Hecke operators on de Rham cohomology (Q1884577)

The author introduces the notion of Hecke operators on de Rham cohomology of a compact oriented manifolds. Such an operator is determined by a pair of covering maps of the given manifold. When the manifolds are regarded as quotients of their universal covering space by discrete subgroup of its group of diffeomorphisms, the properties of these operators are discussed. Automorphic forms on a Hermitian symmetric domain for a discrete subgroup of the corresponding semisimple Lie group as well as Hecke operators acting on the space of such automorphic forms are described. A correspondence between the space of modular forms of weight one and a subspace of the de Rham cohomology space is established. It is proved that the Hecke operators on modular forms are compatible with a special type of Hecke operators on de Rham cohomology under this correspondence.