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

这项研究的主要目的是推广Shintani(J.Fac.Science Univ.)的一个结果。东京22(1975)，25-65)推广到一般半单实李群的可积表示的自同构形式的情形。我们还有很长的路要走，我们的研究现在是在科学研究助学金的支持下继续进行的(标题：关于自同构形式理论的离散级数表示的研究，项目号：17540005)。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拟齐次向量空间，它的Zariski开轨定义了抛物子群的一个特殊性质，称为性质(E)。另一方面，准齐次向量空间的一个通用点给出了一个幂零轨道。现在，任何幂零轨道都有一个抛物线子群。我们建立了具有性质(E)的抛物子群与其相关抛物子群具有中心级数为幂零部分的幂零轨道集之间的双射。2)紧Jordan三系自然地定义了半单实李群。这类半单实李群包含了具有离散级数表示的所有经典群。对于这类群，采用离散级数表示法，并考虑其矩阵系数。我们研究了将矩阵系数限制在抛物子群的中心所定义的函数的傅里叶变换的性质，特别是关于该函数的Poisson求和公式。我们已经证明，如果离散级数的Harish-Chandra参数远离Weyl腔的壁面，则泊松求和公式对有关函数是有效的。

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.