Soit un groupe algébrique semi-simple simplement connexe défini sur un corps algébriquement clos de caractéristique positive. Nous donnons une nouvelle preuve de l’existence d’un scindage de Frobenius de la variété des drapeaux de ainsi que de la nature -équivariante de celui-ci. L’outil principal est un scindage de l’endomorphisme de Frobenius défini sur toute l’algèbre des distributions de qui permet de « détordre » la structure des -modules.
Let be a simply connected semisimple algebraic group over an algebraically closed field of positive characteristic. We will give a new proof of the Frobenius splitting of the flag variety of and of its -equivariant nature. The key tool is a newly found splitting of the Frobenius endomorphism on the algebra of distributions of allowing us to “untwist” the structure of -modules.
Mot clés : scindage de Frobenius, variété des drapeaux, variété de Schubert, algèbre des distributions
Keywords: Frobenius splitting, flag variety, Schubert variety, distribution algebra
@article{AIF_2011__61_6_2507_0,
author = {Gros, Michel and Kaneda, Masaharu},
title = {Contraction par {Frobenius} de $G$-modules},
journal = {Annales de l'Institut Fourier},
pages = {2507--2542},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {61},
number = {6},
year = {2011},
doi = {10.5802/aif.2681},
zbl = {1257.14035},
mrnumber = {2976319},
language = {fr},
url = {http://www.numdam.org/articles/10.5802/aif.2681/}
}
TY - JOUR AU - Gros, Michel AU - Kaneda, Masaharu TI - Contraction par Frobenius de $G$-modules JO - Annales de l'Institut Fourier PY - 2011 SP - 2507 EP - 2542 VL - 61 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2681/ DO - 10.5802/aif.2681 LA - fr ID - AIF_2011__61_6_2507_0 ER -
%0 Journal Article %A Gros, Michel %A Kaneda, Masaharu %T Contraction par Frobenius de $G$-modules %J Annales de l'Institut Fourier %D 2011 %P 2507-2542 %V 61 %N 6 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2681/ %R 10.5802/aif.2681 %G fr %F AIF_2011__61_6_2507_0
Gros, Michel; Kaneda, Masaharu. Contraction par Frobenius de $G$-modules. Annales de l'Institut Fourier, Tome 61 (2011) no. 6, pp. 2507-2542. doi : 10.5802/aif.2681. http://www.numdam.org/articles/10.5802/aif.2681/
[1] Representations of quantum algebras, Invent. Math., Volume 104 (1991) no. 1, pp. 1-59 | DOI | MR | Zbl
[2] Frobenius splitting methods in geometry and representation theory, Progress in Mathematics, 231, Birkhäuser Boston Inc., Boston, MA, 2005 | MR | Zbl
[3] Cohomology, hyperalgebras, and representations, J. Algebra, Volume 63 (1980) no. 1, pp. 98-123 | DOI | MR | Zbl
[4] Notes on homological algebra and representations of Lie algebras, Duke Math. J., Volume 47 (1980) no. 1, pp. 1-15 | DOI | MR | Zbl
[5] A splitting of the Frobenius morphism on the whole algebra of distributions of (à paraître dans Algebras and Represention theory)
[6] Algebraic geometry, Springer-Verlag, New York, 1977 (Graduate Texts in Mathematics, No. 52) | MR | Zbl
[7] Introduction to Lie algebras and representation theory, Springer-Verlag, New York, 1972 (Graduate Texts in Mathematics, Vol. 9) | MR | Zbl
[8] Darstellungen halbeinfacher algebraischer Gruppen und zugeordnete kontravariante Formen, Bonn. Math. Schr. (1973) no. 67, pp. v+124 | MR | Zbl
[9] Lectures on quantum groups, Graduate Studies in Mathematics, 6, American Mathematical Society, Providence, RI, 1996 | MR | Zbl
[10] Representations of algebraic groups, Mathematical Surveys and Monographs, 107, American Mathematical Society, Providence, RI, 2003 | MR | Zbl
[11] The Frobenius morphism of Schubert schemes, J. Algebra, Volume 174 (1995) no. 2, pp. 473-488 | DOI | MR | Zbl
[12] Cohomology of infinitesimal quantum algebras, J. Algebra, Volume 226 (2000) no. 1, pp. 250-282 | DOI | MR | Zbl
[13] Frobenius splitting in characteristic zero and the quantum Frobenius map, J. Pure Appl. Algebra, Volume 152 (2000) no. 1-3, pp. 201-216 Commutative algebra, homological algebra and representation theory (Catania/Genoa/Rome, 1998) | DOI | MR | Zbl
[14] Algebraization of Frobenius splitting via quantum groups, Ann. of Math. (2), Volume 155 (2002) no. 2, pp. 491-551 | DOI | MR | Zbl
[15] Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras, J. Amer. Math. Soc., Volume 11 (1998) no. 3, pp. 551-567 | DOI | MR | Zbl
[16] Modular representations and quantum groups, Classical groups and related topics (Beijing, 1987) (Contemp. Math.), Volume 82, Amer. Math. Soc., Providence, RI, 1989, pp. 59-77 | MR | Zbl
[17] Quantum groups at roots of , Geom. Dedicata, Volume 35 (1990) no. 1-3, pp. 89-113 | DOI | MR | Zbl
[18] Introduction to quantum groups, Progress in Mathematics, 110, Birkhäuser Boston Inc., Boston, MA, 1993 | MR
[19] Filtrations of -modules, Ann. Sci. École Norm. Sup. (4), Volume 23 (1990) no. 4, pp. 625-644 | Numdam | MR | Zbl
[20] Generalized -Schur algebras and quantum Frobenius, Adv. Math., Volume 214 (2007) no. 1, pp. 116-131 | DOI | MR | Zbl
[21] Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. of Math. (2), Volume 122 (1985) no. 1, pp. 27-40 | DOI | MR | Zbl
[22] Tangent coalgebras and hyperalgebras. I, Japan. J. Math., Volume 42 (1974), pp. 1-143 | MR | Zbl
[23] Irreducible modules of quantized enveloping algebras at roots of , Publ. Res. Inst. Math. Sci., Volume 32 (1996) no. 2, pp. 235-276 | DOI | MR | Zbl
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