Extensions for finite Chevalley groups II
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- by Christopher P. Bendel, Daniel K. Nakano and Cornelius Pillen PDF
- Trans. Amer. Math. Soc. 354 (2002), 4421-4454 Request permission
Abstract:
Let $G$ be a semisimple simply connected algebraic group defined and split over the field ${\mathbb {F}}_p$ with $p$ elements, let $G(\mathbb {F}_{q})$ be the finite Chevalley group consisting of the ${\mathbb {F}}_{q}$-rational points of $G$ where $q = p^r$, and let $G_{r}$ be the $r$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 $r >2$ and $p\geq 3(h-1)$, the $G(\mathbb {F}_{q})$-extensions between two simple $G(\mathbb {F}_{q})$-modules are isomorphic to the $G$-extensions between two simple $p^r$-restricted $G$-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 $p^r$-restricted. Furthermore, for $p \geq 3(h-1)$, necessary and sufficient bounds on the size of the highest weight of a $G$-module $V$ 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 $p^r$-restricted $G$-modules coincide in all three categories provided the highest weights are “close" together.References
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Additional Information
- Christopher P. Bendel
- Affiliation: Department of Mathematics, Statistics and Computer Science, University of Wisconsin-Stout, Menomonie, Wisconsin 54751
- MR Author ID: 618335
- Email: bendelc@uwstout.edu
- Daniel K. Nakano
- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- MR Author ID: 310155
- ORCID: 0000-0001-7984-0341
- Cornelius Pillen
- Affiliation: Department of Mathematics and Statistics, University of South Alabama, Mobile, Alabama 36688
- MR Author ID: 339756
- Email: pillen@jaguar1.usouthal.edu
- Received by editor(s): November 16, 2001
- Published electronically: July 2, 2002
- Additional Notes: Research of the second author was supported in part by NSF grant DMS-0102225
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 4421-4454
- MSC (2000): Primary 20C33, 20G10; Secondary 20G05, 20J06
- DOI: https://doi.org/10.1090/S0002-9947-02-03073-8
- MathSciNet review: 1926882