On irreducible representations of discrete Heisenberg groups
type: Item
instance of: scholarly article
title: On irreducible representations of discrete Heisenberg groups
zbMATH Open document ID: 1264.43007
DOI: 10.1134/S0001434612090015
author: S. A. Arnal'
author: A. N. Parshin
published in: Mathematical Notes
publication date: 2013-01-09T00:00:00Z
review text: A representation of a discrete group \(G\) is said to be monomial if it is induced from a character (not necessarily unitary) of a subgroup \(H.\) Let \(\pi:G \to {\operatorname{End}}(V_\pi)\) be a representation of a discrete group \(G\). A subspace \(V(H, \chi)\) is called a weight space if \[ V(H, \chi) = \{ v \in V_\pi:~\pi(h)v = \chi(h)v\;\forall\,h \in H\} \] for a subgroup \(H\) of \(G\). The representation \(\pi\) is said to have finite weight if there is \(H\) and \(\chi\) such that \(V(H, \chi) \neq 0\) and \(\dim_{\mathbb C} V (H, \chi) < \infty\). The authors prove that, if \(G\) is a finitely generated nilpotent group ,then unitary irreducible representations are monomial if and only if they have finite weight. Applications are given for representations of discrete Heisenberg groups.
reviewed by: E. K. Narayanan
Mathematics Subject Classification ID: 43A65
Mathematics Subject Classification ID: 43A40
Mathematics Subject Classification ID: 22A25
zbMATH DE Number: 6123739
zbMATH Keywords: discrete Heisenberg group
zbMATH Keywords: discrete nilpotent group
zbMATH Keywords: monomial representation
zbMATH Keywords: representation of finite weight
MaRDI profile type: Publication
cites work: Q3964798
cites work: Representation of finitely generated nilpotent groups
cites work: On Induced Representations of Groups
cites work: On holomorphic representations of discrete Heisenberg groups
full work available at URL: s0001434612090015
OpenAlex ID: W2067598740
description: scientific article; zbMATH DE number 6123739