Hecke operators on linear ordinary differential equations (Q1973298)

The theory of automorphic forms and that of ordinary linear differential equations are closely related. Automorphic forms are common eigenforms of certain Hecke operators. Thus it is natural to consider the Hecke operators acting on the space of ordinary differential equations associated with automorphic forms. The purpose of this aper is to introduce Hecke operators on the space of ordinary differential equations which is compatible with the ordinary Hecke operators on automorphic forms together with the Hermitian inner product on the space of differential equations associated to automorphic forms. Then the author also determines the adjoint of the Hecke operator with respect to the inner product, and proves that the space of ordinary differential equations associated to an automorphic form for a certain discrete subgroup of \(SL_2 (\mathbb{R})\) has a basis consisting of common eigenvectors of a class of Hecke operators.