Let be a connected, reductive algebraic group over an algebraically closed field of zero or good and odd characteristic. We characterize spherical conjugacy classes in as those intersecting only Bruhat cells in corresponding to involutions in the Weyl group of .
Soit 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 comme celles ayant une intersection seulement avec des cellules de Bruhat de correspondantes à des involutions dans le groupe de Weyl de .
Revised:
Accepted:
DOI: 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}, pages = {2329--2357}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {59}, number = {6}, year = {2009}, doi = {10.5802/aif.2492}, zbl = {1195.20051}, mrnumber = {2640922}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2492/} }
TY - JOUR TI - Spherical conjugacy classes and the Bruhat decomposition JO - Annales de l'Institut Fourier PY - 2009 DA - 2009/// SP - 2329 EP - 2357 VL - 59 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2492/ UR - https://zbmath.org/?q=an%3A1195.20051 UR - https://www.ams.org/mathscinet-getitem?mr=2640922 UR - https://doi.org/10.5802/aif.2492 DO - 10.5802/aif.2492 LA - en ID - AIF_2009__59_6_2329_0 ER -
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. https://aif.centre-mersenne.org/articles/10.5802/aif.2492/
[1] Linear Algebraic Groups, W.A. Benjamin, Inc., 1969 | MR: 251042 | Zbl: 0186.33201
[2] Éléments de Mathématique. Groupes et Algèbres de Lie, Chapitres 4,5, et 6, Masson, Paris, 1981 | MR: 647314
[3] Quelques propriétés des espaces homogènes sphériques, Manuscripta Math., Tome 55 (1986), pp. 191-198 | Article | MR: 833243 | Zbl: 0604.14048
[4] Classification des espaces homogènes sphériques, Compositio Math., Tome 63 (1987), pp. 189-208 | Numdam | MR: 906369 | Zbl: 0642.14011
[5] Spherical orbits and representations of , Transformation Groups, Tome 10 (2005) no. 1, pp. 29-62 | Article | MR: 2127340 | Zbl: 1101.17006
[6] 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] Simple Groups of Lie Type, Pure and Applied Mathematics XXVIII, 1972 | MR: 407163 | Zbl: 0248.20015
[8] Finite Groups of Lie Type, Pure and Applied Mathematics, 1985 | MR: 794307 | Zbl: 0567.20023
[9] Quantum coadjoint action, J. Amer. Math. Soc., Tome 5 (1992), pp. 151-190 | Article | MR: 1124981 | Zbl: 0747.17018
[10] 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] 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] Intersection of conjugacy classes with Bruhat cells in Chevalley groups: the cases , , J. Pure Appl. Algebra, Tome 209 (2007) no. 3, pp. 703-723 | Article | MR: 2298850 | Zbl: 1128.20034
[13] 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] Spherical nilpotent orbits in positive characteristic, Pacific J. Math., Tome 237 (2008), p. 241-186 | Article | MR: 2421122 | Zbl: pre05366370
[15] 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] Conjugacy Classes in Semisimple Algebraic Groups, AMS, Providence, Rhode Island, 1995 | MR: 1343976 | Zbl: 0834.20048
[17] 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] Complexity and nilpotent orbits, Manuscripta Math., Tome 83 (1994), pp. 223-237 | Article | MR: 1277527 | Zbl: 0822.14024
[19] 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] The unipotent variety of a semi-simple group, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) (1969), pp. 373-391 | MR: 263830 | Zbl: 0195.50803
[21] Some results on algebraic groups with involutions, Algebraic groups and related topics (Kyoto/Nagoya, 1983), Tome 6 (1985), pp. 525-543 | MR: 803346 | Zbl: 0628.20036
[22] Linear Algebraic Groups, Second Edition Tome 9, Progress in Mathematics Birkhäuser, 1998 | MR: 1642713 | Zbl: 0927.20024
[23] Conjugacy classes, Seminar on algebraic groups and related finite groups (LNM) Tome 131 (1970), pp. 167-266 | MR: 268192 | Zbl: 0249.20024
[24] 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] Complexity of action of reductive groups, Func. Anal. Appl., Tome 20 (1986), pp. 1-11 | Article | MR: 831043 | Zbl: 0601.14038
[26] 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
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