Spherical varieties and Wahl’s conjecture
[Variétés sphériques et conjecture de Wahl]
Perrin, Nicolas
Annales de l'Institut Fourier, Tome 64 (2014), p. 739-751 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

En utilisant les variétés sphériques, nous donnons, en toute caractéristique impaire, une preuve courte et uniforme de la conjecture de Wahl pour les variétés homogènes cominuscules.

Using the theory of spherical varieties, we give a type independent very short proof of Wahl’s conjecture for cominuscule homogeneous varieties for all primes different from 2.

Publié le : 2014-01-01
DOI : https://doi.org/10.5802/aif.2864
Classification:  14M27,  14M15,  20G10
Mots clés: scindage de Frobenius, variétés sphériques, conjecture de Wahl 
@article{AIF_2014__64_2_739_0,
     author = {Perrin, Nicolas},
     title = {Spherical varieties and Wahl's conjecture},
     journal = {Annales de l'Institut Fourier},
     volume = {64},
     year = {2014},
     pages = {739-751},
     doi = {10.5802/aif.2864},
     zbl = {06387291},
     mrnumber = {3330921},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2014__64_2_739_0}
}

Perrin, Nicolas. Spherical varieties and Wahl’s conjecture. Annales de l'Institut Fourier, Tome 64 (2014) pp. 739-751. doi : 10.5802/aif.2864. http://gdmltest.u-ga.fr/item/AIF_2014__64_2_739_0/

[1] Achinger, P.; Perrin, N. On spherical multiple flags (2013) (Preprint arXiv:1307.7236)

[2] Bourbaki, N. Groupes et algèbres de Lie, Hermann, Paris (1954) | Zbl 0483.22001

[3] Brion, M.; Inamdar, S. P. Frobenius splitting of spherical varieties, Algebraic groups and their generalizations: classical methods (University Park, PA, 1991), Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 56 (1994), pp. 207-218 | MR 1278708 | Zbl 0820.14032

[4] Brion, Michel; Kumar, Shrawan Frobenius splitting methods in geometry and representation theory, Birkhäuser, Boston, MA, Progress in Mathematics, Tome 231 (2005), pp. x+250 | MR 2107324 | Zbl 1072.14066

[5] Brown, J.; Lakshmibai, V. Wahl’s conjecture for a minuscule G/P, Proc. Indian Acad. Sci. Math. Sci., Tome 119 (2009) no. 5, pp. 571-592 | Article | MR 2598420 | Zbl 1192.14036

[6] De Concini, C.; Springer, T. A. Compactification of symmetric varieties, Transform. Groups, Tome 4 (1999) no. 2-3, pp. 273-300 (Dedicated to the memory of Claude Chevalley) | Article | MR 1712864 | Zbl 0966.14035

[7] Donkin, Stephen Invariants of unipotent radicals, Math. Z., Tome 198 (1988) no. 1, pp. 117-125 | Article | MR 938033 | Zbl 0627.14013

[8] Helminck, A. G.; Wang, S. P. On rationality properties of involutions of reductive groups, Adv. Math., Tome 99 (1993) no. 1, pp. 26-96 | Article | MR 1215304 | Zbl 0788.22022

[9] Knop, Friedrich The Luna-Vust theory of spherical embeddings, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), Manoj Prakashan, Madras (1991), pp. 225-249 | MR 1131314 | Zbl 0812.20023

[10] Kumar, Shrawan Proof of Wahl’s conjecture on surjectivity of the Gaussian map for flag varieties, Amer. J. Math., Tome 114 (1992) no. 6, pp. 1201-1220 | Article | MR 1198300 | Zbl 0790.14015

[11] Lakshmibai, V.; Mehta, V. B.; Parameswaran, A. J. Frobenius splittings and blow-ups, J. Algebra, Tome 208 (1998) no. 1, pp. 101-128 | Article | MR 1643983 | Zbl 0955.14006

[12] Lakshmibai, Venkatramani; Raghavan, Komaranapuram N.; Sankaran, Parameswaran Wahl’s conjecture holds in odd characteristics for symplectic and orthogonal Grassmannians, Cent. Eur. J. Math., Tome 7 (2009) no. 2, pp. 214-223 | Article | MR 2506962 | Zbl 1200.14100

[13] Lauritzen, Niels; Thomsen, Jesper Funch Maximal compatible splitting and diagonals of Kempf varieties, Ann. Inst. Fourier (Grenoble), Tome 61 (2011) no. 6, p. 2543-2575 (2012) | Article | Numdam | MR 2976320 | Zbl 1251.14037

[14] Littelmann, Peter On spherical double cones, J. Algebra, Tome 166 (1994) no. 1, pp. 142-157 | Article | MR 1276821 | Zbl 0823.20040

[15] Mehta, V. B.; Parameswaran, A. J. On Wahl’s conjecture for the Grassmannians in positive characteristic, Internat. J. Math., Tome 8 (1997) no. 4, pp. 495-498 | Article | MR 1460897 | Zbl 0914.14021

[16] Oda, Tadao Convex bodies and algebraic geometry, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Tome 15 (1988), pp. viii+212 | MR 922894 | Zbl 0628.52002

[17] Stembridge, John R. Multiplicity-free products and restrictions of Weyl characters, Represent. Theory, Tome 7 (2003), p. 404-439 (electronic) | Article | MR 2017064 | Zbl 1060.17001

[18] Thomsen, Jesper Funch A proof of Wahl’s conjecture in the symplectic case, Transform. Groups, Tome 18 (2013) no. 1, pp. 263-286 | Article | MR 3022765 | Zbl 1271.14024

[19] Vust, Thierry Opération de groupes réductifs dans un type de cônes presque homogènes, Bull. Soc. Math. France, Tome 102 (1974), pp. 317-333 | Numdam | MR 366941 | Zbl 0332.22018

[20] Wahl, Jonathan Gaussian maps and tensor products of irreducible representations, Manuscripta Math., Tome 73 (1991) no. 3, pp. 229-259 | Article | MR 1132139 | Zbl 0764.20022