  According to Sakellaridis, many zeta integrals in the theory of automorphic
forms can be produced or explained by appropriate choices of a Schwartz space
of test functions on a spherical homogeneous space, which are in turn dictated
by the geometry of affine spherical embeddings. We pursue this perspective by
developing a local counterpart and try to explicate the functional equations.
These constructions are also related to the $L^2$-spectral decomposition of
spherical homogeneous spaces in view of the Gelfand-Kostyuchenko method. To
justify this viewpoint, we prove the convergence of $p$-adic local zeta
integrals under certain premises, work out the case of prehomogeneous vector
spaces and re-derive a large portion of Godement-Jacquet theory. Furthermore,
we explain the doubling method and show that it fits into the paradigm of
$L$-monoids developed by L. Lafforgue, B. C. Ngo et al., by reviewing the
constructions of Braverman and Kazhdan (2002). In the global setting, we give
certain speculations about global zeta integrals, Poisson formulas and their
relation to period integrals.
