Abstract
While vector-valued automorphic forms can be defined for an arbitrary Fuchsian group Gamma and an arbitrary representation R of Gamma in GL(n, C), their existence, as far as we know, has been established in the literature only when restrictions are imposed on Gamma or R. In this paper, we prove the existence of n linearly independent vector-valued automorphic forms for any Fuchsian group Gamma and any n-dimensional complex representation R of Gamma. To this end, we realize these automorphic forms as global sections of a special rank n vector bundle built using solutions to the Riemann-Hilbert problem over various non-compact Riemann surfaces and Kodaira's vanishing theorem.