Study on the spherical function of discrete series representations from the point of view of the theory of automorphic forms

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Studies on an explicit expression of the spherical function associated with the minimal K-type of a discrete series representation of a semi-simple real Lie group and the non-zero set of its Fourier transform in terms of the prehomogeneous vector space of parabolic type, and on the parallel problems for supercuspidal representations of reductive p-adic Lie groups. The theory of the Jordan triple system is used as a tool to study reductive real Lie group explicitly