The regular representations of $\mathrm{GL}_{N}$ over finite local principal ideal rings

TL;DR

This paper provides an explicit construction of all regular representations of the finite group isplaystyle G_{r}= ext{GL}_N( ext{o}/ ext{p}^r), expanding understanding of their structure and aiding in the study of supercuspidal representations.

Contribution

It offers the first comprehensive explicit construction of all regular representations of isplaystyle G_{r}= ext{GL}_N( ext{o}/ ext{p}^r), a class crucial for representation theory over finite local rings.

Findings

Explicit construction of all regular representations of G_r.

Clarification of the structure of regular representations.

Facilitation of further studies in supercuspidal representations.

Abstract

Let $\mathfrak{o}$ be the ring of integers in a non-Archimedean local field with finite residue field, $\mathfrak{p}$ its maximal ideal, and $r\geq2$ an integer. An irreducible representation of the finite group $G_{r}=\mathrm{GL}_{N}(\mathfrak{o}/\mathfrak{p}^{r})$ is called regular if its restriction to the principal congruence kernel $K^{r-1}=1+\mathfrak{p}^{r-1}\mathrm{M}_{N}(\mathfrak{o}/\mathfrak{p}^{r})$ consists of representations whose stabilisers modulo $K^{1}$ are centralisers of regular elements in $\mathrm{M}_{N}(\mathfrak{o}/\mathfrak{p})$. The regular representations form the largest class of representations of $G_{r}$ which is currently amenable to explicit construction. Their study, motivated by constructions of supercuspidal representations, goes back to Shintani, but the general case remained open for a long time. In this paper we give an explicit construction of all the regular representations of $G_{r}$.