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

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.