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

本文的主要目的是把Shintani（J.Fac.Sci. Univ. Tokyo 22（1975），25-65）的一个结果推广到一般半单真实的李群的可积表示的自守形式的情形。我们还有很长的路要走，我们的研究现在正在科学研究补助金的支持下继续进行（标题：关于自守形式理论的离散级数表示的研究，项目编号：17540005）。本文给出了迄今为止我们所得到的两个主要结果：1）定义在有理数域上的半单代数群的抛物子群的幂零部分的李代数的中心是一个准齐次向量空间，其Zebrski开轨道定义了抛物子群的一个特殊性质，称为性质（E）.另一方面，预齐次向量空间的一般点给出幂零轨道。现在任何幂零轨道都给出一个抛物子群。我们建立了具有性质（E）的抛物子群的集合与其对应的抛物子群的幂零部分的中心列长度为2的幂零轨道的集合之间的一个双射。2）紧Jordan三系自然地定义了半单真实的李群。这类半单真实的李群包含了所有具有离散级数表示的经典群。对于这样一个群，取一个离散级数表示并考虑它的矩阵系数。我们的研究涉及的性质的傅立叶变换的功能，通过限制矩阵系数的中心的抛物子群，特别是对泊松求和公式的功能。证明了当离散级数的Harish-Chandra参数离Weyl室壁足够远时，Poisson求和公式对有关函数是成立的。

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.