2022 Fiscal Year Final Research Report
The study of differential symmetry breaking operators and minimal representations from an analytic point of view
| Project/Area Number |
18K13432
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 12010:Basic analysis-related
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| Research Institution | Ryukoku University |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Keywords | 絡微分作用素 / 極小表現 / K-type構造 / 超幾何多項式 / Heun多項式 / 三重対角行列式 / Cayley continuant / 関数等式 |
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| Outline of Final Research Achievements |
During the period of research, I mainly studied the K-type structure of the solution space Sol(□(s,3)) of the Heisenberg ultrahyperbolic operator □(s,3) of type A2. There are two main results. One of them is the determination of the K-type structure of Sol(□(s,3)). This result closes a case of the problem that Kable studied before. The other is the the following three discoveries on sequences of polynomials: (1) the discovery of sequences of polynomials that has local Huen functions as their generating functions of sinh type and cosh type, (2) the discovery of the special values of the tridiagonal determinants, and (3) the discovery of the functional equation (palindromic property) of sequences of polynomials. I will use the experience and knowledge obtained in this study for my future research.
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| Free Research Field |
実簡約群の表現論
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| Academic Significance and Societal Importance of the Research Achievements |
本研究成果の主な学術的意義として，Kableの問題の部分的解決が挙げられる．An型ハイゼンベルグ超双極型微分作用素□(s,n+1)の解空間Sol(□(s,n+1))のK-type構造はKableによって調べられているが，その構造を完全に記述するまでには至っていない．本研究では解空間Sol(□(s,3))のK-type構造を完全に決定することにより，n=2の場合にこの問題を解決することができた．さらに本研究によって「多項式列の関数等式」という全く新しい性質を発見することにも成功した．これは古典的によく知られた三重対角行列式に対しても新たな性質を見出すものである．
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