Research on Orbits of Semisimple Lie Algebras and Representations

1.J.Sekiguchi研究了正则多面体群的不变量、带特殊参数的Appell超几何函数和四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，基本不变量是这样选择的，它们被分解成两个因素模的二次不变。这种分解与Appell超几何函数的退化有关。本工作是与加藤光雄的共同工作。2.J.Sekiguchi组织了在富士樱所举行的会议（2004年幂零轨道和表示论会议）（2004年9月），当时D. Z. Djokovic教授报告了与Sekiguchi和Kaiming Zhao的共同工作结果。本文研究了一般线性群在n × n矩阵上的作用.特别是他们表明，nul锥与此行动是不可约的作为一个品种。3.J.Sekiguchi研究了简单的八线安排与一些条件的真实的投影平面。这是和福井铁雄的合作作品。他们的兴趣与E8型Weyl群的作用有关。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，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一个W（E8）-等变映射。这解决了他们的部分猜想。

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.