Counting Irreducible Representations of the Discrete Heisenberg Group Over the Integers of a quadratic number field

TL;DR

This paper computes the representation growth zeta function of the discrete Heisenberg group over quadratic number field integers by classifying and explicitly constructing representatives of twist iso-classes based on eigenspace analysis.

Contribution

It introduces a method to explicitly construct representatives of twist iso-classes for the group's representations, advancing understanding of their growth behavior.

Findings

Calculated the representation growth zeta function for the group.

Developed a classification method for twist iso-classes.

Provided explicit representatives for each class.

Abstract

We calculate the representation growth zeta function of the discrete Heisenberg group over the integers of a quadratic number field. This is done by forming equivalence classes of representations, called twist iso-classes, and explicitly constructing a representative from each twist iso-class. 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 representation.