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

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.