Completely Reducible SL(2)-Homomorphisms

Transactions of the American Mathematical Society (Impact Factor: 1.12). 01/2007; 359(9):4489-4510. DOI: 10.2307/20161784
Source: OAI


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 φ: SL₂ → 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 φ to be geometrically G-cr; this plays an important role in our proof.

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    ABSTRACT: Let G be a connected and reductive group over the algebraically closed field K. J-P. Serre has introduced the notion of a G-completely reducible subgroup H of G. In this note, we give a notion of G-complete reducibility -- G-cr for short -- for Lie subalgebras of Lie(G), and we show that if the closed subgroup H < G is G-cr, then Lie(H) is G-cr as well.
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