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
@unpublished{AIF_0__0_0_A35_0,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.

[1] Andersen, Henning H.; Jantzen, Jens C. Cohomology of induced representations for algebraic groups, Math. Ann., Tome 269 (1984) no. 4, pp. 487-525 | Article | MR 766011

[2] Bendel, Christopher P.; Nakano, Daniel K.; Pillen, Cornelius Extensions for Frobenius kernels, J. Algebra, Tome 272 (2004) no. 2, pp. 476-511 | Article | MR 2028069

[3] Borel, Armand Linear algebraic groups, Springer, Graduate Texts in Mathematics, Tome 126 (1991), xii+288 pages | Article | MR 1102012

[4] Bourbaki, Nicolas Elements of Mathematics: Algebra II. Chapters 4–7, Springer (1990), vii+461 pages (Translated from the French by P. M. Cohn and J. Howie) | MR 1080964 | Zbl 0719.12001

[5] Demazure, Michel Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math., Tome 21 (1973), pp. 287-301 | Article | MR 0342522

[6] Donkin, Stephen On conjugating representations and adjoint representations of semisimple groups, Invent. Math., Tome 91 (1988) no. 1, pp. 137-145 | Article | MR 918240 | Zbl 0639.20021

[7] Hochschild, Gerhard Cohomology of restricted Lie algebras, Am. J. Math., Tome 76 (1954), pp. 555-580 | Article | MR 0063361 | Zbl 0055.26505

[8] Humphreys, James E. Conjugacy classes in semisimple algebraic groups, American Mathematical Society, Mathematical Surveys and Monographs, Tome 43 (1995), xviii+196 pages | MR 1343976

[9] Jantzen, Jens C. First cohomology groups for classical Lie algebras, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991), Birkhäuser (Progress in Mathematics) Tome 95 (1991), pp. 289-315 | MR 1112165

[10] Jantzen, Jens C. Representations of Lie algebras in prime characteristic, Representation theories and algebraic geometry (Montreal, PQ, 1997), Kluwer Academic Publishers (NATO ASI Series. Series C. Mathematical and Physical Sciences) Tome 514 (1998), pp. 185-235 (Notes by Iain Gordon) | MR 1649627 | Zbl 0974.17022

[11] Jantzen, Jens C. Representations of algebraic groups, American Mathematical Society, Mathematical Surveys and Monographs, Tome 107 (2003), xiv+576 pages | MR 2015057

[12] Jantzen, Jens C. Nilpotent orbits in representation theory, Lie theory, Birkhäuser (Progress in Mathematics) Tome 228 (2004), pp. 1-211 | MR 2042689

[13] Van Der Kallen, Wilberd Longest weight vectors and excellent filtrations, Math. Z., Tome 201 (1989) no. 1, pp. 19-31 | Article | MR 990185

[14] Passman, Donald S. A course in ring theory, Wadsworth & Brooks/Cole Advanced Books & Software, The Wadsworth & Brooks/Cole Mathematics Series (1991), x+306 pages | MR 1096302 | Zbl 0783.16001

[15] Premet, Alexander; Stewart, David I. Rigid orbits and sheets in reductive Lie algebras over fields of prime characteristic, J. Inst. Math. Jussieu, Tome 17 (2018) no. 3, pp. 583-613 | Article | MR 3789182

[16] Richardson, Roger W. The conjugating representation of a semisimple group, Invent. Math., Tome 54 (1979) no. 3, pp. 229-245 | Article | MR 553220 | Zbl 0424.20035

[17] Skryabin, Serge Invariants of finite group schemes, J. Lond. Math. Soc., Tome 65 (2002) no. 2, pp. 339-360 | Article | MR 1883187

[18] Springer, Tonny A. Linear algebraic groups, Birkhäuser, Progress in Mathematics, Tome 9 (1998), xiv+334 pages | Article | MR 1642713 | Zbl 0927.20024

[19] Steinberg, Robert Regular elements of semisimple algebraic groups, Publ. Math., Inst. Hautes Étud. Sci. (1965) no. 25, pp. 49-80 | MR 0180554

[20] Steinberg, Robert Torsion in reductive groups, Adv. Math., Tome 15 (1975), pp. 63-92 | Article | MR 0354892