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On zeta functions of prehomogeneous vector spaces

Research Project

Project/Area Number 09640032
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

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)

Co-Investigator(Kenkyū-buntansha) YAMAUCHI Masatoshi  Kyoto Univ., Integrated Human Studies, Professor, 総合人間学部, 教授 (30022651)
YOSHINO Yuji  Kyoto Univ., Integrated Human Studies, Ass.Professor, 総合人間学部, 助教授 (00135302)
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)
KATO Shinichi  Kyoto Univ., Integrated Human Studies, Ass.Professor, 総合人間学部, 助教授 (90114438)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 1998: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1997: ¥2,100,000 (Direct Cost: ¥2,100,000)
Keywordsprehomogeneous vector space / zeta function / functionsl equation / sperical subgroup / double coset decomposition / Bernstein degree / Kawanaka invariant / Cohen-Macaulay module / 軌道上のゼータ関数 / 球等質空間 / カルタン分解 / 特異ユニタリ表現
Research Abstract

(1) Saito gave a proof for the convergence of zeta functions of reduced irreducible regular prehomogeneous vector spaces in full generality, The method used in this proof seems to be applicable to more general cases, for example, to the case of regular prehomogeneous vector spaces.
(2) Saito determined the functional equation explicitly in the case of homogeneous vector spaces over non-archimedean local fields which are related to the representation of symmetric matrices(type (15) in the classification of Sato-Kimura), This result has some applications to the representation of algebraic gorups over non-archimedean local fields.
(3) Kato showed that a subsemigroup of a torus can be chosen as representatives in the doule coset decomposition of a p-adic reductive group with respect to its spherical group and its maximal compact group. This result seems to have some applications in the computation of zeta functions of prehomogeneous vector spaces over non-archimedean local fileds.
(4) Matsuki showed a decomposition theorem, a classification and a theory of root systems in the double coset decomposition of compact Lie groups with respect to two involutions, and calculated several examples of orbit decomposition of flag manifolds in the case of codimension one.
(5) Nishiyama obtained an integral representation for the Bernstein degrees for semi-simple groups. This made it possible to calculate the associated cycles in some special cases.
(6) Nishiyama computed the Kawanaka invariant of the representation of Weyl groups in the case of classical gorups.
(7) Yoshino succeeded in developing the theory of classification of Cohen-Macaulay modules over the local rings of singular points of hypersurfaces by means of linkage, and gave a method to classify general modules by using the Cohen-Macualay approximation.

Report

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

    (24 results)

All Other

All Publications (24 results)

  • [Publications] 斎藤 裕: "Explicit form of zeta functions of prehomogeneous vector spaces" Math.Ann.(未定). (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 斎藤 裕: "On zeta functions associated to symmetric matrices II : Functional equations and special values" 未定. (未定). (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 松木敏彦: "Classification of two involutions on Compact semisimple Lie groups and root systems" 未定. (未定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 西山 享: "Invariants for representations of weyl groups and two-sided cells" J.Math.Soc.Japan. 51巻. 1-34 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 西山 享: "Schur duality for cartan type Lie algebra S W-nS" Journal of Lie Theory. 9巻. 234-248 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 吉野雄二: "Remarks on depth formula, grade inequality and Auslander Conjecture" Communications in Algebra. 26巻. 3793-3806 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Hiroshi, Saito: "Explicit form of zeta functions of prehomogeneous vector spaces" Math.Ann. (to appear). (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Hiroshi, Saito: "On zeta functions associated to symmetric matrices II : Functional equations and special values" (to appear). (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Toshihiko, Matuki: "Classification of two involutions on compact semisimple Lie groups and root systems" (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Kyo, Nishiyama: "Invariants for representations of Weyl groups and two-sided cells" J.Math.Soc.Japan. Vol.51. 1-34 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Kyo, Nishiyama: "Schur duality for Cartan type Lie algebra *W_<-n>*." J.of Lie Theory. Vol.9. 234-248 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Yuji, Yoshino: "Remarks on depth formula, grade inequality and Auslander conjecture" Communications in Algebra. Vol.26. 3793-3806 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 斎藤裕: "Explicit form of zeta functions of prenomogeneous Ve-ctor spaces" Math.Ann.(1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 斎藤裕: "On Zeta functions associated to symmetric matrices II:Functional equatinos and special values" (1999)

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

    • Related Report
      1998 Annual Research Report
  • [Publications] 西山享: "Invariants for representations of weyt 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 S Wn.S." Journal of Lie Theory. 9巻. 234-248 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 吉野雄二: "Remarks on depth formula,grade inequality and Auslandek Conjecture" Communications in Algebra. 26巻. 3793-3806 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 斉藤 裕: "On zeta functions associated to symmetric matrices III :An explicit form of L-functions" Nagoya J. Math. 146巻. 149-183 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 斉藤 裕: "Explicit formula of orbital P-adic Zeta functions associated to symmetric and hermitian matrices" Comm. Math. Univ.St. Pauli. 46巻. 175-216 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 斉藤 裕: "On zeta functions associated to symmetric matrices II :Functional egautions and special values" 未定. 未定. 未定 (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] 斉藤 裕: "Explicit form of zeta functions of prehomogeneous vector spaces" 未定. 未定. 未定 (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] 加藤 信一: "Hecke algebras and quantum general linear groups" J. Math. Kyoto Univ.37巻. 241-249 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 西山 享: "Commutant algebra and harmonic polynomials of the Lie algebra of vector fields" J. Alg.183巻. 545-559 (1996)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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