無限次元表現の分岐則と大域解析

• Project/Area Number: 18H03669
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Review Section: Medium-sized Section 12:Analysis, applied mathematics, and related fields
• Research Institution: The University of Tokyo
• Principal Investigator: 小林 俊行 東京大学, 大学院数理科学研究科, 教授 (80201490)
• Co-Investigator(Kenkyū-buntansha): 関口 英子 東京大学, 大学院数理科学研究科, 准教授 (50281134)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Project Status: Granted (Fiscal Year 2022)
• Budget Amount: ¥42,640,000 (Direct Cost: ¥32,800,000、Indirect Cost: ¥9,840,000); Fiscal Year 2022: ¥9,230,000 (Direct Cost: ¥7,100,000、Indirect Cost: ¥2,130,000); Fiscal Year 2021: ¥8,450,000 (Direct Cost: ¥6,500,000、Indirect Cost: ¥1,950,000); Fiscal Year 2020: ¥8,450,000 (Direct Cost: ¥6,500,000、Indirect Cost: ¥1,950,000); Fiscal Year 2019: ¥8,450,000 (Direct Cost: ¥6,500,000、Indirect Cost: ¥1,950,000); Fiscal Year 2018: ¥8,060,000 (Direct Cost: ¥6,200,000、Indirect Cost: ¥1,860,000)
• Keywords: 解析学 / 幾何学 / 表現論 / リー群 / 分岐則 / 不連続群

Outline of Annual Research Achievements

当該研究代表者が主導している「分岐則の理論」「緩増加等質空間の理論」のプロジェクトを海外の共同研究者等と協力して研究を進めた。以下で本年度の主要な結果を述べる。
(1)（離散的分岐則の理論） 無限次元の既約ユニタリ表現を非コンパクトな簡約部分群に制限するとその分解には連続スペクトラムが現れる場合もあり、また離散的になる場合もある。離散的になるためには、admissible restrictionという概念が鍵となるが、それを判定するという重要な問いに関して、超局所解析の手法で証明した研究代表者の主定理（Annals of Mathematics 1998)の逆が成り立つことを、シンプレクティック幾何の手法を用いて証明した（単著 第１論文)。
(2) (ホログラフィック作用素の構成）研究代表者が導入した「対称性破れ作用素」の随伴作用素として、海外の共同研究者と協力して「ホログラフィック作用素」の概念を導入し、いくつかの基礎的なケースでホログラフィック作用素を明示的に構成し、元の関数の復元公式を証明した（第５論文）。
(3)(緩増加等質空間の分類理論）二乗可積分函数全体のなすヒルベルト空間上に定義されたユニタリ表現が緩増加となる等質空間G/Hを緩増加等質空間という。GとHが簡約リー群の場合、緩増加等質空間G/Hを分類した。その手法は、研究代表者と海外の共同研究者が開発した緩増加性の判定条件を代数的に分析して線型化を行い、次に半単純リー環の有限次元表現に関連する種々の不等式を証明し、さらに凸多面体の組合せ論を用いるというものである。分類を完成し長編の論文を出版した（第３論文）。
(4)その他、保形型式の整数論への応用を意識した分岐則の例（第２論文）や小さい無限次元表現の分岐則における重複度に関する研究を推し進めた（第４論文）。

Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

いろいろな困難はあったが、対面での国際会議や共同研究も少しずつ再開し、予定していた諸課題についてある程度の成果があげられた。今後、これらを基礎に大きく発展させられそうである

Strategy for Future Research Activity

対面での国際会議、対面での会合を積極的に企画し最先端の研究を意識しつつ、適切な課題を選別し、当該研究代表者が主導している
（１）Symmetry Breaking Operatorsの解析
（２）非リーマン局所対称空間上のスペクトル解析
のプロジェクトをフランス、アメリカの研究者等と協力してさらに活発な研究活動を展開したい。特に（１）として、「対称性破れ作用素」を統合的に扱う generating operatorsの概念を導入し、その基盤に関する国際共同研究を行う。（２）の基盤作りとして、非対称な擬リーマン等質空間までを視野に入れた不連続群の作用と軌道の数え上げ問題に焦点を当てて研究を進める。国際連携を保持・強化し、先端研究の情報交換を行い、新しい研究活動が遭遇することによってさらに飛躍が生まれる可能性が見込める場を積極的に作る。

Research Products

• [Int'l Joint Research] IHES/Paris Sud University/Reims University(フランス)
• [Int'l Joint Research] Cornell University/Columbia University(米国)
• [Int'l Joint Research] IHES/Reims University/Paris Sud University(フランス)
• [Int'l Joint Research] Cornell University(米国)
• [Int'l Joint Research] IHES/Universite de Reims/Orsay, Paris-Sud University(フランス)
• [Int'l Joint Research] Cornell University/Yale University/Rutgers Univerity(米国)
• [Int'l Joint Research] Weizmann Institute of Science(イスラエル)
• [Int'l Joint Research] Dhaka University(バングラデシュ)
• [Int'l Joint Research] Charles University(チェコ)
• [Int'l Joint Research] IHES/Reims University/Orsay, Paris-Sud University(フランス)
• [Int'l Joint Research] Dhaka University(バングラデシュ)
• [Int'l Joint Research] Cornell University/Yale University(米国)
• [Int'l Joint Research] Aarhus University(デンマーク)
• [Int'l Joint Research] Charles University(チェコ)
• [Journal Article] Multiplicity in restricting small representations (2022)
  • Author(s): Kobayashi Toshiyuki
  • Journal Title: Proceedings of the Japan Academy, Series A, Mathematical Sciences Volume: 98 Issue: 3 Pages: 19-24
  • DOI: 10.3792/pjaa.98.004
  • Peer Reviewed / Open Access
• [Journal Article] Admissible restrictions of irreducible representations of reductive Lie groups: symplectic geometry and discrete decomposability (2021)
  • Author(s): Kobayashi Toshiyuki
  • Journal Title: Pure and Applied Mathematics Quarterly Volume: 17 Issue: 4 Pages: 1321-1343
  • DOI: 10.4310/pamq.2021.v17.n4.a5
  • Peer Reviewed
• [Journal Article] Distinguished Representations of SO(n+1, 1)×SO(n, 1), Periods and Branching Laws (2021)
  • Author(s): Kobayashi Toshiyuki、Speh Birgit
  • Journal Title: Relative Trace Formulas, Simons Symposia, Volume: NA Pages: 291-320
  • DOI: 10.1007/978-3-030-68506-5_8
  • ISBN: 9783030685058, 9783030685065
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Tempered homogeneous spaces III (2021)
  • Author(s): Toshiyuki Kobayashi
  • Journal Title: Journal of Lie Theory Volume: 31 Pages: 833-869
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Inversion of Rankin-Cohen operators via Holographic Transform (2021)
  • Author(s): Kobayashi Toshiyuki、Pevzner Michael
  • Journal Title: Annales de l'Institut Fourier Volume: 70 Issue: 5 Pages: 2131-2190
  • DOI: 10.5802/aif.3386
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] 緩増加な等質空間 (2021)
  • Author(s): Toshiyuki Kobayashi
  • Journal Title: 日本数学会年会函数解析学分科会 Volume: NA Pages: 1-14
• [Journal Article] Global analysis on manifolds and representation theory of reductive groups (2020)
  • Author(s): Toshiyuki Kobayashi
  • Journal Title: RIMS Kokyuroku Volume: 2139 Pages: 1-20
  • Open Access
• [Journal Article] Spectral analysis on pseudo-Riemannian locally symmetric spaces (2020)
  • Author(s): Kassel Fanny、Kobayashi Toshiyuki
  • Journal Title: Proceedings of the Japan Academy, Series A, Mathematical Sciences Volume: 96 Issue: 8 Pages: 69-74
  • DOI: 10.3792/pjaa.96.013
  • NAID: 40022388988
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Topics on Global Analysis of Manifolds and Representation Theory of Reductive Groups (2020)
  • Author(s): Kobayashi Toshiyuki
  • Journal Title: Springer Proceedings in Mathematics & Statistics Volume: 335 Pages: 3-13
  • DOI: 10.1007/978-981-15-7775-8_1
  • ISBN: 9789811577741, 9789811577758
  • Peer Reviewed / Open Access
• [Journal Article] A Hidden Symmetry of a Branching Law (2020)
  • Author(s): Kobayashi Toshiyuki、Speh Birgit
  • Journal Title: Springer Proceedings in Mathematics & Statistics Volume: 15-28 Pages: 15-28
  • DOI: 10.1007/978-981-15-7775-8_2
  • ISBN: 9789811577741, 9789811577758
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] 表現の分岐則の最近の進展--日本数学会70周年記念特別企画 (2019)
  • Author(s): 小林俊行
  • Journal Title: 数学 Volume: 74 Pages: 388-416
  • Peer Reviewed
• [Journal Article] Invariant differential operators on spherical homogeneous spaces with overgroups (2019)
  • Author(s): Fanny Kassel and Toshiyuki Kobayashi
  • Journal Title: Journal of Lie Theory Volume: 29 Pages: 663-754
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] 多様体上の大域解析と簡約群の表現論 (2019)
  • Author(s): 小林俊行
  • Journal Title: 数理解析研究所講究録 Volume: 2139 Pages: 1-10
  • Open Access
• [Journal Article] Conformal Symmetry Breaking on Differential Forms and Some Applications (2019)
  • Author(s): Kobayashi Toshiyuki
  • Journal Title: Trends in Mathematics Volume: NA Pages: 289-308
  • DOI: 10.1007/978-3-030-01156-7_32
  • ISBN: 9783030011550, 9783030011567
  • Peer Reviewed
• [Journal Article] Double Gegenbauer expansion of |s-t|^α (2019)
  • Author(s): Kobayashi T.、Leontiev A.
  • Journal Title: Integral Transforms and Special Functions Volume: 30 Issue: 7 Pages: 512-525
  • DOI: 10.1080/10652469.2019.1585433
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Residue formula for regular symmetry breaking operators (2018)
  • Author(s): Kobayashi Toshiyuki
  • Journal Title: Contemporary Mathematics, Amer. Math. Soc. Volume: 714 Pages: 175-197
  • DOI: 10.1090/conm/714/14380
  • Peer Reviewed
• [Journal Article] Symmetry Breaking for Orthogonal Groups and a Conjecture by B. Gross and D. Prasad (2018)
  • Author(s): Kobayashi Toshiyuki、Speh Birgit
  • Journal Title: Simons Symposium on the Trace Formula Volume: - Pages: 245-266
  • DOI: 10.1007/978-3-319-94833-1_8
  • ISBN: 9783319948324, 9783319948331
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Global geometry and analysis of locally symmetric spaces with indefiite metric (2018)
  • Author(s): Kobayashi Toshiyuki
  • Journal Title: Proceedings of the 65th Geometry Symmposium Volume: - Pages: 1-10
• [Journal Article] Image of Conformally Covariant Symmetry Breaking Operators for R^{p,q} (2018)
  • Author(s): Kobayashi Toshiyuki、Leontiev Alex
  • Journal Title: Springer Proceedings in Mathematics & Statistics Volume: 263 Pages: 3-35
  • DOI: 10.1007/978-981-13-2715-5_1
  • ISBN: 9789811327148, 9789811327155
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Global geometry and analysis on pseudo-Riemannian locally symmetric spaces (2018)
  • Author(s): Kobayashi Toshiyuki
  • Journal Title: Proceedings of the Symposium on Representation Theory Volume: - Pages: 89-98
  • NAID: 130008005049
• [Journal Article] Geometry of Coadjoint Orbits and Multiplicity-one Branching Laws for Symmetric Pairs (2018)
  • Author(s): Kobayashi Toshiyuki、Nasrin Salma
  • Journal Title: Algebras and Representation Theory Volume: 21 Issue: 5 Pages: 1023-1036
  • DOI: 10.1007/s10468-018-9810-8
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Conformal Symmetry Breaking Operators for Anti-de Sitter Spaces (2018)
  • Author(s): Kobayashi Toshiyuki、Kubo Toshihisa、Pevzner Michael
  • Journal Title: Trends in Mathematis Volume: NA Pages: 75-91
  • DOI: 10.1007/978-3-319-63594-1_9
  • ISBN: 9783319635934, 9783319635941
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] A certain integral formula containing two Gegenbauer polynoimals (2018)
  • Author(s): Kobayashi Toshiyuki、Leontiev Alex
  • Journal Title: RIMS Kokyuroku Volume: 2077 Pages: 22-35
  • Open Access / Int'l Joint Research
• [Journal Article] 解説：リー群の表現論における最近の進展 (2018)
  • Author(s): Kobayashi Toshiyuki
  • Journal Title: 『ユニタリ表現入門』 Volume: - Pages: 214-242
• [Journal Article] Symmetry breaking operators for orthogonal groups O(n,1) (2018)
  • Author(s): Kobayashi Toshiyuki
  • Journal Title: Oberwolfach Reports Volume: 14 Pages: 1572-1575
  • Open Access
• [Presentation] Limit algebras and tempered representations (2021)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: Representation Theory & Noncommutative Geometry. AIM Research Community
  • Int'l Joint Research / Invited
• [Presentation] Limit algebras and tempered representation (2021)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: Lie Groups and Representation Theory Seminar. The University of Tokyo
• [Presentation] Limit Algebras and Tempered Representations (2021)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: XIV. International Workshop: Lie Theory and Its Applications in Physics. Bulgaria
  • Int'l Joint Research / Invited
• [Presentation] Sound of an anti-de Sitter manifold. (2021)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: Inaugural Day of the French-Kazakhstan school of Mathematics, Kazakhstan
  • Int'l Joint Research / Invited
• [Presentation] Limit Algebras and Tempered Representations (2021)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: RIMS Workshop: Lie Theory, Representation Theory and Related Areas
  • Invited
• [Presentation] Global Analysis of Locally Symmetric Spaces with Indefinite-metric (2021)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: Colloquium, National University of Singapore, Singapore
  • Int'l Joint Research / Invited
• [Presentation] Holographic Transform (2021)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: Workshop on Actions of Reductive Groups and Global Analysis
• [Presentation] Tempered Representations and Limit Algebras (2021)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: Research Workshop: Seminar in Representation Theory, Denmark
  • Int'l Joint Research / Invited
• [Presentation] Bounded Multiplicity in the Branching Problems of ''small” Infinite-dimensional Representations (2021)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: Lie Groups and Representation Theory Seminar at the University of Tokyo
• [Presentation] A Foundation of Group-theoretic Analysis on Manifolds. (2021)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: Colloquium di dipartimento. Dipartimento di Matematica, Universita di Roma ''Tor Vergata”
  • Int'l Joint Research / Invited
• [Presentation] Tempered Homogeneous Spaces (2021)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: 日本数学会 関数解析分科会 特別講演
  • Invited
• [Presentation] A Program for Branching Problems in the Representationtheory of Real Reductive Groups: Classification Problem of Symmetry Breaking Operators (2020)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: Representation Theory inspired by the Langlands Conjectures, USA
  • Int'l Joint Research / Invited
• [Presentation] Regular Representations on Homogeneous Spaces (2020)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: Representation Theory of Reductive Groups from Geometric and Analytic Methods
  • Int'l Joint Research / Invited
• [Presentation] Global Analysis of Locally Symmetric Spaces with Indefinite-metric (2019)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: Colloquium. Yale University
  • Invited
• [Presentation] Branching Problems and Symmetry Breaking Operators. (2019)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: Geometry, Symmetry and Physics. Yale University
  • Invited
• [Presentation] Global Analysis of Locally Symmetric Spaces with Indefinite metric (2019)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: Colloquium. Oklahoma State University
  • Invited
• [Presentation] Regular Representations on Homogeneous Spaces (2019)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: Dynamics of Group Actions: a conference in honor of Yves Benoist
  • Int'l Joint Research / Invited
• [Presentation] Global Analysis of Locally Symmetric Spaces with Indefinite-metric (2019)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: Seminar. University of Padova
  • Invited
• [Presentation] Regular Representations on Homogeneous Spaces (opening lplenary ecture) (2019)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: Lie Theory and Its Applications in Physics (LT-13). Bulgaria
  • Int'l Joint Research / Invited
• [Presentation] Symmetry Breaking Operators: General Theory and Concrete Construction for Reductive Groups. (opening lecture). (2019)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: Representation Theory of Lie Groups, Mathematical Physics, and Dynamical Systems. France
  • Int'l Joint Research / Invited
• [Presentation] Regular Representations on Homogeneous Spaces. (opening lecture) (2019)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: Developments in Representation Theory and Related Topics, Japan
  • Invited
• [Presentation] Introduction to Representation Theory of Real Reductive Lie Groups and Branching Problems. (opening lecture). (2019)
  • Author(s): Toshiyuki Kobayashi
  • Organizer: The 2nd International Undergraduate Mathematics Summer School
  • Int'l Joint Research / Invited
• [Presentation] Homomorphisms between Verma Modules and Symmetry Breaking Operators. Representation Theory of Reductive Lie Groups and Algebras (2019)
  • Author(s): Kobayashi Toshiyuki
  • Organizer: Representation Theory of Reductive Lie Groups and Algebras: in honor of Hisayosi Matumoto on the occasion of his 60th birthday. the University of Tokyo, Japan, 27-29 March 2019.
  • Int'l Joint Research / Invited
• [Presentation] Branching Problems and Symmetry Breaking Operators (2018)
  • Author(s): Kobayashi Toshiyuki
  • Organizer: Representation theory, geometry, and quantization: the mathematical legacy of Bertram Kostant. MIT, USA
  • Int'l Joint Research / Invited
• [Presentation] Global Geometry and Analysis on Locally Symmetric Spaces with Indefinite-metric (2018)
  • Author(s): Kobayashi Toshiyuki
  • Organizer: Glances at Manifolds: Conference in celebration of Aleksy Tralle for his 60th birthday. the Jagiellonian University, Krakow, Poland
  • Int'l Joint Research / Invited
• [Presentation] Branching Problems and Symmetry Breaking Operators (2018)
  • Author(s): Kobayashi Toshiyuki
  • Organizer: the 32nd International Colloquium on Group Theoretical Methods in Physics (Group32), Czech Republic
  • Int'l Joint Research / Invited
• [Presentation] Branching Problems and Symmetry Breaking (2018)
  • Author(s): Kobayashi Toshiyuki
  • Organizer: Conference in honor of Joachim Hilgert. Paderborn, Germany,
  • Int'l Joint Research / Invited
• [Presentation] Global Geometry and Analysis on Locally Symmetric Spaces with Indefinite-metric (2018)
  • Author(s): Kobayashi Toshiyuki
  • Organizer: 第65回幾何学シンポジウム
  • Invited
• [Presentation] Global Geometry and Analysis on Locally Symmetric Spaces with Indefinite-metric (2018)
  • Author(s): Kobayashi Toshiyuki
  • Organizer: “Geometric Quantization and Applications” , CIRM Fance
  • Int'l Joint Research / Invited
• [Presentation] From Local to Global-Geometry and Analysis on Locally Symmetric Spaces with Indefinite-metric (2018)
  • Author(s): Kobayashi Toshiyuki
  • Organizer: Colloquium. Hiroshima University, Japan
  • Invited
• [Presentation] Global Geometry and Analysis on Pseudo-Riemannian Locally Symmetric Spaces (2018)
  • Author(s): Kobayashi Toshiyuki
  • Organizer: Symposium on Representation Theory 2018. Tottori, Japan, 13-16 November 2018.
  • Int'l Joint Research / Invited
• [Presentation] Semisimple Symmetric Spaces and Discontinuous Groups: What I Learned from Professor Toshio Oshima (2018)
  • Author(s): Kobayashi Toshiyuki
  • Organizer: Conference in honour of Professor Toshio Oshima's 70th birthday. Josai University, Tokyo, Japan,
  • Invited
• [Book] Symmetry Breaking for Representations of Rank One Orthogonal Groups II (2018)
  • Author(s): Kobayashi Toshiyuki, Birgit Speh
  • Publisher: Springer-Nature
  • ISBN: 9789811329012
• [Remarks] Home Page of Toshiyuki Kobayashi
  • URL: https://www.ms.u-tokyo.ac.jp/~toshi/index.html
• [Remarks] Home Page of Toshiyuki Kobayashi
• [Remarks] Home Page of Toshiyuki Kobayashi
• [Remarks] Kavli IPMU カブリ数物連携宇宙研究機構
  • URL: https://www.ipmu.jp/en/publication/2018
• [Funded Workshop] Geometry, Analysis, and Representation Theory of Lie Groups (2022)
• [Funded Workshop] Geometry, Analysis, and Representation Theory of Lie Groups (2022)
• [Funded Workshop] Representation Theory of Reductive Groups from Geometric and Analytic Methods (2020)
• [Funded Workshop] 第23回高木レクチャー (2019)
• [Funded Workshop] 第24回高木レクチャー (2019)
• [Funded Workshop] 第21回高木レクチャー (2018)
• [Funded Workshop] 表現論シンポジウム (2018)