  Physicists showed that the generating function of orbifold elliptic genera of
symmetric orbifolds can be written as an infinite product. We show that there
exists a geometric factorization on space level behind this infinite product
formula in much more general framework, where factors in the infinite product
correspond to isomorphism classes of connected finite covering spaces of
manifolds involved. From this formula, a concept of geometric Hecke operators
for functors emerges. We show that these Hecke operators indeed satisfy the
usual identity of Hecke operators for the case of 2-dimensional tori.
