  Let G be a reductive group acting on an affine variety X, let x in X be a
point whose G-orbit is not closed, and let S be a G-stable closed subvariety of
X which meets the closure of the G-orbit of x but does not contain x. In this
paper, we study G.R. Kempf's optimal class Omega_G(x,S) of cocharacters of G
attached to the point x; in particular, we consider how this optimality
transfers to subgroups of G.
  Suppose K is a G-completely reducible subgroup of G which fixes x, and let H
= C_G(K)^0. Our main result says that the H-orbit of x is also not closed, and
the optimal class Omega_H(x,S) for H simply consists of the cocharacters in
Omega_G(x,S) which evaluate in H. We apply this result in the case that G acts
on its Lie algebra via the adjoint representation to obtain some new
information about cocharacters associated with nilpotent elements in good
characteristic.
