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Non-commutative harmonic analysis on solvable Lie groups and its applications

Research Project

Project/Area Number 17K05280
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionTottori University

Principal Investigator

Inoue Junko  鳥取大学, 教育支援・国際交流推進機構, 教授 (40243886)

Project Period (FY) 2017-04-01 – 2022-03-31
Project Status Completed (Fiscal Year 2021)
Budget Amount *help
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords非可換調和解析 / 可解リー群 / フーリエ変換 / 複素解析的誘導表現 / ユニタリ表現論 / 表現論 / Lp-Fourier変換 / 関数解析学
Outline of Final Research Achievements

Concerning the norm Fp(G) of the Lp-Fourier transform on a separable unimodular locally compact group G of type 1, we obtained the following estimate. Let N be a unimodular closed normal subgroup of type 1 such that G/N is compact. Then we showed that Fp(G) is less than or equal to Fp(N).
Let G be an exponential solvable Lie group and D be an exponential solvable Lie group of its automorphisms of exponential type containing the group of inner automorphisms of G. By the orbit method, we gave a sufficient condition of the kernel of the D orbit of an irreducible representation to be L1-determined.
Concerning holomorphically induced representation ρ of an exponential solvable Lie group G and the space of associated semi-invariant generalized vectors of each irreducible representation π, we found examples for which the dimension of the space and the multiplicity of π in ρ do not coincide.

Academic Significance and Societal Importance of the Research Achievements

局所コンパクト群・群環の表現論において、Lpフーリエ変換のノルムを求める問題は重要であり、様々な群を対象に多くの研究が行われている。本研究で扱ったコンパクト拡大においては、Russoにより共役指数qが偶数の場合に同様のノルム評価が与えられているが、本研究結果はRussoの結果を一般の指数p(1<p<2)に拡張したものとなる。
誘導表現の既約分解を無限次元表現に拡張されたフロベニウス相互律の形で記述することは基本問題の1つである。本研究の結果から、指数型可解リー群の複素解析的誘導表現で、従来の定義による相互律とは異なる現象が新たに見つかったことになる。

Report

(6 results)
  • 2021 Annual Research Report   Final Research Report ( PDF )
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (11 results)

All 2022 2021 2020 2019 2018 2017

All Journal Article (4 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 4 results,  Open Access: 1 results) Presentation (7 results) (of which Int'l Joint Research: 2 results,  Invited: 2 results)

  • [Journal Article] Semi-invariant Vectors Associated with Holomorphically Induced Representations of Exponential Lie Groups2021

    • Author(s)
      Junko Inoue
    • Journal Title

      Geometric and Harmonic Analysis on Homogeneous Spaces and Applications, Springer Proceedings in Mathematics & Statistics

      Volume: 366 Pages: 79-101

    • DOI

      10.1007/978-3-030-78346-4_6

    • ISBN
      9783030783457, 9783030783464
    • Related Report
      2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The Lp-Fourier Transform Norm on Compact Extensions of Locally Compact Groups2020

    • Author(s)
      Ali Baklouti, Junko Inoue
    • Journal Title

      Journal of Fourier Analysis and Applications

      Volume: - Issue: 2

    • DOI

      10.1007/s00041-020-09739-5

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] An Example of Holomorphically Induced Representations of Exponential Solvable Lie Groups2019

    • Author(s)
      Junko Inoue
    • Journal Title

      Geometric and Harmonic Analysis on Homogeneous Spaces. TJC 2017. Springer Proceedings in Mathematics & Statistics

      Volume: 290 Pages: 111-120

    • DOI

      10.1007/978-3-030-26562-5_5

    • ISBN
      9783030265618, 9783030265625
    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] L1-determined primitive ideals in the C*-algebra of an exponential Lie group with closed non-*-regular orbits2019

    • Author(s)
      Junko Inoue and Jean Ludwig
    • Journal Title

      Kyushu Journal of Mathematics

      Volume: 印刷中

    • NAID

      130007873241

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Presentation] 局所コンパクト群のコンパクト拡大におけるLp-Fourier変換のノルムについて2022

    • Author(s)
      井上順子,Ali Baklouti
    • Organizer
      日本数学会2022年度年会
    • Related Report
      2021 Annual Research Report
  • [Presentation] 可解Lie群の複素解析的誘導表現の既約分解に付随する半不変超関数ベクトルについて2021

    • Author(s)
      井上順子
    • Organizer
      2020年度表現論ワークショップ
    • Related Report
      2020 Research-status Report
  • [Presentation] 群上の L^p-Fourier 変換のノルムについて:局所コンパクト群のコンパクト拡大に関わるノルムの評価2020

    • Author(s)
      井上順子
    • Organizer
      2019年度表現論ワークショップ
    • Related Report
      2019 Research-status Report
  • [Presentation] Semi-invariant vectors associated with holomorphically induced representations of exponential Lie groups.2019

    • Author(s)
      Junko Inoue
    • Organizer
      6th Tunisian-Japanese Conference Geometric and Harmonic Analysis on Homogeneous Spaces and Applications in Honor of Professor Takaaki Nomura
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 指数型可解Lie群の複素解析的誘導表現における半不変超関数ベクトルについて2019

    • Author(s)
      井上順子
    • Organizer
      日本数学会2019年度年会
    • Related Report
      2018 Research-status Report
  • [Presentation] ある指数型可解リー群の複素解析的誘導表現における半不変超関数ベクトルについて2018

    • Author(s)
      井上順子
    • Organizer
      2017年度表現論ワークショップ
    • Related Report
      2017 Research-status Report
  • [Presentation] Some examples of holomorphically induced representations of solvable Lie groups2017

    • Author(s)
      Junko Inoue
    • Organizer
      5th Tunisian-Japanese Conference, Geometric and Harmonic Analysis on Homogeneous Spaces and Applications
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2017-04-28   Modified: 2023-01-30  

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