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Representation theory of algebraic groups via algebraic analysis

Research Project

Project/Area Number 15540041
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionOsaka City University

Principal Investigator

TANISAKI Toshiyuki  Osaka City University, Graduate school of science, professor, 大学院・理学研究科, 教授 (70142916)

Co-Investigator(Kenkyū-buntansha) KASHIWARA Masaki  Kyoto Univ., Research Institute of Mathematical Sciences, professor, 数理解析研究所, 教授 (60027381)
SHOJI Toshiaki  Nagoya Univ., Graduate school of Mathematics, professor, 多元数理科学研究科, 教授 (40120191)
SAITO Yoshihisa  Univ. Tokyo, Graduate school of Mathematical sciences, assistant professor, 数理科学研究科, 助教授 (20294522)
KANEDA Masaharu  Osaka City University, Graduate school of science, professor, 大学院・理学研究科, 教授 (60204575)
TAKEUCHI Kiyoshi  Osaka City University, Graduate school of pure and applied sciences, assistant professor, 数学系, 助教授 (70281160)
Project Period (FY) 2003 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2004: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2003: ¥2,000,000 (Direct Cost: ¥2,000,000)
Keywordsalgebraic groups / representations / algebraic analysis
Research Abstract

1. Tanisaki investigated on the quantized flag manifolds, especially at roots of 1. He has formulated a conjecture which can be regarded as an analogue of the result of Bezrukavnikov-Mirkovic-Ryuminin about the correspondence of representations and D-modules on the flag manifold in positive characteristics. This is different from recent works of Backelin-Kremnitzer and Mirkovic. It should be solved in the near future although there are some problems to be overcome. He also extended a result of Soergel about the ring of differential operators on the partial flag manifold of reductive algebraic groups and obtained similar results for differential operators acting on vector bundles. Furthermore, he considered about parabolic analogue of Soergel's result on the center of category O.
2. Kashiwara showed that the crystal base of some finite dimensional representation of affine quantum group with fundamental weight as its external weight is isomorphic to that of the Demazure module of irreducible module with level 1 highest weight.
3. Ariki has shown that the representation types of the classical Hecke algebras are governed by the Poincare polynomials.
4. Nakajima proved that the first tern of the Necrasov partition function coincides with the pro-potential of Seidberg-Written.
5. Shoji has solved Lusztig's conjecture on the characters of the special linear groups over finte fields. Moreover, he determined the scalar appearing in the conjecture.
6. Kaneda investigated on the correspondence between D-modules on flag varieties in positive characteristics and representations of the corresponding algebraic groups. He formulated a certain derived equivalence in terms of arithmetic differential operators due to Berthelot., and obtained some results in the case of the projective space.
7. Ichino investigated on the diagonal restriction of Saito-Kurokawa lift, and proved the algebraicity of a special value of a certain L-function.

Report

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

    (21 results)

All 2004 2003 Other

All Journal Article (13 results) Book (2 results) Publications (6 results)

  • [Journal Article] Character formulas of Kazhdan-Lusztig type2004

    • Author(s)
      T.Tanisaki
    • Journal Title

      Fields Inst. Commun. 25(8)

      Pages: 261-276

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Subfield symmetric spaces for finite special linear groups2004

    • Author(s)
      T.Shoji
    • Journal Title

      Represent. Theory 8

      Pages: 487-521

    • 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] Extremal weight modules of quantum affine algebras2004

    • Author(s)
      N.Nakajima
    • Journal Title

      Adv. Stud. Pure Math. 40

      Pages: 343-369

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Character formulas of Kazhdan-Lusztig type2004

    • Author(s)
      T.Tanisaki
    • Journal Title

      Fields Inst.Commun. 25(8)

      Pages: 261-276

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Subfield symmetric spaces for finite special linear groups2004

    • Author(s)
      T.Shoji
    • Journal Title

      Represent.Theory 8

      Pages: 487-521

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Extremal weight modules of quantum affine algebras2004

    • Author(s)
      N.Nakajima
    • Journal Title

      Adv.Stud.Pure Math. 40

      Pages: 343-369

    • 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] Character formulas of Kazhdan-Lusztig type2004

    • Author(s)
      T.Tanisaki
    • Journal Title

      Fields Inst.Commun. 40

      Pages: 261-276

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A functional model for the tensor product of level 1 highest and level -1 lowest modules for the quantum affine algebra Uq(sl2)2004

    • Author(s)
      B.Feigin
    • Journal Title

      European J.Combin. 25(8)

      Pages: 1197-1229

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Hecke algebras with a finite number of indecomposable modules2004

    • Author(s)
      S.Ariki
    • Journal Title

      Adv.Stud.Pure Math. 40

      Pages: 17-25

    • Related Report
      2004 Annual Research Report
  • [Journal Article] An introduction to canonical bases2004

    • Author(s)
      Y.Saito
    • Journal Title

      Fields Inst.Commun. 40

      Pages: 261-276

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Radon transforms for quasi-equivariant D-modules on generalized flag manifold2003

    • Author(s)
      C.Marastoni
    • Journal Title

      Differential geometry and its applications 18(2)

      Pages: 147-156

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Realizations of Crystals2003

    • Author(s)
      M.Kashiwara
    • Journal Title

      Contemporary Mathematics 325

      Pages: 133-139

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On tameness of the Hecke algebras of type B2003

    • Author(s)
      S.Ariki
    • Journal Title

      Contemporary Mathematics 325

      Pages: 11-30

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Book] D-rnodules and microlocal calculus2003

    • Author(s)
      M.Kashiwara
    • Total Pages
      254
    • Publisher
      American Mathematical Society, Providence, RI
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Book] D-modules and microlocal calculus2003

    • Author(s)
      M.Kashiwara
    • Total Pages
      254
    • Publisher
      AMS
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] C.Marastoni: "Radon transforms for quasi-equivariant D-modules on generalized flag manifolds"Differential geometry and its applications. 18(2). 147-176 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Kashiwara: "Realizations of Crystals"Contemporary Mathematics. 325. 133-139 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Ariki: "On tameness of the Hecke algebras of type B"Contemp.Mathematics. 325. 11-30 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Takeuchi: "Dimension formulas for the hyperfunction solutions to holonomic D-modules"Adv.Math.. 180. 134-145 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Nakajima: "t-analogs of q-characters of quantum affine algebras of type A_n, D_n"Contemp.Mathematics. 325. 141-160 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Shoji: "Macdonald functions associated to complex reflection groups"J.Algebra. 260. 426-448 (2003)

    • Related Report
      2003 Annual Research Report

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

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