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。T.A.Springer（1971）和I.G. Macdonald（1985）首先讨论了与它们相关的ZETA函数和与它们相关的功能方程。我们首先表明，这些功能方程也适用于Hecke代数h的不可约说明，这是诱导表示的内态代数，并自由多样化。结果如下：（1）可以证明，如果G是有限的还原群，并且如果他的内态代数是Gelfand-Graev代表的代数，则功能方程中出现的Epsilon因子完全是由Saito-Shinoda研究的高斯总和。利用这一事实，我们获得了E的傅立叶变换的明确表达，这是一般线性组的情况。（2）我们表明，在GL（N，Q）的情况下，H在标准基础元素上的不可减至表示的值，该标准基元与Coxeter元素相对应，成为通用的Kloosterman总和。这意味着应该存在Kloosterman总和的Davenport-Hasse类型方程与Hecke代数的标准图之间存在密切的关系A。（3）在GL（4，Q）的情况下，我们在标准基础元素上几乎获得了H的Mordreducibles表示的所有值。它们可以称为Kloosterman较高程度的总和2。（1）Gomi和Shinoda从相关领域研究了该项目。中岛和科加从量子群的角度学习。特别是中岛在舒伯特品种上研究了几何晶体。（3）WADA，Tsunogai和Tsuzuki分别研究了该项目，从数字理论看。