Optimal \(\text{SL}(2)\)-homomorphisms.
Let \(G\) be a semisimple group over an algebraically closed field of `very good' characteristic \(p\). Let \(\varphi\colon\text{SL}_2\to G\) be a homomorphism and put \(X=d\varphi\left(\begin{smallmatrix} 0&1\\ 0&0\end{smallmatrix}\right)\). Then \(X\) is an unstable vector in the sense of geometric invariant theory. Now \(\varphi\) is called optimal for \(X\) if the cocharacter \({\mathbf G}_m\to G\) which one gets by identifying \({\mathbf G}_m\) with the diagonal subgroup of \(\text{SL}_2\) is optimal in the sense of Kempf and Rousseau. It is shown that two optimal \(\text{SL}_2\)-homomorphisms are conjugate under the connected centralizer of \(X\). Applying this to the case that \(X\) is a regular nilpotent one deduces that there is a unique conjugacy class of principal homomorphisms \(\text{SL}_2\to G\). It is shown that the image of an optimal homomorphism is a completely reducible subgroup in the sense of Serre. It is also shown that optimal homomorphisms are good in the sense of Seitz, meaning that all weights of \({\mathbf G}_m\) in \(\text{Lie}(G)\) are \(\leq 2p-2\). Moreover the \(\text{SL}_2\)-module \(\text{Lie}(G)\) is tilting. Finally rationality questions are treated. In an appendix Serre discusses Springer isomorphisms from the variety of unipotent elements in \(G\) to the variety of nilpotent elements in \(\text{Lie}(G)\).
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