Pro-$p$ Iwahori-Hecke modules in semisimple rank one and singularity categories

Let $\mathfrak{F}$ be a non-archimedean local field of residue characteristic $p$ and $G$ be one of the groups ${\mathrm{G}\mathrm{L}}_2(\mathfrak{F})$, ${\mathrm{S}\mathrm{L}}_2(\mathfrak{F})$ or ${\mathrm{P}\mathrm{G}\mathrm{L}}_2(\mathfrak{F})$. Let ${\mathcal{H}}_G$ denote the pro-$p$ Iwahori-Hecke algebra of $G$ over ${\bar{\mathbb{F}}}_p$. We study the homotopy category $\mathrm{H}\mathrm{o}({\mathcal{H}}_G)$ of Hovey’s Gorenstein projective model structure on the category of ${\mathcal{H}}_G$-modules and relate it to the singularity category $\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}(X_{q,\mathbf{G}})$ of an explicit scheme. When $G={\mathrm{G}\mathrm{L}}_2(\mathfrak{F})$, this scheme was first introduced by Dotto-Emerton-Gee \cite{DEG22}. We obtain in that case an equivalence $\mathrm{H}\mathrm{o}({\mathcal{H}}_{{\mathrm{G}\mathrm{L}}_2})\simeq \mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}(X_{q,{\mathrm{G}\mathrm{L}}_2})$ and recover from this Grosse-Klönne’s mod-$p$ Langlands correspondence for Hecke modules \cite{GK20}, building on work of Pépin-Schmidt \cite{PeSch25_2}. We furthermore describe $\mathrm{H}\mathrm{o}({\mathcal{H}}_G)$ completely explicitly when $G={\mathrm{S}\mathrm{L}}_2(\mathfrak{F})$ or ${\mathrm{P}\mathrm{G}\mathrm{L}}_2(\mathfrak{F})$, and make additional computations in the ${\mathrm{G}\mathrm{L}}_2$ case.