Harish-Chandra's Schwartz space

Summary

Harish-Chandra's Schwartz space $\mathcal{C}(G)$ is a space of smooth, rapidly decreasing functions on a semisimple Lie group $G$. It plays the role that the Schwartz space plays on $\R^n$, but the decay is measured against Harish-Chandra's function $\Xi$ rather than against polynomials. It contains the smooth compactly supported functions and lies in $L^2(G)$. Harish-Chandra used it to define tempered distributions on $G$ and to prove the Harish-Chandra Plancherel theorem.