Studies on Group Representations and Related Special Functions

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We studied various kind of special functions associated with aigebraic groups, Lie algebras, symmetric spaces, prehomogeneous spaces, Hecke algebras and so on, form the view point of representation theory. In the course of the research, many interesting results, some of which are related to number theory, or mathematical physics, are obtained.Kato studied Hecke algebras. First he showed how the "dual" of representations of Hecke algebras are given. Next, he constructed a new kind of R-matrix by using Hecke algebras, defined explicitly a quantum Knizhnik-Zamolodchikov equations, a system of q-difference equations, and showed certain relation between eigenfunctions of Macdonald's difference operators and our KZ-equations. Saito investigated the prehomogeneous vector spaces (PV) consisting of symmetric matrices from number theoretical view point. He determined the zeta function of these PV explicitly, and applied this result to the study of Siegel modular forms. Gyoja studied PV in connection with representations and D-modules. Particulary, he studies the relation between reducibility of generalized Verma modules and b-functions of PV.Matsuki investigated orbital decomposition of symmetric spaces and other similar spaces. This research is important in descriving representations geometrically. Nishiyama studied unitary representations of super Lie algebras, especially super version of the theory of dual pairs. Takasaki studies nonlinear integrable systems which appear in mathematical physics. He considered the symmetry hidden in these systems and investigated the relation between these symmetries and infinite dimensional Lie algebras, etc.