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).

16 Pages