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Theory of branching laws of unitary representations of reductive Lie groups and geometric realization of representations

Research Project

Project/Area Number 11440018
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionKYOTO UNIVERSITY (2001)
The University of Tokyo (1999-2000)

Principal Investigator

KOBAYASHI Toshiyuki  Research Institute for Mathematical Sciences, Kyoto University Associate Professor, 数理解析研究所, 助教授 (80201490)

Project Period (FY) 1999 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥5,300,000 (Direct Cost: ¥5,300,000)
Fiscal Year 2001: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2000: ¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 1999: ¥1,900,000 (Direct Cost: ¥1,900,000)
Keywordssemisimple Lie group / unitary representation / conformal geometry / branching law / discontinuous group / minimal representation / highest weight module / discrete spectrum / 簡約 / 離散分解 / 制限 / 極小表現、最高ウェイト加群 / 重複度 / 離散的分岐則 / 簡約リー群 / 等質空間
Research Abstract

The branching law means the irreducible decomposition of an irreducible unitary representation of a group when restricted to a subgroup (e.g. decomposition of tensor products, breaking symmetry in physics,…). It is one of principal subjects in representation theory to find branching laws. Nevertheless, very little has been studied on branching laws of unitary representations, except for some special cases until mid-90s, partly because of analytic difficulties arising from infinite dimensions.
1. Our main results during this period are to establish a basic theory of "discrete branching laws of infinite dimensional representations of semisimple Lie groups. Namely, based on new examples that we had found some years ago, we proposed a formulation of discrete branching laws, and proved a criterion for branching laws to be discretely decomposable by using both micro-local analysis and algebraic representation theory. Furthermore, we found new applications of these representation theoretic res … More ults to the following problems :
i) Non-commutative harmonic analysis. To construct new discrete series representations for homogeneous spaces.
ii) Automorphic forms. To prove a vanishing theorem of modular varieties for locally Riemannian symmetric spaces.
Moreover, we found explicitly branching laws in certain settings in connection with conformal geometry.
On these topics, I gave one-hour lectures in various international conferences, and a plenary lecture at MSJ for the Spring Prize (1999). Also, I gave series of lectures at European School (2000), at Harvard University (2001), and the Winter School at Czech Republic (2002)
2. Since the late 1980s, I have initiated the study of the existence problem of compact CliffordKlein forms of pseudo-Riemannian homogeneous manifolds. Recently, this problem has been studied by different methods such as discrete groups, ergodic theory, symplectic geometry and unitary representation theory, revealing the interactions with other branches of mathematics.
I wrote an expository survey on this area and posed some open problems in "Mathematics Unlimited, 2001 and beyond" as a project of the World Mathematical Year 2000. Less

Report

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

    (52 results)

All Other

All Publications (52 results)

  • [Publications] 小林俊行: "半単純リー群のユニタリ表現の離散的分岐則の理論とその展開"数学(日本数学会), 岩波書店. 51-4. 337-356 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi: "Discretely decomposable restrictions of unitary representations of reductive Lie groups-examples and conjectures"Advanced Study in Pure Mathematics. 26. 99-127 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi: "Adjoint action of a Lie Group"Encyclopaedia Mathematics. Supplement II. 15-16 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 小林俊行: "半単純リー群のユニタリ表現の離散分岐理論とその展開"総合講演・企画特別講演アブストラクト(日本数学会). 1-19 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 小林俊行: "等質空間における不連続群"表現論シンポジウム講演集. 99-110 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi: "Branching laws of unitary highest weight modules with respect to semisimple symmetric pairs"Tangungsbericht, Representation Theory and Complex Analysis. 18. 15-16 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi: "Discontinuous groups for non-Riemannian homogeneous spaces"Mathematics Unlimited-2001 and Beyond, (eds. B. Engquist and W. Schmid), Springer-Verlag. 723-748 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 小林俊行: "名著発掘ポントリャーギン『連続群論』"数学の楽しみ. 23. 110-119 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi, B.Orsted: "Analysis of the Minimal Representation of O(p, q)-I. Realization via Conformal Geometry"RIMS preprint. 1337. 22 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi, B.Orsted: "Analysis of the Minimal Representation of O(p, q)-II. Branching Laws"RIMS preprint. 1338. 26 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi, B.Orsted: "Analysis of the Minimal Representation of O(p, q)-III. Ultrahyperbolic equations on R^<p-1,q-1>"RIMS preprint. 1339. 36 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi: "Introduction to actions of discrete groups on pseudo-Riemannian homogeneous manifolds"Acta Applicandae Mathematicae. Special volume(to appear). (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 小林俊行: "山辺作用素の解空間における不変な内積について"日本工業大学微分幾何学研究集会『種々の幾何構造の発展』講演要旨. 4-5 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 小林俊行: "非リーマン等質空間の不連続群論"数学の最前線,21世紀への挑戦. 第1巻(to appear). (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi: "Theory of discrete decomposable branching laws of unitary representations of semisimple Lie groups and some applications"Sugaku Exposition, Amer. Math. Soc.. (to appear). (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi: "Conformal geometry and global solutions to the Yamabe operators on some classical pseudo-Riemannian manifolds"Rendiconti del Circolo Matematico di Palermo. (to appear). (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 小林俊行-大島利雄: "Lie群とLie環 1 (岩波講座 現代数学の基礎)"岩波書店. 293+16 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 小林俊行: "Lie群とLie環 2 (岩波講座 現代数学の基礎)"岩波書店. 315+14 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi, M.Kashiwara, T.Matsuki, K.Nishiyama, eds.: "Analysis on Homogeneous Spaces and Representation theory of Lie Groups, Okayama-Kyoto"紀伊国屋書店・アメリカ数学会. 359 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi: "Discretely decomposable restrictions of unitary representations of reductive Lie groups - examples and conjectures"Advanced Study in Pure Mathematics. 26. 99-127 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi: "Adjoint action of a Lie Group"Encyclopaedia Mathematics, Supplement. II. 15-16 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi: "Branching laws of unitary highest weight modules with respect to semisimple symmetric pairs"Tangungsbericht, Representation Theory and Complex Analysis. 18. 15-16 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi: "Discontinuous groups for non-Riemannian homogenious spaces"Mathematics Unlimited 2001 and Beyond,(eds. B. Engquist and W. Schmid). Springer-Verlag. 723-748 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi, B. Orsted: "Analysis of the Minimal Representation of O(p,q) - I. Realization via Conforrnal Geometry"RIMS preprint. 1337. 22 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi, B. Orsted: "Analysis of the Minimal Representation of O(p,q) - II. Branching Laws"RIMS preprint. 1338. 26 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi, B. Orsted: "Analysis of the Minimal Representation of O(p,q) - III, Ultrahyperbolic Equations on R^<p-1,q-1>"RIMS preprint. 1339. 36 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi: "Introduction to actions of discrete groups on pseudo-Riemannian homogeneous manifolds"Acta Applicandae Mathematicae. Special volume(to appear). (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi: "Theory of discrete decomposable branching laws of unitary representations of semisimple Lie groups and some applications"Sugaku Exposition, Amer. Math. Soc. (to appear). (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi: "Conformal geometry and global solutions to the Yamabe operators on some classical pseudo-Riemannian manifolds"Rendiconti del Circolo Matematico di Palermo. (to appear). (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi, T.Oshima: "Lie groups and Lie algebras I"Iwanami. 293+16 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi: "Lie groups and Lie algebras II"Iwanami. 315+14 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kobayashi, M. Kashiwara, T. Matsuki, K. Nishiyama, eds: "Analysis on Homogeneous Spaces and Representation theory of Lie Groups, Okayama - Kyoto"Kinokuniya Amer. Math. Soc.. 359 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Toshiyuki Kobayashi: "Discontinuous groups for non-Riemannian homogeneous spaces"Mathematics Unlimited-2001 and Beyond (eds. B.Engquist and W.Schmid). 723-748 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Toshiyuki Kobayashi: "Introduction to actions of discrete groups on pseudo-Riemannian homo-geneous manifolds"Acta Applicandae Mathematicae. Special volume(to appear). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Toshiyuki Kobayashi: "Theory of discrete decomposable branching laws of unitary representations of semisimple Lie groups and some applications"Sugaku Exposition, Amer. Math. Soc.. (to appear). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 小林俊行: "非リーマン等質空間の不連続群論"数学の最前線,21世紀への挑戦,(Springer-Verlag, Tokyo). 1(掲載予定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Toshiyuki Kobayashi: "ポントリャーギン『連続群論』"数学の楽しみ. 23. 110-119 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Toshiyuki Kobayashi: "Conformal geometry and global solutions to the Yamabe operators on some classical pseudo-Riemannian manifolds"Rendiconti del Circolo Matematico di Palermo. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] 小林俊行, 大島利雄 (東京大学): "Lie群とLie環1 岩波講座,現代数学の基礎(改訂版)"岩波書店. 293+10 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 小林俊行(単著): "Lie群とLie環2 岩波講座,現代数学の基礎(改訂版)"岩波書店. 315+14 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Kobayashi: "Discontinuous groups for non-Riemannian homogeneous spaces"Mathematics Unlimited - 2001 and Beyond (eds. B. Engquist and W.Schmid). 723-748 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Kobayashi: "Discretely decomposable restrictions of unitary representations of reductive Lie groups"Advanced Study in Pure Mathematics. 26. 98-126 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Kobayashi: "ポントリャーギン『連続群論』"数学の楽しみ. 23. 110-119 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Toshiyuki Kobayashi: "Analysis on Homogeneous Spaces and Representation Theory of Lie Groups, Okayama - Kyoto."紀伊国屋書店-アメリカ数学会. 359 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 小林俊行: "半単純リー群のユニタリ表現の離散的分岐則の理論とその展開"数学. 51-4. 337-356 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Toshiyuki Kobayashi: "Discretely decomposable Restrictions of Unitary Representations of Reductive Lie Groups - Examples and Conjectueres"Advanced Studies in Pure Mathematics. 26. 99-127 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Toshiyuki Kobayashi: "Adjoint Action of a Lie group"Encyclopaedia of Mathematics,Supplement II. 15-16 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] 小林俊行: "半単純リー群のユニタリ表現の離散分岐理論とその展開"総合講演アブストラクト(日本数学会). 1-19 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 小林俊行: "等質空間における不連続群"表現論シンポジウム講演集. 99-110 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 小林俊行: "Lie群とLie環I(岩波講座、現代数学の基礎)"岩波書店. 293+16 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 小林俊行: "Lie群とLie環II(岩波講座、現代数学の基礎)"岩波書店. 315+14 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Toshiyuki Kobayashi: "Analysison Homogeneous Spaces and Representation Theory of Lie Groups,Okayama-Kyoto"紀伊国屋. 362 (2000)

    • Related Report
      1999 Annual Research Report

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

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