Research on representation theory of algebraic groups and quantum groups via algebraic analysis

2009 Fiscal Year Final Research Report

• Project/Area Number: 19340010
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Osaka City University
• Principal Investigator: TANISAKI Toshiyuki Osaka City University, 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) ; KASHIWARA Masaki 京都大学, 数理解析研究所, 教授 (60027381) ; SHOJI Toshiaki 名古屋大学, 大学院・多元数理科学研究科, 教授 (40120191) ; ASASHIBA Hideto 静岡大学, 理学部, 教授 (70175165) ; FYRUSAWA Masaaki 大阪市立大学, 大学院・理学研究科, 教授 (50294525) ; ARIKI Susumu 京都大学, 数理解析研究所, 教授 (40212641) ; NAKAJIMA Hiraku 京都大学, 数理解析研究所, 教授 (00201666) ; NAITO Satoshi 筑波大学, 大学院・数理物質科学研究科, 准教授 (60252160) ; SAITO Yoshihisa 東京大学, 数理科学研究科, 准教授 (20294522) ; ICHINI Atsushi 大阪市立大学, 大学院・理学研究科, 准教授 (40347480)
• Project Period (FY): 2007 – 2009
• Keywords: 群の表現論

Research Abstract

Qyabtyn griyos are q-deformations of algebraic groups. When the parameter q is not a root of 1, its representation theory is well unerstood ; however, when q is a root of 1, there still remain fundamental problems such as classification of irreducible modules which are not yet solved. The head investigator Tanisaki made research on it using the method of D-modules, and obtained several results like the Azumaya property of certain rings of differential operators. The investigator Kaneda investigated the ring of differential operators in positive characteristics, and made progress in the representation theory of algebraic groups in positive characteristics.

Research Products

• [Journal Article] The structure of Humphreys-Verma modules for projective spaces (2009)
  • Author(s): Kaneda, Masaharu
  • Journal Title: J. Algebra 322 Pages: 237-244
  • 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 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. 148 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 88 Pages: 79-99
  • Peer Reviewed
• [Journal Article] Mirkovic-Vilonen polytopes lying in a Demazure crystal and an opposite Demazure crystal. (2009)
  • Author(s): Naito, Satoshi; Sagaki, Daisuke
  • Journal Title: Adv. Math 221 Pages: 1804-1842
  • Peer Reviewed
• [Journal Article] K-theoretic Donaldson invariants via instanton counting. (2009)
  • Author(s): Gottsche, Lothar; Nakajima, Hiraku; Yoshioka, Kota
  • Journal Title: Pure Appl. Math. 5 Pages: 1029-1111
  • Peer Reviewed
• [Journal Article] On Hecke algebras associated with elliptic root systems and the double affine Hecke algebras. (2009)
  • Author(s): Saito, Yoshihisa; Shiota, Midori
  • Journal Title: Publ. Res. Inst. Math. Sci. 45 Pages: 845-905
  • Peer Reviewed
• [Journal Article] Lusztig's conjecture for finite classical groups with even characteristic. (2009)
  • Author(s): Shoji, Toshiaki
  • Journal Title: Representation theory Pages: 207-236
  • Peer Reviewed
• [Journal Article] Product formulas for the cyclotomic v-Schur algebra and for the canonical bases of the Fock space. (2009)
  • Author(s): Shoji, Toshiaki; Wada, Kentaro
  • Journal Title: J. Algebra 321 Pages: 3527-3549
  • Peer Reviewed
• [Journal Article] Kapranov's tilting sheaf on Grassmannian in positive characteristic, Algebr. (2008)
  • Author(s): Kaneda, Masaharu
  • Journal Title: Represent. Theory 11 Pages: 347-354
  • Peer Reviewed
• [Journal Article] Trilinear forms and the central values of triple product L-functions. (2008)
  • Author(s): Ichino, Atsushi
  • Journal Title: Duke Math. J. 145 Pages: 281-307
  • Peer Reviewed
• [Journal Article] On Maass lifts and the central critical values of triple product L-functions. (2008)
  • Author(s): Ichino, Atsushi; Ikeda, Tamotsu
  • Journal Title: Amer. J. Math 130 Pages: 75-114
  • Peer Reviewed
• [Journal Article] Formal degrees and adjoint gamma-factors. (2008)
  • Author(s): Hiraga, Kaoru; Ichino, Atsushi; Ikeda, Tamotsu
  • Journal Title: J. Amer. Math. Soc. 21 Pages: 283-304
  • Peer Reviewed
• [Journal Article] Equivariant derived category and representation of real semisimple Lie groups. (2008)
  • Author(s): Kashiwara, Masaki
  • Journal Title: Lecture Notes in Math. 1931
  • Peer Reviewed
• [Journal Article] Microlocalization of rational Cherednik algebras. (2008)
  • Author(s): Kashiwara, Masaki; Rouquier, Raphael
  • Journal Title: Duke Math. J. 144 Pages: 525-573
  • Peer Reviewed
• [Journal Article] Affine geometric crystals and limit of perfect crystals. (2008)
  • Author(s): Kashiwara, Masaki; Nakashima, Toshiki; Okado, Masato
  • Journal Title: Trans. Amer. Math. Soc 360 Pages: 3645-3686
  • Peer Reviewed
• [Journal Article] Constructibility and duality for simple holonomic modules on complex symplectic manifolds. (2008)
  • Author(s): Kashiwara, Masaki; Schapira, Pierre
  • Journal Title: Amer. J. Math 130 Pages: 207-237
  • Peer Reviewed
• [Journal Article] Domestic canonical algebras and simple Lie algebras. (2008)
  • Author(s): Asashiba, Hideto
  • Journal Title: Math. Z. 259 Pages: 713-754
  • Peer Reviewed
• [Journal Article] On the tensor product of two basic representations of Uv(sle). (2008)
  • Author(s): Ariki, Susumu; Kreiman, Victor; Tsuchioka, Shunsuke
  • Journal Title: Adv. Math 218 Pages: 28-86
  • Peer Reviewed
• [Journal Article] Lakshmibai-Seshadri paths of level-zero shape and one-dimensional sums associated to level-zero fundamental representations. (2008)
  • Author(s): Naito, Satoshi, Sagaki, Daisuke
  • Journal Title: 144 Pages: 1525-1556
  • Peer Reviewed
• [Journal Article] A modification of the Anderson-Mirkovic conjecture for Mirkovic-Vilonen polytopes in types B and C. (2008)
  • Author(s): Naito, Satoshi; Sagaki, Daisuke
  • Journal Title: J. Algebra 320 Pages: 387-416
  • Peer Reviewed
• [Journal Article] Crystal structure on the set of Lakshmibai-Seshadri paths of an arbitrary level-zero shape. (2008)
  • Author(s): Naito, Satoshi; Sagaki, Daisuke
  • Journal Title: Proc. Lond. Math. Soc. 96 Pages: 582-622
  • Peer Reviewed
• [Journal Article] Instanton counting and Donaldson invariants. (2008)
  • Author(s): Gottsche, Lothar; Nakajima, Hiraku; Yoshioka, Kota
  • Journal Title: J. Differential Geom 80 Pages: 343-390
  • Peer Reviewed
• [Journal Article] Sheaves on ALE spaces and quiver varieties. (2007)
  • Author(s): Nakajima, Hiraku
  • Journal Title: Mosc. Math. J. 7 Pages: 699-722
  • Peer Reviewed
• [Journal Article] Crystal, canonical and PBW bases of quantum affine algebras. (2007)
  • Author(s): Nakajima, Hiraku
  • Journal Title: Algebraic groups and homogeneous spaces Pages: 389-421
  • Peer Reviewed
• [Journal Article] Generalized Green functions and unipotent classes for finite reductive groups. (2007)
  • Author(s): Shoji, Toshiaki
  • Journal Title: II. Nagoya Math. J. 188 Pages: 133-170
  • Peer Reviewed
• [Journal Article] A variant of the induction theorem for Springer representations. (2007)
  • Author(s): Shoji, Toshiaki
  • Journal Title: J. Algebra 311 Pages: 130-146
  • Peer Reviewed
• [Journal Article] J. Ye, Equivariant localization of D-modules on the flag variety of the symplectic group of degree 4 (2007)
  • Author(s): Kaneda, Masaharu
  • Journal Title: J. Algebra 309 Pages: 236-281
  • Peer Reviewed
• [Journal Article] On the Siegel-Weil formula for unitary groups. (2007)
  • Author(s): Ichino, Atsushi
  • Journal Title: Math. Z. 255 Pages: 721-729
  • Peer Reviewed
• [Journal Article] Crystals and affine Hecke algebras of type D. (2007)
  • Author(s): Kashiwara, Masaki; Miemietz, Vanessa
  • Journal Title: Proc. Japan Acad. Ser. A Math. Sci 83 Pages: 135-139
  • Peer Reviewed
• [Presentation] Differential operators on quantized flag manifolds at roots of 1, Algebraic Analysis and Beyond (2010)
  • Author(s): T. Tanisaki
  • Organizer: 京都大学理学研究科
  • Year and Date: 2010-03-16
• [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
  • Year and Date: 2009-09-22
• [Presentation] Variations on a theme of Bezrukavnikov-Mirkovic-Rumynin. (2009)
  • Author(s): T. Tanisaki
  • Organizer: ゲーム理論と表現論稚内北星学園大学
  • 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
  • Place of Presentation: 中国, East China Normal University
  • Year and Date: 2009-07-20
• [Presentation] D-modules on quantized flag manifolds (2008)
  • Author(s): T. Tanisaki
  • Organizer: Taipei workshop in Lie theory 2008
  • Place of Presentation: Academia Sinica, 台湾, 台北市
  • Year and Date: 2008-12-30
• [Presentation] リー代数と量子群の幾何学的表現論 (2008)
  • Author(s): 谷崎俊之
  • Organizer: 日本数学会年会代数分科会特別講演
  • Place of Presentation: 近畿大学理工学部
  • Year and Date: 2008-03-25
• [Presentation] 量子旗多様体上のd加群 (2007)
  • Author(s): 谷崎俊之
  • Organizer: 第52回代数学シンポジウム
  • Place of Presentation: 神戸大学百年記念館
  • Year and Date: 2007-08-06
• [Presentation] Differential operators on quantized flag manifolds at roots of 1 (2007)
  • Author(s): T. Tanisaki
  • Organizer: The 4th international conference on representation theory
  • Place of Presentation: 中国, チベット, Lhasa
  • Year and Date: 2007-06-20
• [Presentation] Symplectic manifolds arising from quantized enveloping algebras at roots of 1 (2007)
  • Author(s): T. Tanisaki
  • Organizer: 第10回代数群と量子群の表現論研究集会上智軽井沢セミナーハウス
  • Year and Date: 2007-06-01
• [Book] Birkhauser Boston, Inc., Boston, MA (2008)
  • Author(s): Hotta, Ryoshi; Takeuchi, Kiyoshi; Tanisaki, Toshiyuki
  • Total Pages: ISBN:978-0-8176-4363-8
  • Publisher: D-modules, perverse sheaves, and representation theory. Progress in Mathematics, 236.