2017 Volume 71 Issue 2 Pages 271-285
While vector-valued automorphic forms can be defined for an arbitrary Fuchsian group Γ and an arbitrary representation R of Γ in GL (n, C), their existence, as far as we know, has been established in the literature only when restrictions are imposed on Γ or R.In this paper, we prove the existence of n linearly independent vector-valued automorphic forms for any Fuchsian group Γ and any n-dimensional complex representation R of Γ. 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.
