Representation-theoretic study of spherical functions arising from number theory

数论中产生的球函数的表示论研究

• 批准号: 10640020
• 负责人: KATO Shinichi
• 金额: $ 1.92万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640020/
• 关键词: Representation theory; Algebraic group; Spherical function; Special function; Symmetric space; Spherical homogeneous space; Orbit; 特殊関係; リー群; ヘッケ環; ワイル群; 軌道分解

Special functions (spherical functions) on algebraic groups play an important role in number theory, especially in the study of automorphic forms. In most cases, these spherical functions are related to spherical homogeneous spaces, such as symmetric spaces. In this research project, Kato (head investigator) studied spherical functions on spherical homogeneous spaces of reductive groups over non-archimedean local fields from a representation theoretic view point. The purpose of this research is two-fold : (1) To understand special functions such as zonal spherical functions or Whittaker functions in a uniform manner from the view point as above. (2) To obtain properties of these functions, including the uniqueness and explicit formulas, for important cases which arise in number theory. As for (1), we studied an orbit decomposition of spherical homogeneous spaces first. Then applying this, we obtained a general formula for spherical functions (at least in the case of symmetric spaces) t … More ogether with a method to compute the coefficients in this formula explicitly. As for (2), we got the uniqueness and an explicit formula for e.g. a symmetric space corresponding to quadratic base change by using the above mentioned method. This research is still under way. Other investigators obtained several results related to representation theory and spherical homogeneous spaces as follows. Saito studied zeta functions of prehomogeneous vector spaces, which is closely related to (spherical functions of) spherical homogeneous spaces, and showed the convergence and explicit formulas (in terms of local orbital zeta functions) in general. Matsuki investigated Weyl groups and Jordan decompositions arising from symmetric spaces. Nishiyama studied multiplicity free actions, which is a characteristic property of spherical homogeneous spaces, and the relation between theta correspondences and nilpotent orbits. Other investigators, Takasaki, Yamauchi et al. carried out researches on mathematical physics, automorphic forms and so on. Less

代数群上的特殊函数（球函数）在数论，特别是自同构形式的研究中起着重要的作用。在大多数情况下，这些球面函数与球面齐次空间有关，例如对称空间。在本课题中，Kato（首席研究员）从表示理论的角度研究了非阿基米德局部域上约化群的球面齐次空间上的球函数。本研究的目的有两个方面：(1)从上述观点统一地理解带状球面函数或惠特克函数等特殊函数。(2)对于数论中出现的重要情况，得到这些函数的性质，包括唯一性和显式公式。对于(1)，我们首先研究了球面齐次空间的轨道分解。在此基础上，我们得到了球面函数的一般公式（至少在对称空间中是如此），并给出了该公式中系数的显式计算方法。对于(2)，利用上述方法，我们得到了如对称空间对应于二次基变换的唯一性和显式公式。这项研究仍在进行中。其他研究者在表示理论和球面齐次空间方面得到了如下几个结果。Saito研究了与球面齐次空间（球函数）密切相关的预齐次向量空间的zeta函数，并给出了一般的收敛性和（局部轨道zeta函数）的显式公式。Matsuki研究了对称空间中产生的Weyl群和Jordan分解。Nishiyama研究了球面齐次空间的特征属性——自由多重性作用，以及对应与幂零轨道之间的关系。其他研究者，Takasaki， Yamauchi等人在数学物理，自同构形式等方面进行了研究。少

项目成果

西山 享: "Invariants for Representations of Weyl Groups,Two-sided Cells, and Modular Representations of Iwahori-Hecke Algebras"Adv.Studies in Pure Math. (未定).

Toru Nishiyama：“Weyl 群表示的不变量、两侧单元和 Iwahori-Hecke 代数的模表示”纯数学高级研究（TBD）。

Shinichi Kato: "Whittaker-Shintani Functions for Orthogonal Groups."(to appear). (2000)

Shinichi Kato：“正交群的 Whittaker-Shintani 函数。”（即将出现）。

Hiroshi Saito: "On the zeta functions associated to symmetric matrices II : Functional equations and special values."(to appear).

Hiroshi Saito：“关于与对称矩阵相关的 zeta 函数 II：函数方程和特殊值。”（即将出现）。

斎藤 裕: "Explicit form of zeta functions of prehomogeneous vector spaccs" Math.Ann.(1999)

Yutaka Saito：“前齐次向量 spacc 的 zeta 函数的显式形式”Math.Ann.(1999)

加藤 信一: "Whittaker-Shintani Functions for Orthogonal Groups"未定. (未定). (2000)

Shinichi Kato：“正交群的 Whittaker-Shintani 函数” 待定（待定）。