  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.
