Lower multiplicity for irreducible representations of nilpotent locally compact groups
The author studies the lower multiplicity \(M_L(\pi)\) for irreducible representations \(\pi\) of a nilpotent locally compact group \(G\). The first result says that if the nilpotent group \(G\) is connected, then for every irreducible representation \(\pi\) of \(G\), such that \(\pi\) is not open in \(\widehat{G}\), the lower multiplicity \(M_L(\pi)= 1\). For an arbitrary non compact nilpotent group \(G\), it is shown that \(M_L(\pi)= 1\) for every finite-dimensional irreducible representation \(\pi\). The next result concerns nilpotent SIN-groups. A locally compact group \(G\) is called a SIN-group if \(G\) has a neighbourhood basis of the identity consisting of sets \(V\) such that \(xVx^{-1}=V\) for all \(x\in G\). The author shows that for a noncompact amenable SIN-group \(G\) with primitive ideal space \(T_1\), the dual space \(\widehat{G}\) has no open points and so \(M_L(\pi)\) is defined for every irreducible representation \(\pi\). Moreover, if \(M_L(\pi)\) is finite, then \(\pi\) is finite-dimensional. As a corollary, it follows that for any noncompact nilpotent SIN-group \(G\) and for every \(\pi\in \widehat{G}\) the number \(M_L(\pi)\) is defined and either \(M_L(\pi)=1\) or \(M_L(\pi)= \infty\). Furthermore \(M_L(\pi)=1\) if and only if \(\pi\) is finite-dimensional.
scientific article; zbMATH DE number 1389439