Representation Theoretic and/or Geometric Research for Theta Series

(1)从表示理论的角度研究了半积分权值的Jacobi形式与Sigel尖形的经典对应关系。最基本的工具是良好的表现。结果发表在“关于半积分权值的Siegel模形式和Jacobi形式”上。A.M.S.351(1999), pp.735 - 780)。(2)通过对限定于对偶对(U（n)， U(1)）的Sp（n, *）的Weil表示的不可约分解的详细研究，用两种方法定义了多变量Hermite多项式。作为K = U(n)的K型向量，我们将得到经典（单变量）埃尔米特多项式的积，它给出了n维谐振子薛定谔方程解的完整系统。另一方面，作为K = U(1)的K型向量，我们将得到另一个不含分离变量的薛定谔方程解的完备体系，结果将发表在“Weil的K型向量表示…More ion和广义Hermite多项式”的论文上。(3)将仅对θ群有效的Weil广义泊松求和公式推广到一般的旁模群。作为应用程序；1)黎曼级数的变换公式的表示理论证明，2)与调和多项式的积分二次型相关的变换公式。研究结果将发表在“关于Weil的广义泊松求和公式的推广及其应用”的论文上。(4)采用T.Shintani （J.Fac）的方法。科学。在*上的一般半简单代数群，并发现附属于可积表示的自同构形式空间的一部分维数公式是由由*上定义的极大抛物子群派生的抛物型预齐次向量空间的ζ函数的一个特殊值给出的。我们还发现在可积表示的球面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着一种有趣的关系。部分研究结果将发表在1998年在Haluba举行的秋季数论研讨会上。少

(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