Spherical conjugacy classes and the Bruhat decomposition
Annales de l'Institut Fourier, Volume 59 (2009) no. 6, pp. 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: 10.5802/aif.2492
Classification: 20GXX,  20E45,  20F55,  14M15
Keywords: Conjugacy class, spherical homogeneous space, Bruhat decomposition
Carnovale, Giovanna 1

1 University of Padova Dipartimento di Matematica Pura ed Applicata via Trieste 63 Padova, 35121 (Italy)
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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://archive.numdam.org/articles/10.5802/aif.2492/

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

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

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

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

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

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

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

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

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

[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 | Zbl

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

[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, Volume 209 (2007) no. 3, pp. 703-723 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

[21] Springer, T.A. Some results on algebraic groups with involutions, Algebraic groups and related topics (Kyoto/Nagoya, 1983), Volume 6 (1985), pp. 525-543 | MR | Zbl

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

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

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

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

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

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