Non-admissibility of some universal supersingular representations

Abstract

Let $K/{\mathbf{Q}}_p$ be an unramified extension of degree $f$ with residue field $k$. Let $\ddot{I}ƒ$ be an irreducible representation of ${\mathrm{GL}}_n(k)$ over ${\overline{\mathbf{F}}}_p$. For $n\ge 3$, we prove that the universal supersingular representation of weight $\ddot{I}ƒ$ is non-admissible and of infinite length when $\ddot{I}ƒ$ is sufficiently generic and satisfies certain technical conditions. This generalizes the previous results for $n=2$ and a non-trivial finite extension $K/{\mathbf{Q}}_p$. Our method employs a weight cycling argument together with recent progress on the Serre weight conjectures.

22 pages, 1 figure; comments welcome!