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 $Γ_1$ and $Γ_2$ 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 $L^{2}(Γ_1 \backslash G)$ and $L^{2}(Γ_2 \backslash G)$. Then $L^{2}(Γ_1 \backslash G) \cong L^{2}(Γ_2 \backslash G)$ as $G$ - modules (resp. the spherical spectra of $L^{2}(Γ_1 \backslash G)$ and $L^{2}(Γ_2 \backslash G)$ are equal).