Studies on Representations of Finite Groups and Applications

(A)我们gave A detailed description of the fiber over the origin of Hilbert-Chow morphism fromG-orbit Hubert scheme, Hilb - D1G - D1(C - D13 - D1)，to C where G is a fnite simple subgroup of SL（3, 2）C) of order 60或168. This work is being clone jointly with I. Nakamura.（B1）我们定义了一个字符sum, called a gaussian sumover a finite reductive group and showed that if the sum is associated with the Delige-Lusztiggeneralized character，那么它是expressed explicitly as a sum over the torus；as an application we also showed that in the case of finite classical groupsthe sum can be expressed using Kloosterman sums and unitary Kloosterman sums. N. Saito works jointly（B2）我们投资Hasse-Davenport type relation for Kloosterman sums andunitary 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 almostcharacters of finite Chevalley groups of type F - D24 - D2 without assumuptions on characteristic ofthe ground field；2.我们应该为symmetric组和孩子们的Hasse princile holdsalternating groups，等等(Wada；3. We gave a method, called the polyhedral realation，to describe the crystal base associated with aim irrreducible highest weight module of a quantum（Nakashimna）4. We classified the case when the parabolic Heck algebras remain semisimpleafter specializing over arbitrary field. (Gomi)5. From the standpoint of Number Theorywe investigated the ranks on the stable derivation algebra related with Dehigne's problem （Tsunogai）与zeta functions associated with the Riemmiannian symmetric space of rank 1 and the discretecoconipact automorphism subgroup （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)