On rings of differential operators derived from automorphic forms

TL;DR

This paper explores the connection between differential equations parametrized on Hermitian symmetric spaces, automorphic forms for symplectic groups, and algebraic structures, extending classical equations like Lamé's.

Contribution

It establishes a novel relation between invariant differential equations on symmetric spaces and automorphic forms for symplectic groups.

Findings

Derived a closed relation between differential equations and automorphic forms.

Connected monodromy, Baker-Akhiezer functions, and algebraic curves to rings of differential operators.

Generalized classical Lamé equations to broader symmetric space contexts.

Abstract

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.