A Tannakian description of the local Kaletha gerbe

TL;DR

This paper constructs a Tannakian category over a p-adic field that captures Kaletha's Galois gerbe, providing explicit classification of its simple objects linked to elliptic twisted Levi subgroups.

Contribution

It introduces a new semisimple Tannakian category over p-adic fields that explicitly models Kaletha's Galois gerbe and classifies its simple objects.

Findings

Constructed an explicit semisimple Tannakian category for p-adic fields.

Recovered Kaletha's Galois gerbe via fiber functors.

Classified simple objects related to elliptic twisted Levi subgroups.

Abstract

We construct, for a $p$-adic field $F$, an explicit semisimple Tannakian category $\text{RigIsoc}_{F}$ whose category of fiber functors recovers Kaletha's Galois gerbe $\mathcal{E}_{\text{Kal}}$. We then classify and write down the simple objects in $\text{RigIsoc}_{F}$, all of which come from elliptic twisted Levi subgroups of $\mathrm{GL}_{n}$.