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Representation-theoretic study of spherical functions arising from number theory

Research Project

Project/Area Number 10640020
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionKYOTO UNIVERSITY

Principal Investigator

KATO Shinichi  Kyoto Univ., Integrated Human Studies, Ass. Professor, 総合人間学部, 助教授 (90114438)

Co-Investigator(Kenkyū-buntansha) NISHIYAMA Kyo  Kyoto Univ., Integrated Human Studies, Ass. Professor, 総合人間学部, 助教授 (70183085)
MATSUKI Toshihiko  Kyoto Univ., Integrated Human Studies, Ass. Professor, 総合人間学部, 助教授 (20157283)
SAITO Hiroshi  Kyoto Univ., Graduate School of Human and Environmental studies, Professor, 大学院・人間・環境学研究科, 教授 (20025464)
YAMAUCHI Masatoshi  Kyoto Univ., Integrated Human Studies, Professor, 総合人間学部, 教授 (30022651)
TAKASAKI Kanehisa  Kyoto Univ., Integrated Human Studies, Ass. Professor, 総合人間学部, 助教授 (40171433)
吉野 雄二  京都大学, 総合人間学部, 助教授 (00135302)
Project Period (FY) 1998 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 1999: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 1998: ¥1,700,000 (Direct Cost: ¥1,700,000)
KeywordsRepresentation theory / Algebraic group / Spherical function / Special function / Symmetric space / Spherical homogeneous space / Orbit / 特殊関係 / リー群 / ヘッケ環 / ワイル群 / 軌道分解
Research Abstract

Special functions (spherical functions) on algebraic groups play an important role in number theory, especially in the study of automorphic forms. In most cases, these spherical functions are related to spherical homogeneous spaces, such as symmetric spaces. In this research project, Kato (head investigator) studied spherical functions on spherical homogeneous spaces of reductive groups over non-archimedean local fields from a representation theoretic view point. The purpose of this research is two-fold : (1) To understand special functions such as zonal spherical functions or Whittaker functions in a uniform manner from the view point as above. (2) To obtain properties of these functions, including the uniqueness and explicit formulas, for important cases which arise in number theory. As for (1), we studied an orbit decomposition of spherical homogeneous spaces first. Then applying this, we obtained a general formula for spherical functions (at least in the case of symmetric spaces) t … More ogether with a method to compute the coefficients in this formula explicitly. As for (2), we got the uniqueness and an explicit formula for e.g. a symmetric space corresponding to quadratic base change by using the above mentioned method. This research is still under way. Other investigators obtained several results related to representation theory and spherical homogeneous spaces as follows. Saito studied zeta functions of prehomogeneous vector spaces, which is closely related to (spherical functions of) spherical homogeneous spaces, and showed the convergence and explicit formulas (in terms of local orbital zeta functions) in general. Matsuki investigated Weyl groups and Jordan decompositions arising from symmetric spaces. Nishiyama studied multiplicity free actions, which is a characteristic property of spherical homogeneous spaces, and the relation between theta correspondences and nilpotent orbits. Other investigators, Takasaki, Yamauchi et al. carried out researches on mathematical physics, automorphic forms and so on. Less

Report

(3 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • Research Products

    (24 results)

All Other

All Publications (24 results)

  • [Publications] 加藤信一: "Whittaker-Shintani Functions for Orthogonal Groups"(未定). (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 斎藤裕: "Convergence of the zeta functions of prehomogeneous vector spaces."(未定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 斎藤裕: "On the zeta functions associated to Symmetric matrices II : Functional equations and special values."(未定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 斎藤裕: "Explicit form of the zeta functions of prehomogeneous vector spaces"Math.Ann.. (未定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 西山 享: "Invariants for Representations of Weyl Groups,Two-sided Cells,and Modular Representations of Iwahori-Hecke Algebrs"Adr.Studies in Pure Math.. (未定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 西山 享: "Kawanaka invariants for representations of Weyl groups"J.Alg.. (未定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Shinichi Kato: "Whittaker-Shintani Functions for Orthogonal Groups."(to appear). (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiroshi Saito: "Convergence of the zeta functions of prehomogeneous vector spaces."(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiroshi Saito: "On the zeta functions associated to symmetric matrices II : Functional equations and special values."(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiroshi Saito: "Eyplicit form of the zeta functions of prehomogeneous vector spaces."Mass. Ann. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kyo Nishiyama: "Invariants for Representations of Neyl Groups, Two-Sided Cells, and Modular Representations of Iwahori-Hecke Algebras."Adv. Studies in Pure Matr.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Kyo Nishiyama: "Kawanaka invariants for representations of weyl groups."J. Alg. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 加藤 信一: "Whittaker-Shintani Functions for Orthogonal Groups"未定. (未定). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 斉藤 裕: "Convergence of the zeta functions of prehomogeneous vector spaces"未定. (未定).

    • Related Report
      1999 Annual Research Report
  • [Publications] 斉藤 裕: "On the zeta functions associated to Symmetric matrices II : Functional equations and special values"未定. (未定).

    • Related Report
      1999 Annual Research Report
  • [Publications] 斉藤 裕: "Explicit form of the Zeta functions of prehomogeneous vector spaces"Math.Ann.. (未定).

    • Related Report
      1999 Annual Research Report
  • [Publications] 西山 享: "Invariants for Representations of Weyl Groups,Two-sided Cells, and Modular Representations of Iwahori-Hecke Algebras"Adv.Studies in Pure Math. (未定).

    • Related Report
      1999 Annual Research Report
  • [Publications] 西山 享: "Kawanaka invariants for representations of weyl groups"J.Alg.. (未定).

    • Related Report
      1999 Annual Research Report
  • [Publications] 斎藤 裕: "Explicit form of zeta functions of prehomogeneous vector spaccs" Math.Ann.(1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 松木敏彦: "Classification of two involutions on Compact semisimple Lie groups and root systems."

    • Related Report
      1998 Annual Research Report
  • [Publications] 西山 享: "Invariants for representations of weyl groups and two-sided cells" J.Math.Soc.Japan. 51巻. 1-34 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 西山 享: "Schur duality for Cartan type Lie algebra 〓 w-n〓" Journal of Lie Theory. 9巻. 234-248 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 吉野雄二: "Auslander's work on Cohen-Macaulay modules and recent developement" Canadian Math.Soc.Conference Proceedings. 23巻. 179-198 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 吉野雄二: "Remarks on depth formula;grade ineqvality and Auslander conjecture" Communications in Algebra. 26巻. 3793-3806 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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