On the first restricted cohomology of a reductive Lie algebra and its Borel subalgebras
Annales de l'Institut Fourier, to appear, 14 p.

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.

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.

Received : 2018-02-19
Revised : 2018-04-28
Accepted : 2018-06-12
Classification:  20G05,  20G10
Keywords: Cohomology, Frobenius kernel, reductive group
     author = {Tange, Rudolf},
     title = {On the first restricted cohomology of a reductive Lie algebra and its Borel subalgebras},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
Tange, Rudolf. On the first restricted cohomology of a reductive Lie algebra and its Borel subalgebras. Annales de l'Institut Fourier, to appear, 14 p.

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