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 with spherical quotients. 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.

在本文的第一部分中，我们回顾了附加到球面约简前齐次向量空间的局部 zeta 积分的形式，它通过合并可容许表示的广义矩阵系数部分扩展了 M. Sato 的理论。我们总结了这些积分的基本性质，例如收敛性、亚纯延拓和抽象函数方程。在第二部分，我们证明了一个泛化，它适用于某些具有球商的非球面空间。作为应用，由此产生的理论适用于 Bhargava 立方体下的前齐次向量空间，F. Sato 和 Suzuki-Wakatsuki 在他们对复曲面周期的研究中也考虑了这一点。