Counting irreducible representations of the Heisenberg group over the integers of a quadratic number field

We calculate the representation zeta function of the Heisenberg group over the integers of a quadratic number field. In general, the representation zeta function of a finitely generated torsion-free nilpotent group enumerates equivalence classes of representations, called twist-isoclasses. This calculation is based on an explicit description of a representative from each twist-isoclass. Our method of construction involves studying the eigenspace structure of the elements of the image of the representation and then picking a suitable basis for the underlying vector space.