Representation Equivalence of Lattices in Lie Groups

Let ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ be two lattices of finite covolume in a semisimple Lie group $G$. We prove a spectral rigidity result for the representation spectra of the right regular representations $L^2({\mathrm{\Gamma }}_1\mathrm{\backslash }G)$ and $L^2({\mathrm{\Gamma }}_2\mathrm{\backslash }G)$ of $G$. This can be thought of as an analogue of the strong multiplicity one theorem and it generalises a result by the first author and Rajan to the case of non-uniform lattices.