Harish-Chandra's Schwartz space

In mathematical abstract harmonic analysis, Harish-Chandra's Schwartz space is a space of functions on a semisimple Lie group whose derivatives are rapidly decreasing, studied by Harish-Chandra (1966, section 9). It is an analogue of the Schwartz space on a real vector space, and is used to define the space of tempered distributions on a semisimple Lie group.