On irreps of a Hecke algebra of a non-reductive group

We study irreducible representations of the Hecke algebra of the pair $({\mathrm{P}\mathrm{G}\mathrm{L}}_2(F[\epsilon ]\mathrm{/}({\epsilon }^2)),{\mathrm{P}\mathrm{G}\mathrm{L}}_2(\mathcal{O}[\epsilon ]\mathrm{/}({\epsilon }^2)))$ where $F$ is a local non-Archimedean field of characteristic different than $2$ and $\mathcal{O}\subset F$ is its ring of integers. We expect to apply our analysis to the study of the spectrum of Hecke operators on the space of cuspidal functions on the space of principal ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$-bundles on curves over rings ${\mathbb{F}}_q[\epsilon ]\mathrm{/}({\epsilon }^2)$.