| Project/Area Number |
09640190
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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 |
解析学
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| Research Institution | HIROSHIMA UNIVERSITY |
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Principal Investigator |
EGUCHI Masaaki FAC.INT.and SCI.HIROSHIMA UNIVERSITY PROF., 総合科学部, 教授 (30037220)
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| Co-Investigator(Kenkyū-buntansha) |
YOSHIDA Kiyoshi FAC.INT.and SCI.PROF., 総合科学部, 教授 (80033893)
FURUSHIMA Mikio FAC.INT.and SCI.PROF., 総合科学部, 教授 (00165482)
SHIBATA Tetutaro FAC.INT.and SCI.ASS.PROF., 総合科学部, 助教授 (90216010)
KONNO Hitoshi FAC.INT.and SCI.ASS.PROF., 総合科学部, 助教授 (00291477)
KOIZUMI Shin ONOMICHI JUNIOR COLL.PROF, 教授 (90205310)
阿賀岡 芳夫 広島大学, 総合科学部, 助教授 (50192894)
吉田 敏男 広島大学, 総合科学部, 教授 (10033854)
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| Project Period (FY) |
1997 – 1998
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| Project Status |
Completed (Fiscal Year 1998)
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| Budget Amount *help |
¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 1998: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 1997: ¥1,700,000 (Direct Cost: ¥1,700,000)
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| Keywords | C Function / Intertwining Operator / Representation Theory / Fourier Transform / Lie Group / Homogeneou Space / Hardy Theorem / Heisenberg-Weyl / 表理論 / ハリシュ・チャンドラC関数 / SU(n,1) / 不確定性原理 / ハ-ディーの定理 / 自由場表現 / 量子アフィン代数 / ヴィラソロ代数 |
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| Research Abstract |
1. The Harish-Chandra C-function plays an essential role in harmonic analysis on semisimple Lie groups, because it closely relates to the Plancherel measure, the reducibility of the principal series representations and gives many information for the analysis. After a time, many peoples studied the Harish-Chandra C-function. However, even now, the explicit expressions of the Harish-Chandra C-functions are not known except for a few semisimple Lie groups and special cases.
In this research we succeeded to give the explicit formulae of the Harish-Chandra C-functions for Spin(m, 1) and SU(n, 1). By the product formula for the Harish-Chandra C-function, the problem of computing the Harish-Chandra C-functions of semisimple Lie groups of general rank is reduced to the real rank one case. For this reason, it is crucial to compute the Harish-Chandra C-function for Spin(n, 1) and SU(n, 1). The reason for restricting our attention to the cases Spin(n, 1) and SU(n, 1) is that no multiple irreducible unitary representations of M occur in any irreducible unitary representation of K.
2. In Euclidean space, various forms of the uncertainty principle between a function and its Fourier transform are known. The Hardy theorem asserts that if a measurable function f on R satisfies |f(x)| <less than or equal> C exp{-ax^2} and |f(y)| <less than or equal> C exp{-by^2} then f = O(a.e.) whenever ab> 1/4. This result is generalized to some semisimple Lie groups by A.Sitaram and M.Sundari, and M.Sunclari, M.0. Cowling and J.F.Price We get an analogue of the Hardy theorem for the Cartan motion group and also an L^p version of the Hardy theorem for the motion group.
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