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

Herb, Rebecca A.

doi:10.1090/S0002-9947-1991-1014250-6

The Schwartz space of a general semisimple Lie group. II. Wave packets associated to Schwartz functions
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Trans. Amer. Math. Soc. 327 (1991), 1-69 Request permission

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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Unitary representations of partially holomorphic cohomology spaces