  We study the relation of the notion of weak admissibility in families of
filtered phi-modules, as considered in a companion paper, with the adjoint
quotient. We show that the weakly admissible subset is an open subvariety in
the fibers over the adjoint quotient. Further we determine the image of the
weakly admissible set in the adjoint quotient generalizing earlier work of
Breuil and Schneider.
