On the structure of modular principal series representations of $\mathrm{GL}_2$ over some finite rings

Abstract

The submodule structure of mod $p$ principal series representations of $\mathrm{GL}_2(k)$, for $k$ a finite field of characteristic $p$, was described by Bardoe and Sin and has played an important role in subsequent work on the mod $p$ local Langlands correspondence. The present paper studies the structure of mod $p$ principal series representations of $\mathrm{GL}_2(\mathcal{O} / \mathfrak{m}^n)$, where $\mathcal{O}$ is the ring of integers of a $p$-adic field $F$ and $\mathfrak{m}$ its maximal ideal. In particular, the multiset of Jordan-Hölder constituents is determined. In the case $n = 2$, more precise results are obtained. If $F / \mathbb{Q}_p$ is totally ramified, the submodule structure of the principal series is determined completely. Otherwise the submodule structure is infinite. When $F$ is ramified but not totally ramified, the socle and radical filtrations are determined and a specific family of submodules, providing a filtration of the principal series with irreducible quotients, is studied; this family is closely related to the image of a functor of Breuil. In the case of unramified $F$, the structure of a particular submodule of the principal series is studied; this provides a more precise description of the structure of a module constructed by Breuil and Pauskūnas in the context of their work on diagrams giving rise to supersingular mod $p$ representations of $\mathrm{GL}_2(F)$.