Studies on Representations of Finite Groups and Applications

1.（A） We gave a detailed description of the fiber over the origin of Hilbert-Chow morphism from G-orbit Hubert scheme，Hilby D1 G（CイD13イD1）， to CイD13イD1/G in the case where G is a fnite simple subgroup of SL（3， C） of order 60 or 168。这份工作是克隆人和我一起工作的。Nakamura。（B1）我们定义了一个字符和，称为一个高斯和，通过一个有限的减少组，并展示了如果和与Delige-Lusztig通用的字符相关，那么它将被描述为一个在torus上的总和;作为我们也展示了一个应用，在有限的经典组的情况下，总和可以使用Kloosterman sums和unitary Kloosterman sums来展示。N. N. Saito和我们一起在这个问题上工作。（B2）我们研究了Kloosterman sum和单位Kloosterman sum的Hasse-Davenport类型关系;我们也应用于Gelfand-Graev GL （2， q）表示的正常图;与C. W. W. Curtis （俄勒冈州）c We determined the values of 7 unipotent almost？ters of finite Chevalley groups of type F_我们向对称性组和替代组展示了Hasse Princile持有人等（Wada ;与T.联合）（一）三。我们给出了一种方法，称之为多元现实化，并描述了与目标相似的晶体基础，一个量子组的最高重量模块。（Nakashimna）4。我们将把情况归类为“抛物线型Heck代数”，在专门化后，将继续进行半幅。（戈米）5。从数字理论的标准点出发，我们调查了与Dehigne的问题（Tsunogai）有关的稳定衍生代数的等级，也调查了与Riemmiannian 1级对称空间和离散共环自动化亚组有关的zeta函数。（Tsuzuki）

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)