ICM-90 Satellite Conference Proceedings pp 140-153 | Cite as

# A Method of Reduction in Harmonic Analysis on Real Rank 1 Semisimple Lie Groups I

## Abstract

Harmonic analysis on semisimple Lie groups G is deeply related to the unitary dual Ĝ of G, in particular the Fourier transformation of *L* ^{2}(G) of square-integrable functions on G is defined by using the principal series and the discrete series of G. Then, the characterization of Fourier transforms of functions in *L* ^{2}(G) and C_{c} ^{∞}(G) of C^{∞} compactly supported functions on G is one of main problems in harmonic analysis on G, so the Plancherel formula and the Paley-Wiener theorem have been studied by various people. Let K be a maximal compact subgroup of G. Since right *K*-invariant functions on G are regarded as functions on the symmetric space X = G/K, harmonic analysis on X can be regarded as one for right K-invariant functions on G. Especially, for right K- invariant functions on G the Plancherel formula consists of only wave packets defined by the principal series and thus, the discrete series does not contribute to harmonic analysis on X. This fact means that the Paley-Wiener theorem on X is simpler than the one on (7, actually, it can be obtained by restricting the corresponding theorem on G to right K-invariant functions on G.

## Keywords

Wave Packet Symmetric Space Discrete Series Principal Series Real Rank## Preview

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