Hecke operators on cohomology

hecke operators play an important role in the theory of automorphic forms, and automorphic forms are closely linked to various cohomology groups. this paper is mostly a survey of hecke operators acting on certain types of cohomology groups. the class of cohomology on which hecke operators are introduced includes the group cohomology of discrete subgroups of a semisimple lie group, the de rham cohomology of locally symmetric spaces, and the cohomology of symmetric spaces with coefficients in a system of local groups. we construct canonical isomorphisms among such cohomology groups and discuss the compatibility of the hecke operators with respect to those canonical isomorphisms.