Spherical conjugacy classes and the Bruhat decomposition
Annales de l'Institut Fourier, Volume 59 (2009) no. 6, p. 2329-2357

Let G be a connected, reductive algebraic group over an algebraically closed field of zero or good and odd characteristic. We characterize spherical conjugacy classes in G as those intersecting only Bruhat cells in G corresponding to involutions in the Weyl group of G.

Soit G un groupe algébrique réductif connexe, sur un corps algébriquement clos de caractéristique zéro ou bonne et impaire. Nous caractérisons les classes de conjugaison sphériques de G comme celles ayant une intersection seulement avec des cellules de Bruhat de G correspondantes à des involutions dans le groupe de Weyl de G.

DOI : https://doi.org/10.5802/aif.2492
Classification:  20GXX,  20E45,  20F55,  14M15
Keywords: Conjugacy class, spherical homogeneous space, Bruhat decomposition
@article{AIF_2009__59_6_2329_0,
     author = {Carnovale, Giovanna},
     title = {Spherical conjugacy classes and the Bruhat decomposition},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {6},
     year = {2009},
     pages = {2329-2357},
     doi = {10.5802/aif.2492},
     mrnumber = {2640922},
     zbl = {1195.20051},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2009__59_6_2329_0}
}
Carnovale, Giovanna. Spherical conjugacy classes and the Bruhat decomposition. Annales de l'Institut Fourier, Volume 59 (2009) no. 6, pp. 2329-2357. doi : 10.5802/aif.2492. http://www.numdam.org/item/AIF_2009__59_6_2329_0/

[1] Borel, A. Linear Algebraic Groups, W.A. Benjamin, Inc. (1969) | MR 251042 | Zbl 0186.33201

[2] Bourbaki, N. Éléments de Mathématique. Groupes et Algèbres de Lie, Chapitres 4,5, et 6, Masson, Paris (1981) | MR 647314

[3] Brion, M. Quelques propriétés des espaces homogènes sphériques, Manuscripta Math., Tome 55 (1986), pp. 191-198 | Article | MR 833243 | Zbl 0604.14048

[4] Brion, M. Classification des espaces homogènes sphériques, Compositio Math., Tome 63 (1987), pp. 189-208 | Numdam | MR 906369 | Zbl 0642.14011

[5] Cantarini, N.; Carnovale, G.; Costantini, M. Spherical orbits and representations of U e (𝔤), Transformation Groups, Tome 10 (2005) no. 1, pp. 29-62 | Article | MR 2127340 | Zbl 1101.17006

[6] Carnovale, G. Spherical conjugacy classes and involutions in the Weyl group, Math. Z., Tome 260 (2008) no. 1, pp. 1-23 | Article | MR 2413339 | Zbl 1145.14040

[7] Carter, R. W. Simple Groups of Lie Type, Pure and Applied Mathematics XXVIII (1972) | MR 407163 | Zbl 0248.20015

[8] Carter, R. W. Finite Groups of Lie Type, Pure and Applied Mathematics (1985) | MR 794307 | Zbl 0567.20023

[9] De Concini, C.; Kac, V. G.; Procesi, C. Quantum coadjoint action, J. Amer. Math. Soc., Tome 5 (1992), pp. 151-190 | Article | MR 1124981 | Zbl 0747.17018

[10] De Concini, C.; Kac, V. G.; Procesi, C. Some Quantum Analogues of Solvable Lie Groups, Geometry and Analysis, Tata Institute of Fundamental Research,(Bombay1992) (1995), pp. 41-65 | MR 1351503 | Zbl 0878.17014

[11] Ellers, E.; Gordeev, N. Intersection of conjugacy classes with Bruhat cells in Chevalley groups, Pacific J. Math., Tome 214 (2004) no. 2, pp. 245-261 | Article | MR 2042932 | Zbl 1062.20050

[12] Ellers, E.; Gordeev, N. Intersection of conjugacy classes with Bruhat cells in Chevalley groups: the cases SL n (K), GL n (K), J. Pure Appl. Algebra, Tome 209 (2007) no. 3, pp. 703-723 | Article | MR 2298850 | Zbl 1128.20034

[13] Fomin, S.; Zelevinsky, A. Double Bruhat cells and total positivity, J. Amer. Math. Soc., Tome 12 (1999) no. 2, pp. 335-380 | Article | MR 1652878 | Zbl 0913.22011

[14] Fowler, R.; Röhrle, G. Spherical nilpotent orbits in positive characteristic, Pacific J. Math., Tome 237 (2008), p. 241-186 | Article | MR 2421122 | Zbl pre05366370

[15] Grosshans, F. Contractions of the actions of reductive algebraic groups in arbitrary characteristic, Invent. Math., Tome 107 (1992), pp. 127-133 | Article | MR 1135467 | Zbl 0778.20018

[16] Humphreys, J. Conjugacy Classes in Semisimple Algebraic Groups, AMS, Providence, Rhode Island (1995) | MR 1343976 | Zbl 0834.20048

[17] Knop, F. On the set of orbits for a Borel subgroup, Comment. Math. Helvetici, Tome 70 (1995), pp. 285-309 | Article | MR 1324631 | Zbl 0828.22016

[18] Panyushev, D. Complexity and nilpotent orbits, Manuscripta Math., Tome 83 (1994), pp. 223-237 | Article | MR 1277527 | Zbl 0822.14024

[19] Panyushev, D. On spherical nilpotent orbits and beyond, Ann. Inst. Fourier, Grenoble, Tome 49 (1999) no. 5, pp. 1453-1476 | Article | Numdam | MR 1723823 | Zbl 0944.17013

[20] Springer, T.A. The unipotent variety of a semi-simple group, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford University Press (1969), pp. 373-391 | MR 263830 | Zbl 0195.50803

[21] Springer, T.A. Some results on algebraic groups with involutions, Algebraic groups and related topics (Kyoto/Nagoya, 1983), Adv. Stud. Pure Math., North-Holland, Amsterdam, Tome 6 (1985), pp. 525-543 | MR 803346 | Zbl 0628.20036

[22] Springer, T.A. Linear Algebraic Groups, Second Edition, Progress in Mathematics Birkhäuser Tome 9 (1998) | MR 1642713 | Zbl 0927.20024

[23] Springer, T.A.; Steinberg, R. Conjugacy classes, Seminar on algebraic groups and related finite groups, Springer-Verlag, Berlin Heidelberg New York (LNM) Tome 131 (1970), pp. 167-266 | MR 268192 | Zbl 0249.20024

[24] Steinberg, R. Regular elements of semisimple algebraic groups, I.H.E.S. Publ. Math., Tome 25 (1965), pp. 49-80 | Numdam | MR 180554 | Zbl 0136.30002

[25] Vinberg, E. Complexity of action of reductive groups, Func. Anal. Appl., Tome 20 (1986), pp. 1-11 | Article | MR 831043 | Zbl 0601.14038

[26] Yang, S-W.; Zelevinsky, A. Cluster algebras of finite type via Coxeter elements and principal minors, Transformation Groups, Tome 13 (2008) no. 3–4, pp. 855-895 | Article | MR 2452619 | Zbl pre05565873