Studies on Representations of Finite Groups and Applications

有限群表示及其应用研究

• 批准号: 09640063
• 负责人: SHINODA Ken-ichi
• 金额: $ 1.54万
• 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: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640063/
• 关键词: character sum; gaussian sum; Kloosterman sum; unitary Kloosterman sum; Gelfand-Graev representation; almost character; Hilbert scheme of G-orbit; coinvariant algebra; Character sum; unitary Kloosterman sum; Gelfand-Gracv representation; 有限簡約代教群

1. (A) We gave a detailed description of the fiber over the origin of Hilbert-Chow morphism from G-orbit Hubert scheme, HilbィイD1GィエD1(CィイD13ィエD1), to CィイD13ィエD1/G in the case where G is a fnite simple subgroup of SL(3, C) of order 60 or 168. This work is being clone jointly with I. Nakamura.(B1) We defined a character sum, called a gaussian sum, over a finite reductive group and showed that if the sum is associated with the Delige-Lusztig generalized character, then it is expressed explicitly as a sum over the torus ; as an application we also showed that in the case of finite classical groups, the sum can be expressed using Kloosterman sums and unitary Kloosterman sums. N. Saito works jointly with us on this problem.(B2) We investigated Hasse-Davenport type relation for Kloosterman sums and unitary Kloosterman sums ; we also applied it to the norm map of Gelfand-Graev representation of GL (2, q) ; jointly studied with C. W. Curtis (Oregon U.)c We determined the values of 7 unipotent almost characters of finite Chevalley groups of type FィイD24ィエD2 without assumuptions on characteristic of the ground field ; some of them were known already.2. We showed that the Hasse princile holds for symmetric groups and alternating groups, etc. (Wada ; joint with T. One)3. We gave a method, called the polyhedral realization, to describe the crystal base associated with aim irrreducible highest weight module of a quantum group. (Nakashimna)4. We classified the case when the parabolic Heck algebras remain semisimple after specializing over arbitrary field. (Gomi)5. From the standpoint of Number Theory, we investigated the ranks on the stable derivation algebra related with Dehigne's problem (Tsunogai) and also zeta functions associated with the Riemmiannian symmetric space of rank 1 and the discrete coconipact automorphism subgroup. (Tsuzuki)

1. (A)在G是SL（3，C）的60阶或168阶有限单群的情况下，详细描述了从G-轨道Hubert方案，Hilb_D_1 G_（13）Hubert D_1（C_（13）Hubert D_1）到C_（13）Hubert D_1/G的Hilbert-Chow态射原点上的纤维.这部作品正在和我联合克隆。中村。(B1)我们在有限约化群上定义了一个特征和，称为高斯和，并证明了如果该和与Delige-Lusztig广义特征有关，则它明确表示为环面上的和;作为应用，我们还证明了在有限经典群的情况下，该和可以用Kloosterman和和酉Kloosterman和来表示。N.齐藤正与我们共同解决这个问题。(B2)研究了Kloosterman和与酉Kloosterman和的Hasse-Davenport型关系，并将其应用于GL（2，q）的Gelfand-Graev表示的范数映射，与C. W.柯蒂斯（俄勒冈州大学）c在不假设基场特征的情况下，我们确定了FイD24 D2型有限Chevalley群的7个幂单几乎特征标的值;其中一些是已知的。2.我们证明了Hasse原理对对称群和交错群等都成立。1）3.给出了一种描述量子群目标可约最高权模对应的晶格基的方法，称之为多面体实现。（Nakashimna）4.对抛物Heck代数在任意域上特殊化后保持半单的情形进行了分类。（五米）5.本文从数论的观点出发，研究了与Dehigne问题（Tsunogai）有关的稳定导子代数的秩，以及与秩为1的黎曼对称空间和离散余紧自同构子群有关的zeta函数.（津月）

项目成果

C. W. Curtis: "Unitary Kloosterman sums and the Gelfand-Graev representation of GL_2"J. of Algebra. 216. 431-447 (1999)

C. W. Curtis：“Unity Kloosterman sums 和 GL_2 的 Gelfand-Graev 表示”J.

T. Ono: ""Hasse principle" for symmetric and alternating groups"Proc. Japan Academy. 75. 61-62 (1999)

T. Ono：“对称群和交替群的“哈斯原理””Proc。

T.Nakashima: "Polyhedaral realizations of crystal bases and braid-type isomorphisms"Contemporary Math.. 248. 419-435 (1999)

T.Nakashima：“晶体基和编织型同构的多面体实现”当代数学.. 248. 419-435 (1999)

T.Nakashima: "Polyhedaral realizations of crystal bases for integrable heighest weight modules"Journal of Algebra. 219. 571-597 (1999)

T.Nakashima：“可积最高重量模块的晶体基的多面体实现”代数杂志。

Wada, Hideo, Takashi Ono: ""Hasse principle" for free groups"Proc.Japan Acad.. 75A. 1-2 (1999)

Wada、Hideo、Takashi Ono：“自由群体的“哈斯原理””Proc.Japan Acad.. 75A。