The Schwartz space of a general semisimple Lie group. I: Wave packets of Eisenstein integrals.

*(English)*Zbl 0711.22005Let G be a semisimple Lie group with infinite center. As in the finite center case, the tempered spectrum of G consists of families of representations induced from cuspidal parabolic subgroups MAN. However the representations of M to be induced are relative discrete series which occur in continuous families. In the reviewed paper the authors construct Schwartz class wave packets of matrix coefficients of induced tempered representations for any connected reductive Lie group G. Note that the authors use the induction not on G but rather on parabolic subgroups of G. This simplifies the constructions. In the special case \(G=M\) the wave packets are exactly the ones constructed in two previous papers [J. Funct. Anal. 73, 1-37 and 38-106 (1987; Zbl 0625.22010 and Zbl 0625.22011)].

Reviewer: V.F.Molchanov

##### MSC:

22E46 | Semisimple Lie groups and their representations |

22E30 | Analysis on real and complex Lie groups |

43A80 | Analysis on other specific Lie groups |

##### Keywords:

semisimple Lie group; tempered spectrum; cuspidal parabolic subgroups; relative discrete series; Schwartz class wave packets; matrix coefficients; induced tempered representations; connected reductive Lie group
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\textit{R. A. Herb} and \textit{J. A. Wolf}, Adv. Math. 80, No. 2, 164--224 (1990; Zbl 0711.22005)

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##### References:

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