权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

A Study on the zeta functions attached with the irreducible representations of finite reductive groups

有限约简群不可约表示的zeta函数研究

基本信息

批准号：
14540042
负责人：
SHINODA Ken-ichi
金额：
$ 2.18万
依托单位：
Sophia University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540042/
关键词：
finite reductive groups irreducible representation zeta function general linear group Gelfand-Graev representation Gauss sum Kloosterman sum 有限群の表現論

项目摘要

1. Zeta functions and functional equations associated with them for representations of a finite group G were first discussed by T.A.Springer (1971) and I.G.Macdonald (1985). We showed first that these functional equations also hold for the irreducible representations of Hecke algebra H which is an endomorphism algebra of an induced representation with multiplicity free.Then we studied the explicit examples and applications ((1)(2) joint work with C.W.Curtis). The results are as follows :(1) It can be shown that if G is a finite reductive group and if His an endomorphism algebra of Gelfand-Graev representation, the epsilon factor which appears in the functional equation is exactly the Gauss sum studied by Saito-Shinoda. Using this fact we obtained an explicit expression of the Fourier transform of e, which is the identity of Hin the case of general linear groups.(2) We showed that in the case of GL(n,q), values of irreducible representations of H on a standard basis element, which corresponds to a Coxeter element, become generalized Kloosterman sums. This implies that there should exist a close relation a between Davenport-Hasse type equations of Kloosterman sums and the norm maps of Hecke algebras.(3) In case of GL(4,q) we obtained almost all values of irreducibles representations of H on standard basis elements. They can be called as Kloosterman sums of higher degree.2. We also studied this project from relating fields.(1) Gomi and Shinoda generalized a result of J.McKay (1999) on coinvariant algebras of finite linear groups (joint with I.Nakamura).(2) Yokonuma studied discrete sets and associated dynamical systems in view of symmetry with a non-commutative setting. Nakashima and Koga studied from the view of quantum groups ; particularly Nakashima studied geometric crystals on Schubert varieties.(3) Wada, Tsunogai and Tsuzuki respectively studied this project in view from the number theory.

1. Zeta函数和与之相关的表示有限群G的泛函数方程是由T.A.Springer（1971）和i.g.m ndonald（1985）首先讨论的。我们首先证明了这些泛函方程也适用于Hecke代数H的不可约表示，Hecke代数H是一个无多重性诱导表示的自同态代数。然后，我们研究了显式实例及其应用(（1)和(2)与C.W.Curtis的联合工作）。结果表明：(1)如果G是有限约化群，如果G是Gelfand-Graev表示的自同态代数，则出现在泛函方程中的ε因子正是Saito-Shinoda研究的高斯和。利用这个事实，我们得到了e的傅里叶变换的显式表达式，它是一般线性群的恒等式。(2)在GL（n,q）的情况下，与Coxeter元对应的标准基元上H的不可约表示的值成为广义Kloosterman和。这意味着Kloosterman和的Davenport-Hasse型方程与Hecke代数的范数映射之间存在着密切的关系。(3)在GL（4,q）的情况下，我们得到了H在标准基元上的不可约表示的几乎所有值。它们可以称为更高次的Kloosterman和。我们也从相关领域研究了这个项目。(1) Gomi和Shinoda推广了J.McKay（1999）关于有限线性群的协不变代数的结果（与i.m akamura联合）。(2) Yokonuma从非交换集对称性的角度研究了离散集和相关动力系统。Nakashima和Koga从量子群的角度进行研究；特别是Nakashima研究了舒伯特品种的几何晶体。(3) Wada、Tsunogai和Tsuzuki分别从数论的角度研究了这个项目。