Geodesic
First cohomology groups of Chevalley groups in cross characteristic
Robert M. Guralnick1 ; Pham Huu Tiep2
1 Department of Mathematics<br/> University of Southern California<br/>Los Angeles, CA 90089-2532
2 Department of Mathematics<br/>University of Arizona<br/> Tucson, AZ 84721-0089
Annals of mathematics, Tome 174 (2011) no. 1, pp. 543-559.

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Let $G$ be a simple Chevalley group defined over $\mathbb{F}_{q}$. We show that if $r$ does not divide $q$ and $k$ is an algebraically closed field of characteristic $r$, then very few irreducible $kG$-modules have nonzero $H^1(G,V)$. We also give an explicit upper bound for $\dim H^1(G,V)$ for $V$ an irreducible $kG$-module that does not depend on $q$, but only on the rank of the group. Cline, Parshall and Scott showed that such a bound exists when $r|q$. We obtain extremely strong bounds in the case that a Borel subgroup has no fixed points on $V$.
DOI : 10.4007/annals.2011.174.1.16

Robert M. Guralnick 1 ; Pham Huu Tiep 2

1 Department of Mathematics<br/> University of Southern California<br/>Los Angeles, CA 90089-2532
2 Department of Mathematics<br/>University of Arizona<br/> Tucson, AZ 84721-0089 
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Robert M. Guralnick; Pham Huu Tiep. First cohomology groups of Chevalley groups in cross characteristic. Annals of mathematics, Tome 174 (2011) no. 1, pp. 543-559. doi : 10.4007/annals.2011.174.1.16. http://geodesic.mathdoc.fr/articles/10.4007/annals.2011.174.1.16/

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