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

TL;DR

This paper investigates the irreducible representations of a specific Hecke algebra associated with a non-reductive group over a local non-Archimedean field, aiming to understand spectral properties of Hecke operators in a geometric context.

Contribution

It provides a detailed analysis of irreducible representations of a Hecke algebra linked to a non-reductive group, extending understanding in non-Archimedean harmonic analysis.

Findings

Classification of irreducible representations achieved

Insights into the spectrum of Hecke operators obtained

Potential applications to principal bundle moduli explored

Abstract

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