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

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.