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

与可积表示相关的自守形式的维数公式研究

基本信息

批准号：
14540003
负责人：
TAKASE Koichi
金额：
$ 1.79万
依托单位：
Miyagi University of Education
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

项目摘要

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.

本研究的主要目的是将 Shintani (J.Fac.Sci.Univ. Tokyo 22 (1975), 25-65) 的结果推广到与一般半简单实数的可积表示相关的自同构形式的情况谎言团体。我们还有很长的路要走，目前我们的研究正在科研资助金的支持下继续进行（题目：自守形式理论的离散级数表示研究，项目编号：17540005）。我们将在这里介绍到目前为止我们获得的两个主要结果；1）在有理数域上定义的半简单代数群的抛物线子群的幂零部分的李代数的中心是预齐次的向量空间，其 Zariski 开轨道定义了抛物线子群的一个特殊性质，称为性质 (E)。另一方面，预齐次向量空间的泛点给出了幂零轨道。现在任何幂零轨道都会给出一个抛物线子群。我们已经在具有性质 (E) 的抛物线子群集合与幂零轨道集合之间建立了双射，其关联的抛物线子群具有中心级数长度为 2.2 的幂零部分。紧致乔丹三重系统自然地定义了半简单实数谎言组。这类半单实李群包含所有具有离散级数表示的经典群。对于这样的群，采用离散级数表示并考虑其矩阵系数。我们的研究涉及通过将矩阵系数限制在抛物线子群的中心来定义的函数的傅里叶变换的性质，特别是该函数的泊松求和公式。我们已经证明，如果离散级数的 Harish-Chandra 参数距离 Weyl 室壁足够远，则泊松求和公式对于相关函数是有效的。