Constructing irreducible representations of discrete groups
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.
scientific article; zbMATH DE number 1157030
scientific article