | Zbl

[21] Panyushev, D. I. On the conormal bundle of a G-stable subvariety, Manuscripta Math., Volume 99 (1999), pp. 185-202 | DOI | MR | Zbl

[22] Panyushev, D. I. On the irreducibility of commuting varieties associated with involutions of simple Lie algebras, Func. Anal. Appl., Volume 38 (2004), pp. 38-44 | DOI | MR | Zbl

[23] Panyushev, D. I. Two results on centralisers of nilpotent elements, J. Pure Appl. Algebra, Volume 212 (2008), pp. 774-779 | DOI | MR | Zbl

[24] Panyushev, D. I.; Yakimova, O. Symmetric pairs and associated commuting varieties, Math. Proc., Volume 143, Cambridge Philos. Soc. (2007), pp. 307-321 | MR | Zbl

[25] Popov, V. L.; Tevelev, E. A. Self-dual projective algebraic varieties associated with symmetric spaces, Algebraic transformation groups and algebraic varieties (Enc. Math. Sci.), Volume 132, Springer Verlag (2004), pp. 131-167 | MR | Zbl

[26] Premet, A. Nilpotent commuting varieties of reductive Lie algebras, Invent. Math., Volume 154 (2003), pp. 653-683 | DOI | MR | Zbl

[27] Richardson, R. W. Commuting varieties of semisimple Lie algebras and algebraic groups, Compositio Math., Volume 38 (1979), pp. 311-327 | Numdam | MR | Zbl

[28] Sabourin, H.; Yu, R. W. T. Sur l’irréductibilité de la variété commutante d’une paire symétrique réductive de rang 1, Bull. Sci. Math., Volume 126 (2002), pp. 143-150 | DOI | MR | Zbl

[29] Sabourin, H.; Yu, R. W. T. On the irreducibility of the commuting variety of the symmetric pair (𝔰𝔬 p+2 ,𝔰𝔬 p ×𝔰𝔬 2 ), J. Lie Theory, Volume 16 (2006), pp. 57-65 | MR | Zbl

[30] Sekiguchi, J. The nilpotent subvariety of the vector space associated to a symmetric pair, Publ. RIMS Kyoto Univ., Volume 20 (1984), pp. 155-212 | DOI | MR | Zbl

[31] Springer, T. A.; Steinberg, R. Conjugacy classes, Seminar on algebraic groups and related finite groups (Lecture Notes in Math.), Volume 131, Springer, 1970, pp. 167-266 | MR | Zbl

[32] Tauvel, P.; Yu, R. W. T. Lie algebras and algebraic groups, Springer Monographs in Mathematics, Springer, 2005 | MR | Zbl

[33] Vinberg, E. B. Classification of homogeneous nilpotent elements of a semisimple graded Lie algebra, Selecta Math. Sovietica, Volume 6 (1987), pp. 15-35 | Zbl

Cité par Sources :