Results on a strong multiplicity one theorem (Q6635597)

In the paper under review, the authors show a so-called \textit{strong multiplicity one property} of the \(\tau\)-spherical representation of the connected component \(G=\mathrm{SO}(2,1)^0\) of the group \(G=\mathrm{SO}(2,1)\). The main theorem states that for an irreducible representation \(\tau\) of the maximal compact subgroup \(K= \mathrm{SO}(2)\) of \(G\) and two torsion-free lattices \(\Gamma_1, \Gamma_2\) in \(G\), if the associated multiplicity functions equal, i.e. \(m(\pi, \Gamma_1) = m(\pi, \Gamma_2)\) for almost all representation \(\pi\) in the \(\tau\)-dual object \(\hat G_\tau\) consisting of all irreducible unitary representations with non-vanishing \(\tau\)-isotopic component \(V^\pi \ne \{0\}\), then they coincide in the whole \(\hat G_\tau\).