  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 $G_{r}$.
