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The Plancherel theorem for general semisimple groups. (English) Zbl 0587.22005
In this paper the authors extend Harish-Chandra’s Plancherel theorem for semisimple Lie groups with finite center to a class of reductive Lie groups which contains all connected semisimple Lie groups and was introduced by the second author in ”Unitary representations on partially holomorphic cohomology spaces” [Mem. Am. Math. Soc. 138 (1974; Zbl 0288.22022)]. The proof avoids Harish-Chandra’s analysis of the Schwartz space of rapidly decreasing smooth functions on the group. It consists of two parts: the first one is a reduction to a special case, by replacing the center of a group by a circle group (Wolf, loc. cit.), and the second one follows closely previous work of the first author on the Fourier inversion and the Plancherel formula for linear semisimple Lie groups [Trans. Am. Math. Soc. 249, 281-302 (1979; Zbl 0419.22015); Am. J. Math. 104, 9-58 (1982; Zbl 0499.43007); Lect. Notes Math. 880, 197-210 (1981; Zbl 0467.43005); Trans. Am. Math. Soc. 277, 241-262 (1983; Zbl 0516.22007); Lect. Notes Math. 1020, 73-79 (1983; Zbl 0523.43006)].
Reviewer: D.Miličić

MSC:
22E46 Semisimple Lie groups and their representations
43A32 Other transforms and operators of Fourier type
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References:
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