Multiplicative Functions with Matrix Characters on Compact Riemann Surfaces

The theory of multiplicative functions and Prym differentials for special (scalar) characters on compact Riemann surfaces has many applications in the theory of functions, in analytic numbers theory and in mathematical physics.
In this article we proove existence of matrix multiplicative functions and Prym $m$-differentials for every matrix characters with values in $GL(n;\mathbb C)$ on compact Riemann surfaces.