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Representation Theoretic and/or Geometric Research for Theta Series

Research Project

Project/Area Number 09640005
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionMiyagi University of Education

Principal Investigator

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

Co-Investigator(Kenkyū-buntansha) 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
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 1998: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 1997: ¥800,000 (Direct Cost: ¥800,000)
KeywordsNumber 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
Research Abstract

(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

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report

Research Products

(17 results)

All Other

All Publications (17 results)

  • [Publications] Koichi Takase: "On the caserics with Harmonic polynomtals or Hermite polynomtals" Commentaris Math. Vniv. Sc. Panli. 46. 57-91 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Koichi Takase: "On Siegel Modulav Forms of Half-Incegral Weights and Jacebi Forms" The Transactions of A. M, S.351. 735-780 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 高瀬幸一: "T., Ibubiyama, M, Saito 「On Zeta Functions Associated to Sym. Mat.」の紹介" 整数論オータムワークショップ報告集. to appear.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Masaki and Tabemoto: "The numerical radius of infinite directed regular graph" Math. Japonica. 45. 337-343 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Hideo Takemoto: "On Saito's problem for the generations of von Neumam alg. by power partial isometries" Nihonkai Math. J.9. 97-104 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Hideo Takemoto: "A characteri zation of the power partially Isometric operators" Bull. Miyagi Univ. Edu.33to appear. (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Takase, K.: "On theta series with harmonic polynomials or Hermite polynomials" Comment.Math.Univ.St.Pauli. 45. 54-91 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Takase, K.: "On Siegel modular forms of half-integral weights anf Jacobi forms" Trans.A.M.S.351. 735-780 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Masaki and Takemoto: "The numerical radius of infinite directed regular graph" Math.Japonica. 45. 337-343 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Takemoto, H.: "On Saito's problem for the generation of von Neumann algebra by powerpartial isometries" Nihinkai Math.J.9. 97-104 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Takemoto, H.: "A characterization of the power partially isometric operators" Bull.Miyagi Univ.Edu.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Koichi Takase: "On Siegel Modular Forms of Half-Integral Weights and Jacobi-Forms" The Transactions of A.M.S.351. 735-780 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 高瀬幸一: "T.Ibukiyama,H.Saito「On Zeta Functions Associated to Sym.Matrices 2」の紹介" 整数論オータムワークショップ報告集(to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] Hideo Takemoto: "On a Saito's problem for the generations of von Neumarm algebras by powerpartial isometries" Nihonkai Mach.J.9. 97-104 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Hideo Takemoto: "A characterization of the power partially isometric operators" Bulletin of Miyagi Univ.of Edu.(to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] Koichi Takase: "On Theta Series wich Havmonic Polynomtals or Hermite Polynomla" Cmmentary Math. Univ. St. Pauli. 46. 57-91 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Koichi Takase: "On Siegel Modular Forms of Half.Integral weighes and Joeubl Form" to appean on the Tvansaction ofA.M.S.

    • Related Report
      1997 Annual Research Report

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Published: 1997-03-31   Modified: 2016-04-21  

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