# Optimal $SL(2)$-homomorphisms

### George J. McNinch

Tufts University, Medford, USA

## 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 \emph{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 $K$.

## Cite this article

George J. McNinch, Optimal $SL(2)$-homomorphisms. Comment. Math. Helv. 80 (2005), no. 2, pp. 391–426

DOI 10.4171/CMH/19