Representation Equivalence of Lattices in Lie Groups

Abstract

Let ${\Gamma }_1$ and ${\Gamma }_2$ 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 $L^2({\Gamma }_1\backslash G)$ and $L^2({\Gamma }_2\backslash G)$ 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.

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