A Method of Reduction in Harmonic Analysis on Real Rank 1 Semisimple Lie Groups I
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 Cc ∞(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.
KeywordsWave Packet Symmetric Space Discrete Series Principal Series Real Rank
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- [Al]Arthur, J.G., Harmonic analysis of tempered distributions on semisimple Lie groups of real rank one, Ph.D., Yale University (1970).Google Scholar
- [H3]A duality for symmetric spaces with applications to group representations, II. Differential equations and eigenspace representations, Advances in Math. 22 (1976), 187–219.Google Scholar
- [H4]Functions on symmetric spaces, in “Harmonic Analysis on Homogeneous Spaces,” Amer. Math. Soc. Providence, Rhode Island, 1973, pp. 101–146.Google Scholar
- [HC2]Harmonic analysis on real reductive groups, II. Wave packets in the Schwartz space, Invent. Math. 36 (1976), 1–55.Google Scholar
- [K3]Szego operators and a Paley-Wiener theorem on SU(1,1), Keio Univ. Res. Rep. 5. (1990), 1–35.Google Scholar
- K4] A method of reduction in harmonic analysis on real rank 1 semisimple Lie groups II, (preprint).Google Scholar
- [Kn]Knapp, A.W., Weyl group of a cuspidal parabolic, A.n. Ec. Norm. Sup. (4)8 (1975), 275–294.Google Scholar
- [V]Varadarajan, V.S., “Harmonic Analysis on Real Reductive Groups,” Lecture Note in Math., Springer-Verlag, New York, 1977.Google Scholar
- [Wa]Warner, G., “Harmonic Analysis on Semi-Simple Lie Groups II,” Springer-Verlag, New York, 1972.Google Scholar