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Recent development of special functios ----an approach from the representation thery and the complex integrals

特殊函数的最新发展----从表示论和复积分出发的方法

基本信息

批准号：
12440010
负责人：
MIMACHI Katsuhisa
金额：
$ 4.67万
依托单位：
Tokyo Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

项目摘要

Mimachi realized an irreducible representation of the Iwahori-Hecke algebra on the twisted homology group associated with a Selberg type integral. It was first constructed in the context of conformal field theory by Tsuchiya-Kanie. Our construction is based on the study of the homology group under a resonant condition on the exponents of integrals. We stress the importance of the study of integrals under such a resonant condition to the study of hypergeometric type functions and spherical functions. Mimachi with H.Ochiai (Nagoya) and M.Yoshida (Kyushu) formulated the concept of visible cycles and invisible cycles, and determined the dimension of the spaces of visible cycles under a resonant condition in some examples. Mimachi with M.Yoshida calculated some examples of the intersection numbers of twisted cycles associated with a Selberg type integral. It gives a natural interpretation of the coefficients of the four-point correlation function calculated by Dotsenko -Fateev. This is an answer to the long standing problem of clarifying the meaning of such coefficients appearing in correlation functions. In higher dimensional cases, the Terada model (nonsingular model arising from the point configuration) plays an important role. Kurokawa with M.Wakayama (Kyushu Univ.) studied generalized zeta regularizations. It shows that a discrete version of intersection numbers of twisted cycles should be settled. Kaneko studied Atkin's orthogonal polynomial from the viewpoint of automorphic forms and hypergeometric functions. Takata studied the volume conjecture by Kashaev from our viewpoint. She also carried out the numerical experiment to give an affirmative support to the volume conjecture.

Mimachi实现了Iwahori-Hecke代数在与Selberg型积分相关的扭曲同调群上的不可约表示。它最初是由土屋-Kanie在共形场论的背景下构建的。我们的构造是基于对积分指数共振条件下的同调群的研究。我们强调了在这种共振条件下研究积分对超几何型函数和球函数研究的重要性。Mimachi与H.Ochiai（名古屋）和M.Yoshida（九州）制定了可见周期和不可见周期的概念，并确定了一些例子中共振条件下可见周期空间的维数。Mimachi和M.Yoshida计算了与Selberg型积分相关的扭曲圈的交叉数的一些例子。它给出了一个自然的解释的系数的四点相关函数计算的Dotsenko -Fateev。这是一个长期存在的问题，澄清相关函数中出现的这些系数的含义的答案。在高维情况下，Terada模型（由点配置产生的非奇异模型）起着重要的作用。黑川与M.和歌山（九州大学）研究了广义zeta正则化。结果表明，扭圈相交数的离散形式是需要解决的。Kaneko从自守形式和超几何函数的角度研究了Atkin正交多项式。Takata从我们的角度研究了Kashaev的体积猜想。她还进行了数值实验，对体积猜想给予了肯定的支持。