佐藤主题变奏曲
Variations on themes of Sato
摘要：
在本文的第一部分中，我们回顾了球面约化预齐次向量空间的局部zeta积分的一个形式，它部分地推广了M。Sato的理论通过引入容许表示的广义矩阵系数。我们总结了这些积分的基本性质，如收敛性、亚纯延拓和一个抽象的函数方程。在第二部分中，我们证明了一个推广，容纳某些非球形空间。作为一个应用，所得到的理论适用于巴加瓦立方体下的预齐次向量空间，这也被F。佐藤和铃木若月在他们的研究Toric时期。
英文摘要：
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.
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