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According to Sakellaridis, many zeta integrals in the theory of automorphicforms can be produced or explained by appropriate choices of a Schwartz spaceof test functions on a spherical homogeneous space, which are in turn dictatedby the geometry of affine spherical embeddings. We pursue this perspective bydeveloping a local counterpart and try to explicate the functional equations.These constructions are also related to the $L^2$-spectral decomposition ofspherical homogeneous spaces in view of the Gelfand-Kostyuchenko method. Tojustify this viewpoint, we prove the convergence of $p$-adic local zetaintegrals under certain premises, work out the case of prehomogeneous vectorspaces 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 theconstructions of Braverman and Kazhdan (2002). In the global setting, we givecertain speculations about global zeta integrals, Poisson formulas and theirrelation to period integrals.

Zeta integrals, Schwartz spaces and local functional equations