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 be a connected semisimple Lie group. If has finite center, Harish-Chandra used Eisenstein integrals to construct Schwartz class wave packets of matrix coefficients and showed that every -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 has infinite center. In this paper, the Plancherel theorem is used to decompose -compact Schwartz class functions (those with -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 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