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First cohomology for finite groups of Lie type: Simple modules with small dominant weights

Author: University of Georgia VIGRE Algebra Group
Journal: Trans. Amer. Math. Soc. 365 (2013), 1025-1050
MSC (2010): Primary 20G10; Secondary 20G05
Published electronically: October 12, 2012
MathSciNet review: 2995382
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Abstract: Let $ k$ be an algebraically closed field of characteristic $ p > 0$, and let $ G$ be a simple, simply connected algebraic group defined over $ \mathbb{F}_p$. Given $ r \geq 1$, set $ q=p^r$, and let $ G(\mathbb{F}_q)$ be the corresponding finite Chevalley group. In this paper we investigate the structure of the first cohomology group $ \operatorname {H}^1(G(\mathbb{F}_q),L(\lambda ))$, where $ L(\lambda )$ is the simple $ G$-module of highest weight $ \lambda $. Under certain very mild conditions on $ p$ and $ q$, we are able to completely describe the first cohomology group when $ \lambda $ is less than or equal to a fundamental dominant weight. In particular, in the cases we consider, we show that the first cohomology group has dimension at most one. Our calculations significantly extend, and provide new proofs for, earlier results of Cline, Parshall, Scott, and Jones, who considered the special case when $ \lambda $ is a minimal non-zero dominant weight.

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Additional Information

University of Georgia VIGRE Algebra Group
Affiliation: Department of Mathematics, University of Georgia. Athens, Georgia 30602-7403; Department of Mathematics, Statistics, and Computer Science, University of Wisconsin–Stout, Menomonie, Wisconsin 54751

Received by editor(s): October 21, 2010
Received by editor(s) in revised form: July 3, 2011, and July 5, 2011
Published electronically: October 12, 2012
Additional Notes: The members of the University of Georgia VIGRE Algebra Group are Brian D. Boe, Adrian M. Brunyate, Jon F. Carlson, Leonard Chastkofsky, Christopher M. Drupieski, Niles Johnson, Benjamin F. Jones, Wenjing Li, Daniel K. Nakano, Nham Vo Ngo, Duc Duy Nguyen, Brandon L. Samples, Andrew J. Talian, Lisa Townsley, and Benjamin J. Wyser.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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