Results on a Strong Multiplicity One Theorem

We prove an analogue of the strong multiplicity one theorem in the context of ${\tau }_n$-spherical representations of the group $G=SO(2,1{)}^{\circ }$ appearing in $L^2({\mathrm{\Gamma }}_i\mathrm{\backslash }G)$ for uniform torsion-free lattices ${\mathrm{\Gamma }}_i,i=1,2$ in $G$. This is a generalisation of a previous result by the first author and C. S. Rajan in \cite{B-R-2011} for the case of $G=SO(2,1{)}^{\circ }$.