Optimal $\mathrm{SL}(2)$-homomorphisms

Abstract

Abstract. Let G be a semisimple group over an algebraically closed field of very good characteristic for G. In the context of geometric invariant theory, G. Kempf and -independently -G. Rousseau have associated optimal cocharacters of G to an unstable vector in a linear G-representation. If the nilpotent element X ∈ Lie(G) lies in the image of the differential of a homomorphism SL 2 → G, we say that homomorphism is optimal for X, or simply optimal, provided that its restriction to a suitable torus of SL 2 is optimal for X in the sense of geometric invariant theory.We show here that any two SL 2 -homomorphisms which are optimal for X are conjugate under the connected centralizer of X. This implies, for example, that there is a unique conjugacy class of principal homomorphisms for G. We show that the image of an optimal SL 2 -homomorphism is a completely reducible subgroup of G; this is a notion defined recently by J.-P. Serre. Finally, if G is defined over the (arbitrary) subfield K of k, and if X ∈ Lie(G)(K) is a K-rational nilpotent element with X [p] = 0, we show that there is an optimal homomorphism for X which is defined over K. Mathematics Subject Classification (2000). 20G15.

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