Regular characters of groups of type A(n), over discrete valuation rings

Let o be a complete discrete valuation ring with finite residue field k of odd characteristic. Let G be a general or special linear group or a unitary group defined over o and let g denote its Lie algebra. For every positive integer P, let K-l be the l-th principal congruence subgroup of G(o). A continuous irreducible representation of G(o) is called regular of level P if it is trivial on Kl+1 and its restriction to Kl/Kl+1 similar or equal to g(k) consists of characters with G((k) over bar)-stabiliser of minimal dimension. In this paper we construct the regular characters of G(o), compute their degrees and show that the latter satisfy Ennola duality. We give explicit uniform formulae for the regular part of the representation zeta functions of these groups. (C) 2017 Published by Elsevier Inc