Study on the dimension formula of automorphic forms associated with an integrable representation

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The primary purpose of this study is to generalize a result of Shintani (J.Fac.Sci.Univ. Tokyo 22 (1975), 25-65) to the case of the automorphic forms associated with the integrable representaions of general semi-simple real Lie groups. We have still a long way to go, and our study is now continued under the support of the grant-in-aid for scientific research (title : Study of discrete series representations with respect to the theory of automorphic forms, project number : 17540005). We will present here two of the main results that we have gained so far ;1)the center of the Lie algebra of nilpotent part of a parabolic subgroup of a semi-simple algebraic group defined over the field of rational numbers is a pre-homogenenous vector space, whose Zariski open orbit define a special property of the parabolic subgroup, which is named property (E). On the other hand a generic point of the pre-homogeneous vector space gives an nilpotent orbit. Now any nilpotent orbit gives a parabolic subgroup. We have established a bijection between the set of parabolic subgroup with property (E) and the set of the nilpotent orbits whose associated parabolic subgroup has the nilpotent part with central series of length 2.2)a compact Jordan triple system defines naturally a semi-simple real Lie group. This class of semi-simple real Lie groups contains all of the classical group which has discrete series representations. For such a group take a discrete series representation and consider its matrix coefficients. Our study concerns the property of the Fourier transformation of the function defined by restricting the matrix coefficient to the center of a parabolic subgroup, particularly on the Poisson summation formula for that functions. We have showed that if the Harish-Chandra parameter of the discrete series is far enoungh from the wall of Weyl chamber, then the Poisson summation formula is valid for the function concerned.