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

Abstract

We study irreducible representations of the Hecke algebra of the pair $({\mathrm{PGL}}_2(F[\epsilon ]/({\epsilon }^2)),{\mathrm{PGL}}_2(\mathcal{O}[\epsilon ]/({\epsilon }^2)))$ 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 ${\mathrm{PGL}}_2$-bundles on curves over rings ${\mathbb{F}}_q[\epsilon ]/({\epsilon }^2)$.