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Isotropy representations associated with Harish-Chandra modules and nilpotent orbit theory

Research Project

Project/Area Number 14340001
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

YAMASHITA Hiroshi  Hokkaido Univ., Grad.School of Sci., Prof., 大学院・理学研究科, 教授 (30192793)

Co-Investigator(Kenkyū-buntansha) YOSHIDA Tomoyuki  Hokkaido Univ., Grad.School of Sci., Prof., 大学院・理学研究科, 教授 (30002265)
SAITO Mutsumi  Hokkaido Univ., Grad.School of Sci., Asso.Prof., 大学院・理学研究科, 助教授 (70215565)
NISHIYAMA Kyo  Kyoto Univ., Grad.School of Sci., Asso.Prof., 大学院・理学研究科, 助教授 (70183085)
OCHIAI Hiroyuki  Nagoya Univ., Grad.School of Math.Sci., Asso.Prof., 大学院・多元数理科学研究科, 助教授 (90214163)
SEKIGUCHI Jiro  Tokyo Univ.of Agr.and Tech., Inst.of Symbiotic Sci.and Tech., Prof., 大学院・共生科学技術研究部, 教授 (30117717)
Project Period (FY) 2002 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥9,300,000 (Direct Cost: ¥9,300,000)
Fiscal Year 2005: ¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 2004: ¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 2003: ¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 2002: ¥2,700,000 (Direct Cost: ¥2,700,000)
Keywordssemisimple Lie group / nilpotent orbit / unitary representation / Harish-Chandra module / isotropy representation / discrete series representation / moment map / Howe duality correspondence / 既約ユニタリ表現 / 単単純リー群 / 無限次元表現 / 不変微分作用素 / 随伴多様体 / エルミート対象空間 / ユニポテント表現 / ユニタリ最高ウェイト加群
Research Abstract

The principal purpose of this project is to develop nilpotent orbit theory for Harish-Chandra modules corresponding to irreducible admissible representations of real semisimple Lie groups, in order to get deep understanding on generalized Whittaker models in relation to various nilpotent invariants of representations. We focused our attention to the isotropy representations which give the multiplicities in the associated cycles of Harish-Chandra modules.
(1)The isotropy representations have been described explicitly for all singular unitary highest weight modules of simple Hermitian Lie algebras by using the projection to the PRV-components. As a result we have proved the irreducibility of such isotropy representations. A new proof of the Howe duality correspondence for reductive dual pairs is obtained via isotropy representations. For EVII, we have shown that the isotropy representations for certain unitary highest weight modules of non-scalar type give the Dvorski-Sahi correspondence. … More This allows relating the isotropy representations and harmonic analysis on certain compact symmetric spaces of real rank one.
(2)General machinery has been established to describe the isotropy representations of Harish-Chandra modules with irreducible associated varieties, by using the principal symbols of differential operators of gradient-type. Applying this machinery, we have revealed an explicit relationship between the isotropy representations for discrete series and the generic fiber of the moment map for the conormal bundle of closed orbits on the flag variety.
(3)To identify the generic fiber of the moment map, we have studied the Richardson orbits associated with symmetric pairs. It has been an open problem whether the parabolic subgroups for the Richardson orbits act transitively on the set of Richardson elements in the symmetric part of its nilradical. In this project, we have got a progress to this problem, by giving nice sufficient conditions for the transitivity, and also a counterexample for the Lie groups of type A Less

Report

(5 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (30 results)

All 2005 2004 2003 2002 Other

All Journal Article (18 results) Book (2 results) Publications (10 results)

  • [Journal Article] Isotropy representation for Harish-Chandra modules2005

    • Author(s)
      Hiroshi Yamashita
    • Journal Title

      Infinite Dimensional Harmonic Analysis (Eds. H. Heyer et al.) 3

      Pages: 325-351

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] 離散系列に付随した等方表現について2005

    • Author(s)
      山下 博
    • Journal Title

      数理解析研究所講究録 1421

      Pages: 26-37

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Isotropy representation for Harish-Chandra modules2005

    • Author(s)
      Hiroshi Yamashita
    • Journal Title

      Infinite Dimensional Harmonic Analysis (Eds.H.Heyer et al.) 3

      Pages: 325-351

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Isotropy representation associated with the discrete series (in Japanese)2005

    • Author(s)
      Hiroshi Yamashita
    • Journal Title

      Su^rikaisekikenkyusho Ko^kyu^roku 1421

      Pages: 26-37

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Isotropy representation for Harish-Chandra modules2005

    • Author(s)
      Hiroshi Yamashita
    • Journal Title

      Infinite Dimensional Harmonic Analysis III (Eds. H.Heyer et al.)

      Pages: 325-351

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A note on affine quotients and equivariant double fibrations2005

    • Author(s)
      Kyo Nishiyama
    • Journal Title

      Infinite Dimensional Harmonic Analysis III (Eds. H.Heyer et al.)

      Pages: 197-212

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Mahler measures via the crystalization2005

    • Author(s)
      Hiroyuki Ochiai
    • Journal Title

      Comment.Math.Univ.St.Pauli, 54・2

      Pages: 121-137

    • NAID

      110007689252

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Crystal base elements of an extremal weight module fixed by a diagram automorphism2005

    • Author(s)
      Satoshi Naito
    • Journal Title

      Algebr.Represent.Theory 8・5

      Pages: 689-707

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Positive definite class functions on a topological group and characters of factor representations2005

    • Author(s)
      Takeshi Hirai
    • Journal Title

      J.Math.Kyoto Univ. 45・2

      Pages: 355-379

    • Related Report
      2005 Annual Research Report
  • [Journal Article] 離散系列に付随した等方表現について2005

    • Author(s)
      山下 博
    • Journal Title

      数理解析研究所講究録 (発表予定)

      Pages: 12-12

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Crossed Burnside rings. II. The Dress construction of a Green functor2004

    • Author(s)
      Tomoyuki Yoshida
    • Journal Title

      Journal of Algebra 282・1

      Pages: 58-82

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] Finite generation of rings of differential operators of semigroup algebras2004

    • Author(s)
      Mutsumi Saito
    • Journal Title

      Journal of Algebra 278・1

      Pages: 76-103

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] Theta lifting of unitary lowest weight modules and their associated cycles2004

    • Author(s)
      Kyo Nishiyama
    • Journal Title

      Duke Mathematical Journal 125・3

      Pages: 415-465

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] Finite generation of rings of differrential operators of semigroup algebras2004

    • Author(s)
      Mutsumi Saito
    • Journal Title

      Journal of Algebra 278・1

      Pages: 76-103

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Non-commutative harmonic oscillators and the connection problem for the Heun differential equation2004

    • Author(s)
      Hiroyuki Ochiai
    • Journal Title

      Letters in Mathematical Physics 70・2

      Pages: 133-139

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Capelli type identities on certain scalar generalized Verma modules. II.2004

    • Author(s)
      Akihito Wachi
    • Journal Title

      Journal of the Mathematical Society of Japan 56・2

      Pages: 447-473

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Isotropy representation and projection to the PRV-component2002

    • Author(s)
      Hiroshi Yamashita
    • Journal Title

      数理解析研究所講究録 1294

      Pages: 62-71

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Isotropy representation and projection to the PRV-component2002

    • Author(s)
      Hiroshi Yamashita
    • Journal Title

      Su^rikaisekikenkyusho Ko^kyu^roku 1294

      Pages: 62-71

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Book] 表現論入門セミナー : 第II部 リー代数と表現論2003

    • Author(s)
      山下 博
    • Total Pages
      146
    • Publisher
      遊星社
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Book] Introductory seminar on representation theory : Part II, Lie algebras and representation theory (in Japanese)2003

    • Author(s)
      Hiroshi Yamashita
    • Total Pages
      146
    • Publisher
      Yu^seisha
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Publications] Hiroshi Yamashita: "Isotropy representation for Harish-Chandra module"The Transactions of the 3rd Japanese-German Symposium on Infinite Dimensional Harmonic Analysis. (発表予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Satoshi Naito: "Three kinds of extremal weight vectors fixed by a diagram automorphism"Jornal of Algebra. 268・1. 343-365 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Jiro Sekiguchi: "Mapping of nilpotent orbits under embeddings of real forms of exceptional complex Lie algebras"The Asian Journal of Mathematics. 6・3. 409-432 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] Hiroshi Yamashita: "Isotropy representation and projection to the PRV-component"数理解析研究所講究録. 1294. 62-71 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] Mutsumi Saito: "The ring of differential operators for semigroup algebras-its finiteness"数理解析研究所講究録. 1289. 64-80 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] Akihito Wachi: "Capelli type identities on certain scalar generalized Verma modules II"J. Math. Soc. Japan. 55(印刷中). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Keiji Matsumoto: "Invariants for some real hyperbolic groups"Internat. J. Math.. 13. 415-443 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Jiro Sekiguchi: "Mapping of nilpotent orbits under embeddings of real forms of exceptional complex Lie algebras"Asian Journal of Mathematics. 6・3. 409-432 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Hiroshi Yamashita: "Isotropy representation and projection to the PRV-component"数理解析研究所講究録. 1294. 62-71 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Hiroyuki Ochiai: "Multiple trigonometry and zeta functions"J. Ramanujan Math. Soc.. 17・2. 101-113 (2002)

    • Related Report
      2002 Annual Research Report

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

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