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

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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.