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Monomial representation of solvable Lie groups

Research Project

Project/Area Number 11640189
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKinki University

Principal Investigator

FUJIWARA Hidenori  Kyushu School of Engeneering, Professor, 九州工学部, 教授 (50108643)

Project Period (FY) 1999 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 2001: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2000: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 1999: ¥700,000 (Direct Cost: ¥700,000)
Keywordsnilpotent Lie group / solvable Lie group / unitary representation / irreducible decomposition / orbit method / multiplicity / invariant differential operator / monomial representation / 既約公解
Research Abstract

It is well known that there exists a strong parallelism for inducing and restricting representations. In this research, I studied this duality for nilpotent Lie groups in the framework of celebrated orbit method. In collaboration with A. Baklouti, G. Lion and B. Magneron, I obtained the following main results. Let G be a connected, simply connected nilpotent Lie group.
1. (Commutativity conjecture of Duflo, Corwin Greenleaf) Let χ be a unitary character of an analytic subgroup H of G. Then, the monomial representation τ induced by χ up to G is of finite multiplicities if and only if the algebra of invariant differential operators on the line bundle over G/H associated to these data is commutative.
2. (Frobenius reciprocity) Let π be an irreducible unitary representation of G. The multiplicity of π in the canonical central decomposition of τ is equal to the dimension of the space of (H, χ ) semi-invariant generalized vectors.
3. The above result 1 has its counterpart for the restrictions. Namely, let's restrict an irreducible unitary representation of G to an analytic subgroup K. Then, this restriction is of finite multiplicities if and only if the associated algebra of K invariant differential operators is commutative.

Report

(4 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • Research Products

    (16 results)

All Other

All Publications (16 results)

  • [Publications] A.Baklouti, H.Fujiwara: "Analyse harmonique pour certaines representations monomiales d'un groupe resoluble exponentiel"Colloque de SMT 99. 112-135 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Fujiwara, G.Lion, B.Magneron, S.Mehdi: "Un critere de commutativite pour l'algebre des operateurs differentiels invariants sur un espace homogene nilpotent"C. R. Acad. Sci. Paris. Ser. I. 332. 597-600 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Fujiwara, G.Lion, S.Mehdi: "On the commutativity of the algebra of invariant differential operators on certain nilpotent homogeneous spaces"Trans. Amer. Math. Soc.. 353. 4203-4217 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] A.Baklouti, H.Fujiwara: "Harmonic analysis on some exponential homogeneous spaces"Research and exposition in Math.. 25. 127-134 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] A.Baklouti, H.Fujiwara: "Operateurs differentiels associes a certaines representations unitaires dun groupe de Lie resoluble exponentiel"Compositio Math..

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Fujiwara, G.Lion, B.Magneron: "Algebre de function associees aux representations monomiales des groupes de Lie nilpotent"Prepublication de l'Universite Paris 13. 2002-02. (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] A. Baklouti et H. Fujiwara: "Analyse harmonique pour certaines representations monomiales d'un groupe resoluble exponentiel"Colloque de SMT. 99. 112-135 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Fujiwara, G. Lion, B. Magneron et S. Mehdi: "Un critere de commutativite pour L'algebre des operateurs differentiels invariants sur un espace homogene nilpotent"C. R. Acad. Sci. Paris, Serie I. 332. 597-600 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Fujiwara, G. Lion and S. Mehdi: "On the commutativity of the algebra of the algebra of invariant differential operators on certain nilpotent homogeneous spaces"Trans. Amer. Math. Soc.. 353. 4203-4217 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] A. Baklouti and H. Fujiwara: "Harmonic analysis on some exponential homogeneous spaces"Research and exposition in Math.. 25. 127-134 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] A. Baklouti et H. Fujiwara: "Operateurs differentiels associes a certaines representations unitaires h'un groupe de Lie resoluble exponentiel"Compositio Math.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Fujiwara, G. Lion et B. Magneron: "Algebre de fonctions associees aux representations monomiales des groupes de Lie nilpotents"Prepublication de l'Universite Paris. 13. 2002-02 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Fujiwara, G.Lion, B.Magneren, S.Meholi: "UN critere de commutativite pour l'algebra des operateurs differentiels inveriants sur un espace homogene nilpotent"C. R, Acad. Paris, Ser. I, Math.. 332. 597-600 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Fujiwara, G.Lion, S.Mehdi: "On the commutativity of the algebra of invariant differential operators on certain nilpotent homogeneous spaces"Trans. Amer. Math. Sic.. 353. 4203-4217 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Fujiwara,G.Lion,B.Magneron,S.Mehdi: "Un critere de commutativite pour l'algebre des operateurs differentiels invariants sur un espace homogene nilpotent"C.R.Acad.Sci.Paris.Ser.I,Math..

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Fujiwara,G.Lion,S.Mehdi: "On the commutativity of the algebra of invariant differential operators on certain nilpotent homogeneous spaces"Trans.Amer.Math.Soc..

    • Related Report
      2000 Annual Research Report

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

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