Prym differentials with matrix characters on a finite Riemann surface

The theory of multiplicative functions and Prym differentials for scalar characters on a compact Riemann surface has found applications in function theory, analytic number theory, and mathematical physics.
We construct the matrix multiplicative functions and Prym $m$-differentials on a finite Riemann surface for a given matrix character with values in $GL(n,\mathbb C)$ starting from a meromorphic function on the unit disk with finitely many poles. We show that these multiplicative functions and Prym $m$-differentials depend locally holomorphically on the matrix character.