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
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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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. http://archive.numdam.org/articles/10.5802/aif.2681/

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