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）从表示理论的角度研究了雅各比形式和半融合权重的西格尔尖头形式之间的经典对应关系。基本工具是很好的表示。结果在“关于半综合体重和雅各比形式的西格尔模块化形式”（trans。A.M.S.351（1999），第735-780页）。（2）通过两种方式，通过两种方式来定义多种方式，通过详细研究（限制）（YIL的详细研究）（ne n decomptition *n of）（2）（2）（2）（2）（2）（2）（2）（u（n），u（1））。作为K = U（n）的K型向量，我们将获得经典（单变量）Hermite多项式的产品，这些产品为N维谐波验证器的Schrodinger方程提供了完整的溶液系统。 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 separate 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一般的临界组。作为申请； 1）表示Riemann theta系列的转换公式的代表理论证明，以及2）与谐波多项式相关的theta系列的转换公式。结果将发表在“有关Weil及其应用的广义泊松总和公式的扩展”的论文上。连接到可集成代表的代表是由抛物线类型的ZETA函数的特殊值来自抛物线类型的Zeta函数，从定义的最大抛物线亚组来给出。我们还发现，似乎存在有趣的关系，即在均匀矢量空间的可集成代表的球形痕量函数的傅立叶变换之间存在有趣的关系。结果的一部分将在Haluba（1998）的秋季理论研讨会上发表。较少的