Iwahori-Hecke model for the universal supersingular representation

TL;DR

This paper extends the Iwahori-Hecke model for universal supersingular representations to arbitrary non-archimedean local fields and constructs explicit non-split self-extensions with specific invariants.

Contribution

It generalizes existing Iwahori-Hecke models for supersingular representations to all local fields for certain parameters and constructs explicit non-split self-extensions.

Findings

Extended Iwahori-Hecke models to arbitrary local fields for r=0, q-1.

Constructed explicit non-split self-extensions with four-dimensional invariants.

Provided new tools for understanding supersingular representations in local fields.

Abstract

Let $F$ be a non-archimedean local field with residue field $\mathbb{F}_q$. When $F$ is a finite extension of $\mathbb{Q}_{p}$, Anandavardhanan-Borisagar and Anandavardhanan-Jana introduced an Iwahori-Hecke model for the universal supersingular representation $\pi_r$ in the regular case $ 0 < r < q-1$. When $F=\mathbb{Q}_{p}$, the first author introduced an Iwahori-Hecke model for $\pi_r$ when $r = 0, p - 1$. We extend this result to an arbitrary local field $F$ for $r = 0, q - 1$. We also write down an explicit non-split self-extension of $\pi_r$ which has a four-dimensional space of $I(1)$-invariants when $F = \mathbb{Q}_p$ and $r = 0, p-1$.