Recent development of special functios ----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 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.