JournalscmhVol. 80 , No. 2DOI 10.4171/cmh/19

Optimal SL(2)SL(2)-homomorphisms

  • George J. McNinch

    Tufts University, Medford, USA
Optimal $SL(2)$-homomorphisms cover

Abstract

Let GG be a semisimple group over an algebraically closed field of \emph{very good} characteristic for GG. In the context of geometric invariant theory, G. Kempf and -- independently -- G. Rousseau have associated optimal cocharacters of GG to an unstable vector in a linear GG-representation. If the nilpotent element X\Lie(G)X \in \Lie(G) lies in the image of the differential of a homomorphism \SL2G\SL_2 \to G, we say that homomorphism is optimal for XX, or simply optimal, provided that its restriction to a suitable torus of \SL2\SL_2 is optimal for XX in the sense of geometric invariant theory. We show here that any two \SL2\SL_2-homomorphisms which are optimal for XX are conjugate under the connected centralizer of XX. This implies, for example, that there is a unique conjugacy class of \emph{principal homomorphisms} for GG. We show that the image of an optimal \SL2\SL_2-homomorphism is a \emph{completely reducible} subgroup of GG; this is a notion defined recently by J-P. Serre. Finally, if GG is defined over the (arbitrary) subfield KK of kk, and if X\Lie(G)(K)X \in \Lie(G)(K) is a KK-rational nilpotent element with X[p]=0X^{[p]}=0, we show that there is an optimal homomorphism for XX which is defined over KK.