2004 Fiscal Year Final Research Report Summary
Geometry and Harmonic Analysis on Nilpotent Orbits
| Project/Area Number |
13440046
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | KYOTO UNIVERSITY |
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Principal Investigator |
NISHIYAMA Kyo Kyoto University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (70183085)
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| Co-Investigator(Kenkyū-buntansha) |
MATSUKI Toshihiko Kyoto University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (20157283)
SAITO Hiroshi Kyoto University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (20025464)
OHTA Takuya Tokyo Denki University, Faculty of Engineering, Professor, 工学部, 教授 (30211791)
SEKIGUCHI Jiro Tokyo University of Agriculture and Technology, Faculty of Engineering, Professor, 工学部, 教授 (30117717)
YAMASHITA Hiroshi Hokkaido University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (30192793)
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| Project Period (FY) |
2001 – 2004
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| Keywords | nilpotent orbit / associated cycle / unitary representation / semisimple Lie groups / isotropy representation / invariant theory / dual pair / theta lifting |
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| Research Abstract |
(1)We describe the theta lifting of nilpotent orbits in the terms of invariant theory, and prove that (the closures of) some good liftings are normal variety and the action of the algebraic groups on them are multiplicity free. We also obtain a degree formula of the nilpotent orbits expressed by integrals.
A general theory of the lifting of coherent sheaves are constructed, and in particular we prove the preservation of the multiplicities of the support of coherent sheaves.
(2)For the indefinite unitary group U(p,p), we completely classify the spherical nilpotent orbits and at the same time we describe the structure of the function rings on them.
(3) We generalize the notion of theta lifting to general orbits other than nilpotent ones, and shed a light on the research of the complete understanding of the orbit correspondence. This includes an example of the unimodular congruence classes of the bilinear forms studied by Sekiguchi, Djokovic, Zhao.
At last, we summarize the research of each investigator.
Sekiguchi has studied the unimodular congruence classes of the bilinear forms from the view point of the invariant theory.
Ohta has clarified the correspondence between the orbits of complex reductive algebraic groups and its real forms.
More generally, he extends his research to the orbits of symmetric pairs.
Yamashita has investigated the relations between associated cycles of the unitary representations of semisimple Lie groups, their isotropy representations and generalized Whittaker vectors.
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