The regular representations of ${\mathrm{G}\mathrm{L}}_N$ over finite local principal ideal rings

Let $\mathfrak{o}$ be the ring of integers in a non-Archimedean local field with finite residue field, $\mathfrak{p}$ its maximal ideal, and $r\ge 2$ an integer. An irreducible representation of the finite group $G_r={\mathrm{G}\mathrm{L}}_N(\mathfrak{o}\mathrm{/}{\mathfrak{p}}^r)$ is called regular if its restriction to the principal congruence kernel $K^{r-1}=1+{\mathfrak{p}}^{r-1}{\mathrm{M}}_N(\mathfrak{o}\mathrm{/}{\mathfrak{p}}^r)$ consists of representations whose stabilisers modulo $K^1$ are centralisers of regular elements in ${\mathrm{M}}_N(\mathfrak{o}\mathrm{/}\mathfrak{p})$. 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 $G_r$.