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The Schwartz space of a general semisimple Lie group. II. Wave packets associated to Schwartz functions


Author: Rebecca A. Herb
Journal: Trans. Amer. Math. Soc. 327 (1991), 1-69
MSC: Primary 22E46; Secondary 46F05
DOI: https://doi.org/10.1090/S0002-9947-1991-1014250-6
MathSciNet review: 1014250
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Abstract: Let $ G$ be a connected semisimple Lie group. If $ G$ has finite center, Harish-Chandra used Eisenstein integrals to construct Schwartz class wave packets of matrix coefficients and showed that every $ K$-finite function in the Schwartz space is a finite sum of such wave packets. This paper is the second in a series which generalizes these results of Harish-Chandra to include the case that $ G$ has infinite center. In this paper, the Plancherel theorem is used to decompose $ K$-compact Schwartz class functions (those with $ K$-types in a compact set) as finite sums of wave packets. A new feature of the infinite center case is that the individual wave packets occurring in the decomposition of a Schwartz class function need not be Schwartz class. These wave packets are studied to obtain necessary conditions for a wave packet of Eisenstein integrals to occur in the decomposition of a Schwartz class function. Applied to the case that $ f$ itself is a single wave packet, the results of this paper yield a complete characterization of Schwartz class wave packets.


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DOI: https://doi.org/10.1090/S0002-9947-1991-1014250-6
Article copyright: © Copyright 1991 American Mathematical Society

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