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

Mimachi在与Selberg型积分相关的扭曲同调群上实现了Iwahori-Hecke代数的一个不可约表示。它首先是由土屋-卡尼在保形场理论的背景下构造的。我们的构造是基于对积分指数共振条件下同调群的研究。我们强调了在这种共振条件下积分的研究对于超几何型函数和球面函数研究的重要性。Mimachi与H.Ochiai(名古屋)和M.Yoshida(九州)提出了可见循环和不可见循环的概念，并在一些例子中确定了共振条件下可见循环的空间维度。Mimachi和M.Yoshida计算了一些与Selberg型积分有关的扭圈的交数的例子。它对Dotsenko-Fateev计算的四点关联函数的系数给出了自然的解释。这是对一个长期存在的问题的答案，即澄清出现在相关函数中的这些系数的含义。在高维情况下，Terada模型(由点组态产生的非奇异模型)起着重要的作用。Kurokawa与M.Wakayama(九州大学)研究了广义Zeta正则化。结果表明，应确定扭圈交点数的离散化形式。金子从自同构型和超几何函数的角度研究了阿特金的正交多项式。高田从我们的角度研究了卡沙耶夫的体积猜想。她还进行了数值实验，肯定地支持了体积猜想。

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.