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Extensions for finite Chevalley groups II

Authors:	Christopher P Bendel Daniel K Nakano Cornelius Pillen

Institution:	Department of Mathematics, Statistics and Computer Science, University of Wisconsin-Stout, Menomonie, Wisconsin 54751 ; Department of Mathematics, University of Georgia, Athens, Georgia 30602 Cornelius Pillen ; Department of Mathematics and Statistics, University of South Alabama, Mobile, Alabama 36688

Abstract:	Let be a semisimple simply connected algebraic group defined and split over the field ${\mathbb{F} }_p$ with elements, let $G(\mathbb{F} _{q})$ be the finite Chevalley group consisting of the ${\mathbb{F} }_{q}$ -rational points of where , and let $G_{r}$ be the th Frobenius kernel. The purpose of this paper is to relate extensions between modules in $\text{Mod}(G(\mathbb{F} _{q}))$ and $\text{Mod}(G_{r})$ with extensions between modules in $\text{Mod}(G)$ . Among the results obtained are the following: for and $p\geq 3(h-1)$ , the $G(\mathbb{F} _{q})$ -extensions between two simple $G(\mathbb{F} _{q})$ -modules are isomorphic to the -extensions between two simple -restricted -modules with suitably ``twisted" highest weights. For $p \geq 3(h-1)$ , we provide a complete characterization of $\text{H}^{1}(G(\mathbb{F} _{q}),H^{0}(\lambda))$ where $H^{0}(\lambda)=\text{ind}_{B}^{G} \lambda$ and $\lambda$ is -restricted. Furthermore, for $p \geq 3(h-1)$ , necessary and sufficient bounds on the size of the highest weight of a -module are given to insure that the restriction map $\operatorname{H}^{1}(G,V)\rightarrow \operatorname{H}^{1}(G(\mathbb{F} _{q}),V)$ is an isomorphism. Finally, it is shown that the extensions between two simple -restricted -modules coincide in all three categories provided the highest weights are ``close" together.

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