The regular representations of GL_N over finite local principal ideal rings

Let o be the ring of integers in a non-Archimedean local field with finite residue field, p its maximal ideal, and r ≥ 2 an integer. An irreducible representation of the finite group G_r = GL_N (o/p ^r ), for an integer N ≥ 2, is called regular if its restriction to the principal congruence kernel K^r−1 = 1 + p^r−1 M_N (o/p^r ) consists of representations whose stabilisers modulo K¹ are centralisers of regular elements in M_N (o/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.