The Schwartz Space of a General Semisimple Lie Group

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

$$ {f_H}(x) = \int\limits_{{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{H}}}} {\Theta (H:x)(R(x)f)m(H:x){d_X}} $$