On the first restricted cohomology of a reductive Lie algebra and its Borel subalgebras
[Sur la première cohomologie restreinte d’une algèbre de Lie réductif et de ses sous-algèbres de Borel]
Annales de l'Institut Fourier, Tome 69 (2019) no. 3, pp. 1295-1308.

Soit k un corps algébriquement clos de charactéristique p>0 and soit G un groupe réductif connexe sur k. Soit B un sous-groupe de Borel de G et soit 𝔤 et 𝔟 les algèbres de Lie de G et B. Notons les premiers noyaux de Frobenius de G et B par G 1 et B 1 . De plus, notons les algèbres des fonctions régulières sur G et 𝔤 par k[G] et k[𝔤], et de même pour B et 𝔟. Le groupe G agit sur k[G] par conjugaison et sur k[𝔤] par l’action adjointe. De même, B agit sur k[B] par l’action de conjugaison et sur k[𝔟] par l’action adjointe. Nous montrons que, sous certaines hypothèses, les groupes de cohomologie H 1 (G 1 ,k[𝔤]), H 1 (B 1 ,k[𝔟]), H 1 (G 1 ,k[G]) et H 1 (B 1 ,k[B]) sont nuls. Nous étendons aussi nos résultats à la cohomologie pour les noyaux de Frobenius supérieurs.

Let k be an algebraically closed field of characteristic p>0 and let G be a connected reductive group over k. Let B be a Borel subgroup of G and let 𝔤 and 𝔟 be the Lie algebras of G and B. Denote the first Frobenius kernels of G and B by G 1 and B 1 . Furthermore, denote the algebras of regular functions on G and 𝔤 by k[G] and k[𝔤], and similarly for B and 𝔟. The group G acts on k[G] via the conjugation action and on k[𝔤] via the adjoint action. Similarly, B acts on k[B] via the conjugation action and on k[𝔟] via the adjoint action. We show that, under certain mild assumptions, the cohomology groups H 1 (G 1 ,k[𝔤]), H 1 (B 1 ,k[𝔟]), H 1 (G 1 ,k[G]) and H 1 (B 1 ,k[B]) are zero. We also extend all our results to the cohomology for the higher Frobenius kernels.

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DOI : 10.5802/aif.3271
Classification : 20G05, 20G10
Keywords: Cohomology, Frobenius kernel, reductive group
Mot clés : Cohomologie, noyau de Frobenius, groupe réductif
Tange, Rudolf 1

1 University of Leeds School of Mathematics LS2 9JT, Leeds (UK)
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Tange, Rudolf. On the first restricted cohomology of a reductive Lie algebra and its Borel subalgebras. Annales de l'Institut Fourier, Tome 69 (2019) no. 3, pp. 1295-1308. doi : 10.5802/aif.3271. http://www.numdam.org/articles/10.5802/aif.3271/

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