Super-zeta functions and regularized determinants associated with cofinite Fuchsian groups with finite-dimensional unitary representations

Let M be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let chi denote a finite dimensional unitary representation of the fundamental group of M. Let Delta denote the hyperbolic Laplacian which acts on smooth sections of the flat bundle over M associated with chi. From the spectral theory of Delta, there are three distinct sequences of numbers: the first coming from the eigenvalues of L-2 eigenfunctions, the second coming from resonances associated with the continuous spectrum, and the third being the set of negative integers. Using these sequences of spectral data, we employ the super-zeta approach to regularization and introduce two super-zeta functions, Z-(s, z) and Z(+)(s, z) that encode the spectrum of Delta in such a way that they can be used to define the regularized determinant of Delta - z(1- z)I. The resulting formula for the regularized determinant of Delta - z(1 - z)I in terms of the Selberg zeta function, see Theorem 5.3, encodes the symmetry z <-> 1 - z.