Zeta integrals, Schwartz spaces and local functional equations

TL;DR

This paper develops a local theory of zeta integrals using Schwartz spaces on spherical homogeneous spaces, explores their functional equations, and connects these ideas to spectral decomposition, prehomogeneous vector spaces, and the doubling method.

Contribution

It provides a local framework for understanding zeta integrals via Schwartz spaces, proves convergence in p-adic cases, and relates these to existing theories like Godement-Jacquet and L-monoids.

Findings

Proved convergence of p-adic local zeta integrals.

Re-derived a large part of Godement-Jacquet theory.

Connected the doubling method to L-monoids paradigm.

Abstract

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.