  We generalize Abrahamse's interpolation theorem from the setting of a
multiply connected domain to that of a more general Riemann surface. Our main
result provides the scalar-valued interpolation theorem for the fixed-point
subalgebra of $H^\infty$ associated to the action of a Fuchsian group. We rely
on two results from a paper of Forelli. This allows us to prove the
interpolation result using duality techniques that parallel Sarason's approach
to the interpolation problem for $H^\infty$. In this process we prove a more
general distance formula, very much like Nehari's theorem, and obtain relations
between the kernel function for the character automorphic Hardy spaces and the
Szeg\"o kernel for the disk. Finally, we examine our interpolation results in
the context of the two simplest examples of Fuchsian groups acting on the disk.
