Component groups for non-supercuspidal $L$-parameters for $p$-adic $\mathrm{SL}_3$

TL;DR

This paper classifies component groups linked to non-supercuspidal, tempered $L$-parameters of $ ext{SL}_3(F)$ over p-adic fields, extending prior work from $ ext{SL}_2$ through explicit case-by-case analysis.

Contribution

It provides a detailed classification of component groups for $ ext{SL}_3$ $L$-parameters, a significant extension of previous classifications for $ ext{SL}_2$.

Findings

Explicit classification of component groups for $ ext{SL}_3$

Case-by-case computational approach in $ ext{PGL}_3( ext{C})$

Extension of prior $ ext{SL}_2$ results

Abstract

We explicitly classify all the component groups associated to the non-supercuspidal, tempered $L$-parameters of $\mathrm{SL}_3(F)$ for a $p$-adic field $F$ of characteristic $0$ by direct case-by-case computations in $\mathrm{PGL}_3(\mathbb{C})$, following earlier work for $\mathrm{SL}_2$ by Labesse-Langlands and Shelstad.