| Project/Area Number |
26400011
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Kyoto University |
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Principal Investigator |
Kato Shin-ichi 京都大学, 国際高等教育院, 教授 (90114438)
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| Research Collaborator |
TAKANO KEIJI 香川大学, 教育学部, 准教授 (40332043)
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| Project Period (FY) |
2014-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2018: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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| Keywords | 対称空間 / 表現論 / 簡約群 / p進体 / 有限体 / 局所体 / 球等質空間 |
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| Outline of Final Research Achievements |
We studied representations and harmonic analysis of symmetric spaces associated to reductive groups with involution sigma on them over p-adic fields, as a generalization of the representation theory of these groups.
Relatively cuspidal representations for symmetric spaces are the counterparts of cuspidal representations for groups and viewed as the most fundamental tools in the study of the representation theory of symmetric spaces. Under the working hypothesis that relatively cuspidal representations should correspond to anisotropic maximal sigma-split tori, we succeeded in constructing relatively cuspidal representations as induced representations of cuspidal representations from sigma-stable parabolic subgroups for certain types of symmetric spaces associated with general linear groups. We also showed that relatively cuspidal representations cannot appear as induced representations from sigma-split parabolic subgroups if the inducing representations have generic parameters.
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| Academic Significance and Societal Importance of the Research Achievements |
p進体上の対称空間の表現論は，それ自身が代数群の表現論として重要な位置を占めるだけではなく整数論の観点からも研究が欠かせないものである．本研究では作業仮説を提出して，最も基本的な相対尖点表現を誘導表現から構成している．これは，別のタイプの誘導表現からは相対尖点表現が得られないというもう一つの結果と合わせて，作業仮説を補強しており．p進対称空間の表現論の全体像の解明に大きく寄与するものである．またここで与えられた誘導表現による構成法は整数論にも応用されるものと思われる．
なお表現論は対称性を扱う数学的分野であることから，得られた成果は自然現象の対称性の理解にも役立つことが期待される．
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