Contraction par Frobenius de G-modules
[Frobenius contraction of G-modules]
Annales de l'Institut Fourier, Volume 61 (2011) no. 6, pp. 2507-2542.

Let G be a simply connected semisimple algebraic group over an algebraically closed field 𝕜 of positive characteristic. We will give a new proof of the Frobenius splitting of the flag variety of G and of its G-equivariant nature. The key tool is a newly found splitting of the Frobenius endomorphism on the algebra of distributions of G allowing us to “untwist” the structure of G-modules.

Soit G un groupe algébrique semi-simple simplement connexe défini sur un corps algébriquement clos 𝕜 de caractéristique positive. Nous donnons une nouvelle preuve de l’existence d’un scindage de Frobenius de la variété des drapeaux de G ainsi que de la nature G-équivariante de celui-ci. L’outil principal est un scindage de l’endomorphisme de Frobenius défini sur toute l’algèbre des distributions de G qui permet de « détordre » la structure des G-modules.

DOI: 10.5802/aif.2681
Classification: 14M15, 13A35, 17B10, 20G05, 20G10
Mot clés : scindage de Frobenius, variété des drapeaux, variété de Schubert, algèbre des distributions
Keywords: Frobenius splitting, flag variety, Schubert variety, distribution algebra

Gros, Michel 1; Kaneda, Masaharu 2

1 Université de Rennes I IRMAR Campus de Beaulieu 35042 Rennes cedex (France)
2 Osaka City University Department of Mathematics 3-3-138 Sugimoto Sumiyoshi-ku Osaka 558-8585 (Japan)
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     title = {Contraction par {Frobenius} de $G$-modules},
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Gros, Michel; Kaneda, Masaharu. Contraction par Frobenius de $G$-modules. Annales de l'Institut Fourier, Volume 61 (2011) no. 6, pp. 2507-2542. doi : 10.5802/aif.2681. https://aif.centre-mersenne.org/articles/10.5802/aif.2681/

[1] Andersen, Henning Haahr; Polo, Patrick; Wen, Ke Xin Representations of quantum algebras, Invent. Math., Volume 104 (1991) no. 1, pp. 1-59 | DOI | MR | Zbl

[2] Brion, Michel; Kumar, Shrawan Frobenius splitting methods in geometry and representation theory, Progress in Mathematics, 231, Birkhäuser Boston Inc., Boston, MA, 2005 | MR | Zbl

[3] Cline, Edward; Parshall, Brian; Scott, Leonard Cohomology, hyperalgebras, and representations, J. Algebra, Volume 63 (1980) no. 1, pp. 98-123 | DOI | MR | Zbl

[4] Enright, T. J.; Wallach, N. R. Notes on homological algebra and representations of Lie algebras, Duke Math. J., Volume 47 (1980) no. 1, pp. 1-15 | DOI | MR | Zbl

[5] Gros, Michel A splitting of the Frobenius morphism on the whole algebra of distributions of SL 2 (à paraître dans Algebras and Represention theory)

[6] Hartshorne, Robin Algebraic geometry, Springer-Verlag, New York, 1977 (Graduate Texts in Mathematics, No. 52) | MR | Zbl

[7] Humphreys, James E. Introduction to Lie algebras and representation theory, Springer-Verlag, New York, 1972 (Graduate Texts in Mathematics, Vol. 9) | MR | Zbl

[8] Jantzen, Jens Carsten Darstellungen halbeinfacher algebraischer Gruppen und zugeordnete kontravariante Formen, Bonn. Math. Schr. (1973) no. 67, pp. v+124 | MR | Zbl

[9] Jantzen, Jens Carsten Lectures on quantum groups, Graduate Studies in Mathematics, 6, American Mathematical Society, Providence, RI, 1996 | MR | Zbl

[10] Jantzen, Jens Carsten Representations of algebraic groups, Mathematical Surveys and Monographs, 107, American Mathematical Society, Providence, RI, 2003 | MR | Zbl

[11] Kaneda, Masaharu The Frobenius morphism of Schubert schemes, J. Algebra, Volume 174 (1995) no. 2, pp. 473-488 | DOI | MR | Zbl

[12] Kaneda, Masaharu Cohomology of infinitesimal quantum algebras, J. Algebra, Volume 226 (2000) no. 1, pp. 250-282 | DOI | MR | Zbl

[13] Kumar, Shrawan; Littelmann, Peter 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] Kumar, Shrawan; Littelmann, Peter Algebraization of Frobenius splitting via quantum groups, Ann. of Math. (2), Volume 155 (2002) no. 2, pp. 491-551 | DOI | MR | Zbl

[15] Littelmann, Peter 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] Lusztig, G. 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] Lusztig, George Quantum groups at roots of 1, Geom. Dedicata, Volume 35 (1990) no. 1-3, pp. 89-113 | DOI | MR | Zbl

[18] Lusztig, George Introduction to quantum groups, Progress in Mathematics, 110, Birkhäuser Boston Inc., Boston, MA, 1993 | MR

[19] Mathieu, Olivier Filtrations of G-modules, Ann. Sci. École Norm. Sup. (4), Volume 23 (1990) no. 4, pp. 625-644 | Numdam | MR | Zbl

[20] McGerty, Kevin Generalized q-Schur algebras and quantum Frobenius, Adv. Math., Volume 214 (2007) no. 1, pp. 116-131 | DOI | MR | Zbl

[21] Mehta, V. B.; Ramanathan, A. Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. of Math. (2), Volume 122 (1985) no. 1, pp. 27-40 | DOI | MR | Zbl

[22] Takeuchi, Mitsuhiro Tangent coalgebras and hyperalgebras. I, Japan. J. Math., Volume 42 (1974), pp. 1-143 | MR | Zbl

[23] Xi, Nanhua Irreducible modules of quantized enveloping algebras at roots of 1, Publ. Res. Inst. Math. Sci., Volume 32 (1996) no. 2, pp. 235-276 | DOI | MR | Zbl

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