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2005 Fiscal Year Final Research Report Summary

Theory of branching laws of unitary representations and non-commutative harmonic analysis by transformation groups of geometric structures

Research Project

Project/Area Number 14340043
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKYOTO UNIVERSITY

Principal Investigator

KOBAYASHI Toshiyuki  KYOTO UNIVERSITY, Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (80201490)

Co-Investigator(Kenkyū-buntansha) OSHIMA Toshio  The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (50011721)
Project Period (FY) 2002 – 2005
Keywordssemisimple Lie groups / unitary representations / conformal geometry / branching law / multiplicity-free representations / discontinuous groups / minimal representations / homogeneous spaces
Research Abstract

The branching law is the irreducible decomposition of a group representation when restricted to a subgroup (e.g. decomposition of tensor products, breaking symmetry in physics,...). Analysis of branching laws is one of principal subjects in representation theory. Nevertheless, very little has been studied on branching laws of unitary representations of reductive groups, except for some special cases until mid 1990s, partly because of analytic difficulties arising from infinite dimensions.
Our main results during this period are :
1.To publish the basic theory of branching laws of unitary representations with emphasis on discrete decomposable cases and its applications to representation theory itself, non-commutative harmonic analysis, and topology of locally symmetric spaces.
2.To construct the canonical representations of the conformal groups of pseudo-Riemannian manifolds by means of the Yamabe operator. In particular, we use this machinery for the study of minimal representations of in … More definite orthogonal groups. Besides, we find (conformal) conservation laws of solutions to certain ultra-hyperbolic equations. Branching laws when restricted to isometry groups play an important role in our analysis (joint with B.Orsted).
3.To make a unified approach to multiplicity-free theorems of the biholomorphic transformation groups of complex manifolds by means of orbit-preserving anthi-holomorphic maps. In particular, we use this machinery for the study of multiplicity-free tensor product representations of general linear groups.
4.In the late 1980, I initiated the study of discontinuous groups for pseudo-Riemannian homogeneous spaces. In the memorial volume of A.Borel, I published a survey article on this area, and also gave new criteria for the existence of co-compact discontinuous groups for semisimple symmetric spaces and their tangential symmetric spaces (joint with T.Yoshino). Besides, the deformation spaces of discontinuous groups are investigated.
5.On these topics, I gave one-hour invited lectures in various international conferences, including ICM 2002, Pan-African Congress 2004 (plenary address), and Asian Congress 2005. Less

  • Research Products

    (57 results)

All 2006 2005 2004 2003 2002 Other

All Journal Article (53 results) Book (4 results)

  • [Journal Article] O(p,q)の極小表現の反転を与える積分作用素2006

    • Author(s)
      Toshiyuki Kobayashi, Gen Mano
    • Journal Title

      数理解析研究所講究録(RIMS Kokyuroku),「群の表現と調和解析の広がり」(2005年7月25-28日,河添健 編) vol. 1467

      Pages: 51-61

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Deformation space of discontinuous groups Z^k for a nilmanifold R^<k+1>2006

    • Author(s)
      Toshiyuki Kobayashi, Salma Nasrin
    • Journal Title

      数理解析研究所講究録(RIMS Kokyuroku),「群の表現と調和解析の広がり」(2005年7月25-28日,河添健 編) vol. 1467

      Pages: 101-111

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Deformation space of discontinuous groups Z^k for a nilmanifold R^<k+1>2006

    • Author(s)
      Toshiyuki Kobayashi, Salma Nasrin
    • Journal Title

      RIMS Kokyuroku, Representation theory of groups and extension of harmonic analysis (T.Kawazoe, ed.) vol.1467

      Pages: 101-111

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Integral formulas for the minimal representations for O(p, 2)2005

    • Author(s)
      Toshiyuki Kobayashi, Gen Mano
    • Journal Title

      Acta Applicandae Mathematicae vol. 83

      Pages: 103-113

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Multiplicity-free representations and visible actions on complex manifolds2005

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      Publ. Res. Inst. Math. Sci., Special issue commemorating the fortieth anniversary of the founding of RIMS vol. 41

      Pages: 497-549

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Theory of discrete decomposable branching laws of unitary representations of semisimple Lie groups and some applications2005

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      Sugaku Expositions, American Mathematical Society vol. 18

      Pages: 1-37

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] 非リーマン等質空間の不連続群について(論説)2005

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      数学 第57巻

      Pages: 267-281

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Compact Clifford-Klein forms of symmetric spaces - revisited2005

    • Author(s)
      Toshiyuki Kobayashi, Taro Yoshino
    • Journal Title

      Pure and Applied Mathematics Quarterly, Special Issue : In Memory of Armand Borel vol. 1

      Pages: 603-684

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] O(p, 2)の極小表現と反転の積分表示2005

    • Author(s)
      小林俊行, 真野元
    • Journal Title

      数理解析研究所講究録(RIMS Kokyuroku),「表現論および等質空間上の調和解析」(2004年8月9-12日,井上順子 編) vol. 1410

      Pages: 173-187

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Fourier transform of a minimal K-type vector in the minimal representation of O(p+1,q+1)2005

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      数理解析研究所講究録(RIMS Kokyuroku),「Sp(2, R)とSU(2, 2)上の保型形式,III」(2004年9月28日-10月1日,織田孝幸 編) vol. 1421

      Pages: 1-11

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Invariant multipliers and O(p,q)-action2005

    • Author(s)
      Toshiyuki Kobayashi, Andreas Nilsson
    • Journal Title

      表現論シンポジウム講演集(2005年11月15-18日,掛川,青木茂,加藤末広,織田寛 編)

      Pages: 10-21

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] 重複度1の表現と複素多様体上の可視的な作用2005

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      表現論シンポジウム講演集(2005年11月15-18日,掛川,青木茂,加藤末広,織田寛 編)

      Pages: 33-66

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A quantization of conjugacy classes of matrices2005

    • Author(s)
      Toshio Oshima
    • Journal Title

      Adv. Math. 196

      Pages: 124-146

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Integral formulas for the minimal representations for O(p, 2)2005

    • Author(s)
      Toshiyuki Kobayashi, Gen Mano
    • Journal Title

      Acta Applicandae Mathematicae vol.83

      Pages: 103-113

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Multiplicity-free representations and visible actions on complex manifolds2005

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      Publ.Res.Inst.Math.Sci., Special issue commemorating the fortieth anniversary of the founding of RIMS vol.41

      Pages: 497-549

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Theory of discrete decomposable branching laws of unitary representations of semisimple Lie groups and some applications2005

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      Sugaku Expositions (American Mathematics Society) vol.18

      Pages: 1-37

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Compact Clifford-Klein forms of symmetric spaces - revisited2005

    • Author(s)
      Toshiyuki Kobayashi, Taro Yoshino
    • Journal Title

      Pure and Applied Mathematics Quarterly, Special Issue : In Memory of Armand Borel vol.1

      Pages: 603-684

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Fourier transform of a minimal K-type vector in the minimal representation of O(p+1,q+1)2005

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      RIMS Kokyuroku, Automorphic forms on Sp(2,R) and SU(2,2), (T.Oda, ed.) vol.1421

      Pages: 1-11

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Invariant multipliers and O(p,q)-action2005

    • Author(s)
      Toshiyuki Kobayashi, Andreas Nilsson
    • Journal Title

      Proceedings of Symposium of Representation Theory, held at Kakegawa, November 2005 (S.Aoki, S.Kato, H.Oda, eds.)

      Pages: 10-21

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A quantization of conjugacy classes of matrices2005

    • Author(s)
      Toshio Oshima
    • Journal Title

      Adv.Math. 196

      Pages: 124-146

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Restrictions of unitary representations of real reductive groups2005

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      Lie Theory : Unitary Representations and Compactifications of Symmetric Spaces, Progress in Mathematics, ISBN 0-8176-3526-2 (J.-P.Anker and B.Orsted, editors) (Birkhauser) vol.229

      Pages: 139-207

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Geometry of multiplicity-free representations of GL(n), visible actions on flag varieties, and triunity2004

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      Acta Applicandae Mathematicae vol. 81

      Pages: 129-146

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Geometry of multiplicity-free representations of GL(n), visible actions on flag varieties, and triunity2004

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      Acta Applicandae Mathematicae vol.81

      Pages: 129-146

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Analysis on minimal representations of O(p,q), I. Realization and conformal geometry2003

    • Author(s)
      Toshiyuki Kobayashi, Bent Orsted
    • Journal Title

      Advances in Mathematics vol. 180

      Pages: 486-512

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Analysis on minimal representations of O(p,q), II. Branching laws2003

    • Author(s)
      Toshiyuki Kobayashi, Bent Orsted
    • Journal Title

      Advances in Mathematics vol. 180

      Pages: 513-550

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Analysis on minimal representations of O(p,q), III. Ultra-hyperbolic equations on Rp-1, q-12003

    • Author(s)
      Toshiyuki Kobayashi, Bent Orsted
    • Journal Title

      Advances in Mathematics vol. 180

      Pages: 551-595

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Conformal geometry and global solutions to the Yamabe equations on classical pseudo-Riemannian manifolds2003

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      Rendiconti del Circolo Matematico di Palermo, Serie II vol. 71

      Pages: 15-40

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Multiplicity one theorem in the orbit method2003

    • Author(s)
      Toshiyuki Kobayashi, Salma Nasrin
    • Journal Title

      Amer. Math. Soc. Transl., Advances in the Mathematical Sciences, Series 2, Special volume in memory of Professor F. Karpelevic vol. 210

      Pages: 161-169

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] O(p,q)の極小ユニタリ表現のシュレディンガーモデル2003

    • Author(s)
      小林俊行
    • Journal Title

      数理解析研究所講究録(RIMS Kokyuroku),「IV型対称領域上の保型形式の研究」(2002年12月24-26日,織田孝幸 編) vol. 1342

      Pages: 107-116

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Multiplicity one theorem on branching laws and geometry of complex manifolds2003

    • Author(s)
      小林俊行
    • Journal Title

      数理解析研究所講究録(RIMS Kokyuroku),「Lie Theoryのひろがりと新たな進展」(2003年7月22-25日,有木進 編) vol. 1348

      Pages: 1-9

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Characterizing multipliers by relative invariance2003

    • Author(s)
      Toshiyuki Kobayashi, Andreas Nilsson
    • Journal Title

      数理解析研究所講究録(RIMS Kokyuroku),「Lie Theory のひろがりと新たな進展」(2003年7月22-25日,有木進 編) vol. 1348

      Pages: 10-22

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] 非リーマン等質空間の不連続群について2003

    • Author(s)
      小林俊行
    • Journal Title

      日本数学会秋季総合分科会,幾何学分科会特別講演要旨(2003年9月25日,千葉大学)

      Pages: 75-89

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A calculation of c-functions for semisimple symmetric spaces2003

    • Author(s)
      Toshio Oshima
    • Journal Title

      Amer. Math. Soc. Transl., Advances in the Mathematical Sciences, Series 2, Special volume in memory of Professor F. Karpelevic vol. 210

      Pages: 307-330

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Analysis on minimal representations of O(p,q), I. Realization and conformal geometry2003

    • Author(s)
      Toshiyuki Kobayashi, Bent Orsted
    • Journal Title

      Advances in Mathematics vol.180

      Pages: 486-512

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Analysis on minimal representations of O(p,q), II. Branching laws2003

    • Author(s)
      Toshiyuki Kobayashi, Bent Orsted
    • Journal Title

      Advances in Mathematics vol.180

      Pages: 513-550

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Analysis on minimal representations of O(p,q), III. Ultra-hyperbolic equations on Rp-1, q-12003

    • Author(s)
      Toshiyuki Kobayashi, Bent Orsted
    • Journal Title

      Advances in Mathematics vol.180

      Pages: 551-595

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Conformal geometry and global solutions to the Yamabe equations on classical pseudo-Riemannian manifolds2003

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      Rendiconti del Circolo Matematico di Palermo, Serie II vol.71

      Pages: 15-40

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Multiplicity one theorem in the orbit method2003

    • Author(s)
      Toshiyuki Kobayashi, Salma Nasrin
    • Journal Title

      Amer.Math.Soc.Transl., Advances in the Mathematical Sciences, Series 2, Special volume in memory of Professor F.Karpelevic vol.210

      Pages: 161-169

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Multiplicity one theorem on branching laws and geometry of complex manifolds2003

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      RIMS Kokyuroku, Expansion of Lie Theory and New Advances (S.Ariki, ed.) vol.1348

      Pages: 1-9

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Characterizing multipliers by relative invariance2003

    • Author(s)
      Toshiyuki Kobayashi, Andreas Nilsson
    • Journal Title

      RIMS Kokyuroku, Expansion of Lie Theory and New Advances (S.Ariki, ed.) vol.1348

      Pages: 10-22

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A calculation of c-functions for semisimple symmetric spaces2003

    • Author(s)
      Toshio Oshima
    • Journal Title

      Amer.Math.Soc.Transl., Advances in the Mathematical Sciences, Series 2, Special volume in memory of Professor F.Karpelevic vol.210

      Pages: 307-330

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Branching problems of unitary representations2002

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      Proceedings of ICM 2002, Beijing vol. 2

      Pages: 615-627

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Introduction to actions of discrete groups on pseudo-Riemannian homogeneous manifolds2002

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      Acta Applicandae Mathematicae vol. 73

      Pages: 115-131

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] 山辺作用素の解空間における共形不変な内積について2002

    • Author(s)
      小林俊行
    • Journal Title

      微分幾何学研究集会「種々の幾何構造の発展」講演要旨集

      Pages: 4-5

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] 山辺作用素の解空間における不変な内積について2002

    • Author(s)
      小林俊行
    • Journal Title

      数理解析研究所講究録(RIMS Kokyuroku)「可換代数系の表現と調和解析」(2002年7月23-26日,太田琢也 編) vol. 1294

      Pages: 76-86

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Branching problems of unitary representations2002

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      表現論シンポジウム講演集(2002年11月12-15日,富士ハイツ,佐野茂 編)

      Pages: 39-52

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Branching problems of unitary representations2002

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      Proceedings of ICM 2002, Beijing vol.2

      Pages: 615-627

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Introduction to actions of discrete groups on pseudo-Riemannian homogeneous manifolds2002

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      Acta Applicandae Mathematicae vol.73

      Pages: 115-131

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Branching problems of unitary representations2002

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      Proceedings of Symposium of Representation Theory, held at Fuji Haitsu, November 2002 (S.Sano, ed.)

      Pages: 39-52

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Deformation of properly discontinuous actions of Z^k on R^<k+1>

    • Author(s)
      Toshiyuki Kobayashi, Salma Nasrin
    • Journal Title

      International Journal of Mathematics (to appear in )

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On discontinuous group actions on non-Riemannian homogeneous spaces

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      Sugaku Expositions, American Mathematical Society (to appear in )

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Deformation of properly discontinuous actions of Z^k on R^<k+1>

    • Author(s)
      Toshiyuki Kobayashi, Salma Nasrin
    • Journal Title

      International Journal of Mathematics. (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On discontinuous group actions on non-Riemannian homogeneous spaces

    • Author(s)
      Toshiyuki Kobayashi
    • Journal Title

      Sugaku Expositions (American Mathematical Society). (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Book] リー群と表現論,ISBN 4-00-006142-92005

    • Author(s)
      小林俊行, 大島利雄
    • Total Pages
      610
    • Publisher
      岩波書店
    • Description
      「研究成果報告書概要(和文)」より
  • [Book] Lie Theory : Unitary Representations and Compactifications of Symmetric Spaces, Progress in Mathematics, vol. 229, ISBN 0-8176-3526-2(Restrictions of unitary representations of real reductive groups. In J.-P. Anker and B. Orsted, editors)2005

    • Author(s)
      Toshiyuki Kobayashi
    • Total Pages
      139-207
    • Publisher
      Birkhauser
    • Description
      「研究成果報告書概要(和文)」より
  • [Book] 表現論シンポジウム講演集 平成16年度 (2004年11月16-19日,淡路島), ISBN 4-9902328-0-12004

    • Author(s)
      小林俊行, 落合啓之, 田川裕之(編)
    • Total Pages
      164
    • Description
      「研究成果報告書概要(和文)」より
  • [Book] 非リーマン等質空間の不連続群論 『数学の最先端 21世紀への挑戦』,第1巻,ISBN 4-431-70962-22002

    • Author(s)
      小林俊行
    • Total Pages
      18-73
    • Publisher
      シュプリンガー・フェアラーク東京
    • Description
      「研究成果報告書概要(和文)」より

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Published: 2007-12-13  

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