On a spectral analogue of the strong multiplicity one theorem

TL;DR

This paper establishes a spectral analogue of the strong multiplicity one theorem for automorphic forms, showing that if two lattices have matching multiplicities for all but finitely many representations, their associated spectral spaces are essentially identical.

Contribution

It introduces a spectral version of the strong multiplicity one theorem for lattices in semisimple groups, extending classical results to the spectral domain.

Findings

Spectral equivalence of lattices with finitely many mismatched representations

Equality of spherical spectra under specified conditions

Generalization of classical multiplicity one results to spectral setting

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).