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Infinite dimensional quaternionic representations and nilpotent orbits

Research Project

Project/Area Number 12640001
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

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

Co-Investigator(Kenkyū-buntansha) NISHIYAMA Kyo  Kyoto Univ., Fac. of Int. Hum. St., Asso. Prof., 総合人間学部, 助教授 (70183085)
SHIBUKAWA Youichi  Hokkaido Univ., Grad. School of Sci., Inst., 大学院・理学研究科, 助手 (90241299)
SAITO Mutsumi  Hokkaido Univ., Grad. School of Sci., Asso. Prof., 大学院・理学研究科, 助教授 (70215565)
WACHI Akihito  Hokkaido Institute of Technology, Div. of Gen. Edu., Lec., 総合教育研究部, 講師 (30337018)
OHTA Takuya  Tokyo Denki Univ., Fac. of Eng., Asso. Prof., 工学部, 助教授 (30211791)
山田 裕史  岡山大学, 理学部, 教授 (40192794)
Project Period (FY) 2000 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2001: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2000: ¥2,000,000 (Direct Cost: ¥2,000,000)
Keywordssemisimple Lie group / Harish-Chandra module / nilpotent orbit / quaternionic symmetric space / unitary highest weight module / discrete series / isotropy representation / invariant differential operator / 無限次元表現 / 随伴サイクル
Research Abstract

The associated variety of an irreducible Harish-Chandra module gives a fundamental nilpotent invariant for the corresponding irreducible admissible representation of a real reductive group. Moreover, the multiplicity in the Harish-Chandra module of an irreducible component of the associated variety can be regarded as the dimension of a certain finite-dimensional representation, called the isotropy representation.
The head investigator, Yamashita, has already shown that, in many cases, the isotropy representation can be described, in principle, by means of the principal symbol of a differential operator of gradient-type whose kernel realizes the dual Harish-Chandra module. In this research project, we have begun a systematic study of the isotropy representations attached to Harish-Chandra modules with irreducible associated varieties, including quaternionic representations, discrete series and unitary highest weight modules.
The results are summarized as follows:
We developed a general the … More ory for the isotropy representations, starting from the Vogan theory on associated cycles. In particular, a criterion for the irreducibility of an isotropy representation is presented. Also, we looked at when the isotropy representation can be described in terms of a differential operator of gradient-type.
As for the discrete series, a nonzero quotient of the isotropy representation has been constructed in a unified manner. It seems that this quotient representation is large enough in the whole isotropy module. We have shown that this is the case if the theta-stable parabolic subgroup canonically determined from the discrete series in question admits a Richardson nilpotent orbit with respect to the complexified symmetric pair.
The isotropy representation is explicitly described for every singular unitary highest weight module of Hermitian Lie algebras BI, DI and EVII. This allows us to deduce that the isotropy modules are irreducible for all singular unitary highest weight modules of arbitrary simple Hermitian Lie algebra.
Principal contribution by the investigators : Saito developed his research on A-hypergeometric system, which is closely related to a realization of unitary highest weight modules. He has established a formula for the rank of a homogeneous A-hypergeometric system. Wachi constructed an analogue of the Capelli identity for generalized Verma modules of scalar type. Nishiyama and Ohta gave a correspondence of nilpotent orbits associated to a symmetric pair, by menas of the moment map with respect to a reductive dual pair. Less

Report

(3 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • Research Products

    (37 results)

All Other

All Publications (37 results)

  • [Publications] Hiroshi Yamashita: "Two dual pair methods in the study of generalized Whittaker models for unitary highest weight modules"Transactions of Japanese-German Symposium "Infinite Dimensional Harmonic Analysis(Kyoto, 1999)". 373-387 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hiroshi Yamashita: "Associated cycles of Harish-Chandra modules and differential operators of gradient type"数理解析研究所講究録. 1183. 157-167 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hiroshi Yamashita: "Cayley transform and generalized Whittaker models for irreducible highest weight modules"Asterisque. 273. 81-137 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hiroshi Yamashita: "Isotropy representations attached to the associated cycles of Harish-Chandra modules"数理解析研究所講究録. 1238. 233-247 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Mutsumi Saito: "Isomorphism classes of A-hypergeometric systems"Compositio Mathematica. 128・3. 323-338 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Mutsumi Saito: "Differential algebras on semigroup algebras"AMS Contemporary Mathematics. 286. 207-226 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Youichi Shibukawa: "Classification of R-operators"Journal of Mathematical Physics. 42・6. 2725-2745 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kyo Nishiyama: "Multiplicity-free actions and the geometry of nilpotent orbits"Mathematische Annalen. 318・4. 777-793 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kyo Nishiyama: "Bernstein degree and associated cycles of Harish-Chandra modules-Hermitian symmetric case"Asterisque. 273. 13-80 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kyo Nishiyama: "Theta lifting of holomorphic discrete series. The case of U(p, q)xU(n, n)"Transactions of AMS. 353・8. 3327-3345 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Takuya Ohta: "On θ-stable Borel subalgebras of large type for real reductive groups"Tohoku Mathematical Journal. 52. 127-152 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Akihito Wachi: "Capelli type identities on certain scalar generalized Verma modules"Journal of Mathematics of Kyoto University. 40・4. 705-727 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hiroshi Yamashita: "Two dual pair methods in the study of generalized Whittaker models for unitary highest weight modules"Transactions of Japanese-German Symposium "Infinite Dimensional Harmonic Analysis (Kyoto, 1999)". 373-387 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hiroshi Yamashita: "Associated cycles of Harish-Chandra modules and differential operators of gradient type"RIMS Kokyuroku. Vol. 1183. 157-167 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hiroshi Yamashita: "Cayley transform and generalized Whittaker models for irreducible highest weight modules"Asterisque. 273. 81-137 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hiroshi Yamashita: "Isotropy representations attached to the associated cycles of Harish-Chandra modules"RIMS Ko^kyu^roku. Vol. 1238. 233-247 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Mutsumi Saito: "Isomorphism classes of A-hypergeometric systems"Compositio Mathematica. 128-3. 323-338 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Mutsumi Saito: "Differential algebras on semigroup algebras"AMS Contemporary Mathematics. 286. 207-226 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Youichi Shibukawa: "Classification of R-operators"Journal of Mathematical Physics. 42-6. 2725-2745 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kyo Nishiyama: "Multiplicity-free actions and the geometry of nilpotent orbits"Mathematische Annalen. 318-4. 777-793 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kyo Nishiyama: "Bernstein degree and associated cycles of Harish-Chandra modules -- Hermitian symmetric case"Asterisque. 273. 13-80 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kyo Nishiyama: "Theta lifting of holomorphic discrete series. The case of U(p, q) X U(n, n)"Transactions of AMS. 353-8. 3327-3345 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Takuya Ohta: "On θ-stable Borel subalgebras of large type for real reductive groups"Tohoku Mathematical Journal. 52. 127-152 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Akihito Wachi: "Capelli type identities on certain scalar generalized Verma modules"Journal of Mathematics of Kyoto University. 40-4. 705-727 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hiroshi Yamashita: "Cayley transform and generalized Whittaker models for irreducible highest weight modules"Asterisque. 273. 81-137 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Hiroshi Yamashita: "Isotropy representations attached to the associated cycles of Harish-Chandra modules"数理解析研究所講究録. 1238. 233-247 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Mutsumi Saito: "Isomorphism classes of A-hypergeometric systems"Compositio Mathematica. 128・3. 323-338 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Mutsumi Saito: "Differential algebras on semigroup algebras"AMS Contemporary Mathematics. 286. 207-226 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Youichi Shibukawa: "Classification of R-operators"Journal of Mathematical Physics. 42・6. 2725-2745 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Kyo Nishiyama: "Bernstein degree and associated cycles of Harish-Chandra modules-Hermitian symmetric case"Asterisque. 273. 13-80 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Kyo Nishiyama: "Theta lifting of holomorphic discrete series. The case of U(p,q)×U(n,n)"Transactions of AMS. 353・8. 3327-3345 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 和地 輝仁: "一般バーマ加群上のカペリ恒等式の類似物"数理解析研究所講究録. 1190. 153-181 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Hiroshi Yamashita: "Two dual pair methods in the study of generalized Whittaker models for unitary highest weight modules"Transactions of Japanese-German Symposium held from September 20th to 24th, 1999 at Kyoto University. なし. 373-387 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Hiroshi Yamashita: "Associated cycles of Harish-Chandra modules and differential operators of gradient type"数理解析研究所講究録. 1183. 157-167 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Hiroshi Yamashita: "Cayley transform and generalized Whittaker models for irreducible highest weight modules"Asterisque. 273. 81-137 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Kyo Nishiyama: "Multiplicity-free actions and the geometry of nilpotent orbits"Mathematische Annalen. 318・4. 777-793 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Kyo Nishiyama: "Bernstein degree and associated cycles of Harish-Chandra modules-Hermitian symmetric case"Asterisque. 273. 13-80 (2001)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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