| Co-Investigator(Kenkyū-buntansha) |
HATA Masayoshi Kyoto University., Integrated Human Studies, Ass.Professor, 総合人間学部, 助教授 (40156336)
MATSUKI Toshihiko Kyoto University., Integrated Human Studies, Ass.Professor, 総合人間学部, 助教授 (20157283)
KATO Shinichi Kyoto University., Integrated Human Studies, Ass.Professor, 総合人間学部, 助教授 (90114438)
SAITO Hiroshi Kyoto University., Graduate School of Human and Environmental Studieds, Professor, 大学院・人間・環境学研究科, 教授 (20025464)
ASUNO Kiyoshi Kyoto University., Graduate School of Human and Enviromental Studieds, Professor, 大学院・人間・環境学研究科, 教授 (90026774)
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| Research Abstract |
We developed certain machinery for studying associated cycles of unitary representations of semisimple Lie groups by means of theta correspondence.
To be more explicit, let (G_1 , G_2) be an irreducible, reductive dual pair of type I, which is in the stable range with G_2 as the smaller member. Take an irreducible representation π_2 of G_2 (or of its metaplectic double cover), and lift it to the representation π_1 of G_1 (or its metaplectic cover), called the theta lift ofπ_2. We study the cases where π_2 is a finite-dimensional unitary representation, or a representation in the holomorphic discrete series. Even if π_2 is the trivial representation, π_1 is an infinite dimensional unitary representation, which is supposed to be a unipotent representation, one of important objects in representation theory. In the following, we shall list up our results, where π_2 is a finite-dimensional unitary representation, or a representation in the holomorphic discrete series.
We get the K-type formula for π_1.
We describe the correspondence of associated varieties of (π_1, π_2) explicitly. Moreover, we succeed to get the multiplicities in their associated cycles and get a simple correspondence (still conjectural in general) between them.
An associated variety can often be irreducible, and is the closure of a single nilpotent orbit. Using the results on the theta correspondence of representations, we investigate the corre spondence of nilpotent orbits. The main results are, a description of the structure of the function ring, an integral formula of degree of nilpotent orbits.
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