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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.有限群G的表示的Zeta函数和与之相关的函数方程首先由T.A.Springer（1971）和I. G. Macdonald（1985）讨论。我们首先证明了这些函数方程对Hecke代数H的不可约表示也成立，Hecke代数H是一个无重数的诱导表示的自同态代数，然后我们研究了显式例子和应用（（1）（2）与C. W. Curtis的联合工作）。（1）若G是有限约化群，且His是Gelfand-Graev表示的自同态代数，则函数方程中出现的Gauss因子恰为Saito-Shinoda研究的Gauss和.利用这一事实，我们得到了一个明确的表达式的傅里叶变换的e，这是身份的H在一般的线性群的情况下。(2)我们证明了在GL（n，q）的情况下，H在标准基元上的不可约表示的值（对应于Coxeter元）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密切的联系。(3)在GL（4，q）的情况下，我们获得了H在标准基元素上的不可约表示的几乎所有值。它们可以称为高次Kloosterman和。并从相关领域对该项目进行了研究。(1)Gomi和Shinoda推广了J.McKay（1999）关于有限线性群的上不变代数的一个结果（与I.中村联合）. (2)横沼研究离散集和相关的动力系统鉴于对称性与非交换设置。中岛和古贺研究从量子群的观点，特别是中岛研究几何晶体舒伯特品种。(3)和田、角井和月分别从数论的角度研究了这一课题。