Transformation groups for geometric structures, global geometric analysis, and theory of branching laws of infinite dimensional representations

2009 Fiscal Year Final Research Report

• Project/Area Number: 18340037
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: The University of Tokyo (2007-2009); Kyoto University (2006)
• Principal Investigator: KOBAYASHI Toshiyuki The University of Tokyo, 大学院・数理科学研究科, 教授 (80201490)
• Co-Investigator(Kenkyū-buntansha): OSHIMA Toshio (50011721) ; SEKIGUCHI Hideko (50281134)
• Project Period (FY): 2006 – 2009
• Keywords: ユニタリ表現 / リー群 / 極小表現 / 無重複表現 / 不連続群 / 分岐則 / 冪零軌道 / 可視的作用

Research Abstract

Minimal representations are building blocks of unitary representations. During this period, we established the Schrodinger model of minimal representations of the indefinite orthogonal group, and determined a closed formula of the unitary inversion operator on the L^2-model on the isotropic cones, that generalizes the Euclidean Fourier transform. A new deformation theory was introduced in [1]. Further, I made systematic and synthetic applications of the original theory of visible actions on complex manifolds to multiplicity-free theorems, in particular, branching problems to symmetric pairs.

Research Products

• [Journal Article] Generalized Fourier transforms F_{k, a} (2009)
  • Author(s): T. Kobayashi, S. Ben Said, B. Orsted
  • Journal Title: C. R. Math. Acad. Sci. Paris 347 (2009) Pages: 1119-1124
  • Peer Reviewed
• [Journal Article] Composition formulas in the Weyl calculus (2009)
  • Author(s): T. Kobayashi, B. Orsted, M. Pevzner, A. Unterberger
  • Journal Title: J.Funct.Anal. 257 Pages: 948-991
  • Peer Reviewed
• [Journal Article] Indefinite higher Riesz transforms (2009)
  • Author(s): T. Kobayashi, A. Nilsson
  • Journal Title: Arkiv for Matematik 47 Pages: 331-344
  • Peer Reviewed
• [Journal Article] Group invariance and L^p-bounded operators (2008)
  • Author(s): T. Kobayashi, A. Nilsson
  • Journal Title: Math. Z. 260 Pages: 335-354
  • Peer Reviewed
• [Journal Article] Visible actions on symmetric spaces (2007)
  • Author(s): T. Kobayashi
  • Journal Title: Transformation Groups 12 Pages: 671-694
  • Peer Reviewed
• [Journal Article] Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs, Representation Theory and Automorphic Forms (2007)
  • Author(s): T. Kobayashi
  • Journal Title: Progr.Math. vol. 255 Pages: 45-109
  • Peer Reviewed
• [Journal Article] The inversion formula and holomorphic extension of the minimal representation of the conformal group, Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory : In Honour of Roger E. Howe (2007)
  • Author(s): T. Kobayashi, G. Mano
  • Journal Title: Singapore University Press and World Scientific Publishing Pages: 159-223
  • Peer Reviewed
• [Journal Article] Generalized Bernstein-Reznikov integrals
  • Author(s): T. Kobayashi, J. -L. Clerc, B. Orsted, M. Pevzner
  • Journal Title: Mathematische Annalen (published on line first) 37 Pages: arXiv : 0906.2874
  • Peer Reviewed
• [Presentation] Branching Problems for Zuckerman's Derived Functor Modules (2009)
  • Author(s): T. Kobayashi
  • Organizer: Representation Theory and Mathematical Physics : (Gregg Zuckerman 教授還暦記念研究集会)
  • Place of Presentation: Yale University, USA
  • Year and Date: 20091024-20091027
• [Presentation] Geometric Analysis on Minimal Representations (2009)
  • Author(s): T. Kobayashi
  • Organizer: Representation Theory of Real Reductive Groups (organized by Jeffrey Adams, Susana Salamanca, John Stembridge, Peter Trapa and David Vogan)
  • Place of Presentation: University of Utah, Salt Lake City, USA
  • Year and Date: 20090727-20090731
• [Presentation] Conformal Geometry and Schrodinger Model of Minimal Representations (2009)
  • Author(s): T. Kobayashi
  • Organizer: (2 lectures), Representation Theory XI
  • Place of Presentation: Dubrovnik, Croatia
  • Year and Date: 20090614-20090621
• [Presentation] Conformal Geometry, Schrodinger Model of Minimal Representations, and Deformation of Fourier Transforms 83eme Rencontre entre physiciens theoriciens et mathematiciens (Encounter between theoretical physicists and mathematicians) (2009)
  • Author(s): T. Kobayashi
  • Organizer: Institut de Recherche Mathematique Avancee
  • Place of Presentation: Strasbourg, France
  • Year and Date: 20090611-20090613
• [Presentation] Discontinuous Groups on pseudo-Riemannian Spaces. Mathematishe Arbeitstagung 2009 (2009)
  • Author(s): T. Kobayashi
  • Organizer: Max-Planck Institut fur Mathematik
  • Place of Presentation: Bonn, Germany
  • Year and Date: 20090605-20090611
• [Presentation] Confromal Geometry and Schrodinger Model of Minimal Representations. (opening lecture) (2009)
  • Author(s): T. Kobayashi
  • Organizer: (B. Orsted 教授還暦記念研究集会) Representations Lie groups, and Conformal Geometry
  • Place of Presentation: Gottingen, Germany
  • Year and Date: 20090406-20090410
• [Presentation] Global Geometry and Analysis on Locally Symmetric Spaces - Beyond the Riemannian case (2009)
  • Author(s): T. Kobayashi
  • Organizer: colloquium
  • Place of Presentation: Yale University, USA
  • Year and Date: 2009-03-25
• [Presentation] Global Geometry on Locally Symmetric Spaces - beyond the Riemannian case (2008)
  • Author(s): T. Kobayashi
  • Organizer: Representations of Lie groups and applications. Institut Henri Poincare
  • Place of Presentation: Paris, France
  • Year and Date: 20081215-20081218
• [Presentation] Holomorphic semigroups for minimal representations of conformal groups. Harmonic Analysis (2008)
  • Author(s): T. Kobayashi
  • Organizer: Operator Algebras and Representation Theory (organized by J. Ludwig and A. To-Ming Lau)
  • Place of Presentation: CIRM, Luminy, France
  • Year and Date: 20081103-20081107
• [Presentation] Visible Actions and Multiplicity-free Representations. (4 lectures) (2008)
  • Author(s): T. Kobayashi
  • Organizer: Functional Analysis X : Representation Theory (organized by D. Adamovic, D. Bakic, D. Butkovic, H. Kraljevic, G. Muic, M. Prime, M. Rao, D. Renard, H. Sikic and D. Vogan).
  • Place of Presentation: Dubrovnik, Croatia
  • Year and Date: 20080630-20080705
• [Presentation] Visible Action, Polar Action and Coisotropic Action. (Jean-Louis Clerc 教授退官記念研究集会) "Hermitian Symmetric Spaces, Jordan Algebras and Related Problems" (organized by W. Bertram and K. Koufany). (2008)
  • Author(s): T. Kobayashi
  • Organizer: Centre International de Recherches Mathematiques (CIRM)
  • Place of Presentation: Luminy, France
  • Year and Date: 20080623-20080627
• [Presentation] Restriction of Unitary Representations of Real Reductive Groups. Locally Symmetric Spaces (organized by S. Kudla, J. Rohlfs, L. Saper and B. Speh). (2008)
  • Author(s): T. Kobayashi
  • Organizer: Banff International Research Station
  • Place of Presentation: Canada
  • Year and Date: 20080518-20080523
• [Presentation] Global Geometry of Locally Symmetric Spaces - beyond the Riemannian case (2008)
  • Author(s): T. Kobayashi
  • Organizer: The Monna Lectures
  • Place of Presentation: Utrecht University, the Netherlands
  • Year and Date: 2008-12-11
• [Presentation] Existence Problem of Compact Locally Symmetric Spaces (2008)
  • Author(s): T. Kobayashi
  • Organizer: colloquium
  • Place of Presentation: Louisiana State University, Baton Rouge, USA
  • Year and Date: 2008-03-27
• [Presentation] Existence Problem of Compact Locally Symmetric Spaces (2008)
  • Author(s): T. Kobayashi
  • Organizer: colloquium
  • Place of Presentation: Harvard University, USA
  • Year and Date: 2008-03-17
• [Presentation] Branchings to Symmetric Pairs and Analysis on Symmetric Spaces (2007)
  • Author(s): T. Kobayashi
  • Organizer: International Conference on Integral Geometry, Harmonic Analysis and Representation theory (Sigurdur Helgason 教授80歳記念研究集会)
  • Place of Presentation: Reykjavik, Iceland
  • Year and Date: 20070815-20070818
• [Presentation] On Locally Symmetric Spaces. (closing lecture), Representation Theory, Complex Analysis and Integral Geometry (organized by Simon Gindikin and Bernhard Krotz) (2007)
  • Author(s): T. Kobayashi
  • Organizer: Max-Planck Institut fur Mathematik
  • Place of Presentation: Bonn, Germany
  • Year and Date: 20070725-20070727
• [Presentation] Multiplicities in the Decomposition of Unitary Representations of Reductive Lie Groups. (opening lecture), Lie Groups, Algebraic Groups and Transformation Groups : (Ernest B. Vinberg 教授70歳記念研究集会) (2007)
  • Author(s): T. Kobayashi
  • Organizer: organized by H. Abels, V. Chernousov, G. Margulis, D. Poguntke, and K. Tent
  • Place of Presentation: Universitat Bielefeld, Germany
  • Year and Date: 20070720-20070724
• [Presentation] Existence Problem of Compact Locally Symmetric Spaces. Journees Solstice dete 2007 : Theorie de Lie, Geometrie et Representations (organized by B. Keller, B. Klingler, R. Rentschler and O. Schiffmann) (2007)
  • Author(s): T. Kobayashi
  • Organizer: Institut de Mahematiques de Jussieu
  • Place of Presentation: Paris, France
  • Year and Date: 20070621-20070623
• [Presentation] Branching Problems of Unitary Representations. (2 lectures) (2007)
  • Author(s): T. Kobayashi
  • Organizer: Sackler Distinguished Lectures in Pure Mathematics
  • Place of Presentation: Tel Aviv University, Israel
  • Year and Date: 20070507-20070509
• [Presentation] The unitary inversion operator on the L^2 model of a minimal representation (2007)
  • Author(s): T. Kobayashi
  • Organizer: Journee-GAG
  • Place of Presentation: Poitiers, France
  • Year and Date: 2007-06-25
• [Presentation] Visible Actions on Complex Manifolds. Workshop on Representation Theory and Prehomogeneous Vector Spaces (organized by Y. Hironaka, I. Mutter, H. Ochiai, H. Rubenthaler, F. Sato) (2006)
  • Author(s): T. Kobayashi
  • Organizer: Institut de Recherche Mathematique Avancee (IRMA)
  • Place of Presentation: Strasbourg, France
  • Year and Date: 20060911-20060914
• [Presentation] Clifford-Klein Forms of Non-Riemannian Homogeneous Spaces. (4 lectures) (2006)
  • Author(s): T. Kobayashi
  • Organizer: International Conference and Instructional Workshop : Geometry Topology, Analysis of Locally Symmetric Spaces and Discrete Groups (organized by S-T. Yau (chair), L. Ji, K. Liu, N. Xi H. Xu, L. Yang, Z-J Zheng and X. Zhou)
  • Place of Presentation: Beijing, Republic of China
  • Year and Date: 20060717-20060804
• [Presentation] Unitary Representations, Restrictions and their Applications. (8 lectures) (2006)
  • Author(s): T. Kobayashi
  • Organizer: Summer School : Microlocal and Geometric Methods in Representation Theory
  • Place of Presentation: Gunzburg, Germany
  • Year and Date: 20060717-20060728
• [Presentation] Compact Clifford-Klein Forms of Non-Riemannian Symmetric Spaces (2006)
  • Author(s): T. Kobayashi
  • Organizer: Mini-Workshop on Lie groups, Algebraic Groups and Transformation Groups (organized by H. Abels and E. Vinberg).
  • Place of Presentation: Universitat Bielefeld, Germany
  • Year and Date: 20060715-20060716
• [Presentation] Branching Problems of Unitary representations. (2006)
  • Author(s): T. Kobayashi
  • Organizer: Perspectives in Representation Theory Strategic Workshop and Round Table Discussion (organized by S. Koenig, H. Krause, P. Littelmann, and G. Malle)
  • Place of Presentation: Physikzentrum Bad Honnef, Germany
  • Year and Date: 20060707-20060709
• [Presentation] Multiplicity-free Repre-sentations and Visible Actions on Complex Manifolds (2006)
  • Author(s): T. Kobayashi
  • Organizer: Analysis Seminar
  • Place of Presentation: University of Aarhus, Denmark
  • Year and Date: 2006-09-18
• [Presentation] Is the Universe Closed? -Existence Problem of Compact Clifford-Klein Forms of Symmetric Spaces (2006)
  • Author(s): T. Kobayashi
  • Organizer: colloquium
  • Place of Presentation: Universitat Paderborn, Germany
  • Year and Date: 2006-07-04
• [Book] Progr. Math.(Representation theory and automorphic forms) (2007)
  • Author(s): T. Kobayashi, W. Schmid, J. -H. Yang, eds.
• [Book] The Schrodinger model for the minimal representation of the indefinite orthogonal group O(p,q), Memoirs of American Mathematical Society (accepted for publication)
  • Author(s): T. Kobayashi, G. Mano
  • Total Pages: 171
  • Publisher: アメリカ数学会
• [Remarks] ホームページ等
  • URL: www.ms.u-tokyo.ac.jp/~toshi