Harmonic analysis for vector bundles on Riemannian symmetric spaces with fine fibers

2020 Fiscal Year Final Research Report

• Project/Area Number: 18K03346
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Takushoku University
• Principal Investigator: Oda Hiroshi 拓殖大学, 工学部, 教授 (20338619)
• Project Period (FY): 2018-04-01 – 2021-03-31
• Keywords: リーマン対称空間 / ベクトル束 / プランシェレル測度 / ヘックマン・オプダム超幾何関数 / ミニスキュルKタイプ

Outline of Final Research Achievements

Fourier transforms, which decompose a function on Euclidean spaces into a sum of exponential functions, is generalized to spherical transforms, which decompose a section of vector bundles on Riemannian symmetric spaces into a sum of elementary spherical functions. The two most basic problems on a spherical transform are to present the inversion formula and to determine the Plancherel measure, though the problems had been solved only in some limited cases. Just before this research program started Shimeno (Kwansei Gakuin Univ) and I solved the problems in the cases where the fiber of the vector bundle is classified as `small.' The method we adopted there is to use a new formula that relates elementary spherical functions to Heckman-Opdam hypergeometric functions. In the research program we extended this result to the cases where the vector bundle has an invariant differential operators of the first order and the fiber of the vector bundle is classified as `minuscule.'

Free Research Field

表現論

Academic Significance and Societal Importance of the Research Achievements

物理系の状態は時空間上の関数ではなく，ベクトルを値に取る関数のようなもの（ベクトル束の切断）で表されることが多い．我々はベクトル束が一階の不変微分作用素を持ち，ファイバーが「ミニスキュル」の場合の調和解析を確立したが，スピンに対するベクトル束はその適用範囲に含まれる（ファイバーはスピン表現でミニスキュル，一階の不変微分作用素はディラック作用素）．
一方，ヘックマンとオプダムの超幾何関数およびそれが満たす量子可積分系の理論は十分成熟しているのにもかかわらず，これまでほとんど応用がなかった．我々が今回用いた手法により，対称空間の調和解析の結果を導く際，これらの理論が強力なツールになることが示された．