Quasi-regular representations of two-step nilmanifolds
type: Item
instance of: scholarly article
title: Quasi-regular representations of two-step nilmanifolds
zbMATH Open document ID: 1406.22007
DOI: 10.1007/978-3-319-65181-1_5
author: Amira Ghorbel
author: Hatem Hamrouni
publication date: 2018-11-28T00:00:00Z
review text: Let \(G\) be a simply connected and connected nilpotent Lie group with \(\Gamma \leq G\) a uniform lattice in \(G\), then we call the quotient space \(G / \Gamma\) a \textit{nilmanifold}, where \(\Gamma\) acts by right translations on \(G\). Nilmanifolds play an important role in both geometry and dynamical systems. For example it was shown in [\textit{M. Gromov}, Publ. Math., Inst. Hautes Étud. Sci. 53, 53--78 (1981; Zbl 0474.20018)] that these are up to finite cover the only manifolds admitting expanding maps (and it is conjectured that the same holds for Anosov diffeomorphisms). Note that the Lie group \(G\) acts on the manifold \(G / \Gamma\) by left translations and hence gives rise to a unitary representation of \(G\) on \(L^2(G/\Gamma)\), called the \textit{quasi-regular representation} of \(G / \Gamma\). The paper under review studies explicit unitary equivalences between quasi-regular representations of nilmanifolds in the special case where the Lie group \(G\) is \(2\)-step nilpotent. In [\textit{R. Gornet}, J. Funct. Anal. 119, No. 1, 121--137 (1994; Zbl 0830.22007)] it was shown that two nilmanifolds \(G / \Gamma_1\) and \(G/ \Gamma_2\) have unitarily equivalent quasi-regular representations if and only if there exists a \(\Gamma_1\)-equivalence \(\Phi\) of \(G\) such that \(\Phi(\Gamma_1) = \Gamma_2\). These \(\Gamma_1\)-equivalences were introduced in the same paper as automorphisms of \(G\) satisfying an extra condition on the elements of \(\Gamma_1\). The main result of the paper, which is Theorem 4.6, gives an explicit form for the intertwining operator depending on the \(\Gamma_1\)-equivalence \(\Phi\). For the entire collection see [Zbl 1392.42001].
reviewed by: Jonas Deré
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 6986093
zbMATH Keywords: nilpotent Lie group
zbMATH Keywords: uniform subgroup
zbMATH Keywords: unitary representation
zbMATH Keywords: polarization
zbMATH Keywords: disintegration
zbMATH Keywords: orbit
zbMATH Keywords: Kirillov theory
zbMATH Keywords: intertwining operator
MaRDI profile type: Publication
full work available at URL: 978-3-319-65181-1 5
OpenAlex ID: W2786025581
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description: scientific article; zbMATH DE number 6986093