On families of weakly admissible filtered phi-modules and the adjoint quotient of GL_d

TL;DR

This paper explores the structure of weakly admissible filtered phi-modules in relation to the adjoint quotient of GL_d, establishing openness and describing the image of weakly admissible sets in the quotient.

Contribution

It demonstrates that the weakly admissible subset forms an open subvariety over the adjoint quotient and generalizes previous results on the image of these sets.

Findings

Weakly admissible sets form open subvarieties in fibers over the adjoint quotient.

The image of the weakly admissible set in the adjoint quotient is explicitly determined.

Generalization of earlier work by Breuil and Schneider on the subject.

Abstract

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.