Studies on discrete series representations and the theory of automorphic forms

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Shintani's studies on the dimension formula of the space of Siegel cusp forms (J.Fac.Sci.Univ.Tokyo, 22(1975),25-65) are quite impressive, and we must understand the general mechanism working behind the beautiful formula of Shinta's, and that is the fundamental motivation of our studies on the spherical function, and Fourier transform of it, of discrete series represe tations of semi-simple real Lie group. The basic problems are 1) estimate of matrix coefficients in order to justify the Poisson summation formula, 2) the determination of the non-zero set of the Fourier transform of the spherical functions. In the case of holomorphic discrete series is the most easy case tfor ten explicit treatment, and that case suggests the general conjectural statements on the non-zero set of the Fourier transform of spherical functions. That statement relates the non-zero set with the connected component of the real orbits of the open orbit of certain pre-homogeneous vector space arising from the par … More abolic subgroup with which we are considering the Fourier transform. Such kind of relation between the theory of unitary representation and the theory of pre-homogeneous vector space is uite new, and give us several interesting problems. We are now at the starting point of the study, and we have several strategies by which we may finally establish the relation between two quite different branch of Mathematics. One of our strategy is to use Jordan triple system to describe the explicit construction of discrete series representaions. The other strategy is to study the decomposition of the restriction of taholomorphic discrete series to certain subgroup. Jordan triple system is also quite useful in the second strategy, because the subgroup to which the holomorphic discrete series is restricted must be carefully chosen, and a good choice of the subgroup is given by the language of Jordan triple system. So we must study intensively the groups coming from Jordan triple system, and also the relation between these groups. These problms will be the theme of the next project supported by this grant in aide. Less