2
$\begingroup$

Let $H(\mathbb{Z})$ be the discrete Heisenberg group. What are the finite dimensional irreducible unitary representations of $H(\mathbb{Z})$? Do they all arise from the coordinate-wise quotient map to a Heisenberg group over a finite field $H(\mathbb{Z}) \to H(\mathbb{F}_p)$?

I am not sure if I can apply a Stone-Von Neumann like result, since $H(\mathbb{Z})$ is over the integers, and not over a field.

Improve this question
$\endgroup$
2
  • 4
    $\begingroup$ No, but they all factor through a virtually abelian quotient. Indeed the Zariski closure of the image is a nilpotent compact Lie group, hence virtually abelian. Conversely every f.g. virtually abelian group has a faithful finite-dim unitary representation, so we can't expect more. $\endgroup$
    – YCor
    Commented Nov 28, 2022 at 15:49
  • 4
    $\begingroup$ The irreducible representations of $H(\mathbb{Z})$ are listed in [Davidson, C*-algebras by Example. Theorem VII.5.1] or in [Boca, Rotation C*-Algebras and Almost Mathieu Operators. Chapter 1]. $\endgroup$
    – Narutaka OZAWA
    Commented Nov 29, 2022 at 0:35

0

Reset to default

You must log in to answer this question.