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Study on the dimension formula of automorphic forms associated with an integrable representation

Research Project

Project/Area Number 14540003
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, Dept. of Math., Professor, 教育学部, 教授 (60197093)

Co-Investigator(Kenkyū-buntansha) SATO Fumihiro  Rikkyo University, Dept. of Math., Professor, 理学部, 教授 (20120884)
NISHIYAMA Kyo  Kyoto University, Dept. of Math., Associated Professor, 理学部, 助教授 (70183085)
OCHIAI Hiroyuki  Nagoya University, Dept. of Math., Associated Professor, 多元数理化学研究科, 助教授 (90214163)
Project Period (FY) 2002 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2004: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2003: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2002: ¥1,000,000 (Direct Cost: ¥1,000,000)
Keywordsautomorphic form / integrable representation / dimension formula / Fourier transformation / Jordan triple system / nilpotent orbit / 巾零軌道 / ユニタリ表現 / 概均質ベクトル空間 / リー群 / 代数群 / 波面集合
Research Abstract

The primary purpose of this study is to generalize a result of Shintani (J.Fac.Sci.Univ. Tokyo 22 (1975), 25-65) to the case of the automorphic forms associated with the integrable representaions of general semi-simple real Lie groups. We have still a long way to go, and our study is now continued under the support of the grant-in-aid for scientific research (title : Study of discrete series representations with respect to the theory of automorphic forms, project number : 17540005). We will present here two of the main results that we have gained so far ;
1)the center of the Lie algebra of nilpotent part of a parabolic subgroup of a semi-simple algebraic group defined over the field of rational numbers is a pre-homogenenous vector space, whose Zariski open orbit define a special property of the parabolic subgroup, which is named property (E). On the other hand a generic point of the pre-homogeneous vector space gives an nilpotent orbit. Now any nilpotent orbit gives a parabolic subgroup. We have established a bijection between the set of parabolic subgroup with property (E) and the set of the nilpotent orbits whose associated parabolic subgroup has the nilpotent part with central series of length 2.
2)a compact Jordan triple system defines naturally a semi-simple real Lie group. This class of semi-simple real Lie groups contains all of the classical group which has discrete series representations. For such a group take a discrete series representation and consider its matrix coefficients. Our study concerns the property of the Fourier transformation of the function defined by restricting the matrix coefficient to the center of a parabolic subgroup, particularly on the Poisson summation formula for that functions. We have showed that if the Harish-Chandra parameter of the discrete series is far enoungh from the wall of Weyl chamber, then the Poisson summation formula is valid for the function concerned.

Report

(4 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (21 results)

All 2006 2005 2004 Other

All Journal Article (10 results) Publications (11 results)

  • [Journal Article] Tsuneo Arakawa and his works : Arakawa's works on Selberg zeta functions2006

    • Author(s)
      K.Takase
    • Journal Title

      Automorphic Forms and Zeta Functions

      Pages: 5-15

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Multiplicity one property and the decomposition of b-functions2006

    • Author(s)
      F.Sato, K.Sugiura
    • Journal Title

      International Journal of Mathematics 17

      Pages: 195-229

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Theta lifting of nilpotent orbits for symmetric pairs2006

    • Author(s)
      K.Nishiyama, H.Ochiai, C-B.Zhu
    • Journal Title

      Trans.Amer.Math.Soc. 358

      Pages: 2713-2734

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Arakawa's works on Selberg zeta functions2006

    • Author(s)
      K.Takase, Tsuneo Arakawa, his works
    • Journal Title

      Automorphic Forms and Zeta Functions

      Pages: 5-15

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Theta lifting of nilpotent orbits for symmetric pairs2006

    • Author(s)
      K.Nishiyama, H.ochiai, C-R.Zhu
    • Journal Title

      Transactions of American Mathematical Society 358

      Pages: 2713-2734

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Fourier coefficients of Eisenstein series of Gln, local densities of square matrices and subgroups of finite abelian groups2005

    • Author(s)
      F.Sato
    • Journal Title

      Commentarii Mathematici Universitatis Sancti Pauli 54

      Pages: 33-48

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Theta lifting of unitary lowest weight modules and their associated cycles2004

    • Author(s)
      K.Nishiyama, C-B.Zhu
    • Journal Title

      Duke Math.J. 125

      Pages: 415-465

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Classification of spherical nilpotent orbits for U(p,p)2004

    • Author(s)
      K.Nishiyama
    • Journal Title

      J.Math.Kyoto Univ. 44

      Pages: 203-215

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] A note on affine quotients and equivariant double fibrations

    • Author(s)
      K.Nishiyama
    • Journal Title

      Infinite dimensional hormonic analysis (to uppear)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Theta lifting of unitary lowest weight module and their associated cycles

    • Author(s)
      K.Nishiyama
    • Journal Title

      Duke Math.J. (To uppear)

    • Related Report
      2004 Annual Research Report
  • [Publications] Koichi Takase: "Is a pavabolic subgroup decermined by the open adjoint orbit on the center of its nilpotent vadical?"Commentaril Math Univ Sancti Punli. 152. 139-163 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Ochiai, M.Fujii: "An aljorithm for solving linear ovdinary differential equation of fuchsion type"Intendisciplinary Information Science. 9. 189-200 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Ochiai, M.Kaneko: "On coetticients of Yablonskii-Vorobev polynomials"J. Math.Sot.Japan. 55. 985-993 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Ochiai, K.Mimachi, M.Yoshida: "Intersection theory for twiseel cycles IV"Math.Nachr. 260. 67-77 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Nishiyama, Chen-bo Zhu: "Theta lifting of unitary lowest weight modules and their associnted cycles"Duke Math J. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Nishiyama: "Classification of sphevical nilpotenc orbits for U(p.p)"J.Math.Kyoto Univ. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] N.Kurokawa, E-M Muller Stuner, H.Ochiai, M.Wakayama: "Kronecker's Jugendtraum and ring sine functions"J. of Ramanujan Math. Soc.. 17. 211-220 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] N.Kurokawa, H.Ochiai, M.Wakayama: "Multiple trigonometry and zeta functions"J. of Ramanujan Math. Soc.. 17. 101-113 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Ochiai, T.Oshima: "Commuting differential operators of type B_2"Funkaialaij Ekvacioj. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Kaneko, H.Ochiai: "On coefficients of Yablonski-Vorob'ev ploynomials"J. Math. Soc. Japan. (in press).

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Minachi, H.Ochiai, M.Yoshida: "Intersection theory for twistel cycles IV"Math. Nachr. (in press).

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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