Representation Equivalence of Lattices in Lie Groups

TL;DR

This paper proves a spectral rigidity result showing that the representation spectra of two lattices in a semisimple Lie group are equivalent under certain conditions, extending previous results to non-uniform lattices.

Contribution

It generalizes a spectral rigidity theorem to non-uniform lattices in semisimple Lie groups, expanding the scope of representation equivalence results.

Findings

Spectral rigidity for non-uniform lattices established

Representation spectra determine lattice equivalence

Generalization of strong multiplicity one theorem

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.