D-modules on infinite-dimensional algebraic varieties and their application to representation theory

• Project/Area Number: 21654005
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Osaka City University
• Principal Investigator: TANISAKI Toshiyuki 大阪市立大学, 大学院・理学研究科, 教授 (70142916)
• Co-Investigator(Kenkyū-buntansha): KANEDA Masaharu 大阪市立大学, 大学院・理学研究科, 教授 (60204575)
• Co-Investigator(Renkei-kenkyūsha): KASHIWARA Masaki 京都大学, 数理解析研究所, 教授 (60027381)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥3,270,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥270,000); Fiscal Year 2011: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2010: ¥1,000,000 (Direct Cost: ¥1,000,000); Fiscal Year 2009: ¥1,100,000 (Direct Cost: ¥1,100,000)
• Keywords: 代数解析 / 群論

Research Abstract

The main purpose of this research was to give a proof of the character formula for irreducible highest weight modules over affine Lie algebras at the critical levels. We could not solve the problem itself ; however, we investigated the related problems concerning the quantum groups at roots of 1, and obtained several results such as the localization theorem on flag manifolds and the structure theorem for the center.

Research Products

• [Journal Article] On quantization of complex symplectic manifolds (2011)
  • Author(s): D' Agnolo, Andrea ; Kashiwara, Masaki
  • Journal Title: Comm. Math. Phys Volume: 308, no.1 Pages: 81-113
  • Peer Reviewed
• [Journal Article] Rigidity of tilting modules (2011)
  • Author(s): Andersen, Henning Haahr ; Kaneda, Masaharu
  • Journal Title: Mosc. Math. J Volume: 11, no.1 Pages: 1-39
  • Peer Reviewed
• [Journal Article] Rigidity of tilting modules (2011)
  • Author(s): Andersen, H.H.
  • Journal Title: Mosc.Math.J. Volume: 11 Pages: 1-39
  • Peer Reviewed
• [Journal Article] Quantum queer superalgebra and crystal bases (2010)
  • Author(s): Grantc arov, Dimitar ; Jung, Ji Hye ; Kang, Seok-Jin, Kashiwara, Masaki ; Kim, Myungho
  • Journal Title: Proc. Japan Acad. Ser. A Math. Sci Volume: 86, no.10 Pages: 177-182
  • Peer Reviewed
• [Journal Article] On the codimension-three conjecture (2010)
  • Author(s): Kashiwara, Masaki ; Vilonen, Kari
  • Journal Title: Proc. Japan Acad. Ser. A Math. Sci Volume: 86, no.9 Pages: 154-158
  • NAID: 40017436326
  • Peer Reviewed
• [Journal Article] Tropical R maps and affine geometric crystals (2010)
  • Author(s): Kashiwara, Masaki ; Nakashima, Toshiki ; Okado, Masato
  • Journal Title: Represent. Theory Volume: 14 Pages: 446-509
  • Peer Reviewed
• [Journal Article] On the structure of parabolic Humphreys-Verma modules (2010)
  • Author(s): M.Kaneda
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B11 Pages: 117-123
  • Peer Reviewed
• [Journal Article] Geometric construction of crystal bases for quantum generalized Kac-Moody algebras (2009)
  • Author(s): Kang, Seok-Jin ; Kashiwara, Masaki ; Schiffmann, Olivier
  • Journal Title: Adv. Math Volume: 222, no.3 Pages: 996-1015
  • Peer Reviewed
• [Journal Article] Equivariant K-theory of affine flag manifolds and affine Grothendieck polynomials (2009)
  • Author(s): Kashiwara, Masaki ; Shimozono, Mark
  • Journal Title: Duke Math. J Volume: 148, no.3 Pages: 501-538
  • Peer Reviewed
• [Journal Article] Modules over deformation quantization algebroids : an overview (2009)
  • Author(s): Kashiwara, Masaki ; Schapira, Pierre
  • Journal Title: Lett. Math. Phys Volume: 88, no.1-3 Pages: 79-99
  • Peer Reviewed
• [Journal Article] Algebraic analysis and around (2009)
  • Author(s): Tanisaki, Toshiyuki, Masaki Kashiwara
  • Journal Title: Adv. Stud. Pure Math Volume: 54 Pages: 11-18
  • Peer Reviewed
• [Journal Article] On the structure of parabolic Humphreys-Verma modules (2009)
  • Author(s): Kaneda, Masaharu Kaneda, Masaharu
  • Journal Title: New trends in combinatorial representation theory Volume: B11 Pages: 117-123
  • Peer Reviewed
• [Journal Article] The structure of Humphreys-Verma modules for projective spaces (2009)
  • Author(s): Masaharu, Kaneda
  • Journal Title: J. Algebra Volume: 322, no.1 Pages: 237-244
  • Peer Reviewed
• [Journal Article] The structure of Humphreys-Verma modules for projective spaces (2009)
  • Author(s): M.Kaneda
  • Journal Title: J.Algebra 322 Pages: 237-244
  • Peer Reviewed
• [Presentation] The center of a quantized enveloping algebra at a root of 1 (2012)
  • Author(s): T. Tanisaki
  • Organizer: Representaitons of Chevalley groups and related topics
  • Place of Presentation: 名古屋大学多元数理科学研究科
  • Year and Date: 2012-03-15
• [Presentation] Differential operators on quantized flag manifolds Perspectives in algebraic Lie theory (2011)
  • Author(s): T. Tanisaki
  • Organizer: Isaac Newton Institute for Mathematical Sciences
  • Place of Presentation: Cambridge, England
  • Year and Date: 2011-09-12
• [Presentation] Differential operators on quantized flag manifolds (2011)
  • Author(s): T.Tanisaki
  • Organizer: Perspectives in algebraic Lie theory
  • Place of Presentation: Isaac Newton Institute for Mathematical Sciences(招待講演)
  • Year and Date: 2011-09-12
• [Presentation] Manin tripleと量子群 (2011)
  • Author(s): 谷埼俊之
  • Organizer: 稚内代数セミナー
  • Place of Presentation: 稚内北屋学園大学
  • Year and Date: 2011-02-21
• [Presentation] 有限W代数入門 (2011)
  • Author(s): 谷崎俊之
  • Organizer: 第14回代数群と量子群の表現論研究集会国民宿舎
  • Place of Presentation: 小豆島
• [Presentation] Symplectic manifolds arising from quantized enveloping algebras (2010)
  • Author(s): T. Tanisaki
  • Organizer: The 5^<th> International Conference on Representation Theory
  • Place of Presentation: Xian, China
  • Year and Date: 2010-08-11
• [Presentation] Symplectic manifolds arising from quantized enveloping algebras (2010)
  • Author(s): T.Tanisaki.
  • Organizer: The 5^<th> International Conference on Representation Theory
  • Place of Presentation: 中国, 西安(招待講演)
  • Year and Date: 2010-08-11
• [Presentation] Differential operators on quantized flag manifolds at roots of 1 (2010)
  • Author(s): T. Tanisaki
  • Organizer: Algebraic Analysis and Beyond
  • Place of Presentation: 京都大学理学研究科
• [Presentation] Variations on a theme of Bezrukavnikov-Mirkovic-Rumynin (2009)
  • Author(s): 谷崎俊之
  • Organizer: ゲーム理論と表現論
  • Place of Presentation: 稚内北星学園大学
  • Year and Date: 2009-08-24
• [Presentation] D-modules on quantized flag manifolds at roots of 1 (2009)
  • Author(s): T. Tanisaki
  • Organizer: Workshop on Lie theory and representation theory II中国
  • Place of Presentation: East China Normal University
  • Year and Date: 2009-07-20
• [Presentation] D-modules on quantized flag manifolds at roots of 1 (2009)
  • Author(s): T. Tanisaki
  • Organizer: International workshop on combinatorial and geometric approach to representation theory
  • Place of Presentation: Seoul National University