Recent development of special functins … an approach from the representation thery and the complex integrals

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被积函数的指数在共振条件下的同调群的研究。我们强调了研究在这种共振条件下的积分对于研究超几何型函数和球函数的重要性。Mimachi与H。Ochiai（名古屋）和M.Yoshida（九州）提出了可见圈和不可见圈的概念，并在一些例子中确定了共振条件下可见圈空间的维数。Mimachi和M.Yoshida计算了与Selberg型积分相关的扭曲圈的相交数。它给出了一个自然的解释系数的四点相关函数，在共形场理论，计算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。在高维情形下，Terada模型（由点构型产生的非奇异模型）起着重要作用。Kurokawa和M.和歌山（九州）研究了广义zeta正则化。Takata研究了与扭纽结或环面纽结相关的作为n色Jones多项式因子的q-超几何级数，导出了与扭纽结或环面纽结相关的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.