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The Schwartz Space of a General Semisimple Lie Group

  • Rebecca A. Herb
Chapter
Part of the Progress in Mathematics book series (PM, volume 101)

Abstract

Let G be a connected semisimple Lie group. The tempered spectrum of G consists of families of representations unitarily induced from cuspidal parabolic subgroups. Each family is parameterized by unitary characters of a θ-stable Cartan subgroup. The Schwartz space C(G)is a space of smooth functions decreasing rapidly at infinity and satisfying the inclusions \( C_c^{\infty }(G) \subset C(G) \subset {L^2}(G) \). The Plancherel theorem expands Schwartz class functions in terms of the distribution characters of the tempered representations. Very roughly, we can write \( f(x) = \sum\nolimits_H {{f_H}(x)} \),\( f \in C(G) \),\( x \in G \), where
$$ {f_H}(x) = \int\limits_{{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{H}}}} {\Theta (H:x)(R(x)f)m(H:x){d_X}} $$

Keywords

Wave Packet Discrete Series Principal Series Cartan Subgroup Weyl Chamber 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Rebecca A. Herb
    • 1
  1. 1.Deprt. of MathematicsUniversity of MarylandCollege ParkUSA

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