Research on Orbits of Semisimple Lie Algebras and Representations

1. j。Sekiguchi研究了正多面体群的不变量、带特殊参数的apell超几何函数与4阶不可约反射群的基本不变量之间的关系。特别是，他们发现基本不变量被如此选择，以至于它们被分解成两个因子对二次不变量进行模化。这种分解与阿佩尔超几何函数的退化有关。这部作品是与加光夫的合作作品。2004年9月，关口在富士山组织召开了“零轨道与表示理论会议”（conference on Nilpotent orbit and Representation Theory 2004），当时德约科维奇教授报告了与关口和赵开明共同工作的结果。本文研究了一般线性群对n × n矩阵的作用。他们特别指出，与这一作用相关的空锥作为一个品种是不可约的。关口在真实的投影平面上研究了一些条件下的简单八线排列。这是与Tetsuo Fukui合作的作品。他们的兴趣与E8型Weyl基团的作用有关。他们首先处理一个由十个节点组成的图，类似于Dynkin图，并将e8根附加到节点上。对于每一个这样的图，都可以在一个真实的投影平面上构造一个简单的八条线的排列。他们证明了这种对应实际上是W(E8)-等变映射。这在一定程度上解决了他们的猜想。

1.J.Sekiguchi studied the relation among the invariants of regular polyhedral groups, Appell's hypergeometric functions with special parameters and basic invariants of irreducible reflection groups of rank four. In particular, they find that basic invariants are so chosen that they are decomposed into two factors mod the quadratic invariant. And this decomposition is related with the degeneration of Appell's hypergeometric function. This work is a joint work with Mitsuo Kato.2.J.Sekiguchi organized the conference (Conference on Nilpotent Orbits and Representation Theory 2004) held at Fuji-Sakura-So (September,2004) and at that time Professor D.Z.Djokovic reported the result of the joint work with Sekiguchi and Kaiming Zhao. This work is concerned with an action of general linear group on n by n matrices. In particular they showed that the nul-cone associated with this action is irreducible as an variety.3.J.Sekiguchi studied simple eight-line arrangements with some conditions on a real projective plane. This is a joint work with Tetsuo Fukui. Their interest is related with the action of Weyl group of type E8. They first treat a diagram consisting of ten nodes similar to the Dynkin diagram and attach E8-roots to nodes. To each of such a diagram, it is possible to construct a simple arrangement of eight lines on a real projective plane. They showed that this correspondence is, in fact, a W(E8)-equivariant map. This solves a part of their conjecture.