On families of weakly admissible filtered phi-modules and the adjoint quotient of ${GL}_d$

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We study the relation of the notion of weak admissibility in families of filtered \phi -modules, as considered in [He], 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.