On supersingular representations of $GL_2(D)$ with a division algebra $D$ over a $p$-adic field

TL;DR

This paper studies mod-$p$ supersingular representations of $GL_2(D)$ over a $p$-adic field, providing a basis for invariants of certain quotients, extending previous results in the field.

Contribution

It generalizes prior work by Hendel and Schein to a broader class of division algebras over $p$-adic fields, offering new computational tools.

Findings

Computed a basis for the invariants of pro-$p$ Iwahori subgroups.

Extended the classification of supersingular representations.

Provided explicit descriptions of certain quotient spaces.

Abstract

We investigate the mod-$p$ supersingular representations of $GL_2(D)$, where $D$ is a division algebra over a $p$-adic field with characteristic 0, by computing a basis for the vector space of the pro-$p$ Iwahori subgroup invariants of a certain quotient of a compact induction. This work generalizes the results of Hendel and Schein.