Surprising properties of centralisers in classical Lie algebras
Annales de l'Institut Fourier, Volume 59 (2009) no. 3, p. 903-935

Let 𝔤 be a classical Lie algebra, i.e., either 𝔤𝔩 n , 𝔰𝔭 n , or 𝔰𝔬 n and let e be a nilpotent element of 𝔤. We study various properties of the centralisers 𝔤 e . The first four sections deal with rather elementary questions, like the centre of 𝔤 e , commuting varieties associated with 𝔤 e , or centralisers of commuting pairs. The second half of the paper addresses problems related to different Poisson structures on 𝔤 e * and symmetric invariants of 𝔤 e .

Soit 𝔤 une algèbre de Lie classique, i.e., 𝔤𝔩 n , 𝔰𝔭 n , ou 𝔰𝔬 n , et soit e un élément nilpotent de 𝔤. Nous étudions dans cet article diverses propriétés du centralisateur 𝔤 e de e. Les quatre premières sections concernent des problèmes assez élémentaires portant sur le centre de 𝔤 e , la variété commutante de 𝔤 e , ou encore les centralisateurs des paires commutantes. La seconde partie aborde des questions liées aux différentes structures de Poisson sur 𝔤 e * et aux invariants symétriques de 𝔤 e .

DOI : https://doi.org/10.5802/aif.2451
Classification:  17B45
Keywords: Nilpotent orbits, centralisers, symmetric invariants
@article{AIF_2009__59_3_903_0,
     author = {Yakimova, Oksana},
     title = {Surprising properties of centralisers in classical Lie algebras},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {3},
     year = {2009},
     pages = {903-935},
     doi = {10.5802/aif.2451},
     mrnumber = {2543656},
     zbl = {1187.17008},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2009__59_3_903_0}
}
Yakimova, Oksana. Surprising properties of centralisers in classical Lie algebras. Annales de l'Institut Fourier, Volume 59 (2009) no. 3, pp. 903-935. doi : 10.5802/aif.2451. http://www.numdam.org/item/AIF_2009__59_3_903_0/

[1] Arzhantsev, I. V. On the actions of reductive groups with a one-parameter family of spherical orbits, Mat. Sb., Tome 188 (1997) no. 5, pp. 3-20 | MR 1478627 | Zbl 0895.14015

[2] Brown, J.; Brundan, J. Elementary invariants for centralisers of nilpotent matrices (arXiv:math.RA/0611024)

[3] Collingwood, David H.; Mcgovern, William M. Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Co., New York, Van Nostrand Reinhold Mathematics Series (1993) | MR 1251060 | Zbl 0972.17008

[4] Cushman, Richard; Roberts, Mark Poisson structures transverse to coadjoint orbits, Bull. Sci. Math., Tome 126 (2002) no. 7, pp. 525-534 | Article | MR 1931184 | Zbl 1074.53070

[5] Gan, Wee Liang; Ginzburg, Victor Quantization of Slodowy slices, Int. Math. Res. Not. (2002) no. 5, pp. 243-255 | Article | MR 1876934 | Zbl 0989.17014

[6] Ginzburg, Victor Principal nilpotent pairs in a semisimple Lie algebra. I, Invent. Math., Tome 140 (2000) no. 3, pp. 511-561 | Article | MR 1760750 | Zbl 0984.17007

[7] De Graaf, Willem A. Computing with nilpotent orbits in simple Lie algebras of exceptional type, LMS J. Comput. Math., Tome 11 (2008), pp. 280-297 | Article | MR 2434879

[8] Jantzen, Jens Carsten Nilpotent orbits in representation theory, Lie theory, Birkhäuser Boston, Boston, MA (Progr. Math.) Tome 228 (2004), pp. 1-211 | MR 2042689 | Zbl pre02160654

[9] Kac, V. G. Some remarks on nilpotent orbits, J. Algebra, Tome 64 (1980) no. 1, pp. 190-213 | Article | MR 575790 | Zbl 0431.17007

[10] Kostant, Bertram Lie group representations on polynomial rings, Amer. J. Math., Tome 85 (1963), pp. 327-404 | Article | MR 158024 | Zbl 0124.26802

[11] Kurtzke, John F. Jr. Centralizers of irregular elements in reductive algebraic groups, Pacific J. Math., Tome 104 (1983) no. 1, pp. 133-154 | MR 683733 | Zbl 0477.20025

[12] Mustaţă, Mircea Jet schemes of locally complete intersection canonical singularities, Invent. Math., Tome 145 (2001) no. 3, pp. 397-424 (With an appendix by David Eisenbud and Edward Frenkel) | Article | MR 1856396 | Zbl 1091.14004

[13] Neubauer, Michael G.; Sethuraman, B. A. Commuting pairs in the centralizers of 2-regular matrices, J. Algebra, Tome 214 (1999) no. 1, pp. 174-181 | Article | MR 1684884 | Zbl 0924.15015

[14] Ooms, A. I.; Van Den Bergh, M. A degree inequality for Lie algebras with a regular Poisson semi-center (arXiv:0805.1342v1 [math.RT])

[15] Panyushev, D. I.; Premet, A.; Yakimova, O. S. On symmetric invariants of centralisers in reductive Lie algebras, J. Algebra, Tome 313 (2007) no. 1, pp. 343-391 | Article | MR 2326150 | Zbl pre05166838

[16] Panyushev, Dmitri I. On the coadjoint representation of 2 -contractions of reductive Lie algebras, Adv. Math., Tome 213 (2007) no. 1, pp. 380-404 | Article | MR 2331248 | Zbl pre05166638

[17] Panyushev, Dmitri I.; Yakimova, Oksana S. The argument shift method and maximal commutative subalgebras of Poisson algebras, Math. Res. Lett., Tome 15 (2008) no. 2, pp. 239-249 | MR 2385637 | Zbl pre05310628

[18] Richardson, R. W. Commuting varieties of semisimple Lie algebras and algebraic groups, Compositio Math., Tome 38 (1979) no. 3, pp. 311-327 | Numdam | MR 535074 | Zbl 0409.17006

[19] Sekiguchi, Jirō A counterexample to a problem on commuting matrices, Proc. Japan Acad. Ser. A Math. Sci., Tome 59 (1983) no. 9, p. 425-426 | Article | MR 732601 | Zbl 0566.17002

[20] Shafarevich, Igor R. Basic algebraic geometry. 1, Springer-Verlag, Berlin (1994) (Varieties in projective space, Translated from the 1988 Russian edition and with notes by Miles Reid) | MR 1328833 | Zbl 0797.14001

[21] Steinberg, Robert Conjugacy classes in algebraic groups, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Vol. 366 (1974) (Notes by Vinay V. Deodhar) | MR 352279 | Zbl 0281.20037

[22] Vinberg, E. B.; Yakimova, O. S. Complete families of commuting functions for coisotropic Hamiltonian actions (arXiv:math.SG/0511498)

[23] Weinstein, Alan The local structure of Poisson manifolds, J. Differential Geom., Tome 18 (1983) no. 3, pp. 523-557 | MR 723816 | Zbl 0524.58011

[24] Yakimova, O. S. The index of centralizers of elements in classical Lie algebras, Funktsional. Anal. i Prilozhen., Tome 40 (2006) no. 1, p. 52-64, 96 | Article | MR 2223249 | Zbl 1152.17001