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Representation Theoretic and/or Geometric Research for Theta Series

Theta 级数的表示理论和/或几何研究

基本信息

批准号：
09640005
负责人：
TAKASE Koichi
金额：
$ 0.9万
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：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640005/
关键词：
Number Theory Automorphic Forms Automorphic Representation Theta Series Weil Representation Abelian Varieties Jacobi Forms Pre-Homogeneous Vector Space 概均質ベクトル空間 Weil表現データ級数ブエイユ表現ヤゴビ形式

项目摘要

(1) The classical correspondence between Jacobi forms and Sigel cusp forms of half-integral weights is studied from representation theoretic point of view. The basic tool is Well representation. The results are published on "On Siegel modular forms of half-integral weights and Jacobi forms" (Trans. A.M.S.351 (1999), pp.735-780).(2) Hermite polynomials of multi-variables are defined in two ways through a detailed study of the irreducible decomposition of the Weil representation of Sp(n, *) restricted to the dual pair (U(n), U(1)). As K-type vectors for K = U(n), we will get products of the classical (one-variable) Hermite polynomials which give a complete system of the solutions of the Schrodinger equation of n-dimennsional harmonic ascillator. On the other hand, as K-type vectors for K = U(1), we will get another complete system of the solution of the Schrodinger equation which is not of separated variables, The results will be published on the paper "K-type vectors of Weil representat … More ion and generalized Hermite polynomials".(3)Weil's generalized Poisson summation formula, which is valid only for theta group, is extended to the general paramodular groups. As applications ; 1) a representation theoretic proof of the transformation formula of Riemann's theta series, and 2) the transformation formula of theta series associated with a integral quadratic form with harmonic polynomials. The results will be published on the paper "On an extension of generalized Poisson summation formuls of Weil and its applications".(4) We applied the method of T.Shintani (J.Fac. Sci. Univ. Tokyo 22 (1975), pp. 25-56) to the general semi-simple algebraic group over *, and found that a part of the dimmension formula of the space of the automorphic forms attached to an integrable representaton is given by a special values of the zeta functions of pre-homogeneous vector space of parabolic type srising from a maximal parabolic subgroup defined over *. Also we found that there seems to exist an interesting relationship between the non-zero set of the Fourier tranform of the spherical trace function of the integrable representaiton and the Zariski open orbit of the pre-homogeneous vector space. A part of the results will be published on the proceeding of the Autumn Workshop on Number Theory at Haluba (1998). Less

(1)从表示论的角度研究了半积分权的Jacobi形式和Sigel尖点形式之间的经典对应关系。基本工具是井表示。结果发表在“On Siegel modular forms of half-integral weights and Jacobi forms”上（Trans. A.M.S.351 (1999), pp.735-780）。 (2) 通过详细研究限制于对偶对的 Sp(n, *) Weil 表示的不可约分解，以两种方式定义多变量的 Hermite 多项式（U（n），U（1））。作为 K = U(n) 的 K 型向量，我们将得到经典（一变量）Hermite 多项式的乘积，它给出了 n 维谐振子薛定谔方程的完整解系。另一方面，作为K = U(1)的K型向量，我们将得到另一个完整的非分离变量薛定谔方程解的系统，其结果将发表在论文“K-type向量的Weilrepresentat… More ion and Generalized Hermite polynomials”上。 (3)推广仅对theta群有效的Weil的广义泊松求和公式到一般的副模块群。作为应用程序； 1）黎曼theta级数变换公式的表示理论证明，2）与调和多项式积分二次形式相关的theta级数变换公式。其结果将发表在论文《On an extension ofgeneralized Poisson summation Formulas of Weil and its applications》上。 (4)我们将T.Shintani的方法(J.Fac.Sci.Univ.Tokyo 22 (1975), pp.25-56)应用到*上的一般半简单代数群上，发现附有的自同构空间的维数公式的一部分可积表示由抛物型预齐次向量空间的 zeta 函数的特殊值给出，该向量空间源自 * 上定义的最大抛物型子群。我们还发现，可积表示的球迹函数的傅里叶变换的非零集与预齐次向量空间的 Zariski 开轨道之间似乎存在有趣的关系。部分结果将在 Haluba 秋季数论研讨会（1998 年）的会议记录上发表。较少的