On irreducible representations of discrete Heisenberg groups
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.
scientific article; zbMATH DE number 6123739