The regular representations of GLN 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 Gr=GLN(o/pr), for an integer N⩾2, is called regular if its restriction to the principal congruence kernel Kr−1=1+pr−1MN(o/pr) consists of representations whose stabilisers modulo K1 are centralisers of regular elements in MN(o/p). The regular representations form the largest class of representations of Gr 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 Gr.