On a spectral analogue of the strong multiplicity one theorem

Abstract

We prove spectral analogues of the classical strong multiplicity one theorem for newforms. Let $\Gamma_1$ and $\Gamma_2$ be uniform lattices in a semisimple group $G$. Suppose all but finitely many irreducible unitary representations (resp. spherical) of $G$ occur with equal multiplicities in $L^{2}(\Gamma_1 \backslash G)$ and $L^{2}(\Gamma_2 \backslash G)$. Then $L^{2}(\Gamma_1 \backslash G) \cong L^{2}(\Gamma_2 \backslash G)$ as $G$ - modules (resp. the spherical spectra of $L^{2}(\Gamma_1 \backslash G)$ and $L^{2}(\Gamma_2 \backslash G)$ are equal).