1998 Fiscal Year Final Research Report Summary
Representation Theoretic and/or Geometric Research for Theta Series
Project/Area Number 
09640005

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Miyagi University of Education 
Principal Investigator 
TAKASE Koichi Miyagi University of Education, Faculty of Education Aossiciated Professor, 教育学部, 助教授 (60197093)

CoInvestigator(Kenkyūbuntansha) 
SHIRAI Susumu Miyagi University of Education, Faculty of Education Professor, 教育学部, 教授 (30115175)
URIU Hitoshi Miyagi University of Education, Faculty of Education Professor, 教育学部, 教授 (10139511)
ITAGAKI Yoshio Miyagi University of Education, Faculty of Education Professor, 教育学部, 教授 (30006431)
TAKEMOTO Hideo Miyagi University of Education, Faculty of Education Professor, 教育学部, 教授 (00004408)

Project Period (FY) 
1997 – 1998

Keywords  Number Theory / Automorphic Forms / Automorphic Representation / Theta Series / Weil Representation / Abelian Varieties / Jacobi Forms / PreHomogeneous Vector Space / 概均質ベクトル空間 
Research Abstract 
(1) The classical correspondence between Jacobi forms and Sigel cusp forms of halfintegral 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 halfintegral weights and Jacobi forms" (Trans. A.M.S.351 (1999), pp.735780). (2) Hermite polynomials of multivariables 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 Ktype vectors for K = U(n), we will get products of the classical (onevariable) Hermite polynomials which give a complete system of the solutions of the Schrodinger equation of ndimennsional harmonic ascillator. On the other hand, as Ktype 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 "Ktype vectors of Weil representat
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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. 2556) to the general semisimple 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 prehomogeneous 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 nonzero set of the Fourier tranform of the spherical trace function of the integrable representaiton and the Zariski open orbit of the prehomogeneous vector space. A part of the results will be published on the proceeding of the Autumn Workshop on Number Theory at Haluba (1998). Less

Research Products
(11 results)