Completely reducible $\operatorname {SL}(2)$-homomorphisms
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- by George J. McNinch and Donna M. Testerman PDF
- Trans. Amer. Math. Soc. 359 (2007), 4489-4510 Request permission
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.References
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Additional Information
- George J. McNinch
- Affiliation: Department of Mathematics, Tufts University, 503 Boston Avenue, Medford, Massachusetts 02155
- MR Author ID: 625671
- 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
- MR Author ID: 265736
- Email: donna.testerman@epfl.ch
- 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. - © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 4489-4510
- MSC (2000): Primary 20G15
- DOI: https://doi.org/10.1090/S0002-9947-07-04289-4
- MathSciNet review: 2309195