2023 Fiscal Year Final Research Report
Branching problems of infinite-dimensional representations and global analysis
| Project/Area Number |
18H03669
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Review Section |
Medium-sized Section 12:Analysis, applied mathematics, and related fields
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| Research Institution | The University of Tokyo |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
関口 英子 東京大学, 大学院数理科学研究科, 准教授 (50281134)
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Keywords | 解析学 / 幾何学 / リー群 / 分岐則 / 不連続群 |
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| Outline of Final Research Achievements |
1. In global analysis on a manifold X endowed with a group action, a novel approach, distinct from conventional differential equation methods, was formulated. This approach quantifies the topological condition of "proper actions" by estimating the "intersection volume." We advanced the foundational theory, particularly providing geometric criteria for determining when regular unitary representations on square-integrable spaces of X become tempered. Furthermore, we accomplished the classification for tempered reductive homogeneous spaces.
2. We provided a complete classification with explicit construction of "symmetry-breaking operators" between twisted exterior cotangent bundles for pairs consisting of a conformally flat manifold and its co-dimension one totally geodesic submanifold.
3. We developed the boundedness theorem and finiteness theorem of multiplicities for branching laws of infinite-dimensional representations by extending and refining our previous results.
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| Free Research Field |
数学
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| Academic Significance and Societal Importance of the Research Achievements |
空間の線型対称性において、その対称性が減少したときの数学的記述を「表現の分岐則」という。有限次元のテンソル積表現の分解がその古典例であるが、無限次元表現では分岐則の理論は非常に難しく、手法自身を新しく開発する必要がある。本研究では、「対称性破れ作用素」の構成問題という新しい視点を提示し、共形平坦空間の2つの多様体の上の微分形式の空間に対する対称性破れ作用素の完全な分類を行い、この新領域における基盤整備と未来の発展の方向を与えた。また、非可換調和解析においても「緩増加空間」の概念を提起し、微分方程式を用いる従来の手法とは異なる、新手法を開発し、代数・幾何・解析にまたがる新しい基礎理論を構築した。
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