Since domains on the complex plane might really represent moduli, and we know everything should be about categories at the end of the day, it would be better to replace the eigenfunction $f$ with a more sheaf theoretic concept. As Kapustin and Witten
say in their abstract: The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N=4 super Yang-Mills theory in four dimensions.
Yes, N=4 SUSY Yang-Mills turned up when we were worrying about twistor string theory
and calculating gluon amplitudes. Actually, these days the geometric Langlands
conjecture is about an equivalence of categories, namely derived categories of sheaves. Schreiber
has been blogging about such things. But where
did the number theory go in all this String geometry? Isn't that what this is really about?