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A Study on the zeta functions attached with the irreducible representations of finite reductive groups

Research Project

Project/Area Number 14540042
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionSophia University

Principal Investigator

SHINODA Ken-ichi  Sophia University, Faculty of Science and Technology, Professor, 理工学部, 教授 (20053712)

Co-Investigator(Kenkyū-buntansha) YOKONUMA Takeo  Sophia University, Faculty of Science and Technology, Professor, 理工学部, 教授 (00053645)
WADA Hideo  Sophia University, Faculty of Science and Technology, Professor, 理工学部, 教授 (10053662)
NAKASHIMA Toshiki  Sophia University, Faculty of Science and Technology, Professor, 理工学部, 教授 (60243193)
GOMI Yasushi  Sophia University, Faculty of Science and Technology, Assistant, 理工学部, 助手 (50276515)
TSUZUKI Masao  Sophia University, Faculty of Science and Technology, Assistant, 理工学部, 助手 (80296946)
角皆 宏  上智大学, 理工学部, 講師 (20267412)
後藤 聡史  上智大学, 理工学部, 助手 (00286759)
Project Period (FY) 2002 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2004: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2003: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2002: ¥1,200,000 (Direct Cost: ¥1,200,000)
Keywordsfinite reductive groups / irreducible representation / zeta function / general linear group / Gelfand-Graev representation / Gauss sum / Kloosterman sum / 有限群の表現論
Research Abstract

1. Zeta functions and functional equations associated with them for representations of a finite group G were first discussed by T.A.Springer (1971) and I.G.Macdonald (1985). We showed first that these functional equations also hold for the irreducible representations of Hecke algebra H which is an endomorphism algebra of an induced representation with multiplicity free.
Then we studied the explicit examples and applications ((1)(2) joint work with C.W.Curtis). The results are as follows :
(1) It can be shown that if G is a finite reductive group and if His an endomorphism algebra of Gelfand-Graev representation, the epsilon factor which appears in the functional equation is exactly the Gauss sum studied by Saito-Shinoda. Using this fact we obtained an explicit expression of the Fourier transform of e, which is the identity of Hin the case of general linear groups.
(2) We showed that in the case of GL(n,q), values of irreducible representations of H on a standard basis element, which corresponds to a Coxeter element, become generalized Kloosterman sums. This implies that there should exist a close relation a between Davenport-Hasse type equations of Kloosterman sums and the norm maps of Hecke algebras.
(3) In case of GL(4,q) we obtained almost all values of irreducibles representations of H on standard basis elements. They can be called as Kloosterman sums of higher degree.
2. We also studied this project from relating fields.
(1) Gomi and Shinoda generalized a result of J.McKay (1999) on coinvariant algebras of finite linear groups (joint with I.Nakamura).
(2) Yokonuma studied discrete sets and associated dynamical systems in view of symmetry with a non-commutative setting. Nakashima and Koga studied from the view of quantum groups ; particularly Nakashima studied geometric crystals on Schubert varieties.
(3) Wada, Tsunogai and Tsuzuki respectively studied this project in view from the number theory.

Report

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

    (36 results)

All 2005 2004 2003 2002 Other

All Journal Article (24 results) Publications (12 results)

  • [Journal Article] Geometric crystals on Schubert varieties2005

    • Author(s)
      T.Nakashima
    • Journal Title

      J. Geometry and Physics 53

      Pages: 197-225

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Geometric crystals on Schubert varieties2005

    • Author(s)
      T.Nakashima
    • Journal Title

      J.Geometry and Physics 53

      Pages: 197-225

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Zeta functions and functional equations associated with the components of the Gelfand-Graev representations of finite group2004

    • Author(s)
      C.W.Curtis
    • Journal Title

      Advanced Studies in Pure Mathematics 40

      Pages: 121-139

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Coinvariant algebras of finite subgroups of SL (3, C)2004

    • Author(s)
      Y.Gomi
    • Journal Title

      Canadian J. Math. 56-3

      Pages: 495-528

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Extremal profectors of q-boson algebras2004

    • Author(s)
      T.Nakashima
    • Journal Title

      Comm. in Math. Physics 244

      Pages: 285-296

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Zeta functions and functional equations associated with the components of the Gelfand-Graev representations of a finite group2004

    • Author(s)
      C.W.Crutis
    • Journal Title

      Advanced Studies in Pure Mathematics 40

      Pages: 121-139

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Coinvariant algebras of finite subgroups of SL(3,C)2004

    • Author(s)
      Y.Gomi
    • Journal Title

      Canadian J.Math. 56-3

      Pages: 495-528

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Extremal projectors of q-boson algebras2004

    • Author(s)
      T.Nakashima
    • Journal Title

      Comm.in Math.Physics 244

      Pages: 285-296

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Zeta functions and functional equations associated with the components of the Gelfand-Graev representations of a finite reductive group2004

    • Author(s)
      C.W.Curtis
    • Journal Title

      Advanced Studies in Pure Math. 40

      Pages: 121-139

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Coinvariant algebras of finite subgroups of SL(3,C)2004

    • Author(s)
      Y.Gomi
    • Journal Title

      Canadian J.of Math. 56

      Pages: 495-528

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Certain Poincare series and period-integrals of Eisenstein series on R-rank one classical groups2004

    • Author(s)
      M.Tsuzuki
    • Journal Title

      Commentarii Math.Univ.Sancti Pauli 53

      Pages: 37-75

    • NAID

      110007689259

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the Diophantine equation x(x+1)...(x+n)=y^2, (17【less than or equal】n=odd【greater than or equal】27)2003

    • Author(s)
      H.Wada
    • Journal Title

      Proc. Japan Academy 80

      Pages: 99-100

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The stable derivation algebras for higher genera2003

    • Author(s)
      H.Tsunogai
    • Journal Title

      Israel J. of Math. 136

      Pages: 221-250

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Automorphic Green functions associated with the secondary spherical functions2003

    • Author(s)
      M.Tsuzuki
    • Journal Title

      Publ. of RIMS 39

      Pages: 451-533

    • NAID

      130003585543

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the Diophantine equation x(x+1)…(x+n)=y^2, (17【less than or equal】n=odd【less than or equal】27)2003

    • Author(s)
      H.Wada
    • Journal Title

      Proc.Japan Academy 80

      Pages: 99-100

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The stable derivation algebras for higher genera2003

    • Author(s)
      H.Tsunogai
    • Journal Title

      Israel J.of Math. 136

      Pages: 221-250

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Automorphic Green functions associated with the secondary spherical functions2003

    • Author(s)
      M.Tsuzuki
    • Journal Title

      Publ.of RIMS 39

      Pages: 451-533

    • NAID

      130003585543

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The resolvent trace formula for rank one Lie groups2002

    • Author(s)
      M.Tsuzuki
    • Journal Title

      Asian J. Math. 6

      Pages: 227-252

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The resolvent trace formula for rank one Lie groups2002

    • Author(s)
      M.Tsuzuki
    • Journal Title

      Asian J.Math. 6

      Pages: 227-252

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Representations of the Hecke algebra for GL_4 (q)

    • Author(s)
      K.Shinoda
    • Journal Title

      J. of Algebra and Its Applications To appear

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Discrete sets and associated dynamical systems in a non-commutative setting

    • Author(s)
      T.Yokonuma
    • Journal Title

      Canadian Mathematical Bulletin To appear

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Representations of the Hecke algebra for GL_4(q)

    • Author(s)
      K.Shinoda
    • Journal Title

      J.of Algebra and Its Applications (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Discrete sets and associated dynamical systems in a non-commutative setting

    • Author(s)
      T.Yokonuma
    • Journal Title

      Canadian Mathematical Bulletin (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Discrete sets and associated dynamical systems in a non-commutative setting

    • Author(s)
      Y.Yokonuma
    • Journal Title

      Canadian mathematical Bulletin (To appear)

    • Related Report
      2004 Annual Research Report
  • [Publications] C.W.Curtis: "Zeta functions and functional equations associated with the components of the Gelfand-Graev representations of a finite group"Advanced Studies in Pure Mathematics. (To appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Gomi: "Coinvariant algebras of finite subgroups of SL (3,C)"Canadian J. of Mathematics. (To appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Wada: "On the Diophantine equation x(x+1)【triple bond】(x+n)+1=y^2(17≦n=odd≦27)"Proc. of the Japan Academy. 79-5. 99-100 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Nakashima: "Extremal projectors of q-boson algebras"Comm. in Mathematical Physics. 244. 285-296 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Tsunogai: "The stable derivation algebras for higher genera"Israel J. of Math.. 136. 221-250 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Kaneda: "On certain maximal cyclic modules for the quantized special linear algebras at a root of unity"Pacific J. of Math.. 211-2. 273-282 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] C.W.Curtis: "Zeta functions and functional equations associated with the components of the Gelfand-Graev representations of a finite reductive group"Advanced Studies in Pure Mathematics. (To appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Nakashima: "Polytopes for crystallized Demazure modules and extremal vectors"Communications in Algebra. 30. 1349-1367 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Nakashima: "Irreducible modules of finite dimensional quantum algebras of type A at roots of unity"J. of Mathematical Physics. 43. 2000-2014 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Tsunogai: "The stable derivation algebras for higher genera"Israel J. Mathematics. (To appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Kaneda: "On certain maximal cyclic modules for the quantized special linear algebras at a root of unity"Pacific J. of Mathematics. (To appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Gon: "The resolvent trace formula for rank one Lie groups"Asian J. of Mathematics. 6. 227-252 (2002)

    • Related Report
      2002 Annual Research Report

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

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