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

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 $X\in \mathrm{Lie}(G)$ lies in the image of the differential of a homomorphism ${\mathrm{SL}}_2\to G$, we say that homomorphism is optimal for $X$, or simply optimal, provided that its restriction to a suitable torus of ${\mathrm{SL}}_2$ is optimal for $X$ in the sense of geometric invariant theory. We show here that any two ${\mathrm{SL}}_2$-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 \emph{principal homomorphisms} for $G$. We show that the image of an optimal ${\mathrm{SL}}_2$-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 $X\in \mathrm{Lie}(G)(K)$ is a $K$-rational nilpotent element with $X^{[p]}=0$, we show that there is an optimal homomorphism for $X$ which is defined over $K$.