• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Geometric invariants of representations and the Whittaker models

Research Project

Project/Area Number 15540183
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionAoyama Gakuin University

Principal Investigator

TANIGUCHI Kenji  Aoyama Gakuin University, College of Science and Engineering, Associate Professor, 理工学部, 助教授 (20306492)

Co-Investigator(Kenkyū-buntansha) KOIKE Kazuhiko  Aoyama Gakuin University, College of Science and Engineering, Professor, 理工学部, 教授 (70146306)
ITO Masahiko  Aoyama Gakuin University, College of Science and Engineering, Associate Professor, 理工学部, 助教授 (30348461)
YANO Kouichi  Aoyama Gakuin University, College of Science and Engineering, Professor, 理工学部, 教授 (60114691)
KIMURA Isamu  Aoyama Gakuin University, College of Science and Engineering, Assistant, 理工学部, 助手 (40082820)
KAWAMURA Tomomi  Aoyama Gakuin University, College of Science and Engineering, Assistant, 理工学部, 助手 (40348462)
伊原 信一郎  青山学院大学, 理工学部, 教授 (30012347)
Project Period (FY) 2003 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2005: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2004: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2003: ¥1,200,000 (Direct Cost: ¥1,200,000)
Keywordscommuting differential operators / Calogero models / Jackson integrals / ワイル群の不変式
Research Abstract

During the research period, we obtain many results about the symmetry of commuting differential operators, which is relevant to the main subjects of this research, namely Whittaker models and invariants of representations.
The completely integrable system called Calogero-Moser-Sutherland model (which we abbreviate as CMS model in the following of this abstract) is closely related to root systems and Weyl groups. The Hamiltonian operator of this model is invariant under the action of a Weyl group and the potential function of it is expressed by means of the corresponding root system. Note that the potential function of it possesses inverse square singularities along the walls of a Weyl chamber. We study the commutants of a Hamiltonian operator whose potential function possesses inverse square singularities along some hyperplanes passing through the origin. It is shown that the Weyl group symmetry of the potential function and the commutants naturally results from such singularities and the generic nature of the coupling constants. Moreover, we have obtained the following results :
(1)If this symmetry is of the classical type, the potential function must be one of the known ones.
(2)In the rank two cases, the potential function must satisfy some linear relations.
(3)In the rank two cases, the order of the commutant is at least the number of singular lines.
(4)Deformation of A_2 type CMS model is possible if and only if two of the coupling constants are 1.
(5)Deformation of B_2 type CMS model is possible even if no coupling constant is 1.
(6)A new deformation of the B_2 type CMS model is constructed.
Besides the above investigation, we study an elementary method of constructing F_4 type invariant polynomials.

Report

(4 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (28 results)

All 2005 2004 2003 Other

All Journal Article (25 results) Publications (3 results)

  • [Journal Article] On the structure of Jackson integrals of BC_n type and holonomic q-difference equations2005

    • Author(s)
      Kazuhiko Aomoto, Masahiko Ito
    • Journal Title

      Japan Academy.Proceedings.Series A.Mathematical Sciences 81 No.9

      Pages: 145-150

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Links and Gordian numbers associated with certain generic immersions of circles2005

    • Author(s)
      Tomomi Kawamura
    • Journal Title

      Pacific Journal of Mathematics 220 No.2

      Pages: 341-357

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the structure of Jackson integrals of BC_n type and holonomic q-difference equations2005

    • Author(s)
      Kazuhiko Aomoto, Masahiko ITO
    • Journal Title

      Japan Academy. Proceedings. Series A. Mathematical Sciences 81, no.9

      Pages: 145-150

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Links and Gordian numbers associated with certain generic immersions of circles2005

    • Author(s)
      Tomomi KAWAMURA
    • Journal Title

      Pacific Journal of Mathematics 220, no.2

      Pages: 341-357

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the structure of Jackson integrals of BC_n type and holonomic q-difference equations2005

    • Author(s)
      Kazuhiko Aomoto, Masahiko Ito
    • Journal Title

      Proceedings of the Japan Academy 81(A) No.9

      Pages: 145-150

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On the symmetry of commuting differential operators with singularities along hyperplanes2004

    • Author(s)
      Kenji Taniguchi
    • Journal Title

      International Mathematics Research Notices 36

      Pages: 1845-1867

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Links associated with beneric immersions of graphs2004

    • Author(s)
      Tomomi Kawamura
    • Journal Title

      Algebraic and Geometric Topology 4

      Pages: 571-594

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the symmetry of commuting differential operators with singularities along hyperplanes2004

    • Author(s)
      Kenji TANIGUCHI
    • Journal Title

      International Mathematics Research Notices 36

      Pages: 1845-1867

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Links associated with generic immersions of graphs2004

    • Author(s)
      Tomomi KAWAMURA
    • Journal Title

      Algebraic and Geometric Topology 4

      Pages: 571-594

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the Symmetry of Commuting Differential Operators with Singularities along Hyperplanes2004

    • Author(s)
      Kenji Taniguchi
    • Journal Title

      International Mathematics Research Notices 36

      Pages: 1845-1867

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Symmetry classification for Jackson integrals associated with The root system BC_n2003

    • Author(s)
      Masahiko Ito
    • Journal Title

      Compositio Mathematica 136

      Pages: 209-216

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Convergence and asymptotic behavior of Jackson integrals associated with irreducible reduced root systems2003

    • Author(s)
      Masahiko Ito
    • Journal Title

      Journal of Approximation Theory 124

      Pages: 154-180

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Symmetry classification for Jackson integrals associated with the root system BC_n2003

    • Author(s)
      Masahiko ITO
    • Journal Title

      Compositio Mathematica 136

      Pages: 209-216

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Convergence and asymptotic behavior of Jackson integrals associated with irreducible reduced root systems2003

    • Author(s)
      Masahiko ITO
    • Journal Title

      Journal of Approximation Theory 124

      Pages: 154-180

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] F_4型Weyl 群不変式の初等的構成法

    • Author(s)
      谷口 健二
    • Journal Title

      数理解析研究所講究録 (掲載予定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Askey-Willson integrals associated with root systems

    • Author(s)
      Masahiko Ito
    • Journal Title

      The Ramanujan Journal (To appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Another proof of Gustafson's C_n-type summation formula via 'elementary' symmetric polynomials

    • Author(s)
      Masahiko Ito
    • Journal Title

      Publications of Research Institute for Mathematical Sciences Kyoto University (To appear)

    • NAID

      110004626193

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] q-difference shift for a BC_n-type Jackson integral arising from 'elementary' symmetric polynomials

    • Author(s)
      Masahiko Ito
    • Journal Title

      Advances in Mathematics (To appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] An Elementary Method of Constructing W(F_4) Invariant Polynomials (in Japanese)

    • Author(s)
      Kenji Taniguchi
    • Journal Title

      Surikaisekikenkyusho Kokyuroku (To appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Askey-Willson integrals associated with root systems

    • Author(s)
      Masahiko ITO
    • Journal Title

      The Ramanujan Journal (To appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Another proof of Gustafson's C_n-type summation formula via 'elementary' symmetric polynomials

    • Author(s)
      Masahiko ITO
    • Journal Title

      Publications of Research Institute for Mathematical Sciences, Kyoto University (To appear)

    • NAID

      110004626193

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] q-difference shift for a BC_n-type Jackson integral arising from 'elementary' symmetric polynomials

    • Author(s)
      Masahiko ITO
    • Journal Title

      Advances in Mathematics (To appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] F_4型ワイル群不変式の初等的構成法

    • Author(s)
      谷口 健二
    • Journal Title

      数理解析研究所講究録 (掲載決定)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] q-difference shift for a BC_n-type Jackson integral arising from 'elementary' symmetric polynomials

    • Author(s)
      Masahiko Ito
    • Journal Title

      Advances in Mathematics (To appear)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Askey-Wilson integrals associated with root systems

    • Author(s)
      Masahiko ITO
    • Journal Title

      The Ramanujan Journal (To appear)(印刷中)

    • Related Report
      2004 Annual Research Report
  • [Publications] Kenji Taniguchi: "On the symmetry of commuting differential operators with singularities along hyperplanes"International Mathematics Research Notices. To appear.

    • Related Report
      2003 Annual Research Report
  • [Publications] Masahiko Ito: "Symmetry classification for Jackson integrals associated with the root system BCn"Compositio Mathematica. 136. 209-216 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Masahiko Ito: "Convergence and asymptotic behavior of Jackson integrals associated with irreducible reduced root systems"Journal of Approximation Theory. 124. 154-180 (2003)

    • Related Report
      2003 Annual Research Report

URL: 

Published: 2003-04-01   Modified: 2016-04-21  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi