Optimal -homomorphisms

  • George J. McNinch

    Tufts University, Medford, USA


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

Cite this article

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

DOI 10.4171/CMH/19