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Theta correspondence and associated cycles

Research Project

Project/Area Number 11640025
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

NISHIYAMA Kyo  Kyoto University., Integrated Human Studies, Ass.Professor, 総合人間学部, 助教授 (70183085)

Co-Investigator(Kenkyū-buntansha) HATA Masayoshi  Kyoto University., Integrated Human Studies, Ass.Professor, 総合人間学部, 助教授 (40156336)
MATSUKI Toshihiko  Kyoto University., Integrated Human Studies, Ass.Professor, 総合人間学部, 助教授 (20157283)
KATO Shinichi  Kyoto University., Integrated Human Studies, Ass.Professor, 総合人間学部, 助教授 (90114438)
SAITO Hiroshi  Kyoto University., Graduate School of Human and Environmental Studieds, Professor, 大学院・人間・環境学研究科, 教授 (20025464)
ASUNO Kiyoshi  Kyoto University., Graduate School of Human and Enviromental Studieds, Professor, 大学院・人間・環境学研究科, 教授 (90026774)
岩井 斉良  京都大学, 総合人間学部, 教授 (70026764)
Project Period (FY) 1999 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2000: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 1999: ¥1,900,000 (Direct Cost: ¥1,900,000)
Keywordssemisimple Lie group / unitary representation / associated variety / associated cycle / nilpotent orbit / theta correspondence / geometric invariant theory / 半単純群の表現論 / reductive dual pair / 概均質ベクトル空間 / 球等質空間
Research Abstract

We developed certain machinery for studying associated cycles of unitary representations of semisimple Lie groups by means of theta correspondence.
To be more explicit, let (G_1 , G_2) be an irreducible, reductive dual pair of type I, which is in the stable range with G_2 as the smaller member. Take an irreducible representation π_2 of G_2 (or of its metaplectic double cover), and lift it to the representation π_1 of G_1 (or its metaplectic cover), called the theta lift ofπ_2. We study the cases where π_2 is a finite-dimensional unitary representation, or a representation in the holomorphic discrete series. Even if π_2 is the trivial representation, π_1 is an infinite dimensional unitary representation, which is supposed to be a unipotent representation, one of important objects in representation theory. In the following, we shall list up our results, where π_2 is a finite-dimensional unitary representation, or a representation in the holomorphic discrete series.
We get the K-type formula for π_1.
We describe the correspondence of associated varieties of (π_1, π_2) explicitly. Moreover, we succeed to get the multiplicities in their associated cycles and get a simple correspondence (still conjectural in general) between them.
An associated variety can often be irreducible, and is the closure of a single nilpotent orbit. Using the results on the theta correspondence of representations, we investigate the corre spondence of nilpotent orbits. The main results are, a description of the structure of the function ring, an integral formula of degree of nilpotent orbits.

Report

(3 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • Research Products

    (24 results)

All Other

All Publications (24 results)

  • [Publications] 西山享: "Invariant for Representations of Weyl Groups Two-sided Cells, and Modular Reoresentations of lwahori-Hecke Algebrds"Adv.Studies in Pure Math.. 28巻. 103-112 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 西山享: "Kawanaka invariants for representations of Weyl groups"J.Alg.. 225巻. 842-871 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 西山享: "Multiplicity free actions and the geometry of nilpotent orbits"Math.Ann.. 318巻. 777-793 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 畑政義: "C^2-saddle method and Beukers'integral"Trans.Amer.Math.Soc.. 352巻. 4557-4583 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 畑政義: "A new irrationality measure for ζ(3)"Acta.Arithmetica. 92巻. 47-57 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 西山享: "Theta lifting of holomorphic discrete series, The case of U(piq)×U(nin)."Trans.AMS.. (未定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Kyo Nishiyama: "Invariants for Representations of Wey1 Groups. Two-sided Colls and Modular Representations of Iwahori-hocke algebras"adv. Studies in Pure Math.. Vol. 28. 103-112 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Kyo Nishiyama: "Kawanaka invarian for representations of wey1 groups"J.Mg.. Vol. 225. 842-871 (2000)

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

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Masayoshi Hata: "C_2-saddle method and Beukese integsal"Trane, Amers Math. Soc.. Vol. 352. 4557-4583 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Masayoshi Hata: "a near irrationality measure for S(3)"acta arithmetica. Vol. 92. 47-57 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Kyo inshiyama: "Theta lifting of holomorphic discrete series. The case of U(p. q)×U (n. m)"Trans AMS.. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 西山享: "Invariants for Representations of Weyl Groups, Two-Sided Cells, and Modular Representations of lwahni-Hecke Alglbras"Adv.Studies in Pure Math.. 28巻. 103-112 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 西山享: "Kawanaka invariants for representations of Weyl groups"J.Alg.. 225巻. 842-871 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 西山享: "Multiplicity-free actions and the geometry of nilpotent orbits"Math.Ann.. 318巻. 777-793 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 畑政義: "C^2 saddle method and Beukers' integral"Trans.Amer.Math.Soc.. 352巻. 4557-4583 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 畑政義: "Anew irrationality measure for ζ(3)"Acta.Arithmetica. 92巻. 47-57 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 西山享: "The ta lifting of holomorphic discrete series.The case of U (p.q) × (n.n)."Trans.AMS.. (未定).

    • Related Report
      2000 Annual Research Report
  • [Publications] 西山 享: "Invariants for representations of Weyl Groups and Two-sided Cells"J,Math.Soc,Japan. 51巻. 1-34 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 西山 享: "Bernstein degree of singular unitary highest weight representations of metaplectic group"Proc.Japan Acad.. 75巻. 9-11 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 西山 享: "Schur duality for Cartan Type Lie algebra Wn"Journal of Lie Theory. 9巻. 234-248 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 西山 享: "Dipolarizations in Semisimple Lie algebras and homogeneous parakahler manifolds"Journal of Lie Theory. 9巻. 215-232 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 西山 享: "Invariants for Representations of Weyl Groups,Two-sided Cells,and Modular Representations of Iwahori-Hecke Algebras"Adv.Studies in Pure Math.. (未定).

    • Related Report
      1999 Annual Research Report
  • [Publications] 西山 享: "Kawanaka invariants for repersentations of Weyl groups"J,Alg.. (未定).

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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