Recent development of special functins … 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被积指数的共鸣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和H.Ochiai(名古屋)和M.Yoshida(九州)提出了可见圈和不可见圈的概念，并在某些例子中确定了共振条件下可见圈的空间的维度，Mimachi和M.Yoshida计算了与Selberg型积分相关的扭圈的交数。它给出了Dotsenko-Fateev计算的保形场论中四点关联函数系数的自然解释。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(由点构型产生的非奇异模型)扮演着重要的角色。黑川和M.Wakayama(九州)研究了广义Zeta正则化。高田研究了一个q-超几何级数，它表现为与扭结或环结相关的n色Jones多项式的一个因子，并导出了与之相关的A-多项式。

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 integrands. 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 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, in conformal field theory, calculated by Dotsenko-Fateev. This is an answer to a 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) studied generalized zeta regularizations. It shows that a discrete version of intersection numbers of twisted cycles should be settled.Takata studied a q-hypergeometric series which appears as a factor of the n-colored Jones polynomial associated with a twisted knot or a torus knot and derived the A-polynomials associated with them.