Motivated by known examples of global integrals which represent automorphic
L-functions, this paper initiates the study of a certain two-dimensional array
of global integrals attached to any reductive algebraic group, indexed by
maximal parabolic subgroups in one direction and by unipotent conjugacy classes
in the other. Fourier coefficients attached to unipotent classes,
Gelfand-Kirillov dimension of automorphic representations, and an identity
which, empirically, appears to constrain the unfolding process are presented in
detail with examples selected from the exceptional groups.