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

Abstract

Let $\mathcal{O}$ be a discrete valuation ring with maximal ideal $\mathfrak{p}$ and with finite residue field $\mathbb{F}_{q}$, the field with $q$ elements where $q$ is a power of a prime $p$. For $r \ge 1$, we write $\mathcal{O}_r$ for the reduction of $\mathcal{O}$ modulo the ideal $\mathfrak{p}^r$. An irreducible ordinary representation of the finite group $\mathrm{GL}_{N}(\mathcal{O}_{r})$ is called stable if its restriction to the principal congruence kernel $K^l=1+\mathfrak{p}^{l}\mathrm{M}_{N}(\mathcal{O}_r)$, where $l=\lceil \frac{r}{2} \rceil$, consists of irreducible representations whose stabilizers modulo $K^{l'}$, where $l'=r-l$, are centralizers of certain matrices in $\mathfrak{g}_{l'}=\mathrm{M}_{N}(\mathcal{O}_{l'})$, called stable matrices. The study of stable representations is motivated by constructions of strongly semisimple representations, introduced by Hill, which is a special case of stable representations. In this paper, we explore the construction of stable irreducible representations of the finite group $\mathrm{GL}_{N}(\mathcal{O}_{r})$ for $N \ge 2$.