Study on symplectic quotients concerned with decompositions of representations of Lie groups

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K03475
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Chuo University
• Principal Investigator: Takakura Tatsuru 中央大学, 理工学部, 教授 (30268974)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: シンプレクティック商 / 余随伴軌道 / ルート系 / ウェイト多様体 / 体積関数 / 多変量スプライン

Outline of Final Research Achievements

We studied the multiplicity varieties and multiple weight varieties associated with a compact Lie group, and obtained the results as follows. First, we published the paper on a characterization of the special vector volume function of type A, by means of a system of differential equations. Also, we applied it to the cohomology rings of special multiple weight varieties. Second, we generalized a theorem by Lidskii on the vector volume functions of type A to an identity of the family, defined by the action of the Weyl group, of bases for the root system. Third, for a general sequence of vectors, we obtained a conjecture about a presentation of a system of differential equations which characterizes the vector volume function over any chamber. We gave presentations about the results as above in some domestic conferences.

Free Research Field

位相幾何学

Academic Significance and Societal Importance of the Research Achievements

リー群の表現の分解に関連する重要な空間族の組織的構成、それらの各種不変量の決定と同型類や大域的構造の深い理解、またその過程における表現論や組合せ論への寄与、等が、学術的意義として挙げられる。また、個々の具体例における計算過程や計算結果に特徴的な簡明さがある点は、本研究の特色・独創性の一つと考えられる。さらに、他分野との関連が判明し、派生する問題と予想もいくつか得られたことは、今後の研究の広がりを示唆している。