Completely reducible SL(2)-homomorphisms

Details Completely reducible SL(2)-homomorphisms Completely reducible SL(2)-homomorphisms

2005-10-18

arxiv.org/abs/math/0510377v1

Transact. AMS 2007 (359) 4489-4510.

Mathematics

Representation Theory

AMS LaTeX 20 pages

Scientific paper

10.1090/S0002-9947-07-04289-4

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:SL(2) --> 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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