  We revisit miscellaneous linear differential operators mostly associated with
lattice Green functions in arbitrary dimensions, but also Calabi-Yau operators
and order-seven operators corresponding to exceptional differential Galois
groups. We show that these irreducible operators are not only globally
nilpotent, but are such that they are homomorphic to their (formal) adjoints.
Considering these operators, or, sometimes, equivalent operators, we show that
they are also such that, either their symmetric square or their exterior
square, have a rational solution. This is a general result: an irreducible
linear differential operator homomorphic to its (formal) adjoint is necessarily
such that either its symmetric square, or its exterior square has a rational
solution, and this situation corresponds to the occurrence of a special
differential Galois group. We thus define the notion of being "Special
Geometry" for a linear differential operator if it is irreducible, globally
nilpotent, and such that it is homomorphic to its (formal) adjoint. Since many
Derived From Geometry n-fold integrals ("Periods") occurring in physics, are
seen to be diagonals of rational functions, we address several examples of
(minimal order) operators annihilating diagonals of rational functions, and
remark that they also seem to be, systematically, associated with irreducible
factors homomorphic to their adjoint.
