The Selberg trace formula for non-unitary representations of the lattice

TL;DR

This paper extends the Selberg trace formula to non-unitary representations of lattice groups in compact locally symmetric spaces, linking spectral data of a non-self-adjoint operator to geometric properties.

Contribution

It introduces a version of the Selberg trace formula applicable to non-unitary representations, broadening the scope of spectral analysis in geometric contexts.

Findings

Derived the trace formula for non-unitary representations

Connected the spectrum of the flat Laplacian to geometric data

Established properties of the non-self-adjoint operator involved

Abstract

For a compact locally symmetric space X, we establish a version of the Selberg trace fromula for a non-unitary representation of the fundamental group of X. On the spectral side appears the spectrum of the "flat Laplacian", acting in the space of sections of the associated flat bundle. In general, this is a non-self-adjoint operator with the same principal symbol as the Laplace operator associated to any choice of a Hermitian fiber metric in the flat bundle.