Representation Equivalence of Lattices in Lie Groups

TL;DR

The paper proves a spectral rigidity result for lattices Γ_1, Γ_2 of finite covolume in a real rank-one semisimple Lie group G by using a simple trace formula restricted to test functions supported on the regular set G_reg. It shows that if the lattices are isocuspidal and their M-restricted spectra match while discrete multiplicities m(π,Γ) agree for all but finitely many π, then the lattices are representation equivalent in G, with the continuous spectrum equality following from cusp data. The approach extends prior cocompact results to non-uniform lattices, clarifying how cusp structure and M-spectra control the full right-regular spectrum through Harish-Chandra characters. This advances understanding of when lattice spectra are determined by cusp configurations and M-spectral data, with potential implications for arithmeticity and trace-formula techniques in lattice theory.

Abstract

Let $Γ_1$ and $Γ_2$ 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 $L^2(Γ_1 \backslash G)$ and $L^2(Γ_2 \backslash G)$ 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.

Theorems & Definitions (6)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Remark 3.1
• Remark 3.2