Harmonic analysis on weakly spherical homogeneous spaces and its application to number theory.

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We investigate Sp_2 as a spherical homogeneous Sp_2×(Sp_1)^2-space intensively. First we give explicit formulas of spherical functions explicitly as an application of the previous general formula which was given by the author. Then we determine the Hecke module structure of the space of Schwartz-Bruhat functions on Sp_2 and parametrization of all spherical functions, where the space of spherical functions attached to the same parameter has dimension 4.Extending the calculation of functional equations of spherical functions on Sp_2, we give a general method to obtain functional equations of spherical functions on certain spherical homogeneous p-adic spaces. This is based on the uniqueness of the relatively invariant distributions on a homogeneous space, and it guarantees the existance of functional equations attached to elements of Weyl group.As a joint research with F.Sato, we give an integral representation of the p-oart of the Siegel series, which is the main part of the Fourier coefficients of Siegel Eisenstein series. By using this representation, Siegel series is expresses as a finite sum of spherical functions on SO(n, n)/S(O(n)×O(n)) with respect to Siegel parabolic subgroup, which gives us a new proof of functional equations of Siegel series. The explicit calculation is now in progress.