ON THE EXISTENCE OF VECTOR-VALUED AUTOMORPHIC FORMS
Kyushu Journal of Mathematics
While vector-valued automorphic forms can be defined for an arbitrary Fuchsian group and an arbitrary representation R of in GL(n, C), their existence, as far as we know, has been established in the literature only when restrictions are imposed on or R. In this paper, we prove the existence of n linearly independent vector-valued automorphic forms for any Fuchsian group and any n-dimensional complex representation R of . To this end, we realize these automorphic forms as global sections of a
... al 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. where J γ (z) = cz + d if γ = * * c d . In addition, we require that at each cusp of , F has a meromorphic behaviour to be made explicit in the next section. Also, the question of multivaluedness of (cz + d) k has to be addressed via multiplier systems. The theory of vector-valued automorphic forms has been around for a long time, first, as a generalization of the classical theory of scalar automorphic forms, then as natural objects appearing in mathematics and physics. For instance, Selberg suggested vector-valued forms as a tool to study modular forms for finite index subgroups of the modular group  and they also connected to Jacobi forms [6, 34] . In physics, they appear as characters in rational conformal field theory [4, 5, 27] . In the last decade, there has been a growing interest in the 2010 Mathematics Subject Classification: Primary 11F12, 35Q15, 32L10.