On rings of differential operators derived from automorphic forms

We study linear ordinary differential equations which are analytically parametrized on Hermitian symmetric spaces and invariant under the action of symplectic groups. They are generalizations of the classical Lam\'e equation. Our main result gives a closed relation between such differential equations and automorphic forms for symplectic groups. Our study is based on techniques concerning with the monodromy of complex differential equations, the Baker-Akhiezer functions and algebraic curves attached to rings of differential operators.