# ANNALES DE L'INSTITUT FOURIER

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 $k\left[G\right]$ and $k\left[𝔤\right]$, and similarly for $B$ and $𝔟$. The group $G$ acts on $k\left[G\right]$ via the conjugation action and on $k\left[𝔤\right]$ via the adjoint action. Similarly, $B$ acts on $k\left[B\right]$ via the conjugation action and on $k\left[𝔟\right]$ via the adjoint action. We show that, under certain mild assumptions, the cohomology groups ${H}^{1}\left({G}_{1},k\left[𝔤\right]\right)$, ${H}^{1}\left({B}_{1},k\left[𝔟\right]\right)$, ${H}^{1}\left({G}_{1},k\left[G\right]\right)$ and ${H}^{1}\left({B}_{1},k\left[B\right]\right)$ 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 $k\left[G\right]$ et $k\left[𝔤\right]$, et de même pour $B$ et $𝔟$. Le groupe $G$ agit sur $k\left[G\right]$ par conjugaison et sur $k\left[𝔤\right]$ par l’action adjointe. De même, $B$ agit sur $k\left[B\right]$ par l’action de conjugaison et sur $k\left[𝔟\right]$ par l’action adjointe. Nous montrons que, sous certaines hypothèses, les groupes de cohomologie ${H}^{1}\left({G}_{1},k\left[𝔤\right]\right)$, ${H}^{1}\left({B}_{1},k\left[𝔟\right]\right)$, ${H}^{1}\left({G}_{1},k\left[G\right]\right)$ et ${H}^{1}\left({B}_{1},k\left[B\right]\right)$ sont nuls. Nous étendons aussi nos résultats à la cohomologie pour les noyaux de Frobenius supérieurs.

Revised : 2018-04-28
Accepted : 2018-06-12
Classification:  20G05,  20G10
Keywords: Cohomology, Frobenius kernel, reductive group
@unpublished{AIF_0__0_0_A35_0,
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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