On irreducibility of an induced representation of a simply connected nilpotent Lie group
type: Item
instance of: scholarly article
title: On irreducibility of an induced representation of a simply connected nilpotent Lie group
zbMATH Open document ID: 1343.22007
author: Adjiey Jean-Luc Koffi
author: Kinvi Kangni
published in: African Diaspora Journal of Mathematics
publication date: 2016-01-26T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.adjm/1446472393
review text: Let \(G\) be a simply connected nilpotent Lie group with Lie algebra \(\mathfrak g\), \(V\) a finite dimensional \(\mathbb{K}\)-vector space (\(\mathbb{K}=\mathbb{R}\), or \(\mathbb{C}\)), \(\pi: \mathfrak g\to \mathfrak{gl}(V)\) a representation of \(\mathfrak g\), where \(\mathfrak{gl}(V)\) is the Lie algebra of endomorphisms of \(V\). Let \(\mathfrak h\) be a Lie subalgebra of \(\mathfrak g\) subordinate to \(\pi\) (that is, \(\pi([\mathfrak h, \mathfrak h])=\{0\}\)) and let \(H\) be the connected Lie subgroup of \(G\) associated to \(\mathfrak h\). We define a representation \(\chi_\pi\) of \(H\) in the space \(V\) by the rule \(\chi_\pi(\exp X) =e^{i\pi(X)}\) for every \(X\in \mathfrak h\). Let \(\rho(\pi, H, G)= \mathrm{ind}_H^G\chi_\pi\) be the unitary representation of \(G\) induced by the representation \(\chi_\pi\) of \(H\). In the paper under review, the authors study the irreducibility of the representation \(\rho(\pi, H, G)\) in the setting of simply connected nilpotent Lie groups \(G\). More precisely, the authors prove that for a non-abelian simply connected nilpotent Lie group \(G\) such that \(\mathcal{Z}\cap \ker(\pi)\neq\{0\}\), where \(\mathcal{Z}\) denotes the center of the Lie algebra \(\mathfrak g\), the representation \(\rho(\pi, H, G)\) is irreducible if and only if \(\mathfrak h\) is a polarization at \(\pi\) (Theorem 3.3).
Mathematics Subject Classification ID: 22E27
zbMATH DE Number: 6535925
zbMATH Keywords: nilpotent Lie group
zbMATH Keywords: induced representation
zbMATH Keywords: irreducible representation
zbMATH Keywords: Schur's lemma
zbMATH Keywords: polarization
reviewed by: Hatem Hamrouni
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6535925