Representation theory of Lie algebras and b-functions of prehomogeneous vector spaces

• Project/Area Number: 23540179
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Hokkaido University of Education
• Principal Investigator: Wachi Akihito 北海道教育大学, 教育学部, 准教授 (30337018)
• Project Period (FY): 2011-04-28 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2015: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 概均質ベクトル空間 / ｂ─関数 / Capelli恒等式 / b-関数

Outline of Final Research Achievements

We have computed b-functions of some reducible prehomogeneous vector spaces by using Lie theory, and have studied properties of b-functions such as relations to group orbits. In addition, we have studied related things as follows. We have clarified combinatorial and representation-theoretic properties of nilpotent orbits on symmetric spaces, and the Lefschetz properties of coinvariant rings of reflection groups.

Research Products

• [Journal Article] Codimension one connectedness of the graph of associated varieties. (2016)
  • Author(s): Kyo Nishiyama, Peter Trapa and Akihito Wachi.
  • Journal Title: Tohoku Math. J. Volume: 68
  • NAID: 130008106072
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] The quadratic complete intersections associated with the action of the symmetric group. (2015)
  • Author(s): Tadahito Harima, Akihito Wachi and Junzo Watanabe.
  • Journal Title: Illinois J. Math. Volume: 59 Pages: 99-113
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Capelli identities of odd type and b-functions (2014)
  • Author(s): Akihito Wachi
  • Journal Title: Commentarii Mathematici Universitas Sancti Pauli Volume: 63 Pages: 291-303
  • Peer Reviewed / Open Access
• [Journal Article] Logarithmic derivative and Capelli identities (2012)
  • Author(s): Akihito Wachi
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B36
  • Peer Reviewed
• [Presentation] 格子多面体の体積公式. (2015)
  • Author(s): 和地輝仁.
  • Organizer: 日本数学会2015年度秋季総合分科会一般講演,
  • Place of Presentation: 京都産業大学
  • Year and Date: 2015-09-13
• [Presentation] 交代行列に対する奇数型Capelli恒等式. (2015)
  • Author(s): 和地輝仁.
  • Organizer: 研究会「不変性と双対性」
  • Place of Presentation: 鹿児島大学
  • Year and Date: 2015-09-06
• [Presentation] 格子多面体の体積公式. (2015)
  • Author(s): 和地輝仁.
  • Organizer: 数理解析研究所研究集会「幾何学・組合せ論に現れる環と代数構造」
  • Place of Presentation: 数理解析研究所
  • Year and Date: 2015-06-09
• [Presentation] The Lefschetz properties and complex reflection groups (2015)
  • Author(s): Akihito Wachi
  • Organizer: Workshop on Lefschetz properties
  • Place of Presentation: Goettingen大学 (ドイツ)
  • Year and Date: 2015-03-09
• [Presentation] The b-function of a reducible prehomogeneous vector space and a scalar generalized Verma module (2012)
  • Author(s): Akihito Wachi
  • Organizer: International workshop on representation theory and harmonic analysis
  • Place of Presentation: 南開大学 (天津市)
• [Presentation] 可約概均質ベクトル空間のb-関数と一般Verma加群 (2012)
  • Author(s): 和地輝仁
  • Organizer: 数理解析研究所研究集会「表現論と非可換調和解析の展望」
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] q-Analogues of dimension formula and the SLP (2012)
  • Author(s): Akihito Wachi
  • Organizer: Workshop in Hawaii: Aspects of SLP and WLP
  • Place of Presentation: Hawaii Tokai International College (ホノルル市)
• [Presentation] 奇数型Capelli恒等式 (2012)
  • Author(s): 和地輝仁
  • Organizer: 日本数学会2012年度秋季総合分科会一般講演
  • Place of Presentation: 九州大学
• [Presentation] 複素鏡映群の余不変式環の強レフシェッツ性
  • Author(s): 和地輝仁
  • Organizer: 日本数学会2014年度年会一般講演
  • Place of Presentation: 学習院大学
• [Presentation] 複素鏡映群の余不変式環の強レフシェッツ性
  • Author(s): Akihito Wachi
  • Organizer: 第35回可換環論シンポジウム
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] 不変式で生成されるイデアル
  • Author(s): 和地輝仁
  • Organizer: 2013年度表現論シンポジウム
  • Place of Presentation: マホロバ・マインズ三浦
• [Book] The Lefschetz properties. Lecture Notes in Mathematics, 2080 (2013)
  • Author(s): Harima, Tadahito; Maeno, Toshiaki; Morita, Hideaki; Numata, Yasuhide; Wachi, Akihito; Watanabe, Junzo
  • Total Pages: 250
  • Publisher: Springer, Heidelberg