| Project/Area Number |
17540180
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Kinki University |
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Principal Investigator |
FUJIWARA Hidenori Kinki University, School of Humanity-Oriented Science and Engineering, Professor (50108643)
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| Project Period (FY) |
2005 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥2,310,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥210,000)
Fiscal Year 2007: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2006: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2005: ¥700,000 (Direct Cost: ¥700,000)
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| Keywords | representation theory / exponential solvable Lie group / unitary representation / harmonic analysis / nilpotent Lie group / intertwining operator / Frobenius reciprocity / orbit method / 余随伴表現 / フロベニウスの相互律 / 不変微分作用素 / 誘導表現 / 表現の制限 |
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| Research Abstract |
For these three years I have been continuing the research collaboration with Prof. Ali Baklouti of Sfax University in Tunisia and Prof. Jean Ludwig of Metz University in France on unitary representations and harmonic analysis for exponential solvable Lie groups. Inviting them for about one week to the School of Humanity-Oriented Science and Engineering in lizuka or being invited to Sfax and Metz, I have proceeded the joint research. I also invited to Iizuka for a short period Prof. G. Grelaud of Poitiers University and Prof. D. Arnal of Bourgogne University in France, specialists in related fields to learn from them some their methods. In this way, I got the following results.
1. Let's construct by the orbit method irreducible representations of an exponential solvable Lie group G = exp g with Lie algebra g. Then, inducing from the unitary character of two polarizations at f ∈ g* satisfying the Pukanszky condition, we get two irreducible unitary representations which are unitarily equivalent. The author is interested since a long time ago in the problem to construct explicitly an intertwining operator between these two representations. We succeeded to prove the convergence of an integral appearing in the well-known formal intertwining operator and resolved this long-standing problem. The paper is submitted to a mathematical journal.
2. Let's consider a Lie group G of type I with an irreducible unitary representation π and a closed subgroup K with an irreducible unitary representation σ. The reciprocity of Frobenius means that the multiplicity of π in the irreducible decomposition of the induced representation ind^G_K σ is equal to the multiplicity of σ in the irreducible decomposition of the restriction π|_k of π on K . When G is a connected and simply connected nilpotent Lie group, we proved results corresponding in some sense to a distribution version of this reciprocity.
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