THE EXPLICIT EXPRESSION OF C FUNCTION ON SEMISIMPLE LIE GROUPS AND ITS APPLICATION

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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.