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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Completely reducible $ \operatorname{SL}(2)$-homomorphisms


Authors: George J. McNinch and Donna M. Testerman
Journal: Trans. Amer. Math. Soc. 359 (2007), 4489-4510
MSC (2000): Primary 20G15
Published electronically: April 17, 2007
MathSciNet review: 2309195
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Abstract: Let $ K$ be any field, and let $ G$ be a semisimple group over $ K$. Suppose the characteristic of $ K$ is positive and is very good for $ G$. We describe all group scheme homomorphisms $ \phi:\operatorname{SL}_2 \to G$ whose image is geometrically $ G$-completely reducible-or $ G$-cr-in the sense of Serre; the description resembles that of irreducible modules given by Steinberg's tensor product theorem. In case $ K$ is algebraically closed and $ G$ is simple, the result proved here was previously obtained by Liebeck and Seitz using different methods. A recent result shows the Lie algebra of the image of $ \phi$ to be geometrically $ G$-cr; this plays an important role in our proof.


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Additional Information

George J. McNinch
Affiliation: Department of Mathematics, Tufts University, 503 Boston Avenue, Medford, Massachusetts 02155
Email: george.mcninch@tufts.edu

Donna M. Testerman
Affiliation: Institut de géométrie, algèbre et topologie, Bâtiment BCH, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
Email: donna.testerman@epfl.ch

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04289-4
PII: S 0002-9947(07)04289-4
Received by editor(s): October 18, 2005
Published electronically: April 17, 2007
Additional Notes: The research of the first author was supported in part by the US National Science Foundation through DMS-0437482.
The research of the second author was supported in part by the Swiss National Science Foundation grant PP002-68710.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.