A trace formula for non-unitary representations of a uniform lattice

TL;DR

This paper extends the classical Selberg trace formula to include non-unitary finite-dimensional complex representations of uniform lattices in real Lie groups, broadening its applicability.

Contribution

It introduces a generalized trace formula for non-unitary representations of uniform lattices, expanding the theoretical framework of harmonic analysis on Lie groups.

Findings

Generalization of the Selberg trace formula to non-unitary representations

Application to finite-dimensional complex representations of uniform lattices

Potential new tools for spectral analysis in non-unitary settings

Abstract

In this work we shall generalize the Selberg trace formula to a non-unitary finite-dimensional complex representation $\chi:\Gamma\rightarrow\operatorname{GL}(V)$ of a uniform lattice $\Gamma$ of a real Lie group $G$.