Constructing irreducible representations of discrete groups
type: Item
instance of: scholarly article
title: Constructing irreducible representations of discrete groups
zbMATH Open document ID: 0908.22005
DOI: 10.1007/BF02867253
author: Marc Burger
author: Pierre de la Harpe
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1998-05-26T00:00:00Z
review text: Let \(\Gamma\) be a discrete (countable) group. If \(\Gamma\) is not virtually Abelian, it follows from results of Glimm and Thoma that there is no hope to classify the unitary dual of \(\Gamma\). There is, however, an easy way to produce irreducible unitary representations of \(\Gamma\). Indeed, let \(\Gamma_0\) be a subgroup of \(\Gamma\). Suppose that \(\Gamma_0\) is equal to its own commensurator in \(\Gamma\). Then the induced representation to \(\Gamma\) of any finite-dimensional irreducible representation \(\sigma\) of \(\Gamma_0\) is irreducible. This is a well-known fact, due to G. W. Mackey (in the case of a character \(\sigma\) -- the generalization to finite-dimensional \(\sigma\) is due to L. Corwin). In this article, the authors describe constructions of pairs of groups \(\Gamma_0<\Gamma\) with \(\Gamma_0\) equal to its own commensurator. Such pairs arise as follows. Suppose that \(\Gamma\) acts on a set \(A\) with noncommensurable stabilizers, that is, the stabilizers of different points in \(A\) are noncommensurable. Then take as \(\Gamma_0\) the stabilizer of any point \(a\in A\). An example of a (natural) action with noncommensurable stabilizers is given when \(\Gamma\) is a group of isometries of a Gromov hyperbolic space. For a lattice \(\Gamma\) in an algebraic group \(\mathbb{G}\), actions of \(\Gamma\) on maximal tori and on other subgroups of \(\mathbb{G}\) are shown to be with noncommensurable stabilizers.
Mathematics Subject Classification ID: 22D10
Mathematics Subject Classification ID: 22D30
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 1157030
zbMATH Keywords: Gromov hyperbolic groups
zbMATH Keywords: arithmetic lattices
zbMATH Keywords: irreducible unitary representations
zbMATH Keywords: commensurator
zbMATH Keywords: Gromov hyperbolic space
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02867253
OpenAlex ID: W2019995045
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description: scientific article; zbMATH DE number 1157030