Variations on themes of Sato

TL;DR

This paper extends Sato's theory of local zeta integrals to include non-spherical spaces, with applications to Bhargava's cubes and toric periods, advancing the understanding of prehomogeneous vector spaces.

Contribution

It generalizes the formalism of local zeta integrals to non-spherical spaces and applies this to Bhargava's cubes, broadening the scope of Sato's original framework.

Findings

Extended the formalism to non-spherical spaces.

Applied the theory to Bhargava's cubes.

Connected the results to toric periods.

Abstract

In the first part of this article, we review a formalism of local zeta integrals attached to spherical reductive prehomogeneous vector spaces, which partially extends M. Sato's theory by incorporating the generalized matrix coefficients of admissible representations. We summarize the basic properties of these integrals such as the convergence, meromorphic continuation and an abstract functional equation. In the second part, we prove a generalization that accommodates certain non-spherical spaces. As an application, the resulting theory applies to the prehomogeneous vector space underlying Bhargava's cubes, which is also considered by F. Sato and Suzuki-Wakatsuki in their study of toric periods.