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

TL;DR

This paper investigates the detailed structure of mod p principal series representations of GL_2 over finite rings, extending known results from finite fields to more complex local ring settings, with special cases analyzed in depth.

Contribution

It provides a comprehensive analysis of the submodule structure of these representations over various ramification scenarios, including explicit descriptions and filtrations, advancing understanding in mod p representation theory.

Findings

Complete submodule structure for totally ramified extensions.

Infinite submodule structure in non-totally ramified cases.

Explicit filtrations and connections to Breuil's functor in ramified cases.

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-H\"older 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 Pa\u{s}k\=unas in the context of their work on diagrams giving rise to supersingular mod $p$ representations of $\mathrm{GL}_2(F)$.