  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.
